º º | i : º: # 2. º º º, * * * * * * * * º º º & º § º º º } º # ; º º º * * * * * * * * * 3 . : . . . * * * ; : . . . . . º * {} Tº º : ; : ; º º à º . . . . .” “...º.º. º. | “... ºr º: ". . . . . tº - º sº. *** * * *, **** *, *. .* a r = **...* * * * s'º - . . . * * : ". * * * * *:::::: : ! º: º § º º sº º: º : º # ºº : *.*. ... sº *::: § 5 º ºf gºº. *. sº Jº & w - c sº tº sº º sº sº sº. * , . . . ; º 3 g ºf . tº .. §§§ º sº º i: ; w H sº. . . sº ſ | sº f º : # : . º º f ; * . . . . † : * * , º: º § ! . : : s º ea. º. a * a tº ºf 3 ºr 3 - *--- ~~ $Y- * - ſ!!!!!!!!!!!!!!!!!!!！!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!！!!!!！!!!!!!!！!！!!!!!!!!!！!!!!!!!!!!!！!!!！!!!!!!!!！!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!！ ■■■ ■ ■ ■ ■ ■！！！！！！！！！！！！！！！== --！！！！！！！！！！！ • • • • ！=§§§2º（º）（º）（º）（º）（º）（º）（es.： --~~~~！！！！！！！！！！！！！！！！！！！！！！！！！！！ -∞·----•- an ICH º AP º > E - - £a º º ||||I||||Illililllllliſt||||||| srºofm. Hºssº ANZººUANZA FU - º *Nº. s IVERS |- :§§§;&N º bºrº Tuttºn ºr sººººººº serºse ºvºseº º wº Dr. ºº: illfullllliºntº i. ſº i. ºn agº Tºnºmiſſiºn sº cºee cºcºcºa cºcº º fift mºtiºninºffilth * § § § º łºś. *ś 3.x.º.º. *: 。 ¿ §§ Žž §§ §§Y 、 §§§§ *** These portions of the text need be read only, except where marked omit. Page 9, Cordite to line 7, page IO. Page II, E. C. Powder to Proof of Powders. Page 20, Sphere to Multiperforated Cylinder, page 2I. Page 62, Lines I7 to 25 inclusive. Page 64, Mean Specific Heat of Products. - Page 64, Omit from line 19 to last paragraph page 66. Page 71, Line 9 Regarding to include line 2I. - Page 72, Omit Muggle l’elocity for Quick Powders. Page 74, Line 5 to include equation (64). Page 77, Line 7 to Pressures, page 78. Page 88a, Equation (IOS) line 26 to bottom page 88b. Page 95, Welocity of Combustion to line 15, page 96. All foot notes. Problems involving the use of logarithms are included in every lesson. A thorough knowledge of the use of log- arithms is therefore essential and is required. Page 12, line I 5 from bottom, write 275 for 2 II. ORDNANCE AND GUNNERY - PART I - 3’ ERRATA Page 15, line 11, write plaits for paits. line 13, write cap for bag. line 17, write attached for attacked. line 21, write portion for position. line 27, erase of after minimum. Page 16, line 26, write dimension for demension. Page 17, line 2, write dimension for demension. line 14, make last factor of equation, lº/lo”. Page 23, table, write l/lo for l/l". Page 27, line 1, period after 3. Large S following. Page 28, line 1, write an for the after with. Page 30, line 12, write registrar for registar. line 26, write projectile for proectile. Page 32, line 11, write spring catch for catch spring. Page 36, last line, write from for fro. . Page 37, 6 line from bottom, write quill for quil. Page 40, line 19, write cannon for common. line 28, erase first word. , Page 46, 10 line from bottom, write 1874 for 1847. Page 82, line 2, write X; for X5. line 15, write equations (74) for equation (74). Page 97, 3rd equation, write u for w in first member. Page 107, line 16, write 1.1524 for 1.11524. ！-* * … （ — ‘ ·- “№ſſae***--------- … … -- ORDNANCE AND GUNNERY. PART I. Gun Powders, Measurement of Velocities and Pressures, Interior Ballistics. * - / PREPARED FOR THE Cadets of the United States Military Academy. f - - ºW y; b. ? *_ - * BY ORMOND M'LISSAK, Major, Ordnance Department, Instructor of Ordnance and Gannery. - WEST POINT, N. Y. U. S. MILITARY ACADEMY PRESS - 1906 The chapters on Interior Ballistics and on the Burning of Gunpowder under Constant Pressure are derived prin- cipally. from the writings of Colonel James M. Ingalls, U. S. Army.; I desire to acknowledge my indebtedness to him for the use, before publication, of his manuscript on the deduc- tion of his new interior ballistic formulas. These formulas, accurate, simple and of extended application, provide the means of determining the conditions existing in the bore of a gun to an extent and to a degree of accuracy never hitherto attained. - ORMOND M. LISSAK, Major, Ordnance Department, U. S. Army, - Instructor of Ordnance and Gunnery. WEST POINT, N. Y., August 1, 1906. CONTENT S. CHAPTER I. - Page Gun POW ders..................................................... e s > * * * * * * * * * * * * * * * * * * * * * * * * 3 Combustion of Powder under Constant Pressure Determination of the Coustants of Form for Different Powder Grains... 17 Comparison of Manner of Burning of Different Powder Grains............ 23 Considerations as to Best Form of Grain........................ ~~~~ 25 Various Determinations..................."… , 26 Density of Gun Powder…~~~~ 27 CHAPTER II. Measurement Of VeloCities and Pressures......................... 29 Le Boulengé Chronograph................................................................... 29 Other Velocity Instruments ................................................................. 37 Measurement of Pressures.................................................................... 39 CHAPTER III. sº Interior Ballistics..............................................'......................... 44 Properties of Perfect Gases.................................................................. 51 Relations Between Heat and Work...................................................... 55 Noble and Abel's Experiments............................................................ 59 Formulas for Velocities and Pressures in the Gun................................ 66 Principle of the Covolume................................................................... 68 Dissociation of Gases........................................................................... 70 Combustion of Powder under Variable Pressure .................................. 73 Velocity Formulas............................................................................... 76 Pressure Formulas............................................................................... 78 Application of the Ballistic Formulas.................................................. 86 Determination of the Ballistic Formulas from Measured Interior Veloc- ities…~~~~ S9 Determination of the Ballistic Formulas from a Measured Muzzle Ve- locity and Maximum Pressure....................................................... 105 United States Army Cannon. Data..................................................... 117 * Table of the X Functions......................................................... 118, 119, 120 s 421947 MUZZLE. BREECH. - RIM BASE.---, PROJECTLE RIF LED BORE. 12-INCH RIFLE, MODEL OF 1900, 40 CALIBERS, 59.10 TONS. (Diameters Exaggerated.) Gun Powders. DEFINITIONs. Axplosion is the extremely rapid conversion of a solid or a liquid into the gaseous form. It is accompanied by great heat. Explosion of gun powder may be divided into three parts; Ignition, Inflammation and Combustion. Ignition is the setting on fire of a part of the grain or charge. Gunpowder is ignited by heat, which may be produced by elec- tricity, by contact with an ignited body, by friction, shock, or by chemical reagents. An ordinary flame, owing to its slight density, will not ignite powder readily. The time necessary for ignition will vary with the condition of the powder. Thus damp powder ignites less easily than dry; a smooth grain less easily than a rough One; a dense grain less easily than a light one. º Powder charges in guns are ignited by primers, which are fired by electricity, by friction, or by percussion. Inflammation is the spread of the ignition from point to point of the grain, or from grain to grain of the charge. With small grain powders the spaces between the grains are small, and the time of inflammation is large as compared with the time of combustion of a grain; but with modern large grain powders the facilities for the spread of ignition and the time of burning of the grain are so great that the whole charge is supposed to be in- flamed at the same instant, and the time of inflammation is not con- sidered. Combustion is the burning of the inflamed grain from the sur- face of ignition inward or outward or both, according to the form of the grain. Experiment shows that powder burns in the air according to the following laws: I. In parallel layers, with uniform velocity, the velocity being independent of the cross section burning. 2. The velocity of combustion varies inversely with the density of the powdet. 4 When a charge of powder is ignited in a gun inflammation of the whole charge is rapidly completed. The gases evolved from the burning grains accumulate behind the projectile until the pressure they exert is sufficient to overcome the resistance of the projectile to motion. The accumulated gases, augmented by those formed by the continued burning of the charge, expand into the space left behind the projectile as it moves through the bore, exerting a continual pressure on the projectile and increasing its velocity until it leaves the muzzle. -. History.—The Chinese are said to have employed an ex- plosive mixture, very similar to gunpowder, in rockets and other pyrotechny as early as the 7th century. - The earliest record of the use in actual war of the mixture of charcoal, nitre and sulphur called gunpowder, dates back to the four- teenth century. Its use in war became general at the beginning of the sixteenth century. Until the end of the sixteenth century it was used in the form of fine powder or dust. To overcome the difficulty experienced in loading small arms from the muzzle with powder in this form, the powder was at the end of the sixteenth century given a granular form. With the same end in view attempts at breech loading were made; but without success, as no effective gas check, which would prevent the escape of the powder gases to the rear, was devised. - No marked improvement was made in gunpowder until 1860 when General Rodman, of the Ordnance Department, U. S. Army discovered the principle of progressive combustion of powder, and that the rate of combustion, and consequently the pressure exerted in the gun, could be controlled by compressing the fine grained pow- der previously used into larger grains of greater density. The rate or velocity of combustion was found to diminish as the density of the powder increased. The increase in size of grain diminished the surface inflamed, and the increased density diminished the rate of combustion, so that, in the new form, the powder evolved less gas in the first instants of combustion, and the evolution of gas continued as the projectile moved through the bore. By these means higher muzzle velocities were attained with lower maximum pressures. To obtain a progressively increasing surface of combustion General 5 Rodman proposed the perforated grain, and the prismatic form as the most convenient for building into charges. As a result of his in- vestigations powder was thereafter made in grains of size suitable to the gun for which intended, small grained powder for guns of small calibre, and large grained powder for the larger guns. The powders of regular granulation such as the cubical, hexagonal and sphero-hexagonal came into use, and finally for larger guns the pris- matic powder in the form of perforated hexagonal prisms. PRISMATIC. SPHERO-HEXAGONAL. A further control of the velocity of combustion of powder was obtained in 1880 by the substitution of an underburned charcoal for the black charcoal previously used. The resulting powder, called “brown " or “cocoa ‘’ powder from its appearance, burned more slowly than the black powder, and wholly replaced that powder in the larger guns. A still further advance in the improvement of powder was brought about in 1886 by the introduction of smokeless powders. These powders are chemical compounds, and not mechanical mix- tures like the charcoal powders; they burn more slowly than the charcoal powders, and produce practically no smoke. Smokeless powders have now almost wholly replaced black and brown pow- ders for charges in guns. Black powder is used in fuzes, primers, and igniters, in saluting charges, and as hexagonal powder in the Smaller charges for seacoast mortars. Charcoal Powders. -- CoMPOSITION. — Black gunpow- der is a mechanical mixture of nitre, charcoal, and sulphur, in the proportions of 75 parts nitre, I5 charcoal, and Io sulphur. 6 The nitre furnishes the oxygen to burn the charcoal and sul- phur. The charcoal furnishes the carbon, and the sulphur gives den- sity to the grain and lowers its point of ignition. The distinguishing characteristic of charcoal is its color, being brown when prepared at a temperature up to 280°, from this to 340° red, and beyond 340° black. Brown powder contains a larger percentage of nitre than black powder, and a smaller percentage of sulphur. A small percentage of Some carbo-hydrate, such as sugar, is also added. Its color is due to the underburnt charcoal. . MANUFACTURE.-The ingredients, purified and finely pulver- ized, are intimately mixed in a wheel mill under heavy iron rollers. The mixture is next subjected to high pressure in a hydraulic press. The cake from the press is broken up into grains by rollers, and the grains are rumbled in wooden barrels to glaze and give uniform den- sity to their surfaces. The powder is then dried in a current of warm dry air, and the dust removed. The powder is thoroughly blended to overcome as far as possible irregularities in manufacture. For powders of regular granulation the mixture from the wheel- mill was broken up and pressed between die plates constructed to give the desired shape to the grains. Prismatic powder was made by reducing the mill-cake to powder and pressing it into the required form. SmokeleSS PowderS.—There are two classes of smoke- less powders used in our service; nytroglycerine powder in small arms, and nitrocellulose powder in cannon. They are both made from guncotton, to which is added for the small arm powder about 30 per cent. by weight of nitroglycerine. CoMPARISON OF NITROGLYCERINE AND NITROCELLULOSE Pow- DERS.—The temperature of explosion of nitroglycerine powder is higher than that of nitrocellulose powder. As the erosion of the metal of the bore of the gun is found to increase with the tempera- ture of the gases, greater erosion follows the use of nitroglycerine powder. The endurance, or life, of a modern gun is dependent on the condition of the bore, and Ön account of the great cost of cannon, erosion becomes a more serious defect in cannon than in small arms. On this account therefore nitrocellulose powder is more suitable than nitroglycerine powder for cannon. 7 To produce a given velocity a larger charge of nitrocellulose than of nitroglycerine powder is required. This necessitates for nitrocellulose powder a larger chamber in the gun, and the increase in size of the chamber involves increased weight of metal in the gun. This is more objectionable in a small arm than in cannon, for the in- creased weight of the gun and of the charge adds to the burden of the soldier. For this reason nitroglycerine powder is more suitable than nitrocellulose powder in the small arm. In the manufacture of nitroglycerine powders for cannon, a sat- isfactory degree of stability under all the sconditions to which can- non powders are exposed was not obtained. In time the powder de- teriorated, and exudation of free nitroglycerine occurred. Detona- tions and the bursting of guns followed. In the small arm cartridge the powder is hermetically sealed, and as now manufactured appears to possess a satisfactory degree of stability. - For these reasons nitroglycerine powder has been selected for use in small arms in our service, and nitrocellulose powder for use in Ca1111011. . . * A disadvantage attending the use of nitrocellulose powder arises from the fact that, in the explosion, there is not a sufficient amount of Oxygen liberated to combine with the carbon and form CO2. The reaction on explosion is approximately represented by the following equation : . 2 (C.A.O.) O,(AVO.), :- 9 CO + 3 CO, + 7 A.O.-- 3 AV, A large quantity of CO , an inflammable gas, is often left in the bore. On opening the breech more oxygen is admitted with the air, and should a spark be present the CO burns violently, uniting with the oxygen and forming CO2. This burning of the gas is called a flareback. An instance of it has recently occurred with disastrous results in a turret gun aboard one of our men-of-war. Gun Cotton. – Guncotton forms the base of most smoke- less powders. When dry cotton, C, H, O, is immersed in a mix- ture of nitric and sulphuric acids part of the hydrogen of the cotton is replaced by NO, from the nitric acid. The sulphuric acid takes up the water formed during the reaction and prevents the dilution of the nitric acid. The nitrated cotton, or nitrocellulose, may be of sev- 8 eral orders of nitration, depending on the strength and proportions of the acids, and the temperature and duration of immersion; as mono-nitrocellulose, di-nitrocellulose, tri-nitrocellulose, according as One or more atoms of hydrogen are replaced. All nitrocellulose is explosive, and the order of explosion produced is higher as the nitration is higher. Di-nitrocellulose and tri-nitrocellulose are used in the manufacture of smokeless powders. The lower orders of ni- trocellulose, containing less than I2.75 per cent. Of nitrogen, are sol- uble in a mixture of alcohol and ether. Tri-nitrocellulose contains a higher percentage of nitrogen, and is insoluble in alcohol and ether, but is soluble in acetone. MANUFACTURE OF GUN-COTTON FOR SMOKELESS PowDERS.– The process followed is practically the same for all varieties, the ni- tration being stopped at the point desired in each case. The cotton used is the waste or clippings from cotton mills. It is first finely divided and then freed from grease, dirt, and other impurities by boiling with caustic soda. After cleansing it is passed through a centrifugal wringer and then further dried in a dry-house. The dry cotton is immersed in a mixture of about three parts sulphuric acid and two parts nitric acid for about fifteen minutes; after which the cotton is run through a wringer to remove as much acid as possible. It is then thoroughly washed or “drowned.” After this washing the guncotton is reduced to a pulp and further washed to remove any trace of acid which may have been freed in pulping, carbonate of soda being added to neutralize the acid. The water is then partially removed from the pulp by hydraulic pressure and the dehydration is completed by forcing alcohol under high pressure through the compressed cake. Nitroglycerine Small Arm Powder, – Laffin and Rand, W. A. In the manufacture of this powder highly nitrated gun cotton called “insoluble nitrocellulose ’’ is used. It is in- soluble in ether and alcohol, but is soluble in acetone. The powder is composed, of Insoluble nitrocellulose 67.25 per cent. Nitroglycerine 3O.OO per cent. Metallic Salts 2.75 per cent. 9 Forty pounds of acetone serve as solvent for IOO lbs of the above mixture. The nitroglycerine and acetone are first mixed. The acetone makes the nitroglycerine less sensitive to pressure or shock, and therefore less dangerous to handle in the subsequent operations. The dried guncotton is spread in a large copper pan, the finely ground metallic salts are sifted over it, and the mixed nitroglycerine and acetone are sprinkled over both. The whole is mixed by hand by means of a wooden rake for a period of about ten minutes, the object of the mixing being to thoroughly moisten the guncotton for the purpose of eliminating the danger from the presence of dry guncot- ten in the next operation. The mixed mass is put into a mixing machine, where it is mechanically mixed for a period of three hours. It comes from the mixing machine in the form of a colloid or jelly- like paste. It is then stuffed and compressed into brass cylinders, from which it is forced by hydraulic pressure through dies fitted with mandrels. It comes from the die in the form of a long hollow String or tube, and is received on a belt which carries it over steam pipes into baskets. The drying which it receives while on the belt strengthens the tube, and after remaining half an hour in the baskets it becomes sufficiently tough to be cut into grains. This is done in a machine provided with revolving knives. The resulting grains are bead shaped, and have a thickness of about one-twentieth of an inch. The powder is dried for two or three weeks at a temperature not to exceed IIo9 F. It is then thoroughly mixed twice in the blending barrels and graphited at the same time. It is carefully screened to remove large grains, dust, and foreign matter, and is packed in mus- lin bags in metallic barrels holding 100 pounds. Tordite.—This is an English nitroglycerine powder, composed of 58 per cent of nitroglycerine, 37 per cent. of guncotton and 5 per cent of vaseline. The vaseline serves to render the powder water- proof and improves its keeping qualities. For small arms it is made in the form of slender cylindrical rods, the length of the chamber of cartridge. For cannon it is in thicker and longer rods, in tubular form, or in the form of perforated cylinders. For heavy guns a powder called “Cordite M. D.” has lately been introduced. The composition 30 parts nitroglycerine, 65 parts guncotton, 5 parts vase- line, is very similar to that of our small arm powder. The reduction º \, \ S \,--ó-º) i º * t 10 in the percentage of nitroglycerine was made for the purpose of low- ering the temperature of explosion and reducing the erosion in the bore. - Wetteren Powder.—A nitroglycerine powder manufactured at the Royal Belgian Factory at Wetteren. The solvent used is amyl ... acetate. Manufacture of Nitrocellulose Powder. — The gun . cotton used contains 12.65 per cent. of nitrogen, and is soluble in the ether-alcohol mixture. It is prepared as previously de- scribed, the dehydration with alcohol being so conducted that when completed the proper proportion of alcohol for solution remains in the cake. The guncotton cake is broken up and ground until it is free from lumps, and is then placed in a mixing machine with the proper amount of ether, two parts of ether to one of alcohol. During the mixing the temperature is kept at 5°C. to prevent loss of the sol- Vent. - The powder comes from the mixing machine as a colloid, and the remaining processes are similar to those described for nitroglyc- erine powder. After graining, the solvent is recovered by forcing heated air over the powder. The ether and alcohol vapors are collected and afterwards condensed for further use. The powder is dried for a period varying from six weeks to three months depending on the size of the grain. The drying is never complete, a small percentage of the solvent always remaining, but care is taken that the remaining percentage shall be uniform. - In the manufacture of all powders uniformity in the product can only be obtained by the strictest uniformity in the quantities and quality of the substances used, and in the conduct of the various processes. - Cannon powders are, as a rule, not graphited. Form and Size Of Grain. — For most cannon in our service the powder is formed into a cylindrical grain with seven longitudinal perforations, one central and the other six equally dis- tributed midway between the centre of the grain and its circumfer- ence. A uniform thickness of web is thus obtained. The powder is 11 of a brown color and has somewhat the appearance of horn. The length and diameter of the grain vary in powders for different guns, the size of grain increasing with the calibre of the gun. For the 3-inch rifle the grain has a length of about 3% of an inch and a dia- meter of 2-IO of an inch. For the 12-inch rifle the length is I3/3 inches and the diameter 7% of an inch. For some of the smaller guns the grains are in the form of thin flat squares. Experiments are now being made in cannon with powder made in the form of rods with single perforations, and also with powder in the form of sheets from which strips are cut in such a manner as to produce an appearance like that of a comb. The purpose of these variations is to obtain better control of the combustion, and it is claimed for the comb powder that by varying the thickness of the sheet, and the length and breadth of the teeth, the combustion of the powder can be controlled to a highly satisfactory degree. Other SmokeleSS POWſierS.–To make up for the insufficiency of oxygen in nitrocellulose, already referred to, a number of smoke- less powders are made by a combination of nitrocellulose with nitro- glycerine or with the nitrates of barium, potassium and sodium. The nitroglycerine or the metallic nitrates furnish the oxygen which is deficient in the nitrocellulose. E. C. Powder—This powder contains both soluble and insolu- ble nitrocellulose and the nitrates of barium, potassium and Sodium. It is yellow in color and of fine granulation. It is an easily ignited quick burning powder and is used in our service in blank small arm & cartridges. * -_º-Schultze Powder, the type of smokeless sporting powders, is of constitution similar to that of the E. C. powder. - Troisdorf Powder used in the German service and B. N. Pow- der in the French service are other powders similiarly constituted. All these powders differ principally in the proportion of the ingredi- : ents, and also in the organic substance used as a cementing agent. : Marim Powder is composed of nitrocellulose, both soluble and . insoluble, nitroglycerine and a small percentage of sodium carbonate. Proof of PowderS.–All powders used by the Army are fur- nished by private manufacturers. The materials and processes em- 12 ployed in the manufacture are prescribed by the Ordnance Depart- ment in rigid specifications, and the manufacture in all its stages is under the inspection of the Department. The proof of the powder consists of tests made to determine its ballistic qualities, its uni- formity, and its stability under various conditions. Its ballistic qual- ities and uniformity are determined from proof firings made in the gun for which the powder is intended. The required velocity must be obtained without exceeding the maximum pressure specified. The mean variation in velocity in a number of rounds must not exceed, in the small arm IO feet per second, in cannon one per cent. of the required velocity, The stability of the powder under various conditions is deter- mined by heat tests, and by a number of special tests. For small arms powder the heat test consists in subjecting the powder, pulver- ized, to a temperature of I5o° to 154°F. for 20 minutes. It must not in that time emit acid vapors, as indicated by the slightest discoloration of a piece of iodide of potassium starch paper partially moistened with dilute glycerine. The other tests consist in exposing the powder both loose and loaded in cartridges, to heat, cold, and moisture, for periods varying from six hours to one week. When fired the varia- tions in velocities and pressures must not exceed specified limits. - C ſºjose cannon powders are subjected to a heat of 135° C. (ºftºp.) for 5 hours. Acid fumes, as indicated by the reddening of blue litmus paper, must not appear under exposure of an hour and a quarter, nor red nitrous fumes under two hours. Explosion must not occur under five hours. Other tests are made for the determin- ation of the loss of weight when subjected to heat, of the moisture and volatile matter in the powder, of the quantities of nitrogen in the powder, and of ash in the cellulose. For the proper regulation of the evolution of gas in the gun it is important that the grains of smokeless powder retain their general shape while burning. If they break into pieces under the pressure to which they are subjected, the inflamed surface is increased, gas is more quickly evolved, and the pressure in the gun is raised. The powder is therefore subjected to a physical test to de- termine that the grain has sufficient strength and toughness. The grains are cut so that the length equals the diameter, and are then 13 subjected to slow pressure in a press. The grain must shorten 35 per cent. of its length before cracking. Powder grains incompletely burned, that have been recovered after firing, show that the burning proceeds accurately in parallel layers. The outer diameter of the grain is reduced and the diam- eter of the perforations increased in exactly equal amounts. Advantages of Smokeless POWiler.-The advantages ob- tained by the use of smokeless powder are due almost wholly to the fact that the powder is practically completely converted into gas. The experiments of Noble and Able show that the gases evolved by charcoal powders amount to only 43 per cent. Of the weight of the powder, and part of the energy Of this quantity of gas is expended in expelling the solid residue from the bore. A smaller quantity of smokeless powder will therefore produce an equal weight of gas, and with smaller charges we may give to the projectile equal or higher velocities. The smokelessness of the powder and the absence of foul- ing in the bore are also due to the complete conversion of the powder into gas. *. Ignition and Inflammation of Smokeless Powder.-Though the temperature at which smokeless powder ignites, about 180° C. , is much lower than that required for the ignition of black powder, 300° C., the complete inflammation of a charge composed only of smokeless powder takes place more slowly than the inflam- mation of a charge of black powder. This is due to the slower burn- ing of the smokeless powder and the consequent delay in the evolu- tion of a sufficient quantity of the heated gas to completely envelop the grains composing the charge. In the long chamber of a gun the gases first evolved at the rear of the charge may, in expanding, acquire a considerable velocity. The pressure due to their energy is added to the static pressure due to their temperature and volume, thus increasing the total pressure in the gun. The movement of the gases back and forth cause what are called wave pressures, and if the complete ignition of the charge is delayed until the projectile has moved some distance down the bore, there may result at some point in the gun a higher pressure than the metal of the gun at that point can resist. 14 For this reason and in order to insure the practically instantan- eous ignition of the whole charge, a small charge of black powder is added to every smokeless powder charge. In addition, in order to prevent as far as possible the production of wave pressures, the charge of powder, whatever its weight, is given when practicable a length equal to the length of the chamber. Blank Charges.—If the same smokeless powder that is pre- scribed for use with the projectile in any piece is used in a blank charge, the grains are not subjected to the pressure under which they were designed to burn, and consequently they burn very slowly and many of them are ejected from the bore only partially consumed. The report made by the explosion under these circum- stances is unsatisfactory for saluting purposes. To produce a sharper report a more rapid evolution of gas is necessary, which requires, if smokeless powder is employed, the use of a smaller grain, or one that is porous through imperfect colloiding. It has been found that a satisfactory report can be obtained from a blank charge of smokeless powder only by the use of a grain so small or of such a nature that the rate of evolution of the gas becomes excessive. This has resulted, in several instances, in the bursting of the gun. - - - For this reason black powder only is used in saluting charges. Bag filled ready Bag Iaced and provided for lacing. with prinning pro- tector caps. SECTION OF POWDER CHARGE FOR HEAVY GUNS. 15 Powder Charges.—The powder for a charge in the gun is inserted in one or more bags, depending upon the weight of the charge. The bags are made of special raw silk and are sewed with silk thread. The ends of each bag are double, and between the two pieces at each end is placed a priming charge of black powder, quilted in in squares of about two inches and uniformly spread over the surface. The charge is inserted through an unsewed seam at one end, and the seam is then sewed. The bag, purposely made large, is then drawn tight around the charge by lacing drawn with a needle between two paits on the exterior. Two prim- ing protector caps are then drawn over the ends of the bag and fastened by a draw string. In the bottom of each bag is a disk of felt which serves to keep moisture from the priming charge and prevents the loss of the priming through wearing of the bottom of the bag. For convenience in handling the charge a cloth strap is attacked to each protector cap. By means of the straps the protector caps may be pulled off with- out undoing the draw strings when the charge is to be inserted in the gun. - - The weight of each position of the charge should not be greater than can be readily carried by one man. Thus the charge of 360 pounds for the 12-inch rifle is put up in 4 bags each holding 90 pounds. As previously stated the charge whatever its weight is made up if practicable of a length nearly equal to that of the cham- ber, with a minimum of limit of nine-tenths of that length. While raw silk does not readily hold fire it occasionally happens that a fragment of the bag remains burning in the bore, and to this fact is ascribed the flarebacks that have occurred. Experiments are now being made therefore with powder bags of a nitrocellulose cloth which will burn up com- pletely and leave no residue. The powder charge in fixed ammunition is placed loose in the cartridge case. One or two wads of felt placed on top of the powder fill the space in the case behind the projectile. The priming charge of black powder is contained in the primer which is inserted in the head of the cartridge case. 2 16 The illustrations show a bag filled ready for lacing, and a bag filled and laced and provided with the priming protector CapS. combusTION OF POWDER UNDER CONSTANT - PRESSURE. Under constant pressure, as in the air, a grain of powder burns in parallel layers and with uniform velocity, in direc- tions perpendicular to all the ignited surfaces. Under the variable pressure in the gun powder burns with a variable velocity, but, as has been previously stated, modern smokeless powders burn accurately in parallel layers in the gun. A determination of the volume burned when anythick- ness of layer is burned will therefore be useful when we come to consider the burning of the powder in the gun. Powders of irregular granulation may be considered as composed of practically equivalent grains of regular form. Let to be one half the least demensions of the grain, . l the thickness of layer burned in the time t, So the initial surface of combustion, S the surface of combustion at the time t, when a thick- ness l has been burned. S’ the surface of combustion when l-lo, Wo the initial volume of the grain, V the volume burned at the time t, • - F= V/V, the fraction of grain burned in the time t. The least dºmension of the grain, 2b, is called the web of the grain. As the burning proceeds equally in directions perpendicular to all the surfaces, the grain will, in most in- stances, be about to disappear when the thickness of burning lo is approached. The surface S', corresponding to this thick- ness, is therefore called the vanishing surface. The general expression for the burning surface of a grain when a thickness l has been burned is S== So-Ha!-H bl” (1) 17 in which a and b are numerical co-efficients whose values depend on the form and demensions of the grain. For grains that burn with a decreasing surface the sign of a in this equation will be found to be negative, and for those that burn with an increasing surface the sign of b becomes negative. - The volume burned at any time is W= ſo S d!. And substituting for S its value from equation (1), V=S. H.; p-Hºp. - - (2) Dividing both members by Vo and introducing lo by multi- plication and division we have, for the fraction of the grain burned, F= *** -º ##} . . . . * |Vo Wo lo 2So lo 3So lº . . . . and making ... *. -** a=S. ivy, A=alo/2S6 u=bl&/3S, # . (3) we obtain - - - .*--> 2 Cº. F=&# | + A} + pu #} - (4) SThis equation gives the value for the fraction of the grain burned when a length l has been burned, and as each grain in a charge of powder burns in the same manner, the equa- Xtion also expresses ſhe value for the fraction of the whole \ \charge burnedºwºsen l-l, the whole grain is burned; F The quantities oc, A, and u are called the constants of form of the powder grain. Their values depend wholly on the form and relative dimensions of the grain. which may always serve to test the correctness of the values of these constants as determined for any grain. Determination of the Walues of the Constants of Form for Different Grains.—In the values of a, X, and u, equations (3) 18 the quantities So, lo, and Vo are known for any form of grain. we must know in addition the values of a and b. When l-lo the volume burned is the original volume Vo, and equation (2) becomes Wo H So lo–H ; lo” + ; ld”. The burning surface at this time, designated by S', is from equation (1), - S’ = So-H alo–H blo”. - - The values of a and b, if desired, may be derived from these two equations. Combining the equations with equations (3) we obtain the following values for oc, A and p. oc =Solo/Vo. A=3/oc —S"/So-2. (6) - p=S"/So-2/ oc +1. The Vanishing Surface.—The quantity Sº which represents the vanishing surface or surface of combustion when l-lo, requires explanation. A spherical grain burning equally along all the radii becomes a point as l becomes equal to lo. S’ for a sphere is therefore 0, and similarly for a cube. A cylindrical grain, of length greater than its diameter, becomes a line when l-lo. S" is therefore 0 for this cylinder. A flat square grain remains flat throughout the burning, its thick- ness being reduced until as l becomes equal to lo there are two burning surfaces with no powder between them. S", in this case, is the sum of these two surfaces. PARALLELOPIPEDON.—Let 2lo be the least dimension, and m and n the other dimensions of the grain of powder,. .img the longer. S’ = 2 (m—2lo) (n–2lo) Wor= 2lomn We therefore have from equations (6) `----. - 19 Make a and y the ratios of the least dimension to the other dimensions of the grain . a-2l/m - g=210/n. With these values we get from (3) for a a =*=2l/n 421/m-1=1+x+y Eliminating the common factors in the values of S and So we have - S' mn–2lºn–2km-H4l.” So 2lm-H2lºn-Hºmn and dividing each term by mn S' - I–2lo/m—2lo/n–H4lo°/mn _ I–4–y-Hary So 2lo/n–H2lo/m-H I TTI-Ex-Ey. Substituting in equations (6) : *-i-y--ry #- . * * TT 1-H++y 5 º :ry *_ S. = — ". . . . . . . . . . . I —Hir-Hy ſº ~~ * . U0 \. -a- I lºve &Z * ~, t And by giving various values to a, and y, this equation may be applied to any form of the parallelopiped. Cube.—For instance for the cube, m = n = 2h, and a and y are unity. - - Therefore a = 3 x = –1 =1/3 and l l, 1 ºn . l\* ..., F=3; {1– ####} =1-(1-#): . (8) Strip.–For strips or ribbons of square cross section-w-2lo and & = 1 - - * - -- tº –4 •=2+% X = #Tº "T3+y, If the strip is very long in comparison with the edge of CTOSS section, ſis practically zero and / a = 2 X = -1/2 u = 0. 20 ... Square Flat Grains.—For SQuare flat grains a -y and a = 1+2, A = –ººl(2 + æ) _ až TIE ºr “T THE 2, If the grains are very thin a is small compared with unity and - ׺wº- x ----------º C, ºr 1 X = 0 u = 0. ** * - w {, , , JY i - º - º º ºf2 * . - . . SPHERE.—For spherical grains we may deduce in the same Dºla,L]][16].' . - ...' - q = 3 X = –1 p = 1/3 the same as for the cube. SOLID CYLINDER.—Assume the diameter of the cylinder, d, to be the least dimension, and let m be the length of the cyl- inder. Then d = 2i. Make a = 21.7m. Then . . . . ~ – 9 -- . — — 1 + 2* ... – a - a = 2 + a X = 3.T., P - 3I.' If the diameter is very Small compared with the length, as in the slender cylinders or threads of cordite, 2lo is small with respect to m, a is small compared with unity, and approximately - q=2 X= −1/2 p=0. Therefore for cordite F-ol. ſ 1 – 1 ll – 1 –/1 —l Yº F=2}{1 #}=1 (1 #): (9) SINGLE PERFORATED CYLINDER.—Let R be the outer radius of the grain, r the radius of the perforation, and m the length of the grain. Make a = 210/m. By proper substitution we find, for the tubular grain in general * — — —& =n q=1+a; X= 1 + æ p = 0. If the grain is very long compared with its thickness of wall, a is small compared wi th unity. We then have, a=1 \=0 u=0 . and F=l/l). (10) : i ; As the surface and volume of a burning sphere of pow- der vary with the diameter in precisely the same manner that the surface and volume of a cube vary with the edge of the cube, the values & , A and p, see equations (6), will be the same for the sphere as for the cube. And similarly the values of these constants for a cylinder of length greater than its diameter will be the same as for the strips of square cross section, and the values for a flat cylinder will be the same as for the flat square grain. } | ‘...-. ..! A - * * f - 21 f { | If This indicates for long tubes with thin walls a constant emission of gas during the burning of the grain, since F now varies directly with l. - - . FLAT CYLINDER.—Making 2/6 = thickness, and r = radius, a = 210/r and we may deduce _a: (2 + æ) wº TIT2, "f IEEE C = 1 + 2a: M = the Same as for the flat square grain. MULTIPERFORATED CYLINDER.—A section of the service multiperforated grain before burning is shown in Fig. 1. Fig. 1. The perforations are equal in diameter and symmetrically distributed. The web, 210, is the thickness between any two adjacent circumferences. When this thickness has burned the section is as shown in Fig. 2. There remain now six interior and six exterior three cornered pieces, called slivers, which burn with a decreasing surface until completely consumed. . The method previously followed can not be used to find the value of F for the multiperforated grain, because the law of burning for this grain changes abruptly when the grain is but partially consumed. To find the value of F for this grain we proceed as follows: Let R be the radius of the grain, r the radius of each per- foration, m the length of the grain. For the initial volume we have Vo = 1 m (R2–7r”). When a thickness l is burned, R, r, and m become respect- ively R – l, r + l, and m – 21, and the volume remaining is 22 obtained from the above equation by making these substitu- tions. The difference between the two volumes will be the volume burned, and dividing this resulting volume by Vo we have the value of F. This may be reduced to F_2b (R-7, Fºn (R+7r); _l {1+ lo £3m – 2 (R + 7r)} l *== m (R2–7r?) lo R*–7,”--m (R+7, ) lo * 6lo? l? R2–7, 3–H m (R+7, ) #}. (11) For the service multiperforated grain we therefore have 2loº R2–7r” + m (R+7, )} l m (R2–7r?) lo 3m – 2 (R -- 7r)} - X R” – 7,” + m (R, + 7,') (12) º- . 6lo? - pl = — R*-7rº-H m (R+7x) d Equation (11) applies only while the web of the grain is burning and does not apply to the slivers. The thickness, of weh hears the following relation, to R and As may be readily seen by drawing a diameter ºf any three perforations, fig. 1. D–3d R–3r 2/5 F - (13) • v ºs-A. º. ººº-º-º- *-* * * . 4 2 —-- - - - - - -.... Lºs v Lºw UV Ullº, burning of the multiperforated cylinder. The grains of a lot of powder for the 8-inch rifle had the following dimensions: R=0.256° r=0.0255 tº m-1,029. Nº Therefore, from (12), lo–0.044875. Substituting in (11) we obtain for this grain F=0.72867; 14 0.19590%–0.023.8%). (14) lo lo lo When l = lo, that is, when the grain is reduced to slivers, F = 0.85174 from which we see that the slivers form about 15 per cent. of this particular grain. 23 Emission of Gas by Grains of Different Forms.-As the velocity of combustion under constant pressure is uniform, the time of burning will be proportional to the thickness of layer burned. - (, º We may conveniently show the manner of burning of the different grains by dividing the web into five layers of equal thickness, that is, by giving to the ratio l/lo in the value of the fraction burned, the values 1/5, 2/5, etc., in succession, and then tabulating the resulting values of F. The succes- sive values of F obtained will be the fractional parts burned in 1/5, 2/5, etc., of the total time of burning; and the differences of the successive values of F will be the fractions burned in the successive intervals of time. The following table is formed from equations (8), (9) and (14). For the multiperforated grain the fractions l/lo are fractions of the web only: r--A | ? *†, CUBE sues per Cyrisprs. Mºoººººp 72%.” & ~~~~~~. CYLINDER. F Diff. F Diff. F Diff. 0.0 0.000 º 0.00 Q.00 0.49 0.36 0.15 0.2 0.49 0.36 0.15 0.29 0.28 0.16 0.4 0.78 0.64 0.31 0.16 0.20 0.17 0.6 0.94 0.84 0.48 0.05 0.12 0.18 0.8 0.99 0.96 0.66 0.01 0.04 0.19 1.0 1.00 1.00 1.00 1.00 || Web 0.85 0.85 • - 0.15 Whole grain 1.00 1.00 Regarding the columns of differences in the table we see that nearly half of the cubical grain is burned in the first layer, and that the volume burned in the successive layers decreases continuously. The slender cylinder emits at first a less volume of gas than the cube and later a greater volume, that is, its burning is more progressive. We have seen, equation (10), that the long tubular grain burns with a con- stant surface. The quantity of gas given off in the burning 22 obtained from the above equation by making these substitu- tions. The difference between the two volumes will be the Volume burned, and dividing this resulting volume by Vo we have the value of F. This may be reduced to lo 3m — 2 (R -- 77°)} l F-ºllº tºol # 14. *m. T m (R2–7r?) lo R*–7,”--m (R+7, ) lo * 6lo? l? R*—7,” + m (R+7, ) lo” }. - (11) For the service multiperforated grain we therefore have 2lo; R2–7r” + m (R+7, )} * m (R2–7r?) Y * lo 3m – 2 (R -- 77°)} * X = R” – 7,3-H m (R -- 77°) } - (12) = - ... - 6lo” * † T F-7,3-Eyn (RIETr) J Equation (11) applies only while the web of the grain is burning and does not apply to the slivers. 4 The thickness of web bears the following relation to R and r in our service grains A We will take a specific grain for use later to illustrate the burning of the multiperforated cylinder. The grains of a lot of powder for the 8-inch rifle had the following dimensions: R=0.256° r=0.02551, m-1,029. tº Therefore, from (12), lº–0.044875. Substituting in (11) we obtain for this grain F=0.72867; 14 0.19590%–0.02378;}. (14) lo lo lo - When l = lo, that is, when the grain is reduced to slivers, F = 0.85174 from which we see that the slivers form about 15 per cent. of this particular grain. - As may be readily seen by drawing a diameter throug any three perforations, fig. 1. D–3d R–3r (13) 4 2 23 Emission of Gas by Grains of Different Forms.-As the velocity of combustion under constant pressure is uniform, the time of burning will be proportional, to the thickness of layer burned. *- ... * (. as We may conveniently show the manner of burning of the different grains by dividing, the web into five layers of equal thickness, that is, by giving to the ratio l/lo in the value of the fraction burned, the values 1/5, 2/5, etc., in succession, and then tabulating the resulting values of F. The succes- sive values of F obtained will be the fractional parts burned in 1/5, 2/5, etc., of the total time of burning; and the differences of the successive values of F will be the fractions burned in the successive intervals of time. The following table is formed from equations (8), (9) and (14). For the multiperforated grain the fractions l/lo are fractions of the web only: { } £º " T MULTIPERFORATED 7.2*.* CURE. SLENDER CYLINDER. CYLINDER. F Diff. F Diff. F Diff. 0.0 0.000 * 0.00 ().00 0.49 0.36 0.15 0.2 0.49 0.36 0.15 0.29 0.28 0.16 0.4 0.78 0.64 0.31 0.16 0.20 0.17 0.6 0.94 0.84 0.48 0.05 0.12 0.18 0.8 0.99 0.96 0.66 0.01 0.04 0.19 1.0 1.00 1.00 1.00 1.00 Web 0.85 0.85 !" 0.15 Whole grain 1.00 1.00 Regarding the columns of differences in the table we see that nearly half of the cubical grain is burned in the first layer, and that the volume burned in the successive layers decreases continuously. The slender cylinder emits at first a less volume of gas than the cube and later a greater volume, that is, its burning is more progressive. We have seen, equation (10), that the long tubular grain burns with a con- stant surface. The quantity of gas given off in the burning 24 of each layer is therefore the same, and the grain is more progressive than the slender cylinder. The multiperforated cylinder burns with a continually increasing surface until the Web is consumed, and the volume of gas given off increases for each layer burned. Whether the burning surface of the multiperforated grain increases or decreases depends on the relation between the length of the grain and the radii of the grain and of the per- forations. Referring to equation (11) it will be seen that when - * 3m=2 (R+7, (15) the Second term within the brackets disappears. m is the length of the grain. Giving to the multiperforated grain considered in equation (14) the length indicated in the last equation, we get m = 0.29, and the value of F becomes 2 F=0.91892: 1–0.08134}}. A table formed from this equation will show that this grain burns with a continuously decreasing surface; the fractional volumes burned in the successive intervals being 0.189, 0.186, 0.178, 0.167 and 0.152. The sum of these, 0.872, is the frac- tion of the grain burned when the web ceases to burn. It is apparent that since the manner of burning of a multi- perforated grain depends upon the relation expressed in equa- tion (15), a grain may start to burn with an increasing surface, and change, as the length is diminished, to burn with a decreasing surface. - - The multiperforated grains used in our service are of lengths considerably greater than that indicated in equation (15). The length of the grain is about 2% times the outer diameter. The diameter of the perforations is about 1/10 the exterior diameter of the grain. The grains burn with a continuously increasing surface until the web is burned, and then with a decreasing surface. The Fraction of Charge Burned.—Assuming instant igni- tion of the whole charge, equation (4) expresses the value of 25 the fraction of the charge burned when any thickness, l, has burned. Let G be the weight of the charge, g the weight burned at any instant. The fraction of the charge burned is therefore y/35, which we may write for F in equation (4), and obtain =wełłºt #) - (16) Q/ lo l, Tºlſ Consideration as to Best Form of Grain.-It would appear that the most desirable form of powder grain would be one that gives off gas slowly at first, starting the projectile before a high pressure is reached, and then with an increased burn- ing surface and a more rapid evolution of gas maintaining the pressure behind the projectile as it moves down the bore. It is this consideration that has led to the adoption in our service of the multiperforated grain, which in the preceding discussion is shown to be the only practicable form of grain that burns with an increasing surface emitting successively increasing volumes of gas. The facilities for complete inflam- mation of the charge are not as great in this grain as in some others, as the grains assume all positions in the car- tridge bag and do not present unobstructed channels to the flame from the igniter. We have seen, page 13, that when there is delay in the complete inflammation of the charge, excessive pressures, called wave pressures, may arise, due to the velocity acquired by the gases first formed. - The single perforated cylinder, or tubular grain, offers ad- vantages in this respect. This grain when its length is great compared to the thickness of Web, as when cut in lengths to fit the chamber, burns with a practically constant surface, as we have seen, equation (10). The charge is readily prepared by binding the grains in bundles, and when so prepared offers perfect facilities for the prompt spread of ignition through the uniformly distributed longitudinal air spaces within and between the grains. While larger charges of powder in this form may be re- 26 Quired to produce a desired velocity, the advantages of greater uniformity in velocities and pressures, and decreased likeli- hood of excessive pressures, will probably be obtained by its U1S62. In the process of drying the tubular grain in manufacture the grain will warp excessively if too long with reference to its diameters. On this account and in order that the grain may serve for convenient building into charges its length is limited. The requirement of prompt ignition throughout the length of the grain also limits its length. Good results have been obtained with grains whose length was 85times the outer diameter. WARIOUS DETERMINATIONS. To Determine the Number of Grains in a Pound.—Let n be the number of grains in a pound of powder, Vo the volume of each grain in cubic inches, b the density of the powder. The volume occupied by the solid powder in one pound is evidently n Vo; the volume of one pound of water is 27.68 cu. in...; and the volumes being inversely proportional to the den- sities, we obtain W!, F 27.68 8 V, " and when the number of grains in a pound is known, we have for the density (17) _27.68 n Vo To Determine the Demensions of Irregular Grains.—Ir- regular grains may be considered as spheres, and the mean radius may be determined as follows. Retaining the above significations of n and Vo, let r be the mean radius of the grains in inches. b (18) ---….” 27 Then Vo = 4tr°/3 Šubstituting this in the above equation and solving for r we obtain 1.8766 * - (5m) : Comparison of Surfaces.—Let S1 be the total initial surface of the grains in a pound of powder. As So is the initial Sur- face of each grain, S1=n So. Substituting the value of n from (17) and the value of So from the first of equations (3) we obtain is = ′ = "2 Si = 27.68c. s−) D • 1 blo - D. From which it appears that for two charges of equal weight made up of grains of the same density and thickness of web, the initial surfaces of the two charges are to each other as the value of a for each form of grain. For charges of equal weights composed of grains of the same shape and density the initial surfaces will be inversely pro- portional to the least dimensions of the grains. * (19) DENSITY OF GUN POWDER. The density, or specific gravity, of gunpowder is the ratio of the weight of a given volume of solid powder to the weight of an equal volume of water. The density of gunpowder is determined by means of an instrument called the mercury densinneter, in which is obtained the weight of a volume of mercury equal to the volume of the powder. From the known specific gravity of the mercury that of the powder is readily determined. Mercury is used in the instrument instead of water because mercury possesses the property of closely enveloping the grains of powder without being absorbed into their pores, and it does not dissolve the con- stituents of the powder. The densineter is shown in the accompanying figure. The 28 , , , , . . . . glass globe a is connected with the air pump by the rubber tube c. The lower outlet of the globe is immersed in mercury in the dish d. As the globe is exhausted of air by means of the air pump, the mercury is drawn upward until it fills the globe and stands at a certain height in the glass tube e. The globe is then de- tached, full of mercury, and weighed. It is then emptied, and a given weight of powder placed in it. The globe is then returned to its original position, the air again exhausted, and mercury allowed to enter until it stands at the same height as before. The globe now filled with mercury and powder is again detached and weighed. With the difference of the two weights we may arrive at the weight of the mer- cury whose volume is equal to that of the powder, in the following manner: Let w be the weight of the powder; P the weight of the vessel filled with mercury; P the weight of the vessel filled with mercury and powder; - - D the density of the mercury, about 13.56; b the density of the powder. Then P-w-the weight of the mercury and vessel when the latter is partially filled with powder; - P- (P -w)=the weight of the volume of mercury displaced by the powder. Since the weights of equal volumes are proportional to the densities, we have w : P–P+w : ; b : D, QUID P–P'--w’ The density of charcoal powders varies between 1.68 and 1.85. The density of smokeless powders varies from 1.55 to 1,58, Or b = CHAPTER II. MEASUREMENT OF WELOCITIES AND PRESSURES. Measurement of Velocity.—In measuring the velocity of a projectile the time of p assage of the projectile between two points, a known distance apart, is recorded by means of a suitable in- strument. The calculated velocity is the mean velocity between the two points, and is considered as the veloc- ity midway hetween the points. In order that this may be done without material error, the two points must be selected at such a distance apart in the path of the projectile that the motion of the projectile between the points may be considered as uniform- ly varying, and the path a right line. Le Boulengé Chronograph. — The instrument generally employed for measuring the time interval in the determination of Velocity was in- vented by Captain Le Boulengé of the Belgian Artillery, and is called the Le Boulengé Chronograph. It has been modified and improved by Captain Bréger of the French Artil- lery. The brass column, a Fig. 4, supporting two electromagnets b and c, is mounted on the triangular bed plate d which is provided with levels and levelling screws. The magnet b 30 supports the long rod e, called the chronometer, which is enveloped when in use by a zinc or copper tube f, called the recorder. The magnet c which supports the short rod g, called the registrar, is mounted on a frame which permits it to be moved vertically along the standard. Fastened to the base of the standard is the flat steel spring h which carries at its outer end the square knife i. The knife is held retracted or cocked by the trigger j which is acted upon by the Spring k. The chronometer e hangs so that one element of the envel- oping tube or recorder is close to the knife. When the knife is released by pressure on the trigger it flies out under the action of the spring h and indents the recorder. The registar g hangs immediately over the trigger. When the electric cir- cuit through the registrar magnet is broken the registrar falls on the trigger and releases the knife. The tube l supports the registrar after it has fallen through it. Adjustable guides are provided to limit the swing of the two rods when first suspended. The stand or table on which the instrument is mounted is provided with a pocket which receives the chron- ometer when it falls, at the breaking of the circuit that actuates its magnet. A quantity of beans in the bottom of the pocket arrests the fall of the chronometer without shock. Accessory Apparatus.--To use the instrument for the measurement of the velocity of a shot two wire targets, each made of a continuous wire, Fig. 5, are erected in the path of the proectile. The targets form parts of electric circuits which include the electro-magnets of the its own circuit independent of the other. The cir- cuit from the nearer or first target includes the |||||||||||chronometer magnet; the circuit from the second ſ A. target includes the registrar magnet. On the pas- circuit is broken, the chronometer magnet demag- Fig. 5. netized, and the long rod, or chronometer, falls. When the projectile breaks the circuit through the second target the short rod, or registrar falls, and striking the trigger O chronograph. Each magnet has its own target and sage of the projectile through the first target the 31 releases the knife which flies out and marks the recorder at the point which has been brought opposite the knife by the fall of the chronometer. The chronometer circuit is led through a contact piece not shown, carried by the spring h, and so arranged that the chronometer circuit cannot be closed until the knife is cocked. This arrangement prevents the loss of a record through failure to cock the knife when suspending the rods before the piece is fired. The Rheostat.—Both circuits are led independently through A/ rheostats, by means of which the Hºſ resistance in the circuits may be regulated, and the strength of the sº - SãS currents through the two magnets |- 2-S-27 E cº- is shown in Fig. 6. The current passes through the contact spring a, and through a German silver wire wound in grooves on the wooden drum b. By turning the thumb nut c the contact Spring is shifted, and more or less of the wire is included in the circuit. Anotherform of rheostat, through Fig. 6. which both circuits pass indepen- dently, is shown in Fig. 7. Each current passes through a ſº One form of rheostat C ,242 Fig. 7. strip of graphite a, and the resistance in the circuit may be increased or diminished by sliding the contact piece b so as to include a greater or less length of the graphite strip in the circuit. 3 32 The Disjunctor.—Both circuits also pass independently through an instrument called the disjunctor, by means of which they may be broken simultaneously. The disjunctor is shown in elevation and part section in Fig. 8. The two * -]. .# |{H}, == \ H-Hºlº-ET1 lºss E=== Fig. 8. halves of the instrument are exactly similar. The two contact Springs c weighted at their free ends, bear again insulated contact pins e supported in a metal frame d. The frame is pressed upward against the spring catch, h by two other con- tact springs, f. The electric circuit passes from one binding post through the parts f, e, c and a to the other binding post. On the release of the ſcatch/Spring the frame d flies upward under the action of the springs f until stopped by the pin g. At the sudden stoppage of the movement the weighted ends of the contact springs simultaneously leave the contact pins, thus breaking both circuits momentarily. Mounted on a shaft are two hard rubber cams b, which bear against other springs a in the two circuits. On turning the cam shaft the connection between the parts a and c is broken, breaking both electric circuits but not necessarily simultaneously. The circuits are habitually broken in this manner except when taking disjunction or records in firing. Disjunction.—By means of the disjunctor both circuits are broken at the same instant. The mark made by the knife under these circumstances is called the disjunction mark, and its height above a zero mark made by the knife when the 33 chronometer is suspended from its magnet is evidently the height through which a free falling body moves in the time used by the instrument in making a record. This time includes any difference in the times required for demagnetiza- tion of the two magnets, the time occupied by the registrar in falling, and the time required for the knife to act. - From the height as measured we obtain the corresponding time from the law of falling bodies t= (2h)/g)%. Now when the circuits are broken by the projectile the chronometer begins to fall before the registrar. The mark made by the knife will therefore be found above the disjunc- tion mark. If we measure the height of this second mark above the zero, the corresponding time is the whole time that the chronometer was falling before the mark was made, and to obtain the time between the breaking of the circuits we must subtract from this time, the time used by the instru- ment in making a record, or the time corresponding to the disjunction. Let hi and he represent the heights of the dis- junction and record marks, respectively, ti and tº the corres- ponding times. Let t be the time between the breaking of screens, then . t=t2—tl= (2h9/g)}6– (2h1/g)%. It will be seen by the equation that the difference of the times, and not the difference of the heights must be taken. FIXED DISJUNCTION. For the velocity at the middle point between targets we have, representing by s the distance between the targets - - - . was/t. | Substituting for t its value, we have *= S *T (2h,77)} = (ghºſº' From this equation we see that if the value of s, and of (2h1/g)”, or the disjunction, be fixed, the values of v can be calculated for all values of he within the limits of practice 34 I -3 & -!- g º Fig. 9. and tabulated. This has been done for the values s = 100 feet and (2h1/g)* = 0.15 seconds, and this value of (2h1/g)% is called the fixed disjunction. If such a table is not at hand, the fixed value of the disjunction avoids the labor of calculat- ing (2h1/g)% for each shot. - In this case - t=t2–0.15 sec. = (2h9/g)}8–0.15. In Ordinary practice it is better to take the disjunction at each shot, and to keep the disjunction mark near the dis- junction circle but not necessarily on it. The times corres- ponding to the heights of the disjunction and record marks are both read from the table, and with the difference of these times the velocity is taken from another table. Measuring Rule.—For measuring the height of the mark on the recorder above the zero mark there is provided with the instrument a rule graduated in millimeters, and with a sliding index and vernier, the least reading being ºn of a millimeter. The swivelled pin at the end of the rule, Fig. 9, is inserted in the hole through the bob of the chronometer, and the knife edge of the index is placed at the lower edge of the mark whose height is to be measured. The height is then read from the scale. Tables are constructed from which can be directly read the time corresponding to any height in millimeters within the limits of the scale. The maximum time that can be measured with this chronograph is limited by the length of the chronometer rod, and is about 0.15 of a second. Adjustments and Use.—The instrument must be properly mounted on a stand at such a distance from the gun that it will not be affected by the shock of discharge. The electrical connections with the batteries and targets, through the rheostats r and disjunctor d are made as shown in Fig. 10. To adjust the instrument first level it by the leveling screws, cock the knife and suspend the chronometer rod, enveloped by the recorder, from its magnet. See that the recorder hangs close to the knife and that no part of the base of the rod 35 touches any part of the instrument. The guides must be close to, but not touching, the bob of the chronometer. De- press the trigger. The knife will mark the recorder near the bottom. Apply the measuring rule to the chronometer, the zero of the index vernier being at the zero of the scale. Adjust the knife edge of the index to the mark on the recorder. Slide the index to the mark “Disjunction” on the rule, and letting the knife edge bear against the recorder turn the recorder around the chronometer rod. The knife edge will scribe a circle on the recorder, and the mark made at disjunction should fall on or near this circle. - Fig. 10. To regulate the strength of the magnets each of the rods is provided with a tubular weight, one tenth that of the rod. Place the proper weight on each rod and suspend the rods from their magnets. Increase the resistance in each circuit by slowly moving the contact piece of the rheostat until the rod falls. Remove the weights from the rods and again sus- pend the rods. Take the disjunction. If the bottom of the mark made by the knife does not lie on or near the circle previously scribed on the recorder raise or lower the registrar magnet until coincidence is nearly obtained. Test the disjunctor by shifting the two circuits. The height of disjunction should remain the same. Test the circuits by suspending the rods and causing the 36 circuits to be broken successively at the two targets. Note that the proper rod falls as each circuit is broken. Always suspend the chronometer rod with the same side of the bob to the front, and always, before suspending it, press the recorder hard against the bob. After each record turn the recorder slightly on the rod to present a new element to the knife. . Circuits should always be broken at the disjunctor when the rods are not actually suspended, and the rods should be allowed to remain suspended as short a time as possible. Targets.-The first target must always be erected at such a distance from the gun that it will not be affected by the blast. For small arms it is placed three feet from the muzzle and consists of fine copper wire wound backward and forward Over pins very close together. For cannon it is placed from 50 to 150 feet from the muzzle, depending upon the size of the gun. For the measurement of ordinary velocities the targets are usually placed, for small arms, 100 feet apart, and for cannon, 150 feet. - The second target for small arms consists of a steel plate to stop the bullets, having mounted on its rear face, and in- sulated from it by the block w, Fig. 11, a contact spring s, -- A. contact pin p, and their binding screws. When the bullet strikes the plate the shock causes the end of the spring to leave the pin, and thus breaks the circuit, which is immediately reestablished by the reaction of the spring. By means of this device constant re- pairing of the target is avoided. * Fig. 11. Measurement of Very Small Intervals of Time.—For the measurement of very small time intervals the registrar mag- net is raised to near the top of the standard and placed in the circuit with the first target. The chronometer magnet is put in the circuit with the second target. Under this arrange- ment the disjunction mark will be made near the top of the recorder and the record mark under the disjunction. The interval of time measured is obtained by subtracting the * time corresponding to the height of the record mark, front). .| 37 the time of disjunction. The object of this arrangement is to obtain the record when the chronometer has acquired a considerable velocity of fall, so that the scale of time will be extended, and small errors of reading will not produce large errors in time. Schultz Chronoscope.—The LeBoulengé chronograph meas- ures a single time interval only. When it is desired to measure the intervals between several successive events an instrument that provides a more extensive time scale is required. | Ol (). |||ſillº) ---b - ..C. d ſ - d Fig. 12. whole length of the cylinder. The Schultz Chronoscope is an instrument of this class. An electrically sustained tun- ing fork, c, Fig. 12, whose rate of vibration is known, carries On One time a quill point, b, wihch bears against the blackened surface of the re- volving cylinder a and marks on it a sinusoidal curve which is the scale of time. By giving motion of translation to the cylinder past the fork the time scale may be ex- tended helically over the The records of events, such as the passage of the shot through Screens, are made by the breaking of Successive circuits which pass through the Marcel Deprez registers shown at e, Fig. 12, and in Fig. 13. When the circuit is broken the magnet e, Fig. 13, is demagnetized, and the spring g rotates the armature f and the quily h attached to it. This marks a bend or offset in the trace of the quill On the revolving cylinder and the point of the bend referred to the time scale marks the instant of the breaking of the circuit. --~~~~~~~~~ steel ribbon S, Fig. 14, is attach ºn Mººs --> tº: Sºº * - f sº I, y ... x . Gravimetric Density of Powder.—The density of powder is, as has been explained on page 27, the ratio of the weight of a given volume of powder to the weight of an equal volume of water. In determining density the volume considered is the volume actually occupied by solid powder. 47 Gravimetric Density is the mean density of the contents of the volume that is exactly filled by the powder charge. The air spaces between the grains are considered as well as the solid powder in the charge. The gravimetric density is obtained by dividing the weight of the charge by the weight of water that will fill the volume occupied by the charge. It is evident that if a solid block of powder of a given density is brokerſ up into grains, the Volume occupied by the powder will increase and will be dependent on the form and size of the grains. While the actual density of the solid powder does not change, the gravimetric density will depend upon the granulation. - A cubic foot of powder is usually taken in determining gravimetric density. A cubic foot of water weighs 62.425 fos. Let Y be the gravimetric density of the powder, |W the weight of a cubic foot of powder. Then by definition, ^ = W/62.425. If the chamber of the gun were filled with a solid cake of powder the value of Y would be the density of the powder. In practice the value of Y usually lies between 0.875 and 1.0, though sometimes the value unity is exceeded. The space actually occupied by the solid powder in a given volume is determined as follows: Let V be the total volume occupied by the powder, o the volume of the solid powder. - Since volumes of equal weights are inversely as the den- sities, we have V; o;:8 : Y or n = . V. * (20) It is evident from this equation that when the gravimetric density of the powder is unity, the volume of the solid powder is equal to the volume of the charge multiplied by the reciprocal of the density of the powder. If Y=1.00 and b-1.76 (a mean value for charcoal powders) we have = .57 V. (21) 48 That is, the volume occupied by the solid powder in a charge is about .57 of the total volume of the charge. Density of Loading.—Density of loading is the mean density of the contents of the whole powder chamber. In addition to the solid powder and the air spaces between the grains, the space in the chamber not occupied by the powder charge is also considered. The density of loading is obtained by dividing the weight of the charge by the weight of the water that will fill the powder chamber. Let A be the density of loading, gº the weight of the charge in pounds, C the volume of the chamber in cubic inches. Then since one pound of water occupies 27.68 cubic inches, C/27.68 will be the weight of water that will fill the chamber, and - A= G-: C/27.68=27.68 g/C. (22) If instead of the cubic inch and pound we use as units the cubic decimeter and kilogram, the number of cubic deci- meters in the volume of the chamber will express at once the weight of water in kilograms that will fill the chamber, since one kilogram of water occupies a volume of one cubic deci- meter. Using metric units the above expression therefore becomes A = G3/C. (23) The density of loading may also be expressed in terms of the density of the powder as follows: Let O’ be the volume in cubic inches occupied by the solid powder of the charge; b the density of the powder. bo' will then be the volume of an equal weight of water, and W=80/27.68, (24) which substituted in equation (22) gives * A=öC')/C. (25) The accompanying figure will serve to illustrate the differ- ence between density, gravimetric density and density of 49 loading. The figure represents a section of the whole cham- ber of a gun charged with powder to the line A. The density of loading is in this case the weight of powder below the line A divided by the weight of water that will fill the whole chamber. The gravimetric density is the weight of the powder divided by the weight of water that will fill all that part of the chamber below the line A. Now considering the powder charge as com- pressed into a solid mass at the bottom of the chamber, represented by the black portion, the density of the powder will be its weight divided by the weight of water that will fill this black portion. As the weight of water that will fill each volume is equal to the volume in cubic inches divided by 27.68, we have: g º - 27.68 gº Density of Loading, A=y. chamber . . . . º 27.68 & Gravimetric Density, Y= vol. of charge 27.68 g) Density, b= vol. of solid powder Using metric units the factor 27.68 will be omitted. Reduced Length of Powder Chamber.—For convenience in the mathematical deductions the volume of the powder cham- ber is reduced to an equal volume whose cross section is the same as the cross section of the bore. The length of this volume is called the reduced length of the powder chamber. Let Mo be the reduced length of the chamber, to the area of cross section of the bore, C the volume of the chamber, d the diameter of the bore. Then C= woo) = Moſt d”/4 and wo- 4 O/td.” (26) 50 Reduced Length of Initial Air Space.—The air space in the loaded chamber, which includes all the space in the cham- ber not occupied by solid powder, is also reduced to a volume whose cross section is that of the bore. The length of this volume is called the reduced length of the initial air space. Let 20 be the reduced length of the initial air space, in inches. Then since C is the volume of the chamber and O' the volume of the solid powder, . C – O' CO 30 F Substituting for C and O' their values from equations (22) and (24) 1 1 2.0-27.687(4– #). - b – A Make 0.2 = A3 (27) Then 200–27.68aº, (28) and since - to = Tº dº/4, 20=35.2441a2a3/d?=|[1.54709] a”g/d? (29) the number in square brackets being the logarithm of 35.2441. Problems.-The volume of the chamber of the 3-inch field rifle is 66.5 cu. in. The weight of the charge is 26 oz. Density of the powder 1.56. What is the density of loading, and what is the reduced length of the initial air space # Ans. A=0.6764 20=5.33 inches. 2. If the gravimetric density of the powder in the last ex- ample is unity, how many pounds will the chamber hold 3 2.4 lbs. 3. The reduced length of the initial air space in the 8-inch rifle loaded with 80 lbs. of powder, density 1.56, is 43.72 inches. What is the capacity of the chamber 3 O=3617 cu. in. 51 4. The 5-inch siege gun has a chamber capacity of 402.5 c.u. in. What is the density of loading with a charge of 5.37 lbs 3 A=0.3693. 5. The 4-inch rifle when loaded with 12 lbs. of sphero-hex- agonal powder has a density of loading of 0.915. What is the chamber capacity ? - - O=363 cu. in. 6. The 12-inch rifle has a chamber capacity of 17487 cu. in. The density of loading is 0.5936. What is the weight of the charge, and what is the volume of the solid powder in the charge 3 b-1.56. - co–375 lbs. Solid volume=6654 cu. in. 7. What is the reduced length of the initial air space in the last example 3 - 20=95.79 inches. 8. The chamber capacity of the 6-inch rifle is 2114 cu. in. What is the reduced length of the chamber 2 wo-74.77 inches. PROPERTIES OF PERFECT GASES. Marriotte’s Law.—At constant temperature the tension, or pressure, of a gas is inversely as the volume it occupies. As the density of a gas is inversely as its volume, this law may also be expressed: At constant temperature the pressure of a gas is proportional to its density. Let v be the volume of a given mass of gas, p its pressure in pounds per unit of area. Then if the volume occupied by the gas be changed to vo, the temperature of the gas being kept constant, the pressure Will change according to the law. 'Up = COnstant. Let p0 represent the normal atmospheric pressure, barome- ter 30 inches. 52. p0=14.6967 pounds per square inch, or 103.33 kilograms per square decimeter. wo the volume of unit weight of a gas at 0°C. under normal atmospheric pressure. Then by Marriotte’s law, at 0°C., . Oſ)=?)0'00. (30) Specific Wolume.—The specific volume of a gasis the volume of unit weight of the gas at zero temperature and under nor- mal atmospheric pressure. v0 in the above equation is the Specific volume of the gas. In English units the specific volume of a gas is the number of cubic feet occupied by a pound of the gas under the above conditions. Specific Weight.—The specific weight of a gas is the weight of a unit volume of the gas at zero temperature and under normal atmospheric pressure. It is the reciprocal of the specific volume. Gay-Lussac’s Law.—The coefficient of expansion of a gas is the same for all gases; and is sensibly constant for all temperatures and pressures. - Let vo be the specific volume of a gas, v, its volume at any temperature t, and a the coefficient of expansion. Then the variation of volume under constant pressure by Gay-Lussac's law will be expressed by the equation Q); — vo - divo, Ol' * Q); F 2)0 (1+qt). The value of a is approximately 1/273 of the specific volume for each degree centigrade. The above equation may there- fore be written v=x(i+3). (31) Characteristic Equation of the Gaseous State.—The last equation, which expresses Gay-Lussac’s law, may be combined with Marriotte's law introducing the pressure p. 53 Let a, be the volume that v, would become at 0°C., under the pressure pe. Then by Gay-Lussac’s law v º w , V . . . . . . ^), - 4 (1+ at), y & ~ : * ^ V Nº. `--~ ~~\º. *A*-4- º . . . . . .- but by Marriotte’s law * pea = povo, --, -º **** s “. …º. `--~~~~ - whence, eliminating a . pºve-povo (1+ ot) =# (273–H t). Since povo/273 is constant for any gas, put R=pov/273, (32) whence, dropping the subscripts as no longer necessary, pº) = R (273-H t). - The temperature (273+t) is called the absolute temperature of the gas. It is the temperature reckoned from a zero placed 273 degrees below the zero of the centigrade scale. Calling the absolute temperature T there results finally pº) = RT, (33) which is called the characteristic equation of the gaseous state, and is simply another expression of Marriotte’s law in which the temperature of the gas is introduced. Equation (33) expresses the relation that always exists between the pressure, volume, and absolute temperature of a unit weight of gas. To apply it to any gas, substitute for wo in the value of R, equation (32), the specific volume of the particular gas. . For any number w units of weight occupying the same vol- Wme the relation evidently becomes pw = w RT. (34) A gas supposed to obey exactly the law expressed in equa- tion (33) is called a perfect gas, or is said to be theoretically in the perfectly gaseous state. This perfect condition repre- Sents an ideal state towards which gases approach more nearly as their state of rarefaction increases. A N 54 Thermal Unit.—The heat required to raise the temperature of unit weight of water at the freezing point one degree of the thermometer, is called a thermal unit. Mechanical Equivalent of Heat.—The mechanical equiva- lent of heat is the work equivalent of a thermal unit. It will be designated by E. The unit weight of water being one pound, the value of E for the Fahrenheit scale is 778 foot pounds; and for the centigrade scale, 1400.4 foot pounds. Specific Heat.—The quantity of heat, expressed in thermal units, which must be imparted to unit weight of a given sub- stance in order to raise its temperature one degree of the thermometer above the standard temperature, is called the specific heat of the substance. The specific heat of a gas may be determined in two ways: under constant pressure, and under constant volume. Suppose heat to be applied to a unit weight of gas retained in a constant volume whose walls are impermeable to heat. The whole effect of the heat will be to raise the temperature of the gas. If, however, the gas is inclosed in an elastic envelope, supposed to maintain a constant pressure on the gas, the gas will expand on the application of heat, and part of the heat applied will perform the work necessary to expand the envelope. Therefore to raise the temperature of the gas one degree, a greater amount of heat must be applied when the gas is under constant pressure than when under constant volume; and the difference of these quanti- ties, that is, the difference between the specific heat under constant pressure, co, and the Specific heat under constant volume, co, will be the heat that performs the work of expan- sion. The mechanical equivalent of a heat unit being repre- sented by E, we may write Work of Expansion = (co-co) E. (35) Actually, part of the work that we have included in the work of expansion is internal work, used in overcoming the attractions between the molecules; but the quantity of work so absorbed is small and is omitted in the discussions. 55 The work of expansion is equal to the constant resistance multiplied by the path. We will assume the constant resist- ance to be the atmospheric pressure, po. The path is measured by the increase of volume of the gas. To deter- mine the latter we have from Gay-Lussac’s law, for the cen- tigrade scale, equation (31), - Q); - ?)0 = tvo/273, and therefore for an increase of temperature of one degree there is an increase of volume equal to vo/273. The work of expansion for one degree is therefore povo/273. Referring to equation (32) tº povo/273 = R. (32) The quantity R is therefore the external work of expansion performed under atmospheric pressure by unit weight of gas when its temperature is raised one degree centigrade. But this work of expansion has been found above to be equal to (co-co) E. Therefore we may write (co-co) E=R=p000/273. (36) From the definition of specific heat we deduce, that the Quantity of heat necessary to raise the temperature of unit Weight of gas any number of degrees, as t, will be Q=ct, (37) c representing either co or cy. Ratio of Specific Heats.—In the study of interior ballistics the particular values of cp and c, for the different gases which are formed by the explosion of gunpowder are of little im- portance. It suffices to know their ratio, which is constant for perfect gases, and approximately so for all gases at the high temperature of combustion of gunpowder. The ratio of the specific heats coſc, is subsequently desig- nated by n. - Relations Between Heat and Work in the Expansion of . Gases.—The relation which exists between the variations of -the-Volume-and-the-pg 3e-et-ar be determined from equation 33) pv=RT, (33) 56 which contains the three variables p, v and T. If we suppose an element of heat, da, to be applied to the gas, the effect will be generally an increase in the temperature, accompanied by an increase in the pressure, or in the volume, or in both the pressure and the volume. Considering p constant, and differentiating, we get dT=pdv/R, and the quantity of heat communicated to the gas will be dq=c,d T=cºpdv/R. Considering v constant we obtain similarly dq=cºvdp/R. If p and v both vary, we obtain from the sum of the partial differentials, Act ºf . . . . . . .”— — . | - -- ^ dq=#(opdo-Head). - (38) The differential of equation (33) is Rd T=pdy-Hodp. (39) Eliminating vap between the last two equations we have ...W. & dq=codT-H ***pdo. - (40) The quantity pav represents the elementary work of the elastic force of the gas while its volume increases by dw. The integral of pdv is therefore the total external work be- tween the limits considered. Represent by Ti and T the initial and final temperatures, and by W the whole external work. Integrating equation (40) between the limits Ti and T we obtain, since co, co, and R are constant for the same gas a = c (T-T)+*** W. (41) Isothermal Expansion.—If we suppose the initial tempera- ture Ti to remain constant, that is, that just sufficient heat is imparted to the gas while it expands to maintain its initial 57 $º temperature, the quantity Ti-T' in equation (41) becomes 0, and solving with respect to W we obtain W= { * - Q. Cp- Co We see that in this case, since R, co, and c, are constant for the same gas, the external work done is proportional to the Quantity of heat absorbed by the gas. Making q equal to one thermal unit, W becomes E, and we obtain, as before in equation (36), - E (co-co) = R. (36) Adiabatic Expansion.—If a gas expands and performs work in such a manner that it neither receives nor gives out heat, the transformation is said to be adiabatic. In this case the temperature and pressure of the gas both diminish, and the work performed will be less than for an isothermal expan- SIOIl. Since no heat is gained or lost, q becomes 0 in equation (41), and we have W= R-4-(1,– T). Cp - Co Make CoAce = n. _ R - Then - W=;*H (T-T). (42) This equation gives the value of the external work done by a unit weight of gas whose temperature is reduced from Ti to T in an adiabatic expansion. It will be seen that the ex- ternal work done is proportional to the fall of temperature. LAW CONNECTING THE VOLUME AND PRESSURE.-In the adia- batic expansion the heat in the gas remaining unchanged, dq in equation (38) becomes 0, and we have º - 0 = Copdo + c, vap. Dividing through by copy we find, since co/Co - n do dp_ 0 n; +; 58 and integrating n logov -- logap = log ec, whence Q)” p = COnstant = v1"p1, Q) 7?, Ol' 70 = 01 (...) g (43) This equation expresses the relation between the volumes and pressures of a gas in an adiabatic expansion. Problems.—Equations (30) to (34) are used in solving the following problems. Specific volumes: Air - - vo = 12.391 cu. ft. Hydrogen vo = 178.891 cu. ft. Coal gas - v0 = 24.6 cu. ft. Water gas vo = 18.09 cu. ft. 1. A volume of 3 cubic feet of air, confined at 59°F. (15°C.) and 30" barometer, is heated to a temperature of 300° C. What pressure does it exert Ž Vol. of 1 lb. air at 15°, equation (31), v, = vo.288/273. 3/v, = w. Equation (34), p = whº T'/v = 29.24 lbs. per sq. in. 2. Two pounds of air confined in a volume of 1 cubic foot exerts a gauge pressure of 679.76 lbs. per square inch. What is its temperature by the Centigrade and Fahrenheit scales 3 The total pressure p is the guage pressure plus the atmos- pheric pressure, p = 679.76 + 14.70 = 694.46. Equation (34), T = p^)/w R = 520.54, t = 247°.54 C. = 477°.57 F. 3. A spherical balloon 20 feet in diameter is to be inflated with hydrogen at 60° F., barometer 30.2 inches, so that gas may not be lost on account of expansion when the balloon has risen until the barometer is at 19.6 inches and the temper- ature 40°F. How many cubic feet of gas must be put in the balloon ? The gas pressure in the balloon is in equilibrium with the - & - ; , , X----' A $.” & *—” so 59 atmospheric pressure. The weight of gas occupying the bal- loon must be such that at 40° F. the pressure will be in equilibrium with a barometric pressure of 19.6 inches. p = p × 19.6/30 ^ = volume of balloon. Equation (34), w = p^)/RT = 15.05 lbs. Volume of w at 60° F. and 30".2 barometer: p = po x 30.2/30, w = w RT/p = 2827.4 cubic feet. 4. What is the lifting power at 70° F. (21°.11C.) and 30 in. barometer of 1000 cubic feet of each of the gases whose specific volumes are given. Vol. 1 lb at 70°. Pounds in Lifting power Equation (31). 1000 cu. ft. wj ft. Air tº- tº- - 13.35 74.91 Hydrogen tº- 192.73 5.19 69.72 Coal gas - - 26.5 37.73 37.18 Water gas E_ 19.49 51.31 23.60 5. The balloon in which Wellman intends to seek the North Pole has a capacity of 224,244 cubic feet, and weighs with its car and machinery 6600 lbs. What will be its lifting capacity when filled with hydrogen at 10° C. and 30 inches barometer ? Ans. 9647 lbs. NOBLE AND ABEL’S EXPERIMENTS. In 1874 and again in 1880, Captain Noble of the English Army and Sir Frederick Abel published the results of their experiments on the explosion of gunpowder in closed vessels. The purpose of their experiments was to determine definitely the nature of the products of combustion, the volume and temperature of the gases, and the pressures with different densities of loading. Apparatus.-The steel vessel in which the powder was ex- ploded was of great strength and capable of resisting Very high pressures. 60 The charge of powder was introduced through the opening a which was then closed with a taper screw plug. A pressure gauge d was inserted in the plug c and an outlet was pro- vided at e through which the gas could be drawn offif desired. The charge was fired by electricity. t The vessels were of two sizes. In the larger one, a charge of 2.2 pounds of powder was fired, and the gases wholly re- tained. Black powder was used in the experiments. The gravimetric density of the powder used was unity, so that when the chamber was completely filled the density of loading was also unity. Results of the Experiments.-Character of the Products. The products of combustion were found to consist of about 43 per cent by weight of permanent gases, and about 57 per cent of non-gaseous products. The non-gaseous products ultimately assume the solid form, but are liquid at the mo- ment of the explosion. This was determined by tilting the vessel at an angle of 45 degrees, one minute after the explo- sion. 45 seconds later it was returned to its original position. On opening the vessel the solid residue was found inclined to the walls at the angle of 45 degrees. The permanent gases are principally CO2, N, and CO, and the solids K2CO3, K, S, K2 SO4 and S. With the exception of the K2 S and the free sulphur, the products agree in character 61 * with those expressed in the formula generally adopted as approximately representing the reaction of black powder on explosion. 4KNO,-- C, -i- S = K, CO; + K, SO, + Na+ 2CO2 + CO. The formula however gives 64% per cent by weight of per- manent gases and 35% per cent of Solids. - It was found, as was to be expected, that in a closed vessel variations in the size, form or density of the grains had practi- cally no effect on the composition of the products of combus- tion, or on the pressures. Wolume of Gases.—Noble and Abel found that the gases, when brought to a temperature of 0° C. and under atmos- pheric pressure, occupied a volume about 280 times the volume of the unexploded powder. Specific Wolume of Gunpowder Gases.—To simplify some- what the discussions concerning the gases of fired gunpowder we will use as the specific volume the volume, at 0°C. and under atmospheric pressure, of the gases produced by the combustion of unit weight of powder. That is, we will con- sider this weight of gas as unit weight. Relation Between Pressure and Density of Loading.—The relation between the pressure, volume, and absolute tempera- ture of the gases from gº units of weight of powder at the moment of explosion is given by equation (34) pv = q R.T. (34) Make f = R T' (44) and we obtain from (34) for the pressure exerted by the gases from ſº pounds of powder occupying the volume v at the tem- perature of explosion p = f&/v. (45) FORCE OF THE POWDER.—If we make both & and v unity in this equation p becomes equal to f. f is therefore the pres- sure per unit of surface exerted by the gases from unit weight 62 of powder, the gases occupying unit volume at the tempera- ture of explosion. f is called the force of the powder. . Let Q be the volume of the residue from unit weight of powder. C the volume of the chamber. Then the volume occupied by the gas from gº units of powder will be - ^) = 0–c.35. We may introduce the density of loading, using metric units, by substituting for Cin this equation its value Ö/A from equation (23), and obtain * v=º (1–a A). | Substituting this value of v in (45) we obtain This equation expresses the relation between the pressure of the gases and the density of loading. When A 1 – CA 1. 1 + C. = 1 that is, when A = (47) f = p. . Comparing the value of A in equation (47) with the general value, A=GAC, we see that in (47) the weight of powder is unity, and the volume of the chamber 1 + &. The volume occupied by the gas is therefore also unity. The pressure therefore becomes in this case the force of the powder as defined above. Tº By substituting in equation (46) two observed values of p corresponding to different values of A, the values of a and f were determined. As the mean of many observations Noble and Abel finally adopted the values: C = 0.57, = 18.49 tons per sq. in., = 29.1200 kilograms per sq. decim. 63 The pressure for any density of loading is given by the equation p = 18.49 tons per Sq. in. –*— 1 – 0.57 A As the constants in this equation were determined for powders whose gravimetric density was unity, the equation applies only to such powders. Under that condition, when A = 1 the charge completely fills the chamber, and the equa- tion gives p = 43 tons per Sq. in. . The value of C., 0.57, means that the volume occupied at the temperature of explosion by the liquid residue from one kilo- gram of powder is 57/100 of one º decimeter. With gravi metric density unity one kilogram of powder occupies OIle já. Referring now to equation (21), we see that the solid powder, of ordinary density and of gravime- tric density unity, occupies 57/100 of the volume of the charge in granular form. The volume of the liquid residue at the temperature of explosion is therefore practically equal to the volume of the solid powder in the charge. - —Temperature of Explosion.—The temperature of explosion may now be determined from equation (44), which with (32) gives • - f = R T =% T. oo is the volume occupied by the gas from unit weight of powder. Since the volume of this quantity of gas is 280 times the volume of the powder, and one kilogram of powder Occupies one cubic decim., v0 = 280 cubic decims. po, the atmospheric pressure, is 103.33 kgs. per sq. decim. Substi- tuting these with the value of f, 29.1200 kilos per sq. decim., we find T = 2748° C. As this is the absolute temperature, Subtracting 273 we find the temperature of explosion to be 24.75° C. Captain Noble later considered the absolute temperature as 2505° C. The approximate correctness of these temperatures was verified by the introduction of pieces of fine platinum wire 64 into the explosion chamber. The platinum, which melts at about 2000° C., was partially fused. s ſ T Mean Specific Heat of Products.-The quantity of heat given off by one kilogram of powder was found to be 705 calories, that is, the heat necessary to raise 705 kilograms of º water one degree centigrade. From the relation Q = ct, equa- tion (37), t being the actual temperature of explosion not the absolute, a value was found for the mean specific heat of the \ products 705 *** - - --> --- - , C = 2505 T273 = 0.316. Relations Between Wolume and Pressure in the Gun.— Noble and Abel found, contrary to their expectations, that the pressures in closed vessels did not differ greatly from the pressures in guns when the powder in the gun was wholly consumed or nearly so. They concluded from this that the expansion of the gases in the gun did not take place without the addition of heat; but that the gases received during the expansion the heat stored in the finely divided liquid residue. TSLet C1 be the specific heat of the residue, assumed to be con tant. 9 The elementary quantity of heat given up by each unit weight of residue will then be cid T. If there are wi units Of weight of residue, wicid T units of heat will be yielded to the gases, and if there are we units of weight of gas each unit will receive, in heat units da = - #oat'--Boat. 3 being the ratio w/w2, and the negative sign being used because T decreases while q increases. Substituting this value of da in equation (40) it becomes — (C, -H 3 C1) dT=***pdo. Eliminate Rdſby means of (39); divide through by po, and integrate, considering cº, cº, c, and B constant. We will obtain 5C1 + Cp ~ ( 48 ) * Q) e S ſp=p1 (...) [3C1 + Co `. - *~ “. . *... 65 rº' # 3 ºre º 3. f * | | wº t ſ & | t en there is no residue 5 is 0, and the equation becomes 1 with equation (43) which was deduced for anadiabatic In both these equations v1 and v are the volumes actually occupied by the gases, exclusive of the residue. Assume the gravimetric density and density of loading to be unity, that is, the chamber is filled with powder, and that the powder is Yi burned before the projectile moves. Then w1 in equation (48) will be the volume occupied by the gases in the chamber of the gun, and p1 the corresponding pressure. If we call v' the volume of the chamber, Cºv' will be the volume of the residue, and v' \- ov' = v1 the volume of the gases; and if we call v" the volume behind the projectile at any instant, the volume v occupied Nby the gases becomes v" – av’ = v. Equation (48) therefore becomes - f > ſºci + C - p=p(; gº)†. (49) * v” – ap' º These values for the constants were determined in the ex- periments. º p1 = 43 tons per square inch. C = 0.57 v' = 27.68 65. 3 = 1.2957 Cp = 0.2324. C1 = 0.45 Co F 0.1762, From these values we find the ratio of the specific heats, co/c, = n = 1.32. The value of the exponent in (49) is 1.074. Theoretical Work of Gunpowder.—The general expression for the work done by a gas expanding from a volume v1 to a volume v is \ . _ (?) º | W = J. pdv. \ Substituting for p its value from (43) and integrating - — £101" 1 - ?), l; * W n-i" + C. ſ)12)1 |W = 0 when v = v1, therefore C = 77 – 1 ° and ``'`--- plvi ſi (v1)*T* `-- - W == %–1 {1 (...) } W -- 66 Assuming that the powder is all burned before the projec- tile moves, and that the gravimetric density and density of loading are unity, the values v1 and v in this equation may be replaced as indicated in equation (49), and we obtain w-nº-(º)" 70 – 1 Q)” – ov' This is the expression for work under the adiabatic expan- sion for which n = 1.32. If we substitute for n the value 1.074 which is the value of the exponent in equation (49) the equation will then apply to Noble and Abel's hypothesis. Work at Infinite Expansion.—When the length of the bore is infinite, v", which is the volume behind the projectile, is infinite, and we have . Jy = pit' (1 - a) - n-1 To obtain the work of the gases from one pound of powder make v' = 27.68 cubic inches, the volume occupied by One pound, the gravimetric density being unity. Make n = 1.32, and substitute for the other constants the values given on page 65. Divide by 12 to reduce from inch-tons to foot-tons. We find for the work Of One pound of powder expanding adiabatically to infinity \ - W = 133.3 foot-tons per pºund. Substituting for ºn the value of the exponent in equation (49), 1.074, we obtain under Noble and Aºtºvºi. that the gases received heat from the residue, . - W = 576.35 foot-tons per pound. FORMULAS FOR WELOCITIES AND PRESSURES IN THE GUN. Formulas connecting the velocity of the projectile with its travel in the bore may be deduced from the relations we have established involving the work of the powder; but these 67 formulas, while they include the force of the powder, do not include consideration of the individual characteristics of dif- ferent powders, such as form and size of grain, density, and velocity of combustion in the air; nor consideration of the effect on the combustion of the variable pressure in the gun. M. Emile Sarrau, engineer-in-chief of the French powder factories, was the first to include these elements in ballistic formulas. He considers the progressive combustion of the charge under the influence of the varying pressure in the gun, regarding the powder as a variable in the formulas. The individual characteristics of the powder employed enter the formulas, which thereby become applicable to the deter- mination, in advance, of the proper weight of charge, the kind of powder, the best form and size of grain to produce desired results in a given gun. Sarrau assumes that the time required for complete inflam- mation of the charge is negligible compared with the time of combustion. He also assumes an adiabatic expansion of the ga,Ses. This latter assumption, while incorrect according to the ex- periments of Noble and Abel, is now generally made by writers on interior ballistics; and whatever erroris introduced through the assumption is later corrected in the determina- tion, by experiment, of the constants in the formulas. Differential Equation of the Motion of a Projectile in a Gun.—Let . g be the weight of powder burned in the time t. Ti the absolute temperature of combustion. T the absolute temperature of the gas at the time t. The Work of a unit weight of gas in an adiabatic expansion between the temperatures T, and T is given by equation (42). For a weight of gas y we have W = y R 77 – 1 From equation (44), since T now represents the tempera- ture of explosion, the value for the force of the powder is (Ti — T). 68 f = RT1; and from equation (34), pv = y R.T. With these Substitutions the above equation becomes (n − 1) W = fu — pºp. (50) In this equation v is the volume occupied by the gases at the temperature T and at the time t. Let u be the distance travelled by the projectile at the time t. to the cross section of the bore. - 20 the reduced length of the initial air space. Principle of the Covolume.—Experiment shows that at the high temperature of explosion powder gases do not accu- rately obey Marriotte’s law, and that the introduction of a subtractive term in the expression for the volume occupied by the compressed gases is necessary. The characteristic equation of the gaseous state, equation (33), then becomes for powder gases RT v – c. Theoretical deductions indicate that the volume c is the actual volume of the incompressible molecules in a unit weight of powder gas; that is, it is the limiting volume beyond which a unit weight of gas cannot be compressed. The volume q is called the covolume. Sarrau determined by experiment with different gases that the mean value of the covolume is one one-thousandth of the specific volume of the gas. Other writers take, for convenience, the reciprocal of the density of the powder as the covolume, this value not differing greatly from the other. We have seen equation (20) that when the gravimetric density is unity the volume of the solid powder in unit volume of the charge is the reciprocal of the density of the powder. The assumption of the reciprocal of the density as the covolume is equivalent therefore to considering the covolume as the volume originally occupied by the solid powder. If the powder leaves a non-volatile residue, the volume of this residue at the temperature of explosion should be added to the covolume of the gases formed. . 69 Under the assumption of the volume originally occupied by the solid powder as the covolume, the initial air space in the chamber becomes the volume Occupied by the powder gases in the chamber. - We therefore have *. ^ = Go (20 + M). Substituting this value in equation (50) we have (n − 1) W = fy – Cop (20 + M), (51) an equation expressing the relation at each instant between the weight of powder burned, the pressure, the travel of the projectile, and the external work performed. In introducing the velocity of the projectile we will assume that the whole work of the gas is expended in giving motion of translation to th projectile, ſº ---. . . * - / →--> .** s: , ), C-: ** ºw..., w (du '2 Q \ . W=;” T2g (#) e p in (51) is the pressure per unit of area; cop the total pres- sure on the base of the projectile. The acceleration of the projectile is dºw/dt”. The total pressure on the base of the projectile is equal to the product of the mass by the accelera- tion. Therefore * 9 **'. z: w dºw - - Substituting these values of W and cop in (51) we have dºw n – 1 (du)*_ ºn y which is Sarrau's differential equation of the motion of a pro- jectile in the bore of a gun. - In deducing this equation there were neglected the follow- ing energies: The heat communicated by the gases to the walls of the gun. The work expended on the charge, on the gun, and in giving rotation to the projectile. - spºjeºlº. Making w the weight of ſhe ^ . # , - ? !, t | f jºy". ) ) ...! A A vs. * * ſ w . -*' * ! ...? : * > . , i J "... . . . . . . . ... 2 70 The work expended in overcoming passive resistances, such as the forcing of the band, the friction along the bore, and the resistance of the air. - Dissociation of Gases.—The error committed by the omis- sion of these energies may not be as great as would at first appear, for we have also omitted from consideration the heat supplied by the phenomenon called dissociation. According to Berthelot the composition of the complex gases from fired gunpowder is not permanent, and at the high tempera- ture during the first instants of explosion these gases decompose into more simple combinations, perhaps into their elements. The increase in volume due to the displacement of the projectile causes a reduction in the temperature which permits the dissociated gases to combine again with a conse- Quent development of heat. The theory of dissociation forms the basis for the assumption of some writers on ballistics, notably Colonel Mata of the Spanish Artillery, that by reason of this phenomenon the expansion of the gases in the gun takes place as though the gases received heat from the ex- terior, and not adiabatically. It will be seen however from the form of equation (53) that the errors of assumption may be allowed for by giving to f a suitable value, and this without changing the form of the dif- ferential equation of motion. The force of the powder as it appears in equation (53) can therefore be considered only as a coefficient whose value must be determined by experiment. Sarrau deduced from the differential equation of motion formulas for the velocity and pressure as functions of the travel of the projectile. We will now follow Colonel Ingalls in the deduction of his formulas. These formulas are considered as giving more accurate results than Sarrau's formulas, for the velocity and pressures produced by modern powders in the bore of the gun; and the use of Sarrau’s formulas is generally limited to the determination of muzzle velocities and maximum preSSures. - . . " 71 Let v be the velocity of the projectile in the bore at the time t. Then - ſº du di = 0. d°w day valºv d(v*) d;2 dº do, 2 dºſ Substituting these values in equation (53) it becomes 2 dº + (?? – 1)^j} = * , The true value of n, the ratio of the 'specific heats, coyºc, is uncertain. For perfect gases its value is 1.41. Regarding iſ the powder gases at the high temperature of éxplosion a.S perfect gases earlier writers assumed this value for n. A Recent investigations have shown that the value of 1.41 is too ygrºat. Some recent writers adopt the value unity for n. As have seen, equation (35), the work of expansion is directly | proportional to the difference of the specific heats; and if their ratio is unity and the difference between them zero, there can be no external work performed. The assumption of the value unity is made for convenience, and the error due to the assumption is compensated for, with the other errors, in the experimental determination of the values of the º, *COnstants. º “; assumes the value n = 4/3 which is practically the value deduced from the experiments of Noble and Abel, see . page 65. == | \ Making n = 4/3 in equation (55) we obtain 3(2,4-u)"º-º-ºw And (54) (20+ wy (55) du 20 (56) w = w/20. (57) Under the assumption made as to the covolume the initial air space is the volume occupied by the gases in the powder chamber. Considering 20, which is the reduced length of the initial air space, as the measure of this volume, a, in equation (57), a = u/20, becomes the number of expansions of the Make 72 wolume occupied by the powder gases in the chamber, when the projectile has traveled the distance M. It is important to bear in mind that a represents a number of expansions, and w the distance traveled by the projectile. Making a = w/20, equation (55) becomes 3 (1+z)^{º} + 9 =9ſº R Ø in 9. (58) gy, the weight of powder burned, is a function of the time and also of the travel u, and of a. The integration of this equation even when the simplest admissible form of y as a function of a, is assumed has not yet been possible. Considering y constant the equation may be integrated. Rearranging it d (v2) d” – 0 , 6 fau 3(1+w) * (M) And integrating {2–0|| (1 + æ)}6 = C. (M) When a = 0, 0) = 0, and C = – 6 fgy/w. Therefore –6 ſqu 1 - * =**{1-ſtºn). (59) Making y constant in equation (58) is equivalent to assum- ing instantaneous combustion for that part of the charge that has burned at the time t. We know this to be in error since the combustion of the charge is progressive. If however we determine the values of the constants in the equations by substituting measured values of v, we obtain an equation that is true for the measured values, and may be true for other values of v at other points in the bore. Only by experiment can we determine whether results obtained under this suppo- sition are correct; and experiment, as stated by Colonel Ingalls, is the final test of nearly all physical formulas. Muzzle - Velocity for Quick Powders. —If the powder is wholly burned in the gun, as is practically the case—in small 73 arms, equation (59) may be used to determine the muzzle ``varocities for charges of different weights after the value of f has been determined by experiment:~~~...~... Welocities in the Bore.—To make equation (59) applicable to points in the bore we must determine a relation between the quantity of powder burned at any instant and the corres- ponding travel of the projectile; that is, we must determine the value of y as a function of u or ac. Then substituting for gy in the equation this value, which for any powder will con- tain a as the only variable, we will have the desired equation expressing the relation between the velocity of the projectile and its travel in the bore. - Combustion under Wariable Pressure.—We have previously deduced, page 25, an expression for the quantity of the pow- der burned under constant pressure as a function of the thickness of layer burned. This relation is given by equation (16) on that page. v= &# {1+\} + #} (60) lo I, T P iſ in which y is the weight of the powder burned when a thick- ness of layer l has been burned, 6 is the weight of the charge, lo is half the least dimension of the grain, and C., \, and u are constants of form of the grain. Representing by tthe time of combustion in the air of the whole grain, or charge, the uniform velocity of combustion will be lo/t. In the gun the powder burns under variable pressure, and the velocity of combustion is expressed by d!/dt. Assuming ...that the velocity of combustion varies as some power of the pressure, and representing by po the pressure of the atmos- phere under which the velocity of combustion is lo/t, we obtain the equation dl lo (...)" (61) di T \O0 in which p represents the pressure on the base of the projec- tile at any instant. 74 The exponent q is given different values by different writ- ers. Sarrau assumes q = 1/2. Recent experiments indicate a mean value of 0.8. The value unity is assumed by other writers. Ingalls assumes the value 1/2, with Sarrau. The pressure per unit of area on the base of the projectile is, from equation (52) w dºw Substituting this value of p in equation (61) and using equation (54) and the relations, and dt da; dt da 20' equation (61) may be brought to the form #-G#)"(*)". Integrating and dividing by lo #=}(; )"ſ(*)"; a. Make - * 1. 2030 % *- - R = t (º) (63) *º-ºº: d(v2)\%. 1 - x- a ſ(*#)"; i. (64) Then l/lo = KX0. (65) Substituting this value in (60) we have • gy = 65 a KX0 {1 + \R^X0 + p (KX0)?}. (66) The value of K in this equation is composed wholly of constants. C., \, and u are the constants of form of the powder 75 grain. By the differentation of equation (59) and substitu- tion in (64), see foot note, we find for the value of Xo da, Yo = 4- 0 ſ W(1 + x) {1 + æ)% – 1} Yo is therefore a function of a only, and a from its value, a = w/20, is itself a function of the travel of the projectile. Equation (66) therefore expresses, for powder of any partic- ular granulation, the relation between the weight burned at any instant and the corresponding travel of the projectile. This equation may be put into another form. At the instant that the powder is all burned in the gun, ty = 65 and l = lo. We will distinguish the particular values of the various quantities at the instant that the burning of the powder is completed by putting a dash over the symbol. When y = 65 and l = lo equations (65) and (66) then become KXo = 1 - (68) 1 = C, (1 + X + pu). This last relation has been previously established in equa- tion (5). * Substituting the value of K from (68) in (66), we obtain 9 – 9 L \ P. Y_2 4=#x (1++, ×, t , x*} (69) We have now, in Xo, introduced into the value of y the travel of the projectile at the specific instant that the burning of the charge is complete. (67) 1 (1 + æ)# } (59) 6 fgy/w- A e-A(l sm (Hºn) =–4– —- ſ1 + æ)3 + T d (v2) ă (THz), dº v = 4/A *###! d (v2 % I …~~ ~~~~~~~~ s:-º:** ( £) # = y; 1 + æ) {(\+ æ)3 – 1} da: W(1 + x) {(I-Faº); FI} : W( From equation (64) Xo = ſ 76 Make 1. X=x, {i-diº). (70) and Y1/Xo F X2, - (71) – || ––– whence - Y2 = | (1+ *} e (72) From equation (59) we obtain for the velocity at the instant that the burning of the charge is complete, * – c...” v - 752 = 6gſ; Y2. (73) Velocity of the Projectile While the Powder is Burning.— Substituting in equation (59) the value of 69f from (73) and the value of y from (69), using equation (71), and making - F2 M M = ** N = } N = + , 74 NG1 Yo Yo” (74) equation (59) reduces to the form w? = MX1 {1 + NXo + N' X62%. (75) This equation expresses the value of the velocity of the projectile at any instant while the powder is burning, in terms of the variable travel of the projectile, and of its velocity and travel at the instant of the complete burning of the charge. Welocity After the Powder is Burned.—Distinguish with the subscript a the values of v and p after the charge is com- pletely burned. y is then equal to 65, and equation (59) when combined with (73) and (72) becomes w.” = *X2/X2, (76) and making V12 = 5?/X2, (77) we have o,” = V12 X2, (78) which is the formula for the velocity after the powder is all d.SN / g e • / ºr r\ \ ^ a * ºiá, is identical with equation (59), if in the latter we make y= 6. V, -6fgó/w, set (73) and (77), and X, is an abbreviation for the quantity in brackets, see (72). - & © As explained under equation (59), equation (78) is there- fore the equation of the velocity under the supposition that the powder is all burned before the projectile moves. is all burned before the projectile moves. The Velocity V1.—From equation (78) we see that V1 is what v, becomes when X2 is equal to unity; and, equation (72), X2 is unity when a, is infinite. Vi is therefore the velocity corresponding to an infinite travel of the projectile. Relation Between the Velocities Before and After the Burning of the Charge. Make ! k = y/6 = fraction of charge burned. Replacing M, N and N' in equation (75) by their values, and combining with equations (69), (70) and (76) we may establish the relation - ~~~ v = v. Wh. (79) That is, the velocity of the projectile before the charge is , consumed is equal to what the velocity would have been at / the same point if all the charge had been burned before the projectile moved, multiplied by the square root of the fraction . of charge burned. - Relation Between the Weight of Powder Burned and the Welocity and Travel of the Projectile.—Replacing va in equa- tion (79) by its value from (78) we obtain k = v?/ V12X, Or y = gºv”/ V12 X2, (80) equations that will be found convenient for determining the ! fraction of charge or weight of powder burned when the velocity and travel of the projectile are known. By reason of the form assumed by the value of k for certain grains very simple relations may be established, for these ł grains, between the fraction of charge burned and the travel : of the projectile. SUBICAL, SPHERICAL AND SPHEROIDAL GRAINS.–For cubical | grains a = 3, X = -1 and u = 1/3 (see page 19). These | values apply also to spherical and spheroidal grains. Sub- —S, | stituting them in equation (69) we obtain º Xn \8 \º k = 1 — (1 – #" - \\ ( #) (81) Q \and Xo = X0 (1 – (1 – k)%}. \ | | \ *!g .." …” 78 From the first equation we may obtain the fraction of charge burned for any travel of the projectile, and the converse from the second. SLENDER CYLINDRICAL AND PRISMATIC GRAINs.-For long slender cylinders k=1-(1-3) No Xo = X0 (1 – (1 – k)%}. which also apply to grains in the form of long slender prisms of square cross section. For other forms of grain the solution of a complete cubic equation is necessary to determine X0 when k is known. (82) Pressures.—The general expression for the pressure per unit of area on the base of the projectile is given in equation (62). Transforming this equation by means of (54) and (57) we obtain w dºw”) . .* * T 20 oz, da, (83) By substituting in succession the values of d(v*)/da, ob- tained from the equations for velocity before and after the complete burning of the charge we will obtain the values of p that apply before and after the charge is burned. Pressure While the Powder is Burning.—Finding the value of d(v*)/da, from equation (75), (see foot note), and making v2=MX, {1+NXo-H N’Ko?} (75) d(v*) ardx1. dxi, º axº ~...~ * Mºd. +MN(x, dº tººl ºr )+ - d). d x MN’ (x, ă. + 2 X1Xo º Make d2C1 irº - X: d (v*) X, d Xo / 2 Xi Xod Xo .* = MX, (1+ N(x+ Y3 da; )+N (x, +-xià) } 79 X3 = d X1/da, *=. Y10. Xo X = X,+*.i.", 2 Xi Xod Xo 2×3 da; © M =s** 9 0 (0 &0 (84) A 5 - ACO2 + (85) We obtain for the pressure per unit of area on the base of the projectile while the powder is burning, p = M' X 3 #1 + NX4+ N' X5}. (86) It will be observed that X3, X4, and X5 are all functions of a; only. Their values for various values of a will be found in the table at the end of the volume. Pressure After the Powder is Burned.—Finding the value of d(v*)/da, from equation (78), Vi’ being constant, we obtain with the aid of (72), - d (?),”) * V12d X2 gººm V12 dº da 3(1+z); Substituting in (83) and making , w Vi’ (87) 6g020 we obtain for the pressure per unit of area on the base of the projectile after the powder is all burned P’ Pa = (II*) (88) Maximum Pressure.—The maximum pressure in a gun occurs when the projectile has moved but a short distance from its seat, or when u and a are small. The position of maximum pressure is not fixed but varies with the resistance encountered. As a rule it will be found that the less the resistance to be overcome by the expanding gases the Sooner will they exert the maximum pressure and the less the maxi- mum pressure will be. By the differentiation of equation (86) 80 we may obtain the value for the maximum, but it is too com- plicated to be of practical use. Examination of the table of the X functions shows that X3 is a maximum when a = 0.65 nearly, while X4 and X5 increase indefinitely. When \, and therefore N, see (74), is negative, that is, when the powder burns with a decreasing surface, p will be a maximum when a is less than 0.65; and when X and N are positive or when the peºwder burns with an increasing surface, p will be a maximum when a is greater than 0.65. A function at or near its maximum changes its value slowly. Therefore a moderate variation of the position of maximum pressure will have no practical effect on the com- puted value of the pressure. It has been found by experi- ment that if we take a = 0.45 for the position of maximum pressure when X is negative, and a = 0.8 when X is positive, no material error results. Therefore to obtain the maximum pressure make a = 0.45 in equation (86) when the powder burns with a decreasing surface, and make a = 0.8 when the powder burns with an increasing surface. The Pressure P".-Combining equations (87), (77) and (73) we obtain - * P = *-f. (89) 3000 Comparing this with equation (45) we see that since 2000 is , the initial air space in the chamber, P’ is the pressure of the gases from 65 pounds of powder occupying the volume behind. the projectile before the projectile has moved from its seat. ... . Equation (88) is therefore the equation of the pressure curve under the supposition that the powder is all burned before the projectile moves. ---...——-> values of the constants in the Equations for velocity, Pressure, and Fraction of Charge Burned.—We have now these equations which express the circumstances of motion of the projectile, and the fraction of charge burned at any instant. The original numbers of the equations are given on the left. : * , Z • . : … N | \\ \ • * 2 : ' f ,7° - AJJ ' ' '. (Yºlº -> "- - * : \ | *- * .” N-Z /* ºv/- ^. *~ }} # '% ..? tº . Rººk - 2: . §: § § *: #. rº '? * * . st 's 2 : * . 81 While the powder burns, (75) w? = MX1 {1 + NX0 -- N' X26% (90) (86) p = M' X3 #1 + NX4 + N' X5} (91) After the powder is burned - * (78) ^j,” = V12X2 (92) - P. - (88) pe = (II*); (93) The fraction of charge burned, substituting N and N' for their values - (69) 9 = 9. Go Xo The quantities M, N, N', M', V1, P’ and Xo in these five equations are constant for any experiment, and their values must be determined before the equations can be used. It will be seen in the equations that express the values of these constants, equations (74) (77) (85) and (87), that the quantities entering the values are of two kinds: the known elements of fire, by which is meant the constants of the powder, of the gun and of the projectile—and quantities such as J, X, X1, etc., that involve the velocity and travel of the projectile at the instant that the powder is all burned. When M and N are known all the constants are known. The value of M given in equation (74) may be reduced by Xo #1 + NXo + N' X?o;. (94) means of (77) and (71) to - M = c Viº/Xo. (95) We have. equation (74) N = X/Xo. (96) M and N being known Xo and VP are determined from these equations, and N’, M' and P' become known from (74), (85) and (87). - Therefore when M and N are known the five equations, (90) to (94), are fully determined, and all the circumstances attending the movement of the projectile become known from them. For any assumed travel of the projectile u, the number . 82 ==" }s of expansions, a = w/20, is obtained, and with this value of a; the functions X0 to X5 are obtained from the table. These substituted with the constants in the equations give , the values of v, p and y. Proceeding in this manner for a number of points along the bore complete curves may be constructed showing the values of v, p and y for any point in the bore of the gun. The value of £ corresponding to Xo is obtained from the table. The value of W follows from the equation W = £20. This value iſ is the distance that the projectile has travelled at the moment that the charge is completely burned. For values of w less than this, equations (90), (91) and (94) apply; for greater values of w equations (92) and (93) apply. Determination of the Constants by Ea:periment.—Regarding equation (90) and noting from equations,(74) that N' is a function of N, it will be seen that if we measure two veloci- ties at known points in the bore of the gun we can determine M and N from equation (90). a, being known for each of the points the X functions are obtained from the table. With the two measured values of v we then form two equations in which M and N are the only unknown quantities. Determ- ining M and N the other constants become known. In using this method care must be exercised that the meas- ured velocities are taken at points passed by the projectile before the powder has completely burned. If the powder is not wholly burned when the projectile leaves the gun one of the measured velocities may be taken at the muzzle. Since M' is also a function of M, equation (85), we may make use of the two equations (90) and (91), or (92) and (91), and with a single measured velocity and a measured pressure determine M and N from these equations. But it has been shown in the chapter on powders that there is room to believe that the pressures as ordinarily measured with the crusher gauge are not reliable. Therefore results obtained in this way are not likely to be as satisfactory as those obtained from measured velocities, which can be determined with a high degree of accuracy. It is found in fact that while the velocities obtained from 83 the formulas agree very closely with those actually measured in practice, there is not as satisfactory an agreement between the pressures. The pressures are obtained in the formulas by the dynamic method and are usually higher than the measured pressures. This is in accord with what has already been said in our previous consideration of the subject of pressures, and adds to the evidence against the accuracy of the crusher gauge. When t and f are known all the constants are known. From equations (63) and (68) we obtain - w20 \% RP t = (º) X0. (97) From equations (73) and (77) - f = Vºw/696, (98) from which can be determined Xo and Viº. M and N follow from equations (95) and (96). t, the time of burning of the whole grain in air, is constant for the same powder. - The value of f, equation (98), is dependent on the value of Vi, a quantity determined by experiment in the gun. f for any powder is therefore constant, within the limits explained below, in the same gun only. It is practically constant for guns that do not differ greatly in caliber. Consequently when t and f have once been determined for a powder and a gun, we may at once form the equations of motion and pres- Sure for different conditions of loading, involving differences in the form and size of grain of the powder, in the weight of the charge, in the weight of the projectile, and in the size of the chamber and length of the gun. The Force Coefficient f.—The quantity f at its first intro- duction, equation (45), was shown to be the pressure exerted by the gases from unit weight of powder, the gases occupy- ing unit volume at the temperature of explosion. It was called the force of the powder, But in the ballistic formulas it has been affected by whatever errors there are in the assumptions made in deducing the formulas. It can conse- 84 Quently be regarded only as a coefficient, and it may conven- iently be called the force coefficient. . Its value, when determined by experiment, may be con- sidered constant in the same gun for charges of the same powder not differing in weight by more than about 15 per cent. from the charge used in determining its value. The effective value of the force coefficient is measured in the for- mulas by projectile energy, and there has been omitted in deducing the formulas all consideration of the force necessary to start the projectile. As the charge decreases the portion of the developed force necessary to start the projectile bears a larger relation to the total force exerted; and if the charge is sufficiently small the projectile will not start at all. The effective force for a small charge must therefore be propor- tionally less than for a large charge, and the value of f determined from one charge must be modified for use with another that differs greatly in weight. The formula used by Ingalls for this modification will be found in equation (137), problem 3 of the applications which follow. - Walues of the X Functions.—We may simplify the value of Yo by means of circular functions. In equation (67) make sec 0 = (1 + æ)*, CŞ iſ we may then deduce, see foot note, ‘…) Xo = 6.ſ'secº 0d8. 2° The value of this integral, designated as (9), is given in ~ : table V of the book of ballistic tables for every minute of arc – up to 87 degrees. We therefore have, simply *- 9-Diºning the equation sec 0 = (1 + ay? cº —s d sec 0 = sec 0 tan 0 d0 = 4 (1 + æ)–3 da = day/6 secº 0. From the second and fourth members, dac = 6 Sec6 0 tan 0 d 0 tan 0 = (sec20–1)} = [ (1 + æ)3 — 1 J3 . Equation (67) becomes 6 000 Xo =ſ* Sec6 6 tan 0d — = 30 d0. secº 0 tan 0 6ſ sec d a.º.º. <-- -------------------- _------- * * A y 3 T- | - 2 1/2 \ ^C. N = No || C. 4 (Y? ) dy–-a is, 7/ From the equatiohaiving the values of the various X functions, equations (70, (71) and (84), first making d N. : X = 1. N = − 1 + 3% X0 cosº. 6 cosec 6 I 3 ~/ | \ we may now deduce the following values: \N - } | } ! d * (i. Y1 = X0 sin” 6 * 4 = * c ºf ...— ... Y2 = Sin? 6 Nº CC/ X = sin 9 cosº/X -- Sſ. 2 | N ty x = x. air, 5 + 7 ſº tº Y5 = X0% (1 + 2 X) X---/ The values of the X functions for various values of a are found in the table at the end of the volume. The argument in the table is ac. The value of a is obtained from the equation a = w/20 in which u is the travel of the projectile and 20 the reduced length of the initial air space. Knowing 20 and assuming the travel we obtain ac, and from the table find the corresponding values of the functions. Interpolation Using Second Differences.—It will often be necessary in determining values of the functions for values of a not given in the table to employ second differences, in Order to get the désired accuracy in the interpolated values of the functions. XThe interpolation may be effected by the In a table containing values of a function, the first dif. fºrences are the differences between the successive Values 9f the function. The second differences are the differences *tween the successive values of the first differences Thus if the successive values of an increasing function are º, *, and "...the first differences are a q= A, and *-* = A 1. The second difference is then A', A.T A 2 the tabular values a, and ach. h = Øb T30a. A1 and A2 are the first and second differences of the function under consideration. Xa the tabular value of the function corresponding to ſºa. - 28 the interpolated value of the function corre- Sponding to a. 86 It will be observed that the difference between successive values of a varies in different parts of the table. In applying the formula we must use the same value of h in getting the two first differences from which the second difference is obtained. The lower sign of the second term of the second member must be used when the function decreases as a, increases. This sign will only be required for the values of the function 23 when the value of a, is greater than 0.65. N | The Characteristics of a Powder.—The quantities f, t, G, \ and p were called by Sarrau the characteristics of the powder, because they determine its physical qualities. Of these fac- tors, f the force coefficient of the powder, depends principally upon the composition of the powder. In the same gun it is practically constant for all powders having the same tem- perature of combustion. It increases with the caliber of the gun and for this reason its value determined for one calibre cannot be depended upon for another. The factor t, the time of combustion of the grain in air, depends upon the least dimension of the grain and upon the density; also in smokeless powders upon the quantity of solvent remaining in the powder. The factors C., X and u depend exclusively upon the form of the grain, and for the carefully prepared powders now employed their values can be determined wit, precision. They are constant as long as the burning grain retains its original form. '. APPLICATION OF THE FORMULAs. For convenience of reference the notation employed in th deduction of the formulas is here repeated, and the unit customarily employed in our service are assigned to the dif ferent quantities. For most of these quantities specific units have not heretofore been designated. a defined by equation (101) below. C volume of powder chamber, cubic inches. d caliber in inches, EXAMPLES IN INTERPOLATION, USING SECOND DIFFERENCES. 1. What is the value of log Xo corresponding to X=1.17? xa-1.15 Xb=1.20 h–.05 x—xa=.02 - 1st diff. 2d diff, Xa=log X, (x=1.15) 0.52860 7.92= A 1 36= A a log X, (x=1.20) 0.53752. 756 log Xa (x=1.25) 0.54508 - - x=052960++792+ #x #x36=052980-81 6.8–H8.6 The parentheses around 0.52960 indicate that this num- ber is to be treated as a whole number in applying the corrections. Therefore 0.52960 316.8 A. 8.6 - t X=log X, (x=1.17)=0.53285 7: 2. What is the value of log X, when x=0.563? Ans. Log X1–9.53337. 3. Log Xa for x=0.275. Log Xs=9.82216. 4. Log Xa for x=2.18. Log Xs=9.76089. 5. Log X, for x=0.772. Log Xs=1.15879. 87 D, outer diameter of powder grain, inches. di diameter of perforation of powder grain, inches. f force coefficient of the powder, pounds per square inch. F' fraction of grain burned. & g 32.16 foot seconds. k=y/G3, fraction of charge burned. l thickness of layer burned at any instant, inches. lo one half least dimension of grain, inches. L constant logarithms in the ballistic equations. m length of powder grain, inches. - M, M' ballistic constants. N, N' ballistic constants. n number of powder grains in one pound. P" ballistic pressure constant, pounds per square inch. p pressure while powder burns, pounds per square inch. pa pressure after powder is burned, pounds per sq. inch. pm maximum pressure, pounds per square inch. po standard atmospheric pressure, 14.6967 lbs. per sq. in. Si initial surface of a pound of powder, square inches. w travel of projectile, inches. U total travel of projectile, inches. v velocity of projectile while powder burns, ft. seconds. va velocity of projectile after powder is burned, ft. secs. V muzzle velocity of projectile, foot seconds. V1 ballistic constant, velocity at infinity, foot seconds. ve velocity of combustion of powder, foot seconds. 00 specific volume of a gas, cubic feet. - Vo initial volume of a powder grain, square inches. w weight of projectile, pounds. º - a number of expansions of volume of initial air space. Xo, Y1, X2, Y3, X4, X5, functions Of 0. g weight of powder burned at any instant, pounds. 20 reduced length of initial air space, inches. C. : constants of form of powder grain. U. - - ô density of powder. A density of loading. - & weight of powder charge, pounds. t time of burning of whole grain in air, seconds. 88 Quantities topped with a bar as Ü, ä, ä, X2, etc., designate the particular values of the quantities at the instant of com- plete burning of the powder charge. With the units assigned above the following working equa- tions are, with the aid of equation (28), derived from the equations whose numbers appear on the left. The numbers in brackets are the logarithms of the numerical constants after reduction to the proper units. (22) A = [1,44217] G3/C. (100) (27) a? = *** e (101) (29) 20 = [1.54708] a” 6/d?. (102) (57) a = w/20. (103) (73) Ú” = [4.44383] f X, 63/w. (104) (85) M’ = [3.82867] Mw/a2 65. (105) (87) P’ = [3.35155] V12 w/a2 65. (106) (89) P’ = [1.79538] f/a2. (107) (97) t = [2.56006] a Wu & Xo/d?. (108) (98) f = [5.55617] V12 w/65. (109) In addition to the above working equations the following formulas are needed or are useful in the solution of most problems: (74) M = a jº/X, N = \/X, N' = u/X 2. (110) (95) - M = a V12/Xo. * (111) (75) v% = MX1 {1 + NXo + N' X62}. (112) (86) p = M' X, {1 + NX, + N' X;}. (113) (78) o,” = V12 X2. (114) (88) p. = a fºr (115) (80) k = y/<=vº Vºx. (116) ooº- ſcº. _-T T- Equation (105). The quantity 2000 in (28) is expressed in cubic inches, and before substituting its value for 2000 cubic * Č/ ſ \ *>asºn /* 3 **** * < .* Transformation of the Formulas into the Forms (104) to (109).-In the deduction of the formulas the quantities have been expressed in general terms, no units having been as- signed. In assigning now to the velocity v the foot second unit and to the weights, the pound unit, we fix the units in the formulas as the foot, the pound and the second. All dimensional quantities in the formulas must then be considered as ex- pressed in feet, square feet, or cubic feet; pressures in pounds per square foot, and time in seconds. As appears On page 87, we intend now to preserve the foot second as the unit of velocity but to express the dimensional quantities, such as d, Co, 20, u, etc., in terms of the inch as the unit, and the pres- sures in pounds per square inch. We must therefore intro- duce into the formulas such factors as will make them applic- able to the new units. This is accomplished as follows: - Equation (104). In the value of Ü%, equation (73), g is already in feet, 6 and w in pounds, X2 is dependent only on a; which is a ratio independent of the unit. f, which we now express in pounds per square inch, must, before being sub- stituted for f pounds per square foot in (73), be converted into pounds per square foot by Jmulti lying ºpy 144. We therefore get for the numerical factor whose logarithm ap- pears in (104) the quantity 6 a 144– J feet in the formulas we must divide the value by 1728. This substitution is made in equation (85). M' is a pressure in pounds per square foot, M being the square of a velocity ex- pressed in feet, see (74). To reduce M' to pounds per square inch in order to convert into pounds per square inch the pressures determined from equation (86) we must divide it by 144. With these two operations we obtain for the numeri- cal factor in (105), 1728/144 × 20.27.68 = 6/g27.68. Equation (106). Substitute for 2.0 in (87) its value from {} d 32%. }roº tº: w ºcco."9 - “… 5 2- * 93 (+ 4*, *, * * * *> - 880, W **-* < *, r* ~~~ rººf - * A tº ºr f ſ 3. º 88b (28) divided by 1728, and divide the value of P’ by 144 to re- duce P' to pounds per square inch. The numerical factor is 2/g27.68. . - Equation (107). Substitute for 2000 in (89); multiply f now in pounds per square inch by 144, and divide by 144 to reduce P" to pounds per square inch. The numerical factor is 1728/27.68. ſº Equation (108). From (97), multiplying and dividing by co’é (**)" º 1728 / ..., x nº/4*" 144 (6g Po 144)?% The numerical factor becomes | 4(27.68)}8/T (72g po)”. = (27.68/4.5 Tºg po)”. Equation (109). Reduce (98) to pounds per square inch by dividing by 144. The numerical factor is 1/69 144. The length of a gun when expressed in calibers ordin- arily means the length measured from the front face of the closed breech block to the muzzle of the gun. The travel of the projectile is the distance passed over by the base of the projectile, measured from its position in the gun when loaded. The length of the gun in calibers is therefore equal to the travel of the projectile plus the length of the powder chamber. 89 —b p\% - v. =: (...) (123) l = lo XoAX0. (124) Gj \% f=f(...) (137) DETERMINATION OF THE BALLISTIC FORMULAS FROM MEASURED INTERIOR, WELOCITIES. As a test of the formulas that have been determined, and at the same time to illustrate their extensive use, we will fol- low Colonel Ingalls in his application of these formulas to the experiments made by Sir Andrew Noble in 1894 with a six- inch gun. The normallength of the gun was 40 calibers but it could be lengthened as desired to 50, 75 or 100 calibers.-- By means of a chronoscope not differing in principle from the Schultz chronoscope that has been described, the velocity of the shot could be measured at sixteen points in the bore. Noble gives the mean instrumental error of the chronoscope as three one-millionths of a second. Problem 1.—A 100 pound projectile was fired from this 6-inch gun with a charge of 27% lbs. of cordite. Diameter of grain 0".4, density 1.56. Velocities measured at points cor- responding to the different positions of the muzzle were as follows: % = 199.2 inches Q) = 2794 f. S. 259.2 “ 2940 “ 409.2 ** 3166 “ 559.2 “ 3284 “ The volume of the chamber was 1384 cu. in. Determine all the circumstances of motion. Constants of the gun. Constants of the powder. C = 1384 65 = 27.5 d = 6 b = 1.56 U = 559.2 C = 2 Sé6) ^0 = 100 sº a tº u = 20 l 0 89 % - Øc =% #) (123) l = lo Xo/Xo. (124) Gj \% f=f(...) - (137) DETERMINATION OF THE BALLISTIC FORMULAS FROM MEASURED INTERIOR, WELOCITIES. As a test of the formulas that have been determined, and at the same time to illustrate their extensive use, we will fol- low Colonel Ingalls in his application of these formulas to the experiments made by Sir Andrew Noble in 1894 with a six- inch gun. The normallength of the gun was 40 calibers but it could be lengthened as desired to 50, 75 or 100 calibers. ;- By means of a chronoscope not differing in principle from the Schultz chronoscope that has been described, the velocity of the shot could be measured at sixteen points in the bore. Noble gives the mean instrumental error of the chronoscope as three one-millionths of a second. Problem 1.-A 100 pound projectile was fired from this 6-inch gun with a charge of 27% lbs. of cordite. Diameter of grain 0".4, density 1.56. Velocities measured at points cor- responding to the different positions of the muzzle were as follows: % = 199.2 inches Q) = 2794 f. S. 259.2 “ 2940 “ 409.2 ** 3166 “ 559.2 “ 3284 “ The volume of the chamber was 1384 cu. in. Determine all the circumstances of motion. Constants of the gun, Constants of the powder. C = 1384 G3 = 27.5 U= 559.2 C = 2 See My = 100 := x : * p = 0 20 lo - 0.2 9 0 0.55 0.07084 D * From equation. (là 9. A & 4 £ 6 ſ log a? { % 6 6 f(6%)og &0 \ - 30 1.500.96 31.693 -*** We have to determine the ballistic cons tants for use in the velocity and pressure formulas. # *. Since u = 0 we see from equation (110) that N' = 0, and that since X is negative N is also negative. Velocity formula (112) therefore becomes for this powder, : from which with two measured values of v and the corres- ponding values of w, and hence of X1 and Xo, we may deter- mine M and N. We must use for this purpose two values of ty while the powder is burning. - . We will take the two measured values 2794 and 3166 and determine afterwards whether we are right in the selection. The X functions for u = 199.2 corresponding to v = 2794 are found as follows: Equation (103), a = 6.2853 for u = 199.2. From the table of X functions using first differences only log X0 = 0. 82110. In the same way the other functions for this value of ac, and the functions for the values of a corresponding to the other given values of u are obtained from the table. (M, 00 Q) inches - f. s. log Xo log XI log X2 199.2 6.2853 2794 0.82110 0.50606 I.68496 259.2 8.1784 2940 0.86213 V 0.58011 I.71799 409.2 12.9112 3166 0.931.17 0.69774 I.76657 559.2 17.6446 3284 0.97710 0.77150 I.79440 In equation (117) using two values v and v'.and the values of Xo and X1 corresponding to each, and solving for N and M, we obtain - - N - ^)/2 Y1 — 092 Y'1 w/2 Yo X1–v? X'o X'i Page 90, insert after line 4, METHOD OF PROCEDURE. With go we may determine from equation (103) the value of a corresponding to any travel of the projectile : and with a we may obtain from the table the correspond- • ing values of the X functions. . We have now all the necessary data for the solution of the problem, and from this data we must determine the values of the constants in the five formulas (112) to (116). º: is as follows: 1 .YSelect two of the measured Velocities and the corre- spofiding values of the travel u and assume that the ve- 2 locities were reached before the powder was all burned. | º Substitute successively in (112) the selected values of v *" with the values of the X functions obtained with the corre- sponding travels. § We have then two equations in which only the constants ! are unknown. As N' is a function of N there are but two : constants, M and N, to be determined from the two equa- , , tions. ; : Determine M and N from the two equations. With the value of N find from the second of equations (110) the value of Xo, and with this determine from the * . . . . table the value of £ and from (103) the value of W. . 3) The powder was all burned at this travel iſ and if the ..., \ ^* values of u corresponding to the selected velocities are less than W, we were right in assuming these two velocities as having been reached before the powder burned. • Our determinations of M and N are therefore correct, and, as explained on page 81, all the other constants may be determined from these two. i 2. If however one, or both, of the selected velocities was reached at a travel greater than W, our assumption : that they were both reached before the powder was burn- ed was wrong and our values of M, N and W obtained under that assumption, are wrong. 'We must therefore determine new values of M and N as follows: Substitute the first of the Selected velocities with the - corresponding values of the X functions in (112) as before. i Substitute the second selected velocity in (114) with the - value of Xs corresponding to the travel. Determine V1. Replace N, N' and M in (112) by their values from (110). and (111). Then in (112) X, is the only unknown quantity, and its value can be determined. With X, and V, the values of M and N are readily found. 3. The constants cannot be determined if both the se- lected velocities were reached after the powder was wholly burned. Equation (114) should give the same value of V1 for both the selected Velocities. - Now to revert the problem, which will be solved after the first method. - * 91 Q92 .* - -X1 — N.Xo XI" Making v = 2794 and v' = 3166, we obtain with the corres- ponding values of X0 and X1 - log M = 6.59155 log N = 2.75465. . With these, as has been shown on page 81, all the other ballistic constants are determined. M = ºw. will first determine from the second of equations (110) log X0 = 0.94432, and from the table find the corresponding value of a by in- terpolation using first differences only £ = 14.11. From equation (103) W = 447.19, that is, the burning of the powder was completed at the instant that the shot had travelled 447.19 inches. The values of w for the points selected for the determina- tion of the constants in the equations being less than W. we find ourselves justified in the selection of these points. From equation (105) log M' = 4.91005, (111) log V12 = 7.23484, (106) log P’ = 5.07622. We now have all the constants that enter the equations ( 112) to (116) for velocity and pressure and fraction of charge burned. These equations become for this round w” = [6.59155] X1 (1 – [ 2.75465 | Xo) (118) p = [4.91005] X3 (1 – [2.75465 | X4) (119) o°,+ [7.23484] X2 (120) [5.07622] - O ſpa = (1+z)4/8 (121) Qy = [6.20449] vº/X2. (122) With these five equations we can determine the Velocity, pressure, and weight of powder burned as the projectile 92 passes any point in the bore, by substituting the values of the X functions, determined from the table for the value of a Corresponding to the travel of the projectile at the point. In this way we find from equation (118) for w = 259.2 for which a = 8.1784:-(the symbol L indicates a constant log- arithm in the equation) log Yo 0.86213 L 2.75.465 0.41379 I.61678 0.58621 I.76805 log X1 0.58011 L 6.59155 log v2 6.93971 log v 3.46985 ty = 2950 foot seconds, This differs from the measured velocity by 10 feet. To find the velocity at the muzzle, for comparisou with the measured velocity, we must make use of equation (114), since the powder was all burned before the projectile reached the muzzle. log V12 7.23484 log X2 T.79440 log V2 7.02924 log V 3.51462 V = 3270.5 foot seconds. This differs but 13.5 feet from the measured velocity of 3284 feet. The difference, ſº of one per cent of the measured velocity, is negligible. In the same way the velocity at any point may be deter- mined and the curve v in Fig. 1 plotted. Pressures.—The pressure at any point may be similarly obtained from equations (119) and (121). The pressures so obtained are plotted in the curve p Fig. 1. 93 .*::-ºr-tº-º-º: ...--> - ***---- ********śx. <>- ©<>CN) '-| '--' º 'L'-|-|-|-|-|-|-1$!-!-|-|-|-|-|-|-|-|+|-|-|-|-|-|-|-1,-2-4& i-------（-\º ! '|\ſ) || || || |! }! ~|1 |! -! ! · | ! | ! | |^} ISÈ---+---+---+------------|----------------+-----------|------È |<ſ •! + → ©}!©NQ ©|• |-CÒ \ſ)!<> N•| !<> !★ ! ||- I- (）- -0 ��- :�- 00! �:|- �T|－- !● w ●}}• - și-i-i-i-i-i-i-i-i-si-ſ-r-ſ-t-r-ſ-t-r）！+-+-+-+-+-===============+++==+++==~:= Uſò•） • • • - - - - - - - • → • • • • - - - - - - - - = ！= « - - - - - • = - - - - - - - - - - - - = = ، ^ <>-<> J 1 I ı ı ( ? 1 : 1 . . ， , , , , , ， , , , , - - - - 50,00l. F========= = = = ––– 50+---------- �~ çº) ========= | | | | | | | | | <> i - 1 | | | | | | <> I ı ı ı ı ı ı ı ı <> i - 1 ı ı ı ı ı ı o 1 I ı ı ı ı ı ı ı QNI！={ âi-H-I-T-I-Höö 559.2 –4- —l- 40.3T447.3T Travel, Inches. 259.2 I99.2 Fig. 3.-Comparison. Of Results with Cordite and Ribbon Grains. * * 103 tWO curves rapidly approach each other, and later cross at a travel of about 130 inches. At the instant before the travel W is reached the area of the burning surface of the ribbon grains has a considerable value. It may readily be determined, from the given dimen- sions and density of the ribbon grains, that there are 76 of these grains in the charge of 27% lbs. The initial surface of the charge is 4560 square inches. - The vanishing surface of each grain determined by mensu- ration or by making l = lo in equation (1) is 24.32 square inches; and for the 76 grains, 1848 square inches. This is more than 4/10 of the original surface. At the travel iſ this large burning area suddenly becomes zero. There is a sudden cessation of the emission of gas, and a sharp drop in the pressure. As the burning surface of the cordite grain approaches zero gradually the pressure curve p1 of this grain is continuous. Since at the travel iſ the projectile has the same velocity from the two charges, the work done upon it is the same in each case, and the areas under the pressure curves to this point must be equal. Corresponding with the sudden change in pressure at the travel W. we find in the curve v2 a sudden variation in the rate of change of the velocity of the projectile as a function of the travel, represented by the tangent to the curve. The above considerations apply to the 100 caliber length of the gun. Now if we consider the gun as 40, 50 or 75 calibers in length neither charge would have been wholly consumed in the bore; and we see from the curves that in each case the muzzle velob- ity would be less from the slower burning powder. It is therefore apparent that to produce in the gun of any of these lengths a given muzzle velocity, vi, taken from the cordite curve, a larger charge of the slower powder would be required. The maximum pressure from the larger charge of slow powder would remain less than that from the quicker powder, since the area under the two pressure curves must be equal and the pressure curve of the slow powder would be the higher at the muzzle. 104 As the gun is longer the difference in the weight of the two charges of the quick and slow powder that produce the same muzzle velocity is less, until at some length the difference becomes Zero. The advantage of lower maximum pressure always remains with the slower powder. Quick and Slow Powders.--It is apparent from Fig. 3 that if the maximum pressure and the muzzle velocities obtained from the cordite in the 40 and 50 caliber guns are satisfactory, the muzzle velocities produced by the same charge of powder in the form of ribbons would be too low. This powder would be too slow for guns of those lengths while for the guns of 75 or more calibers it would be satisfactory. It is also found that usually a powder that is satisfactory in a gun of a given caliber is slow for a gun of less caliber and quick for a gun of larger caliber. Therefore as has been shown in the chapter On gun powders, a special powder is provided for each caliber of gun, and for markedly different lengths of the same caliber. Effects of the Powder on the Design of a Gun.—In the design of a gun the caliber, weight of projectile, and muzzle velocity being fixed, consideration must be given to the powder in order that the size of chamber, length of gun, and thickness of walls throughout the length may be determined. We have seen that to produce a given velocity in any gun we require a larger charge of a powder that is slow for the gun than of a quicker powder. The larger charge will require a larger chamber space, and will thus increase the diameter of the gun over the chamber. The maximum pressure being less than with the quicker powder the walls of the chamber may be thinner. The slow powder will give higher pressures along the chase, therefore the walls of the gun must here be thicker. The weight of the gun is increased throughout its length. If we do not wish to increase the diameter of the chamber we must, for the slow powder, lengthen the gun in Order to get the desired velocity. On the other hand with a powder that is too quick for the 105 i º gun very high and dangerous pressures are encountered, re- quiring excessive thickness of walls over the powder chamber. The difficulties of obturation are increased. Excessive ero- sion accompanies the high pressures and materially shortens the life of the gun. The gun may be shorter and thinner walled along the chase. It is evident from the above considerations that each gun must be designed with a particular powder in view, and that a gun so designed and constructed will not be as efficient with any other powder. ____--~~~ ----- ***** _------ DETERMINATION OF THE BALLISTIC FoRMULAs FROM A MEASURED MUZZLE VELOCITY AND MAXIMUM PRESSURE. In the previous problems we determined the constants in the ballistic formulas by means of measured interior veloci- ties. This method will usually not be available, as interior velocities can be measured only by Special apparatus not usually at hand. The usual data observed in firing are the muzzle velocity and the maximum pressure. The method of determining the constants with this data is illustrated in the following problem, and at the same time, the method of applying the formulas to the multiperforated graln. Problem 3.−Five rounds were fired from the Brown 6-inch wire wound gun at the Ordnance proving grounds, Sandy Hook, March 14th, 1905. The projectiles weighed practically 100 lbs, each. The charge was 70 lbs. of nitrocellulose powder in multiperforated grains, with two igniters, each containing 8 ounces of black powder, at the ends of the charge. The multiperforated grains weighed 89 to the pound. They were of the form described on page 21. Their dimensions, cor- rected for shrinkage, were D1 = 0".512 di = 0”.051 m) = 1".029. The mean muzzle velocity of the five rounds was 3330.4 f. s. The measured maximum pressure was 42497 lbs. per sq. in. 106 The capacity of the powder chamber was 3120 cubic inches. The total travel of the shot was 252.5 inches. Determine the circumstances of motion. Before we can proceed with the solution of the problem we must determine the constants of the powder. We will make no distinction between the two different kinds of powder but consider the weight of charge as 71-pounds of multiperforated powder. *- Dimensions of grains, Di == 0".512 di = 0".051 m = 1".029. Weight of grain, 89 to 1 pound. We will first determine the constants of form of the powder grain. . From equation (13) - § 2lo = 0.08975 and from equations (12) we find q = 0.72688, X = 0.19590, p = 0.02378. Equation (11), in which F is the fraction of grain burned when the web is burned, therefore becomes for this grain - - F = 0.72668 # | 1 + 0.19590 # –0.02378 #} (127) Making l = lo, * F = 0.857.14 (128) the fraction of grain burned when the burning of the web is completed. The slivers therefore form 0.14186 of this partic- ular grain. º The body of the grain burns with an increasing surface, while the slivers burn with a decreasing surface. To avoid the difficulties that would follow from the introduction of the two laws of burning into the ballistic formulas, we will substi- tute for the real grain, a fictitious grain with such a thickness of web, that when the web is burned the same weight of . powder is burned as when the whole of the real grain is burned. That is, the body of the fictitious grain is equivalent to the whole of the real grain. For the body of the fictitious grain F in the formula of the The value of l/l, that will make F=1 in equation (127) can be obtained more simply and with sufficient accuracy by trial, as follows: We have determined that when li-lo and l/lo–1, F= 0.85174. This value is less than unity by 0.148. For a first trial we will increase the value of l/lo by 0.148, and obtain from (127) with l/lo–1.148 F=0.99568 an increase in the value of F of 0.144. Therefore if we further increase l/l, by .005 we will get a value of F near unity. - with l/lo–1.153 F=1.0006 Interpolating, by the rule of proportional parts, between these two Sets of values we find that for F=1 7/7 – 1 1 P 9A 107 - fraction burned must be unity when l = le. Making F = 1 in equation (127) and solving the cubic equation by Horner’s Method, as explained in the algebra, we obtain for l/lo kº--- - - - l/l = 1.1524. substituting this value in (127) it becomes | 1=0.837416 (1+0.22573–0.031581). imparing this with equation (5), 1 = a (1 + X + u), which is #ived from the formula for the fraction burned by making ; lo, and which expresses the relations existing between the nstants of form of the powder grain, we see that for the ºitious grain a = 0.837416 X = 0.22573 u = – 0.031581. "The new value of lo must be the former value multiplied rthe above ratio, l/lo = 1.1524, since we have multiplied l, the quantities in equation (127) by the ratio to make F=1. Therefore l = 0.044875 × 1+H594= 0.051714. The volume of the real grain is // /S 22' V = + 1 (D3-7 dº) m = 0.197144. Whence from equation (18) with n = 89, b = 1.5776. We have now all the data necessary for the solution of the problem. For convenience it is repeated here. Constants of the Gun Constants of the Powder C = 3120 G3 = 71 d = 6 b = 1.5776 U = 252.5 d = 0.837416 70 = 100 X = 0.22573 Measured data p = – 0.031581 V = 3330.4 lo = 0.051714 p.m. – 42497 From equation (100) A = 0.6299 (101) log a” = I.97940 (102) log 20 = 1.82144 20 = 66.289 108 2^ /1 On account of the thinness of web of the powder grain, and the high pressure, we may be certain that the charge Wà,S wholly consumed in the bore. Assuming that the maximum pressure was the maximum pressure on the base of the pro- jectile we then have a pressure while the powder was burn- ing and a velocity after the charge was all burned. As explained on page 82 equations (92) and (91), or (114) and METHOD OF PROCEDURE. (113), are applicable in this casexit was shown on page 80 The procedure is as follows: Substitute in (114) the measured muzzle velocity and the Value of X2 taken from the table with the value of x corresponding to the travel of the projectile at the muzzle. - Determine V1. - Substitute in (113) the measured value of the maximum º, pressure and the values of the X functions corresponding ! y to X=0.8 or x=0.45, according as the grain burns with . ....~" an increasino or decreasing surface. - Assume a value for the travel at the moment of com plete combustion and determine for this travel the values of £ and Xo. With this value of Xo and the value of V, previously determined, find values for N, N' and M' from (110) (111) and (105). Substitute these values in the second member of (113). - If the second member has then the same value as the first member, which is the measured maximum pressure, our assumption of the travel ú is correct. If not we must make new assumptions for ü and determine new values for M, N and N' until we find values that will satisfy equa- tion (112). - log X2 = 9.61019. Therefore equation (114) becomes, for the muzzle o,2 = (3330.4)? = V2 [I.61019) from which log V12 = 7.43481. (131) The prover values of M, N and N' must satisfy equation (130). But we see that equations (110) and (111) express 109 fixed relations between these constants and Vi at the moment of complete burning of the charge. Therefore we will assume the travel at the moment of com- plete consumption, and with the corresponding value of £, and therefore of Xo, determine N and N' from equations (110) and M from (111). Then substituting this set of values in equation (130) we will determine whether the values satisfy that equation. If not we will make other assumptions for £ and proceed in the same way until we find satisfactory values of the constants. The value of a at the muzzle is 3.8091. The value à must be less than this since we are assuming that the charge was all consumed in the gun. Let us assume à = 2. Taking : from the table the corresponding value Xo we find from equa- tions (110) and (111) values of M, N, N'; and these substi- tuted in equation (130) make the second member equal to 45746. This is greater by 3249 pounds than the measured maximum pressure, 42497 lbs. ; and we therefore conclude that we have assumed a too rapid combustion of the powder. The true value of £ is therefore greater than 2. Assume next Ž = 2.3. From the table log XD == 0.65647. From equation (111) log M = 6.70307. From equation (110) log N = 2.69892. log N' = 3.19009. With these values in equation (130) we get p,n = 4.2909 lbs. As this differs from the given pressure, 42497 lbs., by less than one per cent, we may without material error use these values of the constants as the true values. The assumed value £ = 2.3, by means of which the con- stants were determined, gives from equation (103) W = 152.5 inches. We have from equations (105) and (106) log M' = 4.70.108 log P’ = 4.95570. 110 We may now from equations (112) to (116) form the five equations applicable to this round ſº v? = [6.70307] X, 1 + [2.69892] X. — [3.19009] X,*}, (132) p = [4.70108] X3 (1 + [2.69892] X. — [3.19009] X58, (133) o,” = [7.43481] X2, (134) _ _[4.95570] ( Pa = (1+z); (135) y = [6.41645] vº/X2. (136) With these equations we may determine the velocity, pres- sure and weight of charge burned, at any point in the bore. For any travel less than 152% inches equations (132) and (133) apply for the velocity and pressure, and equation (136) for the weight of powder burned. For any travel greater than 152% inches, equations (134) and (135) apply. The table and curves which will follow are derived from these equations. . A convenient method of performing the work in construct- ing the table or curves is here shown. It is always best to assume values of a that are given in the table, rather than values of w which would require interpolation in the table to find the values of the X functions. The symbol L in the following work is used to designate the various constant logarithms in equations (132) to (136). We will take for example the value a = 0.8, corresponding to the travel at which we found the maximum pressure. From the table: - log X0 = 0.46075 log X1 = 9.71100 log X2 = 9.25025 log X3 = 9.86027 log X4 = 0.60479 log X5 = 1.17352 Equation (103) log a 1.90309 log 20 1.82144 - log M 1.72453 * = 53.031 inches. 111 t- & Equation (132) log X0 0.46075 log N 2.69892 log Xi I.71100 I.15967 log X0% 0.92150 log M 6.70307 + 1 1.14443 log N' 3.19009 6.41407 0.01293 . . . . . . . . . . . . 2.1.1159 0.05365. . . . . . . . . . . . . 1.13150 log v2 6.46772 log v 3.23386 ^) = 1713.4 foot seconds. Equation (133) log X, 0.60479 I, 2.69892 log X3 I.86027 I.30371 log X5 1.17352 log M' 4.70.108 + 1 1.20124 log N' 3.19009 4.56135 - 0.02310. . . . . . . . . . . . . . 2.36361. 0.07120............. 1.17814 log pm 4.63255 - £9m = 42909 lbs. per Sq. in. Equation (136) log v2 6.46772 I, 6.41645 colog X2 0.74975 - log y 1.63392 g = 43.045 lbs. And if we desire the values of va and pa, Equation (134) log V12 7.43481 Equation (135) log 1.8 0.25527 log X2 I.25025 X 4/3 0.34036 log v.2 6.68586 log P' 4.95570 log va 3.34253 log pa 4.61534 va = 2200.5 f. s. pa = 41242 lbs. per sq. in. These values of va and pa are what the velocity and pressure would have been had the powder all burned before the pro- jectile moved. - The calculations for velocity and pressure at any point of the bore beyond the point of complete combustion of the charge are extremely simple, being limited to the solving of the two equations (134) and (135), which require from the table the function X2 only. e Proceeding as above for different values of a we obtain th 112 data collected in the following table, from which the curves in Fig. 4 are constructed: 2. Travel Powder Velocity Pressure Velocity Pressure burned (C Q0, Q/ 7) p ?)a. 19a inches lbs. f. S. lbs. f. S. lbs. 0.0 0.00 00.00 000.0 00000 000.0 90300 0.1 6.63 15,02 424.3 26018 922.5 795.25 0.2 13.26 21.43 695.9 33698 1266.8 70814 ().4 26.52 30.48 1113.4 40316 1699.2 57657 0.6 39.77 37.35 1440.8 42500 1986.5 48254 0.8 53.03 43.05 1713.4 4.2909 2200.5 41242 1.0 66.29 47.98 1947.9 42500 2369.5 35836 1.3 86.18 54.43 2248.9 41223 2568.6 29742 1.6 106.06 60.04 2505.3 39659 2724.5 25258 1.9 125.95 65.04 2729.1 38052 2851.4 21836 2.3 152.47 71.00 2989.2 36002 2989.2 18380 3.1 205.50 3.195.4 13761 3.8091 252.50. 8330.4 11124 - -l ! . 40 -> - N 30 * * * * * * _T 30 - wº - - / i … - 20 2. N 20 - - Nº. - * p 1000 lbs. – - Verticaliscales, v i 100 fs. - - 2 lbs. mº 10 / 3/ 10 "O 26.5 53.0 Iööö T52.5 TTravel, Inches. Tää5 Fig. 4.—Charge 71 lbs. Multiperforated Grains. 113 Discontinuity of the Pressure Curve.—We have used in the deduction of the equations from which the table is pro- duced a fictitious multiperforated grain the body of which, without the slivers, equals the whole of the real grain. The body of the real grain was, as shown by equation (128), 85.714 per cent of the whole grain, the slivers forming 14.826 per cent of the whole. The table and curve p show discontinuity of the pressure at the travel 152.5 inches when the burning of the whole charge is completed. Actually there is no discontinuity in the true pressure curve. The web of the real grain was burned when 85.2 per cent of the fictitious grain, or of the whole charge, was burned. This portion of the charge, 60.85 lbs., was burned at a travel of about 109-inches, as may be seen from the table. The charge burned with an increasing surface up to this point of travel and then with a decreasing surface which gradually ap- proached the vanishing surface Zero. The pressure would therefore, at a travel of 109 inches, begin to fall off more rapidly, making a point of inflection in the true pressure curve. From this point, as the slivers burn, the pressure curve should gradually approach the curve pa and join it at the point of complete combustion, u = 152.5 inches. The Constant t for this Powder.—From equation (108) t = 0.374.77 seconds. This is the time of burning of the whole grain in air. The velocity of burning of this grain in air, lo/T = 0.138 inches per second. - The velocity of combustion in the gun is given by equation (123), and the thickness of layer burned at any travel by equation (124). The Force Coefficient f.-From equation (109) f = 1379.5 lbs. per sq. in. It has been previously stated that f is constant for any powder in a given gun for charges not differing greatly in 114 weight. The effective value of f, as measured in the formulas by projectile energy, must decrease as the charge decreases, for we have omitted in the formulas all consideration of the force necessary to start the projectile. It is apparent that if the charge were sufficiently reduced the projectile would not move and f in the formula would be zero. Therefore for any charge differing materially in weight from the charge used in the determination of f the value of f must. be modified. • - - Ingalls adopts, provisionally, this relation, f = fo (...)" (137) in which 6% is the weight of charge used in the determination of fo; f is the modified value of foſor the charge 65; 6) is any charge differing in weight from the charge 650 by fifteen per Cent Or more. - & The value of f will be modified also by a marked change in the weight of the projectile. Ingalls uses for f in this case the value - p. (w Yé f=f'(...) and if both 6 and w change sufficiently f=f(...)" (;) With the modified value of f from equation (137) we may now determine the velocities produced by reduced charges. Problem 4.—What muzzle velocities should be expected from the 6-inch gun of problem 3, with charges (including igniters) of 59 and 33% lbs. of the powder used in that problem 3 As these charges differ in weight by more than 15 per cent of the charge of 71 lbs. used in problem 3, we will obtain the value of f from equation (137), using for 650 and fo the values of problem 3. - We have as before C = 31.20 b = 1.5776 U = 252.5 115 The work may be conveniently performed as follows: Charge, 59 lbs. Charge, 33 1/4 lbs. Equation (137) log 3 1.77085 1.52179 . log Gºo 1.85126 1.85126 –– 3 T.91959 I.67053 T.97320 T.89018 log fo 3.13972 3.13972 - log f 3.11292 3.02990 Equation (109) log 63/w 9.77085 9.52.179 L 4.44383 4.44383 log Vº 7.32760 6.99552 Equation (100) A = 0.5234 0.2950 Equation (101) log a”= 0.10605 0.44028 Equation (702) log 20 = 1.86768 1.95285 20 = 73.736 89.712 Equation (103) for the muzzle a = 3.4244 2.8146 From the table, log X2 9.59202 9.55630 Equation (114) log VI* 7.32760 6.99552 log v.” 6.91962 6.55182 log va 3.45981 3.27591 V = 2883 f. S. W = 1888 f. S. The muzzle velocities actually obtained with charges of the above weights were 2879 and 1913 f. s. respectively. The calculated velocities show differences of 4 and 25 f. s. respect- ively. The latter difference, though practically not very great, shows that the modified value of f determined from the value deduced from one charge gives only approximate results when the second charge is, as in this case, less than 47 per cent of the first. Problem 5.-What muzzle velocities should be expected from the 6-inch gun of problem 3, with charges (including igniters) of 68 and 75 lbs. of the powder used in that problem 2 116 As these charges differ but little in weight from the charge of 71 lbs. used in problem 3, the value of f there determined will serve in this problem. f = 1379.5 C = 3120 b = 1.5776 U = 252.5. Charge, 68 lbs. Charge, 75 lbs. Equation (100) A = 0.6033 0.6654 . Equation (101) log a”= 0.01016 T.93901 Equation (102) log2 = 1.83344 1.80486 20 = 68.146 63.806 Equation (103) = 3.7052 3.9573 Equation (109) log V12= 7.41606 7.45861 From the table, log X2= 9.60555 9.61648 Equation (114) V = 3242 f. S. V = 3448 f. s. The measured muzzle velocities with these charges were, respectively, 3236 and 3455 f. s. The differences between the calculated and measured velocities are immaterial. We may make for this powder and gun any desired assumption as to the form of the powder grain, weight of charge, weight of projectile, size of powder chamber or length of gun, and with the values of f and t from problem 3, deter- mine the full circumstances of motion under the assumption. Sufficient illustration has now been given of the remarkable accuracy, the simplicity and extensiveness of application of the ballistic formulas deduced by Colonel Ingalls. By their use we may obtain a more intimate knowledge of the condi- tions existing in the bore of a gun than has heretofore been attainable; and the knowledge so obtained will be applied in the manufacture of powder and of guns, and will result in the production of more efficient weapons. United States Army Cannon.—On the following page will be found a table containing data concerning the principal cannon now in service. The bursting charges for projectiles, as given in the table, are of rifle powder for all guns down to and including the 4.72-inch Armstrong gun. For the 5-inch rifle and following guns the bursting charges are of high explosive. ...~" ," 2.~, (27) (29) (57) (73) (85) (87) (89) (97) (98) A =[1,44217] &/C. are *** * T A 8 ° 2 = [1,54708) wººd. . OC = w/20. *=[4,44383] f {, 3/w. y -v o ' ' ' ' ' ) -- *.. “, - --, * , / {-} (2 / C’ & ...) * f º, , , - INTERIOR BALLISTIC roRMULAs. M’ = [3.82867] Mw/a8 6. P’ =[3.35155] Wºw/a2 c5. P’ =[1.79538] f/a2. t = [2.56006] a V was Xoyd?. f = [5.55617] Vºw/3. (100) (101) (102) (103) (104) (105) / (106) (107) (108) (109) In addition to the above working equations the following formulas are needed or problems: (74) (95) (75) (86) (78) (88) (80) .> …- -* * ;! 3. . #*·ș�* *+ }·|- * ----# '' ?● | → *，{ & * * «;. » }... ±|4- +ņ.#* , ,· *�-± #----- • •}«¿? - ? * }} , ñ } ... *~~~~ą: �-** *� } t- ！► +: &. w… … *• • • &'）.'.*?¿（...）«==&&*）.--→ ·- ·►ae----... ;--~~~~----- * * *****~~y:~--~~~~）;<--.…**… … * →-a^*，…，*<~ ae**<<,·º·:·-*:: -* -**№.………*…***. ：~„...， （******,…,……*…***· ** -‘ “..… ;-） ****-…--- -* … - ~~~~ ………~~~~~~~~）… ， ** * ·，≤） * ... - * -**~~~~...…?******）*- }} -· --/»£ ** → !«? ·și ■ș”. * E PRINCIPAL UNITED STATES ARMY CANNON. & Cº.; of Travel of Weight of Weight of Muzzle | Maximum Bursting Gun Weight. cº, iºr. projectile. charge. projectile. velocity. pressure. Density of charge. th * * .* loading. pounds. cubic inches inches. pounds. pounds. ft. Sec. pounds. pounds. Mountain, Field and Siege. 1.457-inch pompom.................................. 4.86 40.68 1.2 oz. 1 1800 27000 0.4272 270 grs 2.95-inch mountain gun............... 236 34.9 26.4 8 Oz. 12% 920 18000 0.3966 13 Oz 3-inch field rifle, 1902.................. 835 66.5 74.54 26 Oz. 15 1700 33000 0.6764 |............ 3.6-inch mortar, 1890........... .4 ± e º ºs e a 245 33.2 16.065 || 6 Oz. 20 690 17000 0.3127 4 Oz 5-inch siege rifle, 1898................. 3639 402.5 | 119.8 5.37 45 1830 35000 0.3693 2.3 5-inch siege howitzer, 1900........... 1170 100 55.655 25 Oz. 55 1000 23000 0.4325 2 7-inch siege howitzer, 1898........... 3650 3]6.7 81.385 4.6 105 1100 28000 0.3933 8.6 7-inch siege mortar, 1892............. 1715 182.8 44.82 4 125 800 20000 0.6057 || 12.5 Seacoast. - -- 1.457-inch subcaliber tube........... 120 7.7 68.37 2.5 oz. 1.057 2100 || 25000 0.5617 2.5 Oz 2.95-inch subcaliber tube............ - 236 34.9 27.4 7 oz. 18 750 18000 0.3470 || 13 Oz. 2.24-inch rifle (6 pdr.) 1900......... 850 50 101.759 1.25 6 2600 34000 0.6920 3.4 Oz 3-inch rifle (15 par.)................... 1950 200 127.84 5 15 2150 34000 0.6920 0.62 4.72-inch rifle, 50 cals................. 6160 496 205.1 10.5 45 3000 34000 0.5860 4.5 5-inch rifle, 1900......................... 11120 1211 214.605 26 58 3000 36000 0.5943 2.75 6-inch rifle, 1903......................... 19990 2114 256.8 43 106 3000 36000 0.5630 5 8-inch rifle, 1888..... ................... 32218 3617 205.25 80 3.18 2200 38000 0.6122 19 10-inch rifle, 1900........................ 76830 10040 329.62 245 606 2550 38000 0.6755 36 10-inch mortar, 1890................... 16734 1554 82.76 34 575 1150 33000 0.6056 |............ 12-inch rifle, 1900........................ 132380 17487 395 375 1048 2550 38000 0.5936 64 12-inch mortar, C. I., 1886.......... 31920 2021 91.64 43 827 1200 27500 0.5889 35.5 12-inch mortar, steel, 1890........... 29120 2674 98.92 54 1()51 1150 33000 0.5590 64 16-inch rifle, 1895........................ 284.500 29624 451.86 | 660 2400 2300 38000 0.6167 ............ 118 QC . () : : : ii i log X0 9.03899 9.53911 9.88671 0.03494 0.12078 0.18111 0.22750 0.26509 0.29661 0.32372 0.34746 0.36855 0.38750 0.40469 0.42041 0.43489 0.44829 0.46075 0.47241 0.48334 0.49363 0.50334 0.51255 0.52128 0.52960 0.53752 0.54508 0.55234 0.55929 0.56597 0.57238 0.57856 0.584.52 0.59026 0.59582 0.601.19 0.60639 0.61143 0.61632 (). 62106 0.62567 log XI 5.56.162 7.05911 8.09440 8.53009 8.77897 8.95170 9.08.291 9. 18802 9.27522 9.34942 9.4J 375 9.47036 9.52077 9.56610 9.60719 9.64471 9.67918 9.7.1100 9.74052 9.76802 9.79373 9.81784 9.84053 9.86193 9.88217 9.90.136 9.91958 9.93693 9.95346 9.96926 9.98436 9.99884 0.01.272 0.02605 ().03887 0.05122 log X2 6.52263 7.52000 8.20769 8.49515 8.65819 8.77059. 8.88541 8.92293 8.97861 9.02570 9.06630 9. 10181 9. 13327 9. 16141 9. 18678 9.20982 9.23089 9.25025 9.26812 9.28468 9.30010 9.31450 9.32798 9 34065 9.35258 9.36384 9.37449 9.38459 9.394.17 9.40329 9.4.1198 9.42028 9.42820 9.43579 9.44305 9.45003 log Y3 8.73764 9.23296 9.56059 9.68493 9.74798 9.78653 9.81206 9.82962 9.84.191 9.85051 9.85640 9.86028 9.86260 9.86371 9.86386 9.86325 9.86201 9.86027 9.85811 9.85562 9.85284 9.84984 9.84664 9.84329 9.83981 W 9.83623 9.88256 9.82882 9.82503 9.82119 9.81732 9.81343 9.80953 9.80561 9.80169 9.79777 TABLE OF X FUNCTIONS. log X4 9. 16405 9.66437 0.01322 O. 16295 0.25023 0.31194 0.35965 0.39851 0.431.27 0.45956 0.484.44 0.50663 0.52665 0.54488 0.56161 0.57705 0.59140 0.60479 0.61733 . 0.62913 0.64027 0.65081 0.66082 0.67034 0.67942 0.68809 0.69640 0.70436 0.7120.1 0.71936 0.72644 0.73328 0.73988 0.74625 0.75242 0.75840 ſ log X5 8.30001 9.30059 9.99778 0.29663 0.47060 0.59347 0.68834 0.76552 0.83052 0.88660 0.93587 0.97980 1.01937 1.05539 1.08840 1.11887 1. 14715 1.17352 1.19821 1.22143 1.24332 1.26404 1.28369 1.30239 1.32020 1.33721 1.35348 1.36908 1.38406 1.39846 1.41230 1.42569 1.43858 1.45104 1.46310 1.47.478 0.630.15 0.63875 0.64691 0.65467 0.66207 0.06311 0.07459 0.08567 0.09638 0.10675 0.11678 0.13591 0.15395 0.17097 0.18708 9.45672 9.46316 9.46935 9.47532 9.48108 9.48663 9.497.17 9.50704 9.51630 9.52501 9.79386 9.78996 9.78607 9.78219 9.77833 9.77449 9.76687 9.75939 . 9.75193 9.74461 0.76419 0.76981 0.775.27 0.78057 0.78573 0.790.75 0.80040 0.80958 0.81833 0.82668 1.48608 1.49705 1.50770 1.51803 1.52808 1.53788 1.55668 1.57456 1.59158 1.60783 119 i log X0 0.66914 0.67589 0.68287 0.68859 0,69457 0.70032 0.70587 0.71122 0.71639 0.72140 0.72624 0.73093 0.73548 0.73990 0.74419 0.74836 0.75637 0.76398 0.77121° 0.77810 0.78469 0.79099 0.79703 0.80284 0.80842 0.81379 0.81897, 0.82397 0.8288.1 0.83349 0.83801 0.84241 0.84667 0.85081 0.85483 0.85873 0.86254 0.86625 0.86986 0.87338 0.87682 0.88017 0.88.345 0.88665 0.88978 0.89284 0.89584 0.89877 0.90165 0.90447 log X1 0.20236 0.21687 0.23070 0.24389 0.25650 0.26858 0.28014 0.29124 0.30190 0.31217 0.32205 0.33159 0.34079 0.34969 0.35829 O.3666.2 0.38250 0.397.45 0.41157 0.42492 0.43759 0.44963 0.46110 0.47205 0.48251 0.49253 0.50213 0.51136 0.52022 0.52875 0.53698 0.54492 0.55259 0.56000 0,56717 0.57411 0.58084 0.58737 0.59371 0.59986 0.60585 0.61167 0.61734 0.62286 0.62824 0.63349 0.63860 0.64360 0.64848 0.65324 log X2 9.53322 9.54098 9.54833 9.55531 9.56.194 9.56826 9.57427 9.58001 9.58551 9.59077 9.59582 9.60066 9.60532 9.60979 9.61410 9.61825 9.62613 9.63348 9.64036 9.64682 9.65290 9.65864 9.66407 9.66921 9,67409 9.67874 9.68316 9.68738 9.69142 9.69528 9.69897 9,70252 9.70592 9.70919 9.7.1234 9.7.1538 9.71830 9,72112 9,72385 9.7.2648 9,72903 9.73150 9.73390 9,73621 9.73846 9.7.4065 9.74276 9.74482 9.74683 9.74877 log X3 9.78740 9.73031 9.72333 9.71645 9.70969 9.70304 9.69650 9.69007 9.68374 9.67752 9.67140 9.66538 9.65946 9.65363 9.64790 9.64225 9.63122 9:62053 9.61015 9.60008 9,59029 9.58079 9.57153 9.56252 9.55375 9.54521 9.536.87 9.52874 9.52081 9.51306 9.50549. 9.49809 9.49085 9.48377 9.47683 9.47004 9.4634.1 9.45689 9.45050 9.44424 9.43809 9.43206 9.42614 9.42033 9.41462 9.40901 9.40349 9.39807 9,39274 9.38749 log X4 0.83467 0.84232 0.84966 0.85673 0.86353 0.87009 0.87642 0.88252 0.88842 0.89416 0.89970 0.90508 0.91027 0.91537 0.92037 0.92510 0.93432 0.94308 0.95143 0.95939 0.96700 0.97430 0.98.130 0.98803 0.994.50 1.00074 1.006.76 1.01257 1.01819 1.02363 1.02890 1.03402 1.03898 1.04379 1.04848 1.05304 1.05748 1.06180 1.06601 1.07012 1.07413 1.07804 1.08187 1.085.60 1.08926 1.09.283 1.09633 1.09976 1.10312 1.10640 log X5 1.62338 1.63824 1.65250 1.666.23 1.67945 1.69216 1.70442 1.71627 1.72773 1.73882 1.74956 1.75997 I. 77004 1.77989. 1.78955 1.79872 1.81656 1.83349 1.84962 1.86500 1.87971 1.89379 1.90730 1.92028 1.93277 1.94479 1.95640 1.96760 1.978.44 1.98891 1.99905 2.00892 2.01847 2.02776 2.03677 2.04552 2.05408 2.06240 2.07.050 2.07841 2.08612 2.09345 2.10100 2.10819 2. II 502 2. 12209 2.12882 2. 13540 2.14186 2.14818 120 QC log X4 1.10963 1.11279 1.11589 1.11893 1.12.192 1.12485 1.12772 1.13057 1.13335 1,13609 1.13877 1.14142 1. 14402 1.14659 1. 14911 1.15159 1.15403 1.15644 1.15882 1.16115 1.16346 1.16573 1.16797 1.17018 1.17236 1.17450 1.17663 1.17872 1.18078 1.18282 1.18483 1.18682 1.18879 1.19072 1.19264 1.19454 1.19640 1.19825 1.20008 1.20188 1.20367 1.20543 1.20717 1.20891 1.21062 1.21280 ... --~~~ **** * 11.0 11.2 11.4 11.6 11.8 12.0 12.2 12.4 12.6 12.8 13.0 13.2 13.4 13.6 13.8 14.0 14.2 14.4 14.6 14.8 15.0 15.2 15.4 15.6 15.8 16.0 16.2 16.4 16.6 16.8 17.0 17.2 17.4 17.6 17.8 18.0 18.2 18.4 18.6 18.8 19.0 19.2 19.4 19.6 19.8 20.0 log X0 0.90723 0.90993 0.91259 0.91520 O.91776 0.92027 0.92274 0.92516 0.92754 0.92989 0.932.19 0.93446 0.93669 0.93888 0.94.104 0.94317 0.94527 0.94.733 0.94936 0.95137 0.95334 0.95529 0.95721 0.95910 0.96097 0.96282 0.96463 0.96643 0.96820 0.96995 0.97168 0.97.338 0.97507 0.97673 0.97838 0.98001 0.98161 0.98320 0.98477 0.98632 0.98785 0.98937 0.99086 0.99235 0.99882 0.99527 log XI 0.65790 0.66245 0.66691 0.67127 0.67554 0.67972 0.68381 0.68783 0.691.76 0.69562 0.6994.1 0.70313 0.70678 0.71036 0.71388 0.71734 ().72074 0.724.08 0.72736 0.73059 0.73377 0.73689 0.73997 0.74301 0.745.99 0.74892 0.75181 0.75466 0.75747 0.76024 0.76297 0.76566 0.76831 0.770.93 0.77351 0.77606 0.77856 0.78104 0.78349 0.78591 0.78829 0.79065 0.79296 0.79527 0.79754 0.799.78 log X2 9.75067 9.75.252 9.75432 9.75607 9.75778 9.75945 9.76108 9.76267 9.76422 9.76574 9.76722 9.76867 9.77009 9.77148 9.77284 9.77417 9.77547 9.77675 9.77800 9.77922 9.78043 9.78160 9.78276 9.78391 9. 7850] 9.78610 9.78718 9.78823 9.78927 9.79029 9.79.129 9.79227 9.79324 9.794.19 9.79513 9.79605 9.79696 9.79785 9.7.9872 9.79959 9.80044 9.801.28 9.80210 9.80292 9.80372 9.80451 log X3 9.88233 9.87725 9.37225 9.36732 9.36247 9.35770 9.85301 9.34836 9.34379 9.33928 9.33484 9.33045 9.32613 9.32186 9.31766 9.31350 9.30940 9.30535 9.30.136 9.29741 9.29351 9.28966 9.28585 9.28208 9.27837 9.27470 9.27107 9.26748 9.26393 9.26042 9.25695 9.25352 9.25012 9.24676 9.24344 9.24015 9.23689 9.23367 9.23048 9.22732 9.22419 9.22109 9.2 1803 9.21499 9.21.198 9.20900 log X5 2.15437 2. 16045 2.16642 2.17227 2, 17801 2.18364 2.18916 2. 19462 2. 19996 2.20522 2.21039 2.21547 2.22047 2.22539 2.23023 2.23400 2.23970 2.24433 2.24888 2.25337 2.25780 2.26216 2.26647 2.27073 2.27495 2.27912 2.28309 2.287.11 2.29.108 2.29500 2.29886 2.30268 2.30645 2.31017 2.31385 2.31750 2.32108 2.32463 2.32814 2.33161 2.33504 2.33843 2.34.177 2.34510 2.34838 2.35162 3. Ordnance and Gunnery PART II. * V, Metals Used in Ordnance Construction, Guns, Projectiles. Prepared for the Céets of the United States Military Academy. BY ORMOND M. LISSAK, Major, Ordnance Department, U. S. Army. Instructor of Ordnance and Gunnery. r - * * * : * , * . . - .. - & - - - .* - * - - * i ...” i - Y. r º * t - * * * f. - x * - - g- - … & -- º - --~~~ - g ..º - - ^- - - - & - - * - .3 - - e # ; * - § & + * - • * - ** * ! - - ; º ‘. . d - § * { r t - 3. \ - - s * º - - * - º .. - • *... * . - _^7gºG9 · {! 29:† 2{} |×8T 29!# 27 GT-ſt， søT32}} | __/ 9It L2 ú 360 $|-{{ ：º.ſ.63 . '- /1† 983 ： y lºC~!% Țg93I ſied-xegºſ _^● � ežđBÖ.OJ, * O N�X&ſ, Å9ȚÂ№}} � º 91 ) 6T- JC&{\CI&O<} (I24J:9q3:04, đQ S I Á„ſeuurnſ) pua eourgu prº u¡ SU: ) see "I 9? Å 2 9 3 9°C eſſed OL 3 OÙ {3A}\} • c） 18 26 33. 39 53 75 10 Shrin.ºgge—45 Prob . .3–52 67 * Iliş Şak-part II m ſº º -ri-IIs. '''“*** - - - - - , , . . . .- - - - - - --ºf----' ' ' ' ' ' ' ' ' ' 36 12 Prac. Instr. —— 37 14 Pages 105-121 U. Bruff º f! 155-171 s $1 • ºrg g ‘R CONTENTS. er IV, Metals Used in Ordnance Construction. .. 3 - Stress and Strain, 3. Physical qualities of metals, 3. Strength of metals, 4. Copper, Brass, Bronze, 6. Iron and Steel, 6. MANUFACTURE OF STEEL FORGINGS FOR GUNS, Io Open hearth process, Io. Casting, 13. Whitworth’s process of fluid compression, I 5 Forging, I 7. Oil tempering, 29. .er V, Guns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 ELASTIC STRENGTH OF GUNS . . . . . . . . . . . . . . . 22 Elasticity of metals, 22. Stresses and strains in a closed cylinder, 25. Lamé's laws, 26. Basic principle of gun construction, 30. Stresses in a simple cylinder, 30. Com- pound cylinder, Built up guns, 39. Shrink- age, 45. Application of the formulas, 48. Systems composed of three and four cylin- ders, 56. Wire wound guns, 57. CONSTRUCTION OF GUN's . . . . . . . . . . . . . . . . . . . . 58 MEASUREMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . 07 RIFLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 BREECH MECHANISM . . . . . . . . . . . . . . . . . . . . . . 75 Slotted screw breech mechanism, 76. Ob- - turation, 8o. Firing mechanism, 83. Slid- , -, ing wedge breech mechanism, 85, Older forms of breech mechanism, 86. Automatic ~~ and semi-automatic breech mechanism, 89. - º ſter VI, Projectiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 ~~ Old forms of projectiles, 90. Modern projec- tiles, 92. Canister, 95. Shrapnel, 95. Shot and shell, Ioo. Armor piercing projectiles, IoI. Shell tracers, Io9. Volumes of ogival projectiles, Io9. Weights of projectiles, IoT. Thickness of walls, Io". Sectional density of projectiles, I Io. - MANUFACTURE OF PROJECTILES . . . . . . . . . . . . . . I I I Inspection, II 3. Painting, I I6. # CHAPTER IV. METALS USED IN ORDNANCE CONSTRUCTION. StreSS and Strain.-A proper understanding of these terms will be helpful in what follows. Stress is an applied force. Strain is the effect of the force as measured by the change in form of the material to which the stress is applied. - * Stresses are of different kinds depending on the manner of application of the force; as tensile stress, compressive stress, tor- sional stress. A torsional stress is a compound stress and may be resolved into a tensile stress on some elements of the material and a compressive stress on others. Each kind of stress produces a corresponding strain, or effect On the material, the tensile stress producing elongation, the com- pressive stress compression. As all stresses may be resolved into tensile and compressive stresses, all strains may be resolved into 'elongation and compression. Physical Qualities Of Metals.-The following qualities of metals are those with which we are most concerned in ordnance Construction. Fusibility.—The property of being readily converted into the liquid form by heat. Malleability.—The property of being permanently extended in all directions without rupture when hammered or rolled. Ductility.—The property of being permanently extended with- Out rupture by a tensile stress, as in wire drawing. Hardness.-The property of resisting change of form under a compressive stress. A hard metal offers great resistance to such a stress while a soft meta1 yields readily and changes its form without rupture. The terms hardness and softness are of course comparative only. , sº 4 Toughness.-The property of resisting fracture under change of form produced by any stress. . Elasticity.—The property of resisting permanent deformation under change of form. This is one of the most important prop- erties of gun metals, which under the high stresses due to the ex- plosion are subjected to extensive deformation. Through this property the deformations disappear on the cessation of the stress and the metal resumes its original dimensions. Strength of Metals.-The strength of metals is ordinarily determined by physical tests in a testing machine. As the tensile strength of metals is less than the compressive strength, usually a tensile test Only is applied. A test specimen is cut from the metal to be tested and is prepared in suitable form to be inserted in the machine. The area of the cross section of the test specimen is usu- ally a Square inch or some aliquot part of a square inch. In the machine the test piece is subjected to a tensile stress, the amount of which is recorded by a sliding weight on a scaled beam. The test piece stretches under the applied stress. With elas- tic metals it will be found that up to the application of a certain stress the test piece will resume its original length if the stress is removed, but on the application of a stress greater than this the test piece will remain permanently elongated. When permanent distortion occurs the metal is said to have a permanent set. Elastic Limit.--The stress per square inch applied at the moment that the permanent set occurs is called the elastic limit of the metal. Within this limit the metal has practically perfect elasticity and does not suffer permanent deformation. As the stress increases beyond the elastic limit the metal stretches permanently and more rapidly, the cross section at the weakest point reduces, and finally the test piece ruptures. The elastic strength of metals will be found more fully treated in the chapter On the elastic strength of guns. Tensile Strength.-The stress per square inch that pro- duces rupture of the metal is called the tensile strength. Elongation at Rupture and Reduction Of Area. — In Ordnance structures the stresses are not expected to exceed the elas- 5 tic limit of the metal, but should they by any chance exceed this limit the tensile strength of the metal and its capacity to elongate before rupture become of importance. The permanent elongation will serve as a warning that the elastic strength has been exceeded. The reduction of area of cross section is intimately connected with the elongation. In the tests of metals for ordnance purposes these particulars are therefore always noted and limits are prescribed. For the measurement of the elongation the parts of the ruptured test piece are brought together and the distance is measured be- tween two punch marks that were made on the test piece before insertion in the machine. The tensile test therefore includes the determination of the elastic limit, the tensile strength, the elongation at rupture and the reduction of area of cross section. The last two are recorded in percentages of the original dimensions. The following table shows the physical requirements demanded by the Ordnance Department in the principal metals used in ord- nance construction, the requirements varying for each kind of metal according to the use to which it is destined. Elastic Tensile Elongation Contraction Limit Strength at Rupture of Area lbs. per sq. in.|lbs. per Sq. in.) per cent. per cent. Copper 32000 22.0 Bronze, No. 1 28000 IBronze, No. 4 60000 20.0 Tobin Bronze 60000 25.0 Cast Iron, No. 1 22000 Cast Iron, No. 2 #28000 Wrought Iron 22000 50000 25.0 35.0 Cast Steel, No. 1 25000 60000 16.0 24.0 Cast Steel, No. 3 45000 85000 12.0 18.0 Forged Steel, No. 1 27000 60000 28.0 40.0 Forged Steel, (caps) 60000 30.0 45.0 Forged Steel, (tubes) 46000 86000 17, 0 30.0 Forged Steel,(jackets) 48000 90000 16.0 27.0 Forged Steel, (hoops) 53000 93000 14.0 20.0 Forged Steel, D 100000 120000 14.0 30.0 Nickel Steel - 65000 95000 18.0 30.0 Steel Wire, (guns) 100000 160000 * Cast Iron No. 2 must not show a tensile strength of more than 39,000 pounds per square inch. The standard government testing machine is at Watertown Ar- Senal, Mass. It has a testing capacity of 800,000 lbs. 6 Copper, Brass, Bronze.—Pure copper is used for the bands of projectiles. In alloys, as brass and bronze, it enters into the con- struction of parts of guns and gun carriages not usually subjected to great stress. In this form too it is extensively employed in the manufacture of cartridge cases, fuzes, primers, gun sights and in- struments. Brass is an alloy Of copper with zinc. Bronze is an al- loy of copper with tin and usually a small quantity of zinc. The ad- dition of zinc or tin produces a harder and stronger metal better suited than the soft copper for the uses to which these alloys are applied. By the addition of aluminum. Or manganese in the alloy the strong hard bronzes known as aluminum bronze and mangan- ese bronze are produced. IrOn and Steel.—When iron ore is melted in the furnace the product obtained, called pig iron, is an alloy of iron with carbon, the carbon content being about 5 per cent. This alloy may be read- ily fused and cast, and is then called cast iron. By various process- es in the furnace the amount of carbon in the iron may be reduced. When the quantity of contained carbon is between two per cent and two tenths of one per cent the product is steel. When there is less than two tenths of One per cent of carbon we have wrought 11"O11. As the amount of carbon is reduced the qualities of the metal change in a marked degree. Cast iron is easily fusible, is hard and not malleable or ductile, cannot be welded, and has a crystaline structure. Wrought iron on the other hand is practically infusible, is soft and possesses great malleability and ductility. It is easily welded and has a fibrous structure. . . . The properties of steel lie between those of wrought iron and cast iron, and the steel partakes of the characteristics of One or the Other according to the percentage of carbon contained. Thus low steel, that is, steel low in carbon, is comparatively soft and may be readily welded or drawn into wire, while high steels are harder and more brittle and weld with difficulty. Temper.—The distinguishing characteristic of steel when com- pared with cast Or wrought iron is the property it possesses of hardening when cooled quickly after being heated to a red heat, 7 and of subsequently losing some of its added hardness when sub- jected to a lower heat. The double process produces what is called temper in the steel. There is considerable confusion in the use of the terms applied to the two processes necessary in the production of the temper. By some writers the first process, quick cooling from a high heat, is called tempering, and the second process, reheating to a lower heat, is called annealing. By others the first process is called hardening or quenching, and the second is called tempering. As both processes are necessary for the production of temper in the steel it would certainly be more correct to include under the term tempering both the necessary processes. In shops generally and in the specifications of the Ordnance Department the term tempering is used to designate the first pro- cess. For the present therefore we will continue the use of the term tempering to designate the process of quick cooling from a high heat. The degree of hardness attained in tempering depends upon the rapidity of cooling. In addition to increasing the hardness, tempering increases the elastic limit and tensile strength of the steel and reduces its toughness as shown by reduced elongation at rupture and diminished reduction of area of the cross section. The properties of the metal are affected in varying degrees depending on whether the temper is high or low; that is, whether the steel after being highly heated is suddenly cooled or cooled more slow- ly. If suddenly cooled from a red heat the steel becomes very hard and brittle and has little malleability Or toughness. The effect of oil tempering on the elastic limit and tensile strength of steel and on the elongation at rupture is well shown in the following table taken from tests made by the Navy: ELASTIC LIMIT TENSILE STRENGTH ELONGATION Before After IBefore After Before After Tempering | Tempering Tempering | Tempering Tempering | Tempering Tons Tons Tons Tons Inches Inches 13.77 34.15 28.1 49.8 . 596 .260 13 29.2 26, 8 45.2 .737 .433 13 27.8 27.8 46.0 .707 .345 12.18 34.8 26.9 49.4. . 564 .202 12 26.0 27.6 39.6 . 713 .420 12 25.8 28.0 41.0 . 633 .480 8 The elastic limit is more than doubled, the tensile strength is increased more than 60 per cent, while the elongation is reduced over 40 per cent. Annealing.—The process of tempering leaves internal strains in the metal which may be removed by heating the metal to a lower temperature than that used in tempering, and COOling slowly. This process is called annealing, and has the effect of reducing some- what the elastic limit and tensile strength of the steel after temper- ing, while increasing its toughness, as shown by increased elong- ation at rupture and increased reduction of area in the cross sec- tion. If the steel is heated to the temperature at which it was tem- pered and then cooled slowly, the temper is drawn and the steel again becomes soft and loses its increased elasticity and strength. The annealing following the tempering reduces the elastic limit and tensile strength from IO to I5 per cent and increases the elongation from 25 to 40 per cent, thus restoring the toughness in the metal. - - By proper regulation of the processes of tempering and an- nealing an extensive control of the properties of the metal is ob- tained, permitting the production of metal of the quality best suit- ed to any particular purpose. - Constitution of Steels.-The difference in the character- istics of tempered and untempered steel is due to the difference in the condition of the carbon content. In the harder or tempered steels the carbon is more intimately combined with the iron and is called fixed carbon, while in the softer steels the carbon is segre- gated to a greater or less extent, and is called free carbon. By tem- pering and annealing, the carbon may be changed from One condi- tion to the other. In addition to the carbon in the metal, there are other sub- stances, some of which are always present and others that may be added, that affect the qualities of the steel. . Sulphur, phosphorus, manganese and silicon are usually pres- ent to a greater or less extent in all steels. If present in too large a percentage sulphur produces what is called hot shortness in the metal, that is brittleness when hot, while phosphorus makes the 9 metal cold short, or brittle when cold. Manganese and silicon when present in proper percentages improve the qualities of the metal. Chromium and tungsten give hardness to the steel without brittleness. Nickel also greatly increases the toughness of the steel. Nickel steel for guns contains about 3% per cent of nickel. Uses.—Cast iron, wrought iron, cast steel and forged steel are all used in ordnance constructions. Cast iron on account of its cheapness and ease of manufacture in irregular shapes is used when practicable wherever great strength is not required, as in pro- jectiles for the smaller guns and in parts of carriages not subject to wear or to high stresses. Wrought iron is not now extensively used in Ordnance construc- tions. The older sea coast carriages were almost wholly made of this metal. Wherever great strength is required steel is employed. Cast steel is used in those parts that do not require the greater strength of forged steel, or that on account of their irregular shapes can not be readily produced as forgings, such as the chassis and top car- riages of sea coast gun carriages. Cast steel has also been used for projectiles and for guns but without great success. In structures or parts of structures requiring great strength, or subject to wear, forged steel only is used. Guns and armor and armor piercing projectiles are now made of forged steel Only, and the operative parts of gun carriages and of Other structures are principally of this metal. Gun Steel.-Of two steels, one high in carbon and the other low in carbon, the steel with the higher percentage of carbon will, with similar treatment, have the higher elastic limit. Since the elastic limit of the metal is the limit of the strength considered in the construction of guns it would appear that the metal with the highest elastic limit would be the most desirable. But high steel is more difficult to manufacture than low steel, and in large pieces there is much greater liability to flaws, strains and incipient cracks. After passing the elastic limit the hard steel has little re- maining strength and breaks without warning, while the low steel, 10 due to its greater toughness, yields considerably without fracture. For these reasons a low steel containing about one-half of One per cent of carbon is used in the manufacture of guns. --~~ * *** - ...------------ **** .----~~~~ MANUFACTURE OF STEEL FORGINGS FOR GUN s. Open Hearth Process. – All gun-steel at the present day is made by this process, which derives its name from the fact that the receptacle in which the steel is melted is open at the top and exposed to the flame of the fuel, which plays over the surface and performs a principal part in the formation of the steel. The product is called Siemens or Siemens-Martin steel, according to the ingred- ients contained. • * - The open hearth furnace, invented by Dr. Siemens, consists of the following essential parts: I. The gas-producer; 9 2. The regenerators; 3. The furnace proper. THE GAS PRODUCER—The fuel used in the Siemens furnace is gaseous, and is obtained from ordinary fuel by subjecting the lat- ter to a preliminary process in the gas-producer. This apparatus, […E. /. / wº-Y A. F.J.G.. I Fig. I, consists of a rectangular chamber of fire brick, one side, B, being inclined at an angle of from 45 to 60 degrees. A is the grate. The fuel, which may be of any kind, is fed into the producer through 11 the hopper C. As the fuel slowly burns, the CO, rises through the mass above it and absorbs an additional portion of C, becom- ing converted into 2CO. This gas passes out of the Opening D into a flue. In order to cause it to flow toward the furnace it is led through a long pipe, E, where it is partially cooled, and then de- scends the pipe F leading to the furnace. The gas in F being cool- er than that in E and D, a constant flow of gas from producer to furnace is maintained. THE REGENERATORS.—The gas entering the furnace is, as has been stated, CO. To burn it to CO, air must be mixed with it. This mixture is made in the furnace proper, the CO and air being kept separate till they reach the point where they are to burn. The CO is cooled to some extent, as shown, before being admitted to the furnace. - To heat both air and CO before they are mixed and burned, and to accomplish this economically, and raise the gases to a high temperature, the waste heat of the furnace is employed. The heating of the gases is accomplished by means of the regenerat- ors, Fig. 2. They consist of four large chambers below nº = −. ºr º ºs = º º; º; # = º % § #4;%=B}=%=C=%=D=ſº º [. §://EDILILILITILITILITIL. |CITIETIºſ I, III:It [I aſ T.I. º ...? ſºft C III, º * ATâTTTTāTATATTº TATº * * * * * * * % # 3 % % #4 # # £3% §§§aºs::s::::::::=#3:Sºsºsºsºsº Fig. 2. the furnace, filled with fire-brick. The fire-brick is piled so that there are intervals between the bricks to allow the passage of gas and air. When the furnace is started, CO is admit- ted through A and air through B, both A and B being cold. The gasses pass through the fire-bricks in A and B and through flues at the top, and flow into the furnace proper, where they are lighted. The products of combustion are caused to pass through C and D, 12 which are similar chambers. In doing so these products heat the fire-bricks in C and D. After some time—about one hour general- ly,–by the action of valves controlled by the workmen, the CO and air are caused to enter the furnace through D and C respect- ively, and the products of combustion to pass out through A and B. In this case the CO and air, entering the heated chambers D and C, are raised to a high temperature before ignition, and the temper- ature of the furnace is thereby greatly increased. It is also evident that A and B will be more highly heated than C and D were, and therefore when the next change is made, the gas and air passing through A and B will be more highly heated than when they passed through D and C, and so on. - The action of the furnace is therefore cumulative, and its only limit in temperature is the refractoriness of the material. By regu- lating the proportions of gas and air, which is readily done, the tem- perature may be kept constant. THE FURNACE.-The furnace proper, Fig. 3, consists of a dish-shaped vessel D of cast iron, supported SO that the air can circulate freely around it and keep it from melting. It is lined with refractory sand S and constitutes the hearth of the furnace. pºs ZZZ THTii Tài i %2. H-5 Ø % sº . Af ZZ º º %& gº *** 2. º &T * £% GAs.g. FIG. 3 The gaseous fuel and air enter by the flues F, and the products of combustion escape by the flues F/, or the reverse, according to the positiºn of the regulating valves. * The arrows show the direction of these currents. The roof R is lined with fire-brick, and by its shape deflects the flame over the metal in the hearth. At opposite ends of the furnace are a charging- 13 door for admission of the metal, and a tap-hole, closed with a plug of fire clay, for drawing off the finished steel. These are not shown in the drawing. OPERATION.—The principle of the process is that when wrought iron or steel scrap is added to melted cast iron, the percentage of carbon is thereby reduced till it reaches that required for steel. The charge consists of pig-iron heated red-hot in a separate furnace and then placed on the hearth of the Siemens furnace. By the act- ion of the furnace the pig-iron is soon melted. Scrap wrought-iron or steel is then added in suitable proportions, till the percentage of carbon is low. When it has reached the proper point the per- centage is made exact by adding a pig iron containing a known percentage of carbon, such as spiegeleisen or ferro-manganese, or by the addition of ore. The percentage of carbon in the steel at any stage of the process is determined by taking samples from the melt- ed metal, cooling them and observing their fracture on break- ing; and by dissolving portions of the specimens in nitric acid and comparing the color with that of standard solutions of steel containing different percentages of carbon. In this way the com- position of the steel can be exactly regulated; for the metal can be kept in a melted state without damage for a considerable time, and the character of the flame made oxidizing or reducing at will, according to the relative amounts of air and CO admitted. The operation ordinarily lasts about eight hours for each charge. When the steel has attained its proper composition, the fur- nace is tapped and the metal cast into ingots, ready for the suc- ceeding operations. Other Processes.—The crucible process is used to some ex- tent by Krupp in the production of gun steel. The ingredients of the steel are melted in crucibles, and the resulting steel is poured from the crucibles into a common reservoir from which the ingots are CaSt. The Bessemer Process, though important and producing large quantities of steel, is not used in making gun-steel. Casting.—The molten metal is drawn into an iron ladle which depends from a crane in front of the furnace. The ladle, Fig. 4, is 14 • lined with refractory sand. It is provided with trunnions, T', so that it may be tipped for pouring the metal into the mold, or it may have a taphole, T, in the bottom, closed with a plug of fire clay. The plug is lifted and replaced by means of a rod R also encased in re- fractory sand. There is an advantage in drawing the metal from the bottom of the ladle in that the scoria and impurities that float on the surface may be kept out of the mold. The metal if very hot is poured slowly into the mold in a thin stream, thus allowing oppor- tunity for escape of the gases that it contains. If at a lower tem- perature it may be poured more quickly. It is frequently allowed to cool to the desired temperature in the ladle. MoldS.–In the casting of Ordinary ingots, the iron or steel molds into which the metal is poured from the ladle are slightly ſº) ſ) Fig. 5. sold. SPLT. 15 conical in shape, see Fig. 5, to facilitate their removal from the ingot. They are of various cross sections, depending on the shape of the ingot desired. The interior surface is covered with a wash of clay or plumbago. Sinking Head.—In all castings, whether of iron, steel, or other metal, an excess of metal, called the sinking head, is left at the top of the mold. The pressure due to the weight of this metal gives greater density to the lower-portions of ºthé casting. The sinking head also serves to collect the scoria and impurities which rise to the top, and it provides metal to fill any cracks or cavities that may form in the cooling of the ingot. Defects in Ingots. Blow Holes.—The gases in the melted metal, unable to escape from the mold, form holes in the ingot, called blow holes. These can not be detected, nor can they be re- moved by forging. Forging changes their form Only without giving continuity to the metal. Blow holes are more prevalent in Besse- mer than in open hearth steel and are more apt to Occur at low temperatures of casting, when the metal hardens before the gas can escape. Pipes.—The metal in contact with the molds cools first and so- lidifies. As thé cooling and consequent contraction extends toward the center, the liquid metal is drawn away from the center leaving cavities called pipes along the axis of the ingot. Pipes most fre- quently occur when the metal is cast too hot. Thus on the one hand too low a temperature causes blow holes and too high a temper- ature pipes. Segregation.—As the various constituents of the steel, (iron, silicon, manganese, etc.) solidify at different temperatures, it fre- quently happens that they separate from each other as the ingot cools, forming what is called segregation. This gives a different structure to the metal and greatly weakens it. Segregation is more likely to be found toward the center of the ingot and in the upper portions. Whitworth’s Process of Fluid Compression.—The pur- pose of this process, invented by Sir Joseph Whitworth of Eng- land, is to remove as far as possible the blow holes, pipes and other defects from the ingot and to give the metal greater solidity and 16 uniformity of structure than can be obtained in the ordinary meth- od of casting. The object is accomplished, to a large extent, by the application of enormous pressure on the metal while in the fluid state in molds so constructed as to allow free escape of the gases. The flask, f Fig. 6, made of cast steel, is of great strength to with- stand the great pressure. It is built up of cylindrical sections which are bolted togther to the desired p &= length. The interior of the flask is - lined with wrought iron bars b, whose long edges are cut away or bevelled to form channels, a, by means of which the gas may escape; the interior and exterior channels thus formed being connected by grooves, c, cut in the sides of the bars at short intervals. The cylinder formed by the interior surfaces of the bars is lined with refractory sand. A cast iron plate, d, through which are continued the longitudinal gas channels closes the mold at the bottom. The mold rests on a car in the bottom of a pit. When the mold is filled with metal the car is run under a hydrau- lic press. The head, p, of the press, of diameter only slightly less than the interior of the mold, is brought down against the molten metal and and locked in that position. The metal wells up around the head of the press and, quickly cooling, forms a solid mass which with the head completely closes the top of the mold. By pressure on the bottom of the car, gradually applied until a pressure of six tons to the square inch is reached, the car and mold are slowly forced upward. The mol- : º: : * . * * * -, * - . . .” - I • * * b. i .rº -- - -----------. . ten metal is compressed by the applied pressure, and the gas, forced through the sand lining and the channels between the lining bars, issues from the top and bottom of the mold in a violent flow of flame. The pressure is continued until the column of metal has shortened one eighth of its length. A uniform pressure of about 1500 pounds to the square inch is then left on the ingot while it cools, to follow up the metal as it contracts and prevent the forma- tion of cracks. Processes After Casting.—The specifications for gun forg- ings require that the forgings be made from that part of the ingot that remains after 30 per cent by weight has been cut from the top of the ingot and 6 per cent from the bottom. These parts are cut off as they are likely to contain most of the defects in the ingot. For hollow forgings the center of the part selected is then bored out in a heavy lathe, or punched out if the ingot is short. Heating.—The ingot is then heated preparatory to forging. The heating is accomplished in a furnace erected near the forging hammer or press, and is conducted with great care. The cooling of the ingot in the mold has left in the metal strains due to the suc- cessive contraction of the interior layers. Assisted by unequal ex- pansion in heating the strains may cause cracks to develop in the ingot. Great care is therefore exercised that the heating shall pro- ceed slowly and uniformly, thus avoiding the overheating of the exterior layers of metal before the heat has thoroughly penetrated to the interior. - Forging.—The heated ingot is forged either by blows deliver- ed by a steam hammer, or by pressure delivered by a hydraulic forg- ing press. Under the slow pressure of the forging press the metal Of the forging has more time to flow, the effect of the treatment is more evenly distributed and the metal is more uniformly strained. This process is therefore preferred in the manufacture of gun forgings. Fig. 7 is a reproduction from a photograph of a 5,000 ton hydraulic forging press at the works of the Bethlehem Steel Co. The print shows a bored ingot for the tube of a 12-inch gun being 18 forged on a mandrel. The Outer diameter of the ingot is reduced by the forging and the length of the ingot increased. The diameter of the bore remains practically unchanged. The outer end of the ingot is supported from an overhead crane. The ingot is turned on the anvil of the press, and advanced when desired, by means of the chain seen through the press. The method of turning is better shown in the plate following. The movements of the head of the press are controlled by means of levers situated at a short distance to the right of the press. The operator at the levers sees recorded on the dial the pressure exerted at any instant. - Fig. 8 shows a 10 ton steam hammer forging a solid ingot for a 3 inch gun. The ingot is supported from an overhead crane and is nearly balanced in the sling chain by the bar of iron clamped to the ingot and extending to the rear. This bar is called a porter bar, and by its means the ingot may be moved back and forth under the hammer. The ingot is turned under the hammer from the crane by means of the gearing shown in the upper part of the picture. The movements of the hammer are controlled by the man at the left through the levers shown at his hand. , --- Hollow Forgings.-In forging bored ingots a solid steel shaft, called a mandrel, is passed through the bore of the heated ingot, and the method pursued in forging depends upon whether the length of the ingot is to be increased without change of interior diameter, as in forging a gun tube, or whether the diameters of the ingot are to be enlarged, as in forging hoops. In the first case the ingot, on a mandrel of proper diameter, is placed directly on the anvil of the press as shown in figure 7. The effect of forging is then to increase the length of the ingot and decrease the outer diam- eter while maintaining the interior diameter unchanged. The man- drel is withdrawn from the forging by means of a hydraulic press. In forging hoops, the mandrel rests on two supports on either side of the head of the press, figure 9, and is itself the anvil on which the forging is done. By turning the mandrel new surfaces of the hoop are presented to the press. The walls of the hoop are reduced * 000 Ton HYDRAULic Forcing PREss. 5 FIG. 7. 10 Ton Steam HAMMER. FIG. 8. 19 FRONTiELEVATION. - - "SIDE, ELEVATION. Fig. 9 in thickness by the forging, the diameters of the hoop being in- creased while the length is not materially changed. The specifications for gun forgings require that the part of an ingot used for a tube forging shall have an area of cross section at least four times as great as the maximum area of cross section of the rough forging, and for a jacket forging 3% times as great. For forgings for guns I2 inches or more in caliber these figures are reduced to 3% and 3 respectively. Forgings for lining tubes must be reduced 6 times in area. If bored ingots are used the wall of the ingot must be reduced at least one-half in thickness. Annealing.—The working of the ingot in forging and the irregular cooling leaves the metal in a state of strain. The strains are removed by the process of annealing. For this purpose the forg- ing is usually laid in a brick-walled pit or furnace, and slowly and uniformly heated by wood fires, the burning logs being distributed along the pit as required to heat the forging uniformly. When the proper heat, usually a bright red, has been attained, the fires are allowed to die out, or are drawn, and the ingot remains in the fur- nace until both are cold. Three or four days may be required for the slow cooling of a large forging. 20 Oil Tempering.—Annealing removes the internal strains that exist in the forging but, as befºre explained, it greatly reduces the tensile strength and elastic limit of the metal. To restore the strength to the metal and to produce in it the qualities required in gun forg- ings, the forging is next subjected to the process of tempering. Be- fore tempering it is machined in a lathe nearly to finished dimen- sions. Specimens for tests are cut from the ends, and from their behavior in the testing machine the requirements of the subsequent treatment are determined. The forging is then slowly and uniformly heated throughout. Large forgings, such as tubes and jackets, are heated in vertical furnaces, great care being exercised that the heating shall be uni- form throughout the length of the piece in order that un- due warping may not occur in the subsequent cooling. When the forging is at a uniform red heat the side of the furnace is opened and the forging is lifted out by a crane and immersed in a deep tank of oil alongside the fur- nace. The oil tank is surrounded by another tank through which cold water is constantly running. The heat of the forging passes to the oil and thence to the water and is thus gradually conducted away. . The Bethlehem Steel Co. of Bethlehem, Penn., and the Mid- vale Steel Co. of Philadelphia, the two principal manufacturers of gun forgings in this country, use different oils for oil tempering, The Bethlehem Co. uses petroleum oil once refined. The Midvale Co. uses cotton seed oil with flashing point not less than 360 de- grees Centigrade. The temperature of the forging when immersed is very high compared with that of the oil. The cooling is therefore sudden at first, but as oil is a poor conductor of heat the heat of the forging is carried away slowly leaving the metal with greater toughness than it would have if tempered in water and cooled more quickly. Second Annealing.—The process of tempering greatly in- creases the elastic strength of the metal but reduces its toughness. At the same time it produces internal strains due to contraction in cooling. The strains are removed and the toughness restored by a Second annealing, conducted in the same manner as the previous annealing, but at a lower heat, so that the gain in elastic strength is reduced but slightly and not entirely lost. 21 Sepecimens are again taken from the ends of the forging and broken in the testing machine. If the specimens do not fulfil the re- quirements of the specifications the forging is again tempered and annealed, the temperature and conduct of the processes being so regulated as to improve those qualities in which the metal has prov- £d defective in the tests. Strength. Of Parts of the Gun.-The requirements in steel forgings for guns over 8 inches in calibre are shown in the table on page 5. It wil be observed that the strength of the metal in- creases as we proceed outward from the center of the gun. Thus the elastic limit of the tube is 46,000 lbs., of the jacket 48,000, and of the hoops 53,000. It would be better if the strongest metal were in the tube, which has to endure the greatest strain. But the produc- tion of the high qualities required is much more difficult in large forgings than in smaller ones, and for this reason the requirements for the tubes and jackets must be lower than for the hoops. An additional reason for the difference in requirements is found in the fact that the metal of the tube has the advantage of greater elonga- tion before rupture, as may be seen in the table on page 5. The greater elongation is difficult to produce with the higher elastic limit. . . The tubes and jackets of guns under 8 inches in calibre have an elastic limit of 50,000 lbs. Forged steel that has an elastic limit of over IOO,000 pounds is now produced. CHAPTER V. G U N S . ELASTIC STRENGTH OF GUNS. The Elasticity of Metals. – In the chapter on metals the elastic limit of a metal has been defined as the minimum stress per unit of area of cross section that will produce in the metal a per- manent set. For each kind of stress whether of extension or com- pression the metal has a distinct elastic limit. The elastic limit of extension, or the tensile elastic limit, is usually less than the elas- tic limit of compression. In gun steels the difference is not great and the two are considered equal. The tensile elastic limit is ordin- arily used as it is the limit usually measured. Hooke's Law.—A tensile stress applied to a bar of metal caus- es elongation of the bar, and it is found by experiment that under stresses less than the elastic limit of the metal the elongation is pro- portional to the stress. In other words, within the elastic limit of the metal the ratio of the stress to the strain is constant. This law is known as Hooke's Law, and is often expressed ut tensio sic 701S. Modulus Of Elasticity.—If we measure the elongation of a bar caused by a tensile stress, and divide the measured elongation by the original length of the bar, we will obtain the elongation per unit of length, expressed as a numerical fraction. - Now if we divide any stress per unit of area within the elastic limit of the metal by the elongation per unit of length the result will be the constant ratio of stress to strain within the elastic limit. This ratio is called the modulus of elasticity. 23 Let E be the modulus of elasticity of the metal, 6 the elastic limit of the metal, y the elongation per unit of length. By definition we have A = 6|y (I) If we assume that the elasticity of the metal continues indef- initely we see, by making y equal to unity in the above equation, that the modulus of elasticity is the stress per unit of area that would extend a bar to twice its length. When the elastic limit is expressed in pounds per square inch the modulus of elasticity of steel may, without sensible error, be taken as 3O,OOO,OOO. The modulus of elasticity is really a stress per unit of area but it had best be considered as the abstract ratio between stress and strain. Since by Hooke's law the ratio of the stress to the strain is constant within the elastic limit, we may write for 6 and y in equa- tion (1) any other stress within the elastic limit and its corres- ponding strain. - Let S be a stress per unit of area within the elastic limit l the strain per unit of length due to the stress. Then E = S// and / = S/F (2) That is, the strain per unit of length due to any stress per unit of area within the elastic limit is equal to the stress divided by the modulus of elasticity. Strains Perpendicular to the Direction of the Stress.- In the previous paragraphs we have considered only the strain pro- duced in the direction of the stress. But we have seen in the chap- ter on metals that a tensile stress produces a reduction in area of cross section, and it is found by experiment that, for steel, the strain 24 at right angles to the direction of a stress within the elastic limit of b b the metal is equal to one-third of the Cl i-Tai strain in the direction of the stress. If : the cube in Fig. I is subjected to the P--- TTP tensile stress represented by p, the _gi. * *-* * * * * sº * * * d edges aa, bb, etc, parallel to the direc- G --→ G tion of the stress will be elongated, and Fig. I s the edges aff, ac, etc., perpendicular to this direction will be shortened by an amount equal to one-third the elongation of the parallel edges. Equations Of Relation between StreSS and Strain.-If we consider that the cube is subjected at once to tensile stresses applied in the three directions perpendicular to its faces, the strain in each direction due to the stress in that direction will be diminished by the contrary strains due to the perpendicular stresses. . Let X, Y and Z be the three tensile stresses perpendicular to the faces of the cube. le, l, and l, the strains in the directions of X, Y and Z respec- tively. - The strain in the direction X due to the force X is from equa- * & tº tº º I Y I Z. tion (2) X|AE. It is diminished by 3 E and by 3 E. There- fore, for the total strains in the three directions, we have I y 2. 1. – 4 (3-4 – ) A2 l, -}. (Y-5 – ) (3) - _X y 1. – 2 (2-4 – # Problems. I. A steel test specimen has an elastic limit of 59,000 lbs. What will be its elongation per unit of length at the elas- tic limit? **. O.OOI97 inches. 2. The original diameter of the specimen being O.505 inches, what is its diameter when the piece is stretched to its elastic limit? o,5047 inches. 25 3. A vertical steel rod 20 feet long and 9% inch square sustains at its lower end a load of 6000 lbs. The elastic limit of the steel is 72OOO lbs. What will be the elongation caused by the load P - 3. - O.I.92 inches 4. Taking the modulus of elasticity of copper as 16,OOO,OOO what will be the elongation of a copper bar I inch square and Io feet long supporting a load of 5000 lbs O.O375 inches. Principal Stresses and Strains.—Since every stress applied to a solid produces stresses in directions perpendicular to the direc- tion of the applied stress, at any point in a solid under stress there are always three planes at right angles to each other upon each of which the stress is normal. Thus in the cube we have just consider- ed, the stresses at any point in the cube are normal to three planes parallel to the faces of the cube. The normal stresses are called the principal stresses at the point; and it may be shown by the ellipsoid of stress that one of the principal stresses is the greatest stress at the point. The corresponding strains are called the principal strains. Stresses and Strains in a Closed Cylinder.—The following discussion of the elastic strength of cylinders is based on the theory of Clavarino, published in 1879, and modified through the results of experiments by Major Rogers Birnie, Ordnance Department, U. S. Army. Consider a hollow metal cylinder, closed at both ends, to be subjected to the uniform pressure of a gas confined in the cylinder. The pressure acting perpendicularly to the cylindrical walls will tend to compress the walls radially. If we consider a longitudinal Section of the cylinder by any plane through the axis, the pressure acting in both directions perpendicular to this plane will tend to dis- rupt or'pull apart the cylinder at the section, and will therefore pro- duce a tensile stress in a tangential direction on the metal throughout the section. The pressure acting against the ends of the cylinder will tend to pull it apart longitudinally. The metal of any elementary cube in the cylinder is therefore subjected to three principal stresses: a radial stress of compression, a tangential stress of extension and a longitudinal stress of exten- S1O11. 26 If the cylinder be subjected to a uniform exterior pressure stresses will be similarly developed in the three directions. In the following discussion we will always understand by the term stress, the stress per unit of area, and by the term strain, the strain per unit of length, unless these terms are qualified by the word total or other qualifying word. - Assume a closed cylinder affected by uniform interior and ex- terior pressures. At any point of the cylinder let t be the tangential stress, p the radial stress, q the longitudinal stress. d Z, re- º ** ‘. . º º sº salºº::hºreº e _-_*-* spectively, and changing the sign of Y since #is -stress, the interior and exterior pressures acting towar radially, we obtain the following equations: - Substituting these letters in equations (3) for X, Y an d each other º - . --... 2. `s º º ſº º ... . . '...}. : ...' . . . . . . . ... - . . . . . .” ... . .” * , ( : *-4) J/ . . . . A. 3. 3 . . . . . . . . l, - —-F ( p + — + — ) (4) ...” **, *. *. * * I Ž f • which express the values of the strains in the directions of the three ... --...- -- *** *** - - ----, - StreSSeS. ...------~~ T--~~ **----, ---.” - - *-* *-*... -- . *---------- ------" " ... • * * * ---------------------- * * * *T Relations between the StreSSes t, p and q. Lamé's Laws— The stresses and strains in equations (4) form six unknown quan- tities which cannot be determined from the three equations. Lamé, a distinguished investigator in the subject of elasticity of solid bodies, has established relations between the stresses, by means of which the equations may be solved, and the values of the stresses and strains determined. He assumes that the longitudinal stress q and the longitudinal strain lz are constant throughout the cross Sec- tion. The last of equations (4) may then be written t – p = 3 (g — l, E) = COnStant (5) 27 which equation is true whether q has a value or is zero. As t and p apply to any point in the walls of the cylinder, we have Lamé's first law : - - In a cylinder under uniform pressure the difference between the tangential tension and the radial pressure is the same at all points in the section of the cylinder. - Now let us consider a right section of the cylinder, of unit length, Fig. 2. Let Po be the pressure per unit of area acting on the interior of the cylinder. P1 the pressure per unit of area on the exterior. Ro the interior radius of the cyl-, inder. R1 the exterior radius. The total interior pressure acting normally on either side of the diame- tral plane &c is 2 Po R0. The total - pressure acting on the outer circum- ference on either side of the plane and normal to it is 2 P1R1. The difference of these pressures is the resultant pressure acting-On the metal in the sectional plane bc. The total tangential stress on the metal at the section will therefore be Fig. 2 2 (Po R0 – P1 R1) But since t represents this stress per unit of area, the total stress is equal to 2 ſ #. t dr. Therefore - # dr= P, Ro-P, R R0 -*. o 0 I IV-1 Assuming that t is a function of r, it must be such a function that & dr when integrated between the limits R1 and Ko will be equal to A0 Aºo — P. R. . t dr must then be equal to -d (fr) because the integral of this expression taken between the given limits is • Po Ro – P. R. We therefore have - t dra —d(pr] = −p dr-r d ? 28, From which by combination with equation (5) and integration, See foot note, we obtain * - t–H p = C/r? - (6) in which C is a constant, . . Equation (6) expresses Lamé’s second law : In a cylinder under uniform pressure the sum of the tangen- tial tension and the radial pressure varies inversely as the square of the radius. - Both laws are based on the assumption that the longitudinal stress is constant or zero. Stresses in the Cylinder.—By means of Lamé’s laws we may now determine the value for the stresses at all points in the cyl- inder. We may write for t, p and r in equations (5) and (6) the coordinate values referring to any point in the cylinder and thus form the equations t–? = 7% – P = 7, P. (t +?),” = (T + P) Rºº - (T + P.) R.? Eliminating To and Ti from these equations we may obtain *. # = Ao Aºo? — P. R12 + R. R.? (P - P.) I Aºi? – Ro” F.T. , (7) P, R2 – P. R.? R. R.? (P - P.) I - R12 – Ro” A’z — Roº 72 (8) i.e. t dr = —pdr — rap —(? -- p)dr = rap From equation (5) t –H p = 2p + & Therefore — ºr * r T £p + k Integrating log, -ă = -log, (ºp + k) + log. A Replacing 2p + k by its value t + p we obtain 1/r2 = A* (t + p.) OT - £1,14%, 29 From these equations we may obtain the values of the tangential and radial stresses at any point in the section of the cylinder by sub- stituting for Ü" value for the point. Longitudinal StreSS.—The longitudinal stress has been as- sumed as constant over the cross section of the cylinder. Under this assumption when applied to a gun, the total longitudinal stress due to the pressure on the bottom of the bore is distributed uniformly over the cross section of the gun producing a stress per unit of area that is small compared with the tangential and radial stresses. In the present discussion of the stresses acting on the cylinder the longitu- dinal stress will therefore be neglected, and q in equations (4) will be considered as zero. Later the value of the longitudinal stress will be deduced. - Resultant Stresses in the Cylinder.—Making q = 0 in equations (4) and substituting for t and p their values from (7) and (8) we obtain 2 P, AC02 – A Fºr 4. Aſoº R12 (P) — P.) I A /, F S. = — * * ~ : Tº Tº , -º-º-; (9) Ah Aºn? — A7, A312 A202 A.212 (Æh — A7 El-S == 0-1 lo * —t 02 RI” (P) i) (Io) 3 A'12 — A'02 3 R12 — A'02 rº 2 Po Ro” – P. R12 A /, H S. = — — - Q w Q 3 A’l” — A'02 (II) In the above equations the first members are the respective strains multiplied by the modulus of elasticity. Referring to equation (2) we see that each product is equal to the stress which acting alone would produce the strain. The equations then give the values of the simple stresses jhat Would produce the same strains as are caused by the stresses b and t acting together. Their values at any point in the cylinder are obtained from the above equations by giving to r the value for the point. 30 As equation (2) applies only to stresses and strains within the elastic limit of the metal, the substitution of S for El in the above equations is permissible for such stresses and strains only. Basic Principle of Gun Construction.— The following prin- ciple is the foundation of the modern theory of gun construction. No fiber of any cylinder in the gun must be strained beyond the elastic limit of the metal of the cylinder. This principal is strictly adhered to in the construction of guns built up wholly of steel forgings. In the construction of wire wound guns the tube is often compressed beyond its elastic limit by the pressure exerted upon it by the wire. - The principal fixes a limit to the stresses to which any cylinder that forms part of a gun may be subjected. If we represent by 6 the tensile elastic limit of the metal, ſo the compressive elastic limit of the metal, the stresses represented by the first members of equations (9) to (II) may never exceed either 6 or ſo depending on whether the stress is one of extension or of compression; and the interior and exterior pressures, represented by Po and P1 in those equations, must never have such values as to cause the stresses to exceed these limits. Stresses in a Simple Cylinder.—In a cylinder forming a part of a gun we have three cases to consider. There may be a pressure on the interior of the cylinder and none on the exterior, the atmos- pheric pressure being considered zero. There may be a pressure on the exterior of the cylinder and none on the interior. Or both exte- rior and interior pressures may be acting at once, the interior pressure being usually the greater. We will consider the simple cylinder un- der these circumstances. - * Differentiating equation (9) we obtain d (S) 8 AC02 Æ12 (P6 – A) 1 z - T. R12 — Foº 23 (I.2) and differentiating again d? (S) 8 Riº Riº (P – P.) I (13) d72 Aºi.2 — Roº 7-4 31 Similarly from equation (IO) we obtain d (S) 8 Rº Riº (P - P.) dr 3 A22 – AC02 rà (14) d” (S,) 8 Roº Riº (P6 — P) 1 a/2 - Aºi.2 #Ef Aſoº 7-4 (15) First Case. Interior Pressure Only.—Making P = O in equation (9) and remembering that r may vary between the limits Ado and R1 we see that the smaller the value of r the greater will be the value of the resultant tangential stress. This is more readily seen in equation (I2) in which the first differential coefficient of the stress as a function of the radius is negative when P = 0, showing that S, decreases as r increases. Ko being the least value of r the tangential stress is greatest at the interior of the cylinder. Since when P = O S, in equation (9) is positive for all values of r, the stress is one of extension throughout the cross section of the cylinder. When P = O in equation (13) the second member is positive, showing that the curve of stress is concave upwards, the axis of r being taken as hor- izontal. The curve of tangential stress due to an interior pressure only may then be represented in general by the curve ti in Fig. 3, the 57000! 9000 51000 Fig. 3. ordinates being the values of the stress, the abscissas the values of the radius. * - . The numbers at the extremities of the curve are the actual stresses due to an interior pressure P0 = 36OOO pounds per square 32 inch in a cylinder one calibre thick. They are calculated from equa- tion (9) by making P = 0 and Aºi = 3 ſºo. The equation becomes with these substitutions Po o” * S = − (1 + 18–) (16) I 2 7- Making Po- 36, ooo and r = Rowe obtain S. = 57, ooo; and for r = 3A'o, S. = 9ooo. Similarly from equations (IO), (I4) and (I5) we determine for the radial stress produced by an interior pressure the general curve pi Fig. 3, which shows radial compression throughout the cross section with the greatest stress at the interior. Equations (IO) and (II) become for the cylinder one caliber thick, Po Ro” S. = − (1 - 18−) (17) I 2 7- - Po S. = — — - (18) I 2 and comparing these with the equations above we see that for equal values of r the radial stress from an interior pressure is always less than the tangential stress. The 10ngitudinal stress is 1ess than either. 4 The radial stresses produced by a pressure A* = 36OOO are no- . ted in the curve £1. & We may observe from equations (I6), (17) and (18) that, the thickness of the cylinder being expressed in calibers or what is the same thing in terms of the interior radius, the stresses de- veloped by an interior pressure are entirely independent of the cal- iber, and are the same for all cylinders the same number of calibers thick. - Second Case. Exterior Pressure Only.—Making P = O in equations (9) to (I 5) we may determine the curves of stress for an exterior pressure acting alone. In this case the value of S, equation (9), is always negative. The stress is therefore compress- ive throughout the cylinder. d.S./dr, equation (12), is positive. 33 S, therefore increases algebraically with r. d’S/dr", equation (13), is negative. The curve is therefore concave downwards. The gen- eral curve t, in Fig. 4, therefore results. 27,000 Fig. 4. 81000 In the same way the general curve p, is obtained from equations (IO), (I4) and (I5). The numbers on the curves are the values for the stresses caused by an exterior pressure P = 36OOO lbs. on a cylinder one cal- iber thick, for which R1 = 3 Ro We see as before that the greatest stresses are at the interior of the cylinder, and that the tangential stress is greater than the ra- dia1. The tangential stress is one of compression throughout. The radial stress is one of compression on the exterior and of extensio º - \ ** º ł º $: } % *. ôn the interior. ; ... --------—--- Third Case. Interior and Exterior Pressures Acting.— The curves of stress due to interior and exterior pressures acting at Once may be found from the equations or by combination of the curves of stress due to the pressures acting separately. Thus in Fig 5 in which the curves from Figs. 3 and 4 are repeated the lines p3 and tº represent the stresses due to the equal interior pressures, Po = P1 = 36OOO lbs. The position of the resultant curves of stress from interior and exterior pressures acting together will of course depend on the relative values of the two pressures. In Fig. 5 the pressures are 57 000 } \ P. *—P, Pi—. 8000-p —P. 1 - 23.000 |- 24000 *—# à 24000 - 33000 51000 /* 2 81000 Fig. 5. ºr equal. In Fig. 6 are shown the curves resulting when the interior pressure is twice the exterior pressure; P = 360oo, Pi = 18ooo. We may see at once from these figures that the tangential resist- ance of a cylinder to an interior pressure may be greatly increased by the application of an exterior pressure. Assuming that the maximum 57 000 16500K 1 P \s 9000 3–2. ======= <- - tä-----|7500 * * 16500 p- 40500 Fig. 6. ordinates of the curves t1 and t2, in Fig. 5, are the elastic limits 6 and ſo respectively, the interior pressure acting alone would pro- duce the limit of tangential extension. But with the exterior pressure acting the interior pressure has first to overcome the existing com- pression, and as ſo is usually greater than 6 the interior pressure required to produce the stress p + 6 would be more than twice as great as the pressure required to produce the stress 6 alone. That is to say that by the application of an exterior pressure we may more 35 than double the tangential resistance of a cylinder to an interior pres– sure. Similarly it is seen that the tangential resistance of a cylinder to an exterior pressure is increased by the application of an interior pressure. Limiting Interior Pressures.—In determining the maxi- mum safe pressure that can be applied to the interior of a cylinder there are two cases to be considered; for, as we have just seen, a greater interior pressure may be applied when there is an exterior pressure acting than when the interior pressure acts alone. INTERIOR AND ExtERIOR PRESSURES ACTING.-In Figs. 5 and 6 we see that when both interior-end exterior pressures are acting on a given cylinder the maximum values of the resultant tangential and radial stresses depend upon the relative values of the pressures. In Fig. 5 the maximum values of the two resultant stresses are equal. In Fig. 6 the resultant radial stress of compression has a greater maximum value than the resultant tangential stress of extension. Therefore when both pressures are acting, in order to determine the maximum permissible interior pressure we must find the values of the interior pressures that will produce the limiting stresses both of extension and of compression, and then adopt the smaller value as the greatest permissible pressure. The maximum stress in either case occurs when r = Ro. Therefore make this substitution in equa- tions (9) and (IO). Write 6 for S, and - p for S, and solve the equation for Æð. The negative sign is given to ſo since ſo is an abso- lute value only, while S, now represents a stress of compression, which is negative. - 3 (Riº – Rø) 6 + 6 P. Riº Pop 4. R12 + 2 Ro? (19) 3 (R12 – Rož) p + 2 P R12 (20) 4. R12 — 2 Ro” Aºba = Pºo is the interior pressure that acting with the exterior pres- sure P will produce the limiting tangential stress of extension 6; and Pºo is the interior pressure that acting with the exterior pres- 36 sure P will produce the limiting radial stress of compression p. The lesser of these two values must always be used. Assuming that 6 = p we will find by equating the second members of the above equations that Poe will be less than, equal to, or greater than Pop according as R12 < cº, Fº l s 34 6 \ (2 I) INTERIOR PRESSURE ONLY.-We have seen in Fig. 3 that the greatest stress from an interior pressure acting alone is a tangential stress of extension at the interior of the cylinder. This must never exceed 6 the elastic limit for extension. Therefore to find the great- est permissible value of an interior pressure acting alone make S. = 6 in equation (9), P1 = O, r = Ro and solve for P0. 3 (º- º a A 22 * T a Fe T. F.2 (22) If the cylinder is one caliber thick AE1 = 3 AC0 and Aºba F O . 636 - If the cylinder has infinite thickness AE1 = Co and Aºbo = o.750 (23) From which we conclude that the greatest possible safe value for an interior pressure acting alone in a simple cylinder is o. 756; and also that comparatively little benefit is derived by increasing the thickness of the cylinder to more than one caliber. Now if we consider the effect only of an exterior force, assuming this effect to be the stress ſo of compression, the limiting tangential stress to be produced by the interior pressure becomes 6 + p and this being substituted for 6 in equation (22) it becomes _ 3 (R2 — Fº) - 4A212 + 2 Roº A be (6 + p) (24) From equations (22) and (24) the advantage derived by the 37 interior cooling of cast guns formed of a single cylinder becomes ap- parent. The interior cooling produces a stress of compression on the layers of metal immediately surrounding the bore, similar to the stress that would be produced by the application of an exterior pressure. The limiting interior pressure in this case would be ob- tained by substituting for p in equation (24) the value of the stress resulting from the initial compression. Limiting Exterior Pressure.-This is deduced only for the case of an exterior pressure acting alone, as we will have no occa- sion to use the limiting values of the exterior pressure when both interior and exterior pressures are acting. From Fig. 4 we see that the greatest stress from an exterior pressure is a tangential stress of compression at the interior of the cylinder. This must not exceed ſo the elastic limit for compression. Therefore make S. = –0 in equation (9), P = 0, r = Ro and solve for P. - Aºi.2 – Adoº P = −z−0 (25) Pio being the exterior pressure that acting alone will produce the limiting tangential stress of compression p. For the cylinder one calibre thick R1 = 3 Ro in equation (25), and Pia F. O. 44/0 For the cylinder of infinite thickness R1 = co, and Pla F O. 50/0 again showing how little is gained by increasing the thickness of the cylinder beyond one calibre. Thickness of Cylinder.—The thickness H needed in a simple cylinder to withstand an interior pressure Pºo is obtained by solv- ing equation (22) for R, and subtracting Ro from both members. A” — Rh = H = At (Nºtº- ) (26 l 0 0 36 – 4 Poo I (26) 38 Similarly the necessary thickness to withstand an exterior pres- sure P, is obtained from equation (25). Aºi — AP =/=a. (Nº-) l 0 . 0 p – 2 Pia * Longitudinal Strength of a Simple Closed Cylinder.—The ~. total pressure acting on each of the end walls is 71 ſº Po. This is assumed to be uniformly distributed over the cross section of the cylinder, 7 (Riº — A'02). The longitudinal stress per unit of area is therefore * . P, R2 * - FETº Substituting this value of q in the third equation (4), and for t and p their values from (7) and (8) we obtain for the longitudi- nial stress in the cylinder Ao Fº sºme 2 Pi Aziz El, 3(R12 — A'02) Giving E!, its maximum value, 6 or ſo, and solving for Po, using 6 we obtain 3 (A’,” – Adoº) 6 — 2 Adi Pi AC02 A bo == for the interior pressure that will produce the maximum permissi- ble longitudinal stress. If P = O * 3 (AE1 – AC0) 6 - AC02 a value considerably greater than that expressed in equation (22). Problems.-I. What is the maximum permissible interior pressure on a steel gun hoop the interior diameter of which is 20 inches and the exterior diameter 28 inches, the elastic limit of the metal being 60000 pounds per square inch F A '06 - I756I 1bs. per sq. in. 39 | # 2. The steel tubes of a water tube boiler are 2 inches in inte- rior diameter and 2.4 inches in exterior diameter. The elastic limit of the metal is 30OOO lbs. per sq. in. What is the limiting interior water pressure? 5IO3.2 lbs. per Sq. in. 3. Using a factor of safety of 17%, what is the limiting in- terior pressure in an air compressor tank with interior and exte- rior diameters of I5 and 17 inches respectively. The elastic limit of the metal is 30000 lbs. per sq. in 2 2467.9 lbs. per Sq. in. 4. An iron tube 3 inches in interior diameter is subjected to exterior pressure, I326.5 lbs. per sq. in. The elastic limit of the metal is 20000 lbs. per sq. in. What must be the exterior diameter of the tube in order that it may safely withstand the pressure? 3.25 inches. 5. The 6 inch wire wound gun has the following dimensions at the powder chamber; R = 4.5 inches, AE1 = I2 inches. If the gun were constructed of a single forging with an elastic limit of 6OOOO lbs. per sq. in. what would be the maximum permissible pow- der pressure? . 36132 lbs. per Sq. in. 6. A boiler 6 feet in interior diameter is required to withstand a steam pressure of 350 pounds per sq. in. The elastic limit of the metal is 20000 lbs. per sq. in. What is the maximum thickness re- quired in the shell? - o,6408 inches. 7. The cylinder of a hydraulic jack has an interior diameter of IO inches and a maximum working pressure of IOOOO lbs. per sq.. in. The elastic limit of the metal is 40000 lbs. per sq. in. What thickness Of wall is required in order that the factor of safety may be 1/4. 2.9055 inches. ecº sº.” ”’’ * x ~ * ------- - - - - --~~~~~" --- ~~~~-- - --------- - - - - --~~~~~~ ------------------ sº Compound Cylinder, Built Up Guns.-It has been shown that the resistance of a cylinder to an interior pressure may be greatly in- creased by the application of pressure on the exterior of the cylinder. This is accomplished in practice by shrinking a second cylinder over the first. The shrinkage causes a uniform pressure over the exterior Of the inner cylinder and an equal uniform pressure on the interior of the outer cylinder. The exterior pressure strengthens the inner cylinder against an interior pressure, and at the same time weakens the outer cylinder. --,-º-º-º-º- * : *-*---~ : - - ---, --~~~ * 40. That the full strength of the compound cylinder may be utilized it is important that the shrinkage, and therefore the pressure at the surfaces in contact, be so regulated that under the action of an in- terior pressure the interior of the weakened outer cylinder will not be stretched to its elastic limit before the inner cylinder has reached that limit. Otherwise we can not employ the full strength of the inner cylinder. And if the inner cylinder is strained to the elastic limit be- fore the outer cylinder, we cannot employ the full strength of the outer cylinder. - We have seen in Fig. 3 that the tangential stress produced in a single cylinder by an interior pressure diminishes in value as the thickness of the cylinder increases. It is therefore apparent that the stress transmitted to the outer cylinder may, by giving proper thick- ness to the inner cylinder, be so reduced that when added to the init- ial stress existing in the outer cylinder this cylinder will not be strain- ed beyond its elastic limit. And by adjusting the thicknesses of the two cylinders and the pressure produced by the shrinkage the sys- tem may be so constructed that the cylinders composing it will both be strained to the elastic limit at the same time. There is evidently then a relation between the thicknesses of the cylinders and the shrinkage that must be applied in order that the inner and outer cylinders shall be stretched to their elastic limits by the same interior pressure. This relation must be established if we desire to utilize the full elastic strength of the cylinders. And if a third and fourth cylinder is added the proper relation between the thickness and the shrinkage must be established for these as well. A modern gun is built up of a number of cylinders assembled by shrinkage, the number of the cylinders, from two to four, depend- ing upon the size and power of the gun. The shrinkage of each cylinder is so adjusted that under the action of the powder pressure, if the pressure becomes sufficiently great, all the cylinders will be strained to the elastic limit at Once. - When the powder pressure is acting in a compound cylinder the system is said to be in action. When the powder pressure is not acting the system is at rest. In action each elementary cylinder except the outer one is subjected to both interior and exterior pressures. At rest the inner cylinder is subjected to exterior pressure only, the 41 outer cylinder to interior pressure only, and the intermediate cylin- ders to both pressures. System Composed of Two Cylinders.-Assume a system so assembled that under the action of an interior pressure both cylinders will be strained to their elastic limits. Let Ado, Aºi, AE2 be the radii of the successive surfaces from the interior outwards. - Pó, Pi, P2, the normal pressures on the successive surfaces when the system is in action. fo, p1, f2, variations in Po, Pi, P. as the system passes from a State of action to a state of rest. 60, 61, the tensile elastic limits of the inner and outer cylinders respec- tively. ſoo, ſoi, the compressive elastic limits. A, the modulus of elasticity, as- Fig. 7. - ‘sumed the same for both cylinders. Aºi', the normal pressure at the surface of contact when the system is at rest. System in Action.—The outer cylinder is strained to its elastic limit by an interior pressure. The limiting pressure is given by equation (22), changing the subscripts to conform to the nomen- clature above. - 3(RAE – Riº) Ala = 6 19 4A322 + 2 R12 I (27) The pressure Pig will extend the inner layer of the outer cylin- der to its elastic limit. It is therefore the greatest safe pressure which can be applied to the interior of this cylinder. The pressure Pig just found also acts upon the exterior of the inner cylinder, and the pressure P0 upon the interior. For the limit- ing values of the interior pressure we have, under these circumstan- ces, from equations (19) and (2O). 42 R12 — Foº) {} 6 R12 A Poe - 3 ( l 0°) 0 + I 19 (28) - 4A’ī2 + 2 AC02 3 (Riº — A'0°)/00 + 2 Riº Pig P. = 0p 4A212 – 2 AC02 (29) The smaller of these values as determined by the test, equation (21), must be used as the limiting interior pressure. Acting with the pressure Pig it brings the inner layer of the inner cylinder to its elastic limit of tension or compression according as Pog or Pop is the less. At the same time the pressure P, stretches the inner layer of the Outer cylinder to its elastic limit. -- Equation (27), containing in the second member known quan- tities only, is solved first, and the value of Pig obtained is substituted in equation (28) or (29) as determined by the test. The maximum permissible value of Po results. System at Rest.—We have seen in figs. 4 and 5 that an exter- ior pressure acting alone on a cylinder may produce a greater stress than when an interior pressure is also acting. : It may be therefore that the pressure Pig deduced as a safe pres- sure for the system in action, will be a higher pressure than the inner cylinder can safely withstand when the system is at rest, that is, when the interior pressure P0 is zero. This must be determined be- fore we can assume, as safe values for the pressures, the values ob- tained from the consideration of the system in action. As the system passes from a state of action to a state of rest variations occur in the pressures acting, and consequent variations in the stresses at the various surfaces, ſo and #1 represent the vari- ations in the pressures P0 and P1 respectively. Since the interior pressure changes from Po to O we have - fo = — Po - (30) because Po – Po = O; that is, the algebraic sum of the pressure in action and the variation in the pressure is the pressure at rest. The variations in the tangential stresses due to the variations in the pressures may be determined from equation (9). For the exterior 43 of the inner cylinder, the pressures –Po and £1 acting, write –Po for Po, pi for Pi and make r = RI. – 6.R. P. - (2 Riº + 4.Kº) p. S. = I . 3 (AP2 –Rož) (31) For the interior of the outer cylinder, the pressure pi acting alone, write £1 for Po, make P = O, replace the subscripts by those for the Outer cylinder and then make r = R1. S, F (2Riº +4R2%)?, * — g 3(R22 —R12) As the surfaces of contact of the two cylinders form virtually one surface the two values for the variation in the stress at this surface must be equal. Equating the second members of equations (31). and (32) and solving for pi, we obtain - Ro” (R22 – R12) P, * - T TRECEIK.) (32) (33; which expresses the relation between the variations in pressure at the interior and exterior of the inner cylinder. We have designated the pressure at the surface of contact of the two cylinders, system at rest, by P1/. The variation in pressure from the state of action to the state of rest must therefore be p1 = — (P, - P7) = Pºſ – Pia (34) because Pl, - (PI, -P() = Pº. Solving (34) for P/ Pºſ = Po-H pi and substituting the value of pi from equation (33) we obtain Ro” (R22 — R12) Po P1 = Pio — (35) - R12 (R22 — Roº) for the value of the pressure on the exterior of the inner cylinder, system at rest. - This value of Pi' must not exceed the maximum permissible val- ue of an exterior pressure acting alone on the inner cylinder, as given by equation (24) 44 2. Af- * (36) - 1p — T. F. T ſoo . 3 If it does the inner cylinder at rest will be crushed by the pressure applied to strengthen it in action. . . . The condition that Pi' shall not exceed Pip may be expressed Ro? (R22 – R12) P, R.2 (R.? — Rož) Pia — the first member being the value of PIſ from equation (35). If the values of Pio from equation (27) and of Po from (28) or (29) do not fulfill the above condition these values for the pressures cannot be used for the system in action. To find the safe values for the pressures in this case we must re- duce the value of the first member of (37), P1’, until it is equal to the second member, Pip. Pig becomes then P1 and we have Ro” (R22 — R12) Po P = P 8 l lp + R12 ( R2. ---sºme Roº) (3 ) This is the relation that must exist between P, and Po in order that these pressures may be safe for the system at rest. Equations (28) and (29) express the relations between the safe pressures for the system in action. If therefore we substitute the value of Pi from (38) for Pi, in equations (28) and (29) and solve for Po we will obtain the values of Po that will be safe both in action and at rest. 3 (R2 – Rºž)60 + 6R12 P., Pos = - r (4R/2 + 2 R02) 6 Rož R22 — R12 (39) 2 gº *-*===s 4/V1 0 0 R22 tº-sº Ro” Pos = 3(Riº – Rº)po + 2 Riº P, p — - (4R12 – 2 Roº) – 2 R02 R* – Riº (40) Rø – Rº The lesser of these two values will be the limiting safe interior pressure that can be applied to the system. < Pia * (37) 45 - Assuming 6 and p equal, we will find by equating the second members of equations (39) and (40) that Pop will be less than, equal to, or greater than Po, according as R12 < 3 - - ++(Kº — Rºº).P., sº — Rož) 60 (4I) 0 Maximum Value Of the Safe Interior Pressure in a Compound Cylinder.—Equation (24) gives the value of the greatest permissible interior pressure in a single cylinder acted on by interior arºsexte ior pressures wº º 0 .*-----. *~-ºs--- º - ( 6 + p) `--~~~~ * ~ *. -º-, 2-ºxº~.. ****-*. This applies to the inner cylinder in any compound system, and gives the value of the greatest permissible interior pressure. * ~e Making Bºž co and adding the subscripts, we obtain 3 * = + (6+2) which is the greatest possible value of the safe interior pressure in a compound cylinder. -- The same result is obtained by substituting 60 + po for equation (23). * . . . .--~ * * * * * --- 6 in ... <--------es. ------, ** * ~ * - Shrinkage.—The absolute shrinkage is the difference be- een the exterior diameter of the inner cylinder and the interior diameter of the outer cylinder before assembling, 2a5 fig. 8. The relative shrinkage is the abso- lute skrinkage divided by the diameter; or the shrinkage per unit of length, ab|R1. The shrinkages are so small that it is uneccessary to distinguish be- tween the 1engths of the radii as affect- Fig. 8 ed by the shrinkage. The shrinkage diminishes the exterior radius of the inner cylinder and increases the interior radius of the outer cylinder, so ** :0->ºº:****------- -->42” * ***...* - sº, : * 46 that the radius Ri of the surfaces in contact is of a length interme- diate between the lengths of the original radii. The relative shrinkage is, fig. 8, cz + co q = ab/R1 = — (42) º R1 The relative compression ci|^1 is the strain per unit of 1ength produced by the pressure P1’ acting on the exterior of the inner cylinder. As the circumference is proportional to the radius the diminution of the circumference per unit of length will be the same as the unit shortening of the radius, and the value of the tangen- tial strain produced by the pressure Pi' may be obtained from equa- tion (9), by making Po ri- O and r = R1. (2R12 + 4R02) Pſ 3 E(R12 – Rož) ci/R1 - l, = The negative sign is omitted as it simply indicates compression. The tangential strain co |R1 at the interior of the outer cylinder is similarly obtained from equation (9) by making Ro = R1, R1 = R2, P = P', P, = O. - - (2R12 + 4R22) Pſ 3A2(R22 – R12) co/R1 = Z. = Therefore from equation (42) we have for the relative shrink- age - - 2R12(R22 – Rož) Pſ * ~ Zarº Kºj (Kºtze, (43) The absolute shrinkage is 4R/3(R22 – Rož) Pſ F(R12 – Rº) (R22 – R12) The exterior diameter of the inner cylinder before shrinkage should be * 2K1, - 2R, + S. º S1 = 2 Riq = (44) tº . . . . ), *. Aſ ºf : - - The relative tangential compression of the bore due to the ***** * . * *. | . † :º - º- * *§ 47 shrinkage pressure P1 is found from equation (9) by making Po = O, P = P, and r = R. g 2R12 Pſ E(R12 — Roº) /* = Substituting the value of PIſ from equation (44) and reducing we have . - (R2 — R,2)S. l, = — (R2 I?),Sl (46) (R22 – Rož)2R1 from which we may obtain at Once the tangential compression when the absolute shrinkage is known. Since, equation (9), AE!, =S, the tangential stress On the bore in pounds per square inch is found by multiplying the relative com- pression by the modulus of elasticity; 30,000,000 for gun steel. Radial Compression of the Tube. — The value of the pressure on the exterior of the inner cylinder at rest is given by equation (35), - . Ro” (R22 — R12) Po R12(R22 — Rož) P’ = P, - It will be seen from this equation that the larger the value of Po used the less will be the value of P'; and from equation (44) we see that the less the value of Pi' the less will be the shrinkage. There- fore, if when Pop is greater than Po, we use Poo in equation (35), the resulting shrinkage will be less than if Pop were used, and as may be shown by equation (Io) the resulting radial stress at the inner surface of the inner cylinder, system in action, will be in- creased. Now in deducing the value for the shrinkage we have used the pressures calculated to strain the metal to its elastic limit. Therefore with reduced shrinkage the pressure Poe will produce a Stress of radial compression at the inner surface of the tube greater than the elastic limit of the metal. But it is found that the metal of the inner cylinder supported as it is by the outer cylinder has greater strength to resist radial Compression than is indicated by the tests of the detached specimens 48 of the metal used in determining the elastic limits; and as the re- duced shrinkage resulting from the use of Pog in equation (35) re- duces all the stresses on the system in a state of rest, and those on the Outer cylinder in a state of action, it is the practice to use Po, instead of Pop in calculating the shrinkage. - Guns as constructed yield by tangential extension, and the rad- ial Over-compression if it exists does not determine rupture. Con- Sequently the tangential elastic resistance Of the gun, even though frequently greater than the radial elastic resistance, is taken as the elastic strength of the gun. Prescribed Shrinkage.—Equation (44) expresses the relation \ between the shrinkage and the pressure that it produces. When for any reason the compound cylinder is not assembled in such a manner as to offer the maximum elastic resistance, as for instance when a certain shrinkage less than the maximum permissible shrink- age is prescribed, the pressure due to the prescribed shrinkage may be found by solving equation (44) for P1/. The elastic resistance of the compound cylinder assembled with the prescribed shrinkage will then be found from equations (39) and (40) by substituting for Pip, which represents the pressure at rest, the value of Pi' from equation (44) which is the actual pressure applied. The prescribed value of Si will give in equation (46) the resut- ing relative tangential compression of the bore. Application of the Formulas. - Assuming the caliber of the bore and the thicknesses of the cylinders, to determine the shrinkage and the permissible pressures in the compound cylin- der assembled to offer the maximum resistance: The formulas usually required for a system composed of two cylinders are here assembled for convenience. 3 (R2 — R12) 16 = 6 (27) 4R22 + 2R12 R12 — Rož) {} 6.R. 2 P A.- “ l o”) 60 + 1° 4 19 - (28) 4R12 + 2 Roº 49 Pö _3(&º gºmºs R62) p0 + 2R12 Pig p 4R12 cº-Eº 2 Roº Ré(R2–Riº)po * T TR2(R2Trº) Ro’ (R22 — Riº) Pop Riº — Rºº (P/ - ) Pio *-i-º- R12(R22 – R62) < T.K.T P0 (=P).) S = 4.R.?(R*, — R62) Pſ *T E(R2 – Rºy (R2 – R2) *- (R2 — Riº).S. “TT (R2 IRºyar, \ 3 (R12 tº- Rož) 60 + 6R,2P, N Pºs = – . (4R12 + 2 Roº gº & L ** - 4/VI 2 Rož) — 0 A’22 — A'02 \\ * 3 (Riº *º-º-º: Roº) P0 –H 2 R12 Pip Pos = - are Rož) RA2 R22 — R12 4A1* – 2 — 2 *ºmsºmºmº- 0 0 R22 — Ro? - A Rºº — PAR,”, Ro? R.2 (AE, - P 2, s_*(*-***) + 4 &^*(* *) 3 (RI” — Roº) 3 (RI” – Rºº) 72 P, RA2 — P, R2 A’.” R12 ( PA – P E.-S-º 0 I+ \-I ) 4 0 + \l ( 0 1) I 3(RI” — Roº ) 3 (RI” — Rož) 72 (29), (33) (37) (44) (46) (39) \ ... ." . . . \. (40) (9) (Io) In equation ( 33) above, – Pö has been replaced by its value po from equation (30) in order to make the equation general. — Po is a particular value of £0. In the first member of (37) Po, is written for P, to make the equation conform to the practice of using Pos in determining the shrinkage. - Use the values of 6 and p determined in the testing machine. Find P16 from equation (27). Find Poe and Poe from (28) and (29). 3 50 Make the test indicated in (37) and if either of the conditions are met use the value of the first member of (37) for Pi' in (44) and find Si. The values already found for P16 and Pop are then the 1imiting safe pressures. - If the first member of (37) is greater than the second, Find Pop and Poe from (39) and (40). Use Pia from (37) for P/ in (44) to find S1. The stresses and strains produced by any pressures are found by means of equations (9) and (IO); the tangential stresses and strains from equation (9), the radial from equation (IO). Problem 1.-The dimensions of the 4.7 inch experimental siege rifle, at a section through the bore in front of the chamber, are: Ao = 2.35 inches, R1 = 3.86, R2 = 6. The prescribed elastic 1imit for both tube and jacket is 50, ooo lbs. per Sq. in. What will be the shrinkage when the cylinders are assembled to offer the max- Imum resistance, and what will be the maximum permissible in- terior pressure. We have Ko" = 5.5225 Rı” – Ro” = 9.377 I R2 = I4.9 R2% — R12 = 21. Ioo.4 R2 = 36 R2% — Roº = 30.4775 A1* = 57.514 - 3 X 2 I. I Equation (27) Pe = 3 × 4 +...+ 50000 = 18210 I44 + 29.8 - 3 × 9.377 × 50000 + 6 × 14.9 × 1821O (28) Pop = 5 =42955. . 59.6 –H II.O45 — 3 × 9.377 × 50000 + 2 × I4.9 × 1821O (29) Pos = - - = 4OI44 59.6 – II.O45 - . S22 2I. I 2 - * (37) Pºſ = issio-é 5 X X 4 *=7188 29.377 5OOOO - I4.9 × 3O.477 29. 4 × 57.5 × 30.4775 × 7188 (44) S1 = = .OO8489 3O,OOO,OOO X 9.377 X 2I.I *-*- 51 The outer diameter of the tube must therefore be o.oO85 of an inch greater than the inner diameter of the jacket before assemb- ling. If Pop were used in place of Poo in the determination of Pi", equa- tion (37), we would obtain P1’ = 7909, and from (44) S = O.OO934. Problem 2.-What are the stresses on the inner and outer sur- faces of the tube of the gun in the last problem, both at rest and in action, assuming the gun to be assembled with the shrinkage determined in that problem, and using the pressure Pos = 40I44, equation (29), as the interior pressure in action ? The pressure at rest, Pi' = 7188, determined in Problem I, acts alone. Tangential stresses, (9), S, (Ko) =–22843 S, (R1) = –13259 Radial stresses, (IO), S, (Ko) = +7614 S, (R1) = –1970 In Problem I in determining by equation (37) the pressure at rest we used Pos = 42955 lbs. as the pressure in action. The press- ure at the outer surface of the tube in action as given by equation (27), P16 = 18210, will therefore be produced only by the interior pressure Poo. An interior pressure Pos = 40I44 lbs., less than Pop, will produce a pressure on the exterior of the tube less than I82IO lbs. Equation (33) gives the value of the variation in the exterior pressure due to any variation po in the interior pressure. Making Po = 42955–4O144 = 2811 in equation (33) we find pi = 721. The pressure P1 in action, due to the interior pressure Pop is therefore 18210 – 721 = 17489 lbs. Making Po = 40I44 and P1 = 17489 we find, tangential stresses, (9), S, (Ko) = + 43650 S. (R1) = + 13445 radial stresses, (IO), S,(R0) = — 52342 S, (R1) = – 22 135 Had the shrinkage in Problem I been determined by the use of Po, =4O144 in equation (37), that pressure in action would have compressed the inner layer of the tube to its elastic limit, 5oooo 1bs. But with the reduced shrinkage due to the use of Pog in equation (37) the pressure of 40.144 lbs. exerts a stress on the inner layer of the tube of 52342 lbs., which is in excess of the elastic limit. 52 Problem 3.-The shrinkage actually prescribed at the section of the 4.7 inch rifle used in Problem I is O.008 of an inch. What is the elastic resistance of the gun, tangential and radial, at the sec- tion, and what is the relative compression of the bore and the stress of tangential compression at the surface of the bore ? tº O.OO8 × 3O,OOO,OOO X 9.377 X 2I.I Equation (44) P1’ = = 6773 4 × 57.514 × 3O.4775 3 × 9.377 × 50000 + 89.4 × 6773 (39) Pos = = 42178 - 2I. I 59.6 + II.O45 – 33.I.35 3O.4775 3 × 9.377 × 50000 + 29.8 × 6773 - (40) Pos = = 393.18 6 2I. I .O — II.OAL E — II.O4 tº T 59 45 45 3O.4775 2.I.I X.OO8 (46) Z, - — = – O.OOO717 30.4775 × 772 (9) S. = Ed, = 215IO Curves of Elastic Resistance—In the same way the elastic re- sistances are found at various sections of the gun, and the curves of elastic resistance shown in fig. 9 are constructed. By comparing the ordinates of these curves with the corresponding Ordinates of the curve of powder pressures it will be seen that the gun has a factor of safety of about I } over the part of its length that is subjected to the maximum pressure. - Problem 4. What will be the tangential stresses in the system assembled as in problem 3 under a powder pressure of 32,000 lbs. per sq. in P - Ao = 2.35 Aºi = 3.86 R2 = 6 (See Problem I) The pressure at rest Pi' = 6773, determined in Problem 3, pros. duces stresses as follows, equation (9). - ... * f Tube, S, (Ro) = — 21524 S;(R1) = — I2494 Jacket, S, (Ro) = + 18596 S;(R1) = + 9566 • *\s *93 HON !·� :Ale/30 30 WYW08/01/cſw/9|-############d00H 0IN}{00 0331S N09||13XOV^ ’00009 || '00009 || TB31S Nnº)3801 "Ş3&n9S384 {w^WIXWW I ^JL\y JÍT 911SwT3 || ‘nwł82.1W） Q33||MÓS? ’00009 || '0.000g ITBELLS Nnº|# T WW/7c/ 39!/)/W/A/H/S '#06/_/075/00/W （77//A/3937/S A/OAV/ ZY» F-þá '3808 30 NOISS3?dd W00 3A] IWT38 30 S3A800 g'8] –szi—-+ſzgºſ?'|}<-• G’G-> "SH.l. z- QNVSfl0H.J. Nº.1 TSNOISS383 MOO'T38 30 3TVQS © Nº Q Go Zſy 9'09- O ZHZ9 JL3}{0\/ſº !9′24+17'||Zº’62|| 3gni {————#īſ Ģ|-III!143.– A №}#fffff3Hº |-[ ] z9:09 — — — — −=#5. LE -T Ē$ | E. -pE ）;[…] 5 H- ģ|E # HH02 ºſ | H- žįſka ©[ ] fº | E. >.<ſēļ Hoc �� -�J-3E šį ką�� -ÕõŌŌŌ--~~£ | Eſse ^JN*→2 TE * ſ-----------+-------------------------------------}|E ğ)Ľ_‘Noissaudņ00 ividwu jo lları 3H1803 HoNvisissä ollswił 30 īÑñā8E ^）-…-…-- · No�'S8T-30NVISIS3& Tw11N39Nvı 40, 3A8no，?Hi, 'S39vMNI8HS 3ınīOsaw 038180s38.3 LN3S38d38 03Ning30 Nm sNOISN3WIOģŠ 'QN1808 803 3&w'SN313WWIĢ3QISLAQ 1430x3‘SW313ww10 . 54 The stresses within the elastic limit produced by an interior or exterior pressure on a compound cylinder are exactly the same as would be produced by the same pressure on a simple cylinder of the same dimensions. If therefore we consider the gun as a simple cylinder and calculate the stresses due to an interior pressure of 32,000 lbs., these stresses will be the variations in the stresses in the compound cylinder as it passes from rest to action, and the algebraic sums of the stresses at rest and the variations will be the stresses in action. - Considering the gun as a simple cylinder acted on only by the interior pressure 32OOO lbs. we obtain from equation (9) for the stresses at the surfaces for which r = Ro = 2.35, r = 3.86 and r = R1 = 6 - Inner surface of cylinder, S, = + 54263 At r = 3.86, S. = + 22545 Outer surface of cylinder, S = II597 Taking the algebraic sums of these stresses and those above de- termined for the system at rest, we find for the stresses in action, Tube, S, (Ro) = + 32739 S. (R1) = + IOO5I Jacket, S, (Ra) = + 31 I4I S,(R1) = + 2II63 Curves of Stress in Section.—The \ curves of tangential stress in a section of a gun composed of two cylinders assembled to offer the maximum resistance are shown in Fig. Io. The curves s, show the stresses in the cylinders produced loy the shrinkage, the system being at rest. The curves r show the stresses in the cylinders for the system in action. The curve p shows the stress- es that would result from the pressure Po in a single cylinder. In each cylin- der the ordinates of the curve r are the S algebraic sums of the ordinates of the / curves p and S. The gain and loss of strength in the compound cylinder as compared Fig. Io with the single cylinder are shown in ~ P _^V z * e’ * sº 55 Fig. II. The curve t is the curve of tangential stress due to the maximum permissible interior pressure in the single cylinder. The gain in strength in each cylinder of the compound cylinder is shown by the cross-shaded area mark- ~ k-N-9, ed with the plus sign; and the loss in N strength by the single-shaded area marked f }^ with the minus sign. The total tangential .# stress in the single cylinder is the area § between the curve t and the horizontal P axis. The inner cylinder of the compound cylinder gains over an equal portion of the kH single cylinder the shaded area below the - axis, representing the compressive stress due to the shrinkage; and loses the area between the curves t and r, since the ſ 7–/, single cylinder would be under the stress t while the compound cylinder is sub- jected only to the lower stress r. The outer cylinder at rest being under the stress of extension represented by the area under the curve s, that area is lost to it in action, as compared with the single cylinder; while it gains the area lying between the curves r and t. Fig. I I Problems 5. A section of the 2.38 inch experimental field rifle, model of 1905, has the following dimensions: Ro = I.I.9 inches, R1 = 1.95, R2 = 3. What is the elastic resistance of this section as- sembled to offer the maximum resistance, and what is the absolute shrinkage? The elastic limit of the metal, nickle steel, is 65000 lbs. per sq. in. Pig = 23242 lbs. Pos = 55.182 lbs. - Pos = 51875 lbs. P!! - 9158 lbs. Si = 0.00554 in. 6. The prescribed shrinkage for the above section is O.OO5 of an inch. What is the elastic resistance of the section with this shrink- age and what is the stress of tangential compression on the bore? Pºſ = 8271 lbs. Pos = 53525 lbs. I, + O.OOO879 in. S. = 26359 lbs. ºr º, & . QP. ( → *~. * , *~ * f | ". 56 Systems Composed of Three and Four Cylinders.- The construction and elastic strength of the larger guns built up of three or four cylinders are determined by considerations similar to those explained in the foregoing discussion. Precaution is taken by modifying the shrinkages if necessary that the inner cylinders at rest shall not be injured by the shrinkage pressures of the outer cylinders. The elastic strength of the system, that is, the maximum permissible interior pressure, is the pressure that will bring any one of the elementary cylinders to its elastic limit of ex- tension or compression. In a proper construction the tube is subject- ed to the greatest pressures both at rest and in action, and conse- quently if the elastic strength of the gun is exceeded by the powder pressure the tube will yield first. * In fig. I2 are shown the curves of stress in a section through the powder chamber of the 8 inch gun, model of 1888. The curves s1 show the stresses due to the assembling of the jacket on the tube, the curves S2 the stresses due to the shrinkage of the Outer hoop. The curves S, show the resultant stresses due to both shrinkages. ** The numbers on all curves are the actual values of the stresses in sia, tons per Square inch due to an inte- rior pressure Pos = 23.2 tons. 9.5 The curve A shows the stresses that would be produced by this press- ure in a single cylinder of the same * dimensions as the compound cylinder. 36.1 3.2 The curves 7", the stresses in ac- tion, are the resultants of the curves S, and £ in each cylinder. The curve t shows the stresses resulting in a single cylinder from the maximum interior pressure, 12.4 - tons, permissible in a single cylinder Fig. 12 of these dimensions. 57 The area between the curves p and t represents the gain in strength due to the compound construction. Wire Wound Guns.—As shown in Fig. 12 the various cylinders of a built up gun are strained to the elastic limit at the in- terior surfaces only. It is apparent that if the same thickness of wall is composed of a greater number of cylinders, each cylinder being brought to its elastic limit at the interior surface, more of the total strength of the metal will be utilized. It follows that with a greater number of cylinders the gun may be given the same elastic strength with less thickness of wall. The most convenient method of increasing the number of cylin- ders is by winding wire under tension around the tube of the gun. The tension of the successive layers of wire may be so regulated that each layer will be strained to its elastic limit when the system is in action. Usually however the wire is wound with uniform tension. In the form of wire the metal in the gun is much more likely to be free of defects, and can be given a much higher elastic limit than when in the form of forged hoops. An elastic limit of Over IOO,OOO pounds is obtained in steel gun wire. - - But the elastic strength of the gun is determined by the elastic strength Of the tube about which the wire is wound; and if the tube is worked only within its elastic limit the wire wound gun cannot be stronger than the built up gun. In the Brown wire wound gun shown in fig. 5 on page 61, the wire is wound with a tension of 112,- OOO lbs. per sq. in., compressing the inner surface of the tube beyond its elastic limit without apparent injury. This gun is composed of a lining tube about which are wrapped sheets of steel I-7 of an inch thick and of the shape shown in fig. 6 on the same page. The steel sheets form a segmental tube, supon the strength-of-whieh-the- strength--ef-the-gun-principally-dèpends:- The segmental tube is wrapped with wire from breech to muzzle, and over the wire is shrunk a steel jacket with just suffiéient tension to prevent its rota- tion upon the tube. The jacket is not depended upon to add to the tangential strength of the gun. At takes however a part of the longi- tudinal stress. º The Ordnance Department 6 inch wire wound gun is shown in fig. 4, page 6I. The wire, I-IO of an inch square, is wound with a - l . - s .. ‘. . - . - - wºº \ 58 uniform tension of 47,400 lbs. per sq. in., much less than in the Brown gun. The wire winding extends over the breech and half way along the chase of the gun. After 31 rounds had been fired from each of these guns with velocities of about 3280 feet and pressures of about 45,000 pounds, it was reported that the most notable result observed in the test of the guns was the considerable wear of the rifled bore near the seat of the projectile and near the muzzle of the gun. The wear of the bore was much greater than in built up guns of the same caliber fired with velocities of 2,600 and 3,000 feet. This indicates that the life of the wire wound gun will be very short if fired with the higher velocities and pressures. In other words we are unable at present to take economical advantage of the greater strength of these weapons. No wire wound guns have yet been put in service in the United States. They have been extensively used for some years by the British Government. ...sºsy-rrºsº ºxº - zººxazºº - - :: *-*** - * - __s, --rrºrsº" *rs ----------. § --~~~ - ~ *º-s: xxv-xx-aa-ºº:*::: - • *** * * * * --- $xº~... * - *m-m-m-m-m-m-m-m-m-m-s-s-s-s-mº CONSTRUCTION OF GUNS. The Smaller guns in Our Service, such as the mountain guns, the field mortar and the siege mortar, are made from single forg- ings. All other guns are built up. The smaller built up guns of caliber up to 5 inches consist of a central tube, a jacket surround- ing the breech end of the tube, and a locking ring which locks the tube and jacket together. Guns of caliber greater than 5 inches have one or more layers of hoops surrounding the tube and jacket. The bore of the tube forms the powder chamber, the seat of the projectile, and the rifled bore. The jacket embraces the tube from the breech end forward nearly half the length of the tube and ex- tends to the rear of the tube a sufficient distance to allow the seat of the breech block to be formed in the bore of the jacket. Through the bearing of the breech block in the jacket the longitudinal stress 59 due to the pressure of the powder gases is transmitted to the jack- et, and the metal of the tube is thus relieved from this stress. All guns of 6 inch caliber and above are hooped to the muzzle. The 6 and 8 inch guns have a single layer of hoops over the jacket. Guns of caliber larger than 8 inches have two layers of hoops over the jacket. The construction of the several classes of guns and mortars of the latest models may be seen in the illustrations, pages 59 and 68. The forward end of the jacket of the field and siege rifles is threaded with a broad screw thread. The rear end of locking hoop is provided with a similar female thread, and the locking hoop is both screwed and shrunk on to the jacket. The hoop is also shrunk to the tube, and by means of a bearing against a shoulder on the tube just forward of the jacket it holds the tube and jacket firmly together. .. A noteworthy difference will be observed in the construction of the two I2 inch rifles, Figs. I and 2, page 61. While the gun of the older model, 34 calibres long, is composed of a tube and jacket and I7 hoops, the gun of later model, 40 calibres long, is composed of tube and jacket and but 7 hoops. The reduction in the number of the hoops by increasing their lengths has been made possible by the great advances that have been made in recent years in the production of large masses of steel of the requisite high quality. The improvement has been largely due to the demand of the Ordnance Department, and to the stringent and increased requirements in successive specifications for gun forgings. By the increase in the size of the hoops there has been gained, in addition to ease and economy of manufacture, largely increased longitudinal strength and stiffness in the gun, which permits the construction of a longer gun without the tendency to droop at the muzzle. The D hoop shown in Fig. 2, page 61, locks together the jack- et and the C, hoop; and these, bearing against shoulders on the tube, in rear and in front, hold the tube firmly in place. The space behind the D hoop, left to accommodate the increase of length of the hoop when heated for shrinking, is filled with a steel filling ring 60 *. · “IVIHOW HONI-zr •ı.）• → → → → → a aera - -*-xa 2 --~~~~. ** * * * · * * · *** →، ، I **ISIZLIAAOH HONI-1,---> ø---|-]ºsºv-LYON © IIS (INVOITTIJ & '0\,\!� |-ºqq gºzº"Ocz toqs jo ſoavuſ, *uſºno 0，1 € quanțo A { trºuļAA 8sauļAA OIsºsya itº••ıſa zi”ĀJĀĘ7”EȚ77°małoerº ●4 •�&Ķ 91 009 61 ºnWO OG ‘N nº 3801-38|INA ‘NNAO89 HON 1-9* • • Y - ^ y ‘0IJI —ºſ§@₪：15īISTÕTG√ſ“uſºno çiz 8 ommpo A Z*№='：•| 0-7ĒĶIJĀ， №-№ſſzae aerae«sæſº* đoog gođoog10^sºuſAA. 9’Zſz-， TOE S=--~~~~）=（z-z ‘... --★&… ::::：::-:-, -，.…..…！ dooH G’ doopſ ºuſ ſoortsºonſ dømsºliſnoxoep º “ĐIĆI� ºuſ gººgºrº_doºu ſºno yıl z ºtornio A ± RÇ'an 066 61 “Two Og ‘c061 do TB0OW ‘N09 3'8 HONI-9 ）;- ºg ’01），søđoțg 8aţionusoººººººººº^ xogaes ſ-ovog- ºfissions boſsassi： ||euoo ºu ſºuo）-LI LI*uſºqno șºs Liºgą,wqo xºpaoa.Ll G={∈|-3. O– F –! 9S -Œ-�quJ,noxoup —~~~~ærwy suką suſunoeg-- -g）（ 1 y–1z ſyrTJ -* ， ， ， ，（~）ºrLºT dooH uoſuunuſ,ș erreg auſſi . ，uoſuunny, {SNO L 01'6ç “STWO O+ 'O061 JO T3C]OW '37.418 HON|~? | I ’01），- -•đois ºuţiºnaºo,ºvºg xooqo svo |-' , osoa-u, tss ſoºs jo ſeasºl};|~5āīūō， [VI （ſººstrºº#A）, „…, -e=?=~===>---<=-ſ！)º=}~g';ſa I-49zäIES ¡ O -nøyſowe, ’ (~ſuyunoºg8apYI Keyſquț¢8uſiness~^±–###27y & , ，8uſqøng ºzdow ±（√（−] *- uopuutuſ）, ºsnol. ZG “STWO yº‘8881 HO T3GOW （3n+18 HONI'ZI-º - 62 as noted in the 1888 model. The joint between the C1 and C., hoops is coned as shown exaggerated in Fig. I3. Four securing pins pass- * . ing through the C2 hoop near the muz- C2 zle prevent forward movement of the C, U hoops under the vibration set up in the Fig. I3 gun by the shock of discharge. Operations in Manufacture.—The steel forgings from which the parts of the guns are made are manufactured by private con- cerns and are delivered rough bored and turned to within about 3-IO of an inch of finished dimensions. As the parts of the gun are of a general cylindrical form the principal Operations in preparing them for assembling are the op- erations of boring and turning. - In making long bores of comparatively small diameter, as in the tubes of guns, special tools are necessary in order to insure straightness of the bore. The tube is carefully mounted in the lathe and so centered that any bending or warping that may exist in the long forging will be wholly removed in the operations of boring and turning. The bore is started true with a small lathe tool and continued for a length of about three calibers. The tool shown in Fig. I4 is then used to Sºme * * * = ** * = * = a-e = * * - - - - - - - - - - - * * Fig. I4. continue the bore. This tool, called a reamer, has a semi-cylindrical cast iron body, or bit, A, carrying the steel cutting tool B. It is sup- ported in the boring bar C, which is pushed forward by the feed screw of the lathe. The semi-cylidrical bit exactly fits in the bore already started. As the tube rotates the pressure against the cut- ting edge B forces the bit against the bottom of the bore. This together with the length of the bit prevents deviation of the cutting edge as the tool advances down the bore, and makes the bore a true cylinder. In order to make the surface of the bore smooth and uniform 63 the light finishing cuts are made with a packed bit or wood ream- er, shown in Fig. I5. - ~. S GS T GS GSY s A Fig. I5. The cast iron bit A carries two cutters, b, at opposite extrem- ities of a diameter. Two pieces D of hard wood packing are bolt- ed to the bit and serve to guide the cutters accurately. The tool fits tightly in the bore. The light cut taken and the pressure of the oiled wood packing leaves the surfaces of the bore very smooth and uniform and highly polished. Gun Lathe.—The general features of the lathe by means of which the larger forgings are bored and turned, are shown in Fig. I6. The principal parts are: the bed B, made very strong and P £ EEE xy (TDC-07) JB Fig. I6. much larger than for the ordinary lathe; the head-stock and cone- pulley C; the face-plate F.; the slide rest S, carrying a turning-tool; the back rests R, forming intermediate supports for the tube T.; the boring-bed O, supported on the bed proper, B, and carrying the boring-bar P with its tool Q; the feed-screw V, which lies inside the boring bar P; and the gears W, by which the feed-screw is driven. . Motion is communicated to all the parts by the belt X, acting On the cone-pulley. This causes the face plate and tube to rotate and also communicates motion to the long shaft, not shown in the figure, upon the end of which is the lower gear-wheel W. The motion is transmitted through Wr to W, and thence to the ſeed- 64. screw V. By changing the gears any ratio between the velocity of rotation of the tube and that of translation of the tool Q can be obtained. It is necessary that there be only one source of motion, since if the feed-screw or slide-rest were driven independentlv of the cone-pulley which drives the work a change in the speed of one would not cause a corresponding change in the speed of the others, and damage to the tools, the work, or the machine might result. & & The slide-rest S is driven by a second feed-screw not shown. The back rests R can be adjusted to any diameter of forging. The lathe is supplied with an oil-pump, by means of which a stream of oil is forced into the bore while the work is in progress. The chips or cuttings come out at the opposite end of the tube from that at which the tool enters. - Boring and Turning Mill.—The smaller hoops are usually machined on a vertical boring and turning mill, shown in the illus- tration, Fig. 17. The work is bolted to the slotted table t. The cutting tools are carried in the tool holders o at the lower ends of the boring bars a. In the illustration one of the boring bars is shown in a vertical position and the other inclined. The table rotates, car- rying the work with it. By means of the feed mechanism the cut- ting tools are fed either vertically or horizontally or at an angle as desired. r - On account of the greater difficulty of boring than of turning to prescribed dimensions, the bored shrinkage surface is always fin- ished first. Allowance may then be made in turning the male sur- face for any slight error in the diameter of the bored surface. The desired shrinkage is thus obtained. Assembling.—The interior diameter of the jacket, when bored to finished dimensions, is #eater that the exterior diameter of the tube by the amount of the shrinkage prescribed. In order to as- semble the jacket on the tube it is therefore necessary to expand the jacket sufficiently to permit its being slipped over the tube into its place. The expansion is accomplished by heat. The jacket is placed in a vertical furnace heated by oil or other fuel to a temper- ature varying from 600 to 750 degrees Fahrenheit, depending upon º - Lºº FIG. 17. VERTICAL Boring and Turning Mill, 37-Inch. & S. - ; : - • t º , , s ***** ... } t : - # ... - - - - sº - • - - - 4. - - - : - - - 65 the thickness of the forging and the amount of expansion required. Great care is exercised that the heating shall be uniform through- out the length of the forging. The requisite expansion, which in general is about O.OO4 of an inch per inch of diameter, is determin- ed by a gauge set to the exact diameter to which the bore should expand. The gauge held at the end of a long rod is tried in the bore of the forging in the furnace. When it enters the bore proper- ly the requisite expansion has been attained. Care is taken to avoid overheating which might injuriously affect the qualities of the metal. When the desired expansion has been attained the jacket is hoisted vertically from the furnace. It will be seen by reference to the figures on page 61 that the shoulders on the tubes of the I2 inch guns are so arranged that the jacket must be slipped Over the breech end of the tube; while the arrangement of the shoulders on the wire wrapped tubes of the 6 inch guns require that the tube be inserted into the breech end of the jacket. The method of assembling is called breech insertion or muggle insertion according as the breech or muzzle end of the jacket first encircles the tube. For breech insertion, as in wire wrapped guns, - * * the jacket after being lifted from the furnace is placed upright on a strong iron shelf supported at the mouth of a deep pit, Fig. 18. The tube is then carefully lowered into its seat in the jacket. For muzzle insertion, as in the I2 inch guns, the tube is supported upright in the pit, the breech end up, and the jacket is lowered over the tube. Cooling of the heated jacket is accomplished by means of sprays of water directed against the forging from an encircling pipe as shown at D in Fig. 19. The cooling is begun at the section of gº the jacket, which it is desired should take hold of the tube first, as at the shoulder C, Fig. IQ. As the cooling of the remainder of the jacket pro- gresses the metal is drawn toward the section first * cooled, and thus a tight joint at the shoulder is Fig. 18. insured. After the jacket has gripped at the shoulder the cooling pipe is moved very gradually upward toward -TUBE. #E9ACKET. 66 the breech, care being exercised that the jacket shall grip at successive sections in order that lon- gitudinal stresses due to unequal contraction may not be developed in the metal. The shrinking on of hoops is conducted in practically the same manner as the shrinking of the jacket. When the hoops are small and can I be handled quickly they are often assembled to F | | | T' the gun in a horizontal position. Cooling of the - - hoop is begun at the end toward the jacket, or J) toward the hoop already in place, in order that \ contraction shall take place in that direction and Cº make a tight joint between the parts. When the assembling of all the parts is com- pleted the tube is finish smooth-bored and the exterior of the gun turned to prescribed dimen- Sions. Fig. I9. Rifling the Bore.—The rifling of the bore is effected in the rifling machine, which is essentially similar to the boring and turn- ing lathe previously described. The gun does not rotate in the rifling machine, but the cutting tool is given the combined move- ment of translation and rotation necessary to cut the spiral grooves in the bore. The rifling bar m, Fig 20, carrying at its forward end Fig. 20. the rifling tool g provided with cutters for the grooves, is moved forward and backward by means of the feed screw b. The desired motion of rotation is given to the rifling bar by means of the pinion c and the rack d, which engages on a guide bar e bolted to a table made fast to the side of the rifling bar bed. The bar e is flexible and is given the shape of the developed curve of the rifling. As the rack travels forward with the rifling bar it is forced to the left 67 by the guide bar, imparting the proper amount of rotation to the rifling bar and cutting tools. •. Cutting tools are carried at both ends of a diameter of the rifling tool. At the end of a cut the cutting tools are automatically withdrawn toward the center of the bar and the bar retracted for a new Cult. When a number of guns of the same design are to be manu- factured, a spiral groove is cut in the rifling bar itself. A stud fixed in the forward support of the rifling bar works in the groove and gives to the bar the proper movement of rotation. The guide bar with rack and pinion is not then used. MEASUREMENTS. In order that the gun may be assembled with the required shrinkages the surfaces of the various cylinders composing the gun must be accurately turned and bored to the prescribed dimensions. The dimensions of all parts of the gun must be in accord with the design. The tolerances, or allowed variations from prescribed di- mensions, are in general two thousandths of an inch for the diameter of shrinkage surfaces, and one hundredth of an inch in lengths. Accurate measurements of the various dimensions of every part of a gun are therefore essential. The exact length of any dimension of a forging is usually ob- tained by means of one of two instruments, called measuring points and calipers. The points of the instrument used are adjusted until the distance between them is the exact length of the dimen- sion to be determined. The length between the points of the in- strument is then measured in a vernier, caliper. Vernier Caliper.—The vernier caliper is shown in Fig. 21. The steel blade a graduated in inches and decimal divisions is pro- vided with a fixed jaw b and moveable jaw c. By means of the clamp d and small motion screw e the moveable jaw may be brought accurately to any distance from the fixed jaw. The distance be- tween the jaws is read from the scale and vernier. The least reading of the vernier is one thousandth of an inch. The ends of the jaws 6 and c are usually one eighth of an inch wide so that the Ø \-ºr- -------------> ----------- ~~~~ - \""""|| or 5-10-15-35 Zºº | TL Fig. 2 I. measurement between their outer edges is a quarter of an inch greater than the reading of the scale. Measuring Points. – The measuring point consists ordin- arily of a rod of wood into the ends of which are set metal points, Fig. 22. One of these points at least is capable of a small movement out and in. The rod is of wood in order that the heat of the hand ( - ) ( ) Fig. 22. may not affect its length. One of the metal points may be provided with a micrometer head from which the movement of the point out and in from a fixed length may be read at once. Measuring points are used in determining interior diameters, and the distance between surfaces that face each other. In measuring an interior diameter at any point in a bore, as at a Fig. 23, one end OZ, º 69 of the measuring point is placed at a. As the diameter is the long- est line in the cross section, the end b must be moved out until the rod cannot be revolved about the end a in the plane of the cross sec- tion. - To determine, when touch is made at b, that the rod is truly in the cross sectional plane the rod must be revolved in a direction at right angles to this plane, for as seen in Fig. 24, the diameter is the shortest line in the longitudinal plane, and the rod when set to the proper length must be capable of revolution in that plane, touching only at the point b. In other words the measuring point has the length of the diameter when the measuring point is incapable of revolution in the cross sectional plane and at the same time capable of revolution in the longitudinal plane. Similarly when applying the rod to the vernier caliper to read the length of the rod, the movable jaw of the caliper must be brought to such a distance from the fixed jaw that the rod when revolved about One end in two planes at right angles to each other will touch at one point only in each plane of movement. The length of the in- terior diameter may then be correctly read from the scale of the cal- iper. In making measurements the sense of touch is depended upon to determine when contact exists. When the distance that separates a measuring point from a surface is so minute that light can not be seen between the point and the surface, the lack of contact can be unerringly detected by the touch. The Star Gauge.-In the case of long tubes, all parts of which are not readily accessible, some means must be adopted of making the measurements at a distance from the operator. The instrument used for this purpose is called a star gauge. Its general features are shown in Fig. 25. The long hollow rod Or staff a carries at its forward end the head b. Embracing the 2C <- Af ū =ºo *— 70 rear end of the staff is the handle c to which is attached the square steel rod f. The handle has a sliding motion or screw motion on the end of the staff, and any movement of the handle is communi- cated through the rod f to the cone g in which the square rod ter- minates at its forward end. . The head b has three or more sockets, d, which are pressed in- ward upon the cone g by spiral springs not shown in the figure. Into these sockets are screwed the star guage points e. Three points are generally used, I2O° apart. The points are of different lengths for the different calibres to be measured. Any movement of the cone forward or backward causes a cor- responding movement of the measuring points out or in. The cone has a known taper, and the change in its diameter under the meas- uring points due to any movement of the handle is marked on a Scale at the handle end of the staff. The handle carries a vernier . by means of which the scale may be read to a thousandth of an inch. The reading of the scale is the change in 1ength of the di- ameter that is measured by the points when the handle is at the zero mark. “ The staff a and rod f are made in sections, usually 50 inches long, SO that the gauge may be given a length convenient for the measurement of any length of bore. The star gauge is set for any measurement by means of a stand- ard ring of the proper diameter. The standard rings are of steel, . hardened and very carefully ground to the given diameter. If it is desired to measure a IO-inch bore for instance, measuring points of the proper length are inserted in the sockets d of the star gauge. The IO-inch ring is held surrounding the points, and the handle c Of the star gauge is pushed in until the points touch the inner sur- face of the ring. The handle is then adjusted until the reading of the scale is zero. The instrument is now ready for use. The gun or forging whose bore is to be measured is supported so that its axis is horizontal. The star gauge is also carefully support- ed in the axis of the bore prolonged, and in the bore when neces- sary. The distance of the measuring points from the face of the bore is read from a scale of inches marked on the staff. At each selected position of the gauge the handle is pushed forward until the measuring points touch the surface of the bore. The difference 71 between the diameter of the bore at this point and the standard di- ameter for which the gauge is set is then read from the scale at the handle in thousandths of an inch. Calipers.- For the measurement of outside diameters calipers are used. The ordinary calipers for measurement of short ex- terior lengths are shown in Fig. 26. For the measurement of the exterior diameters of gun forgings, calipers as shown in Fig. 27 Fig. 26. Fig. 27. are employed. One of the points a or b is moveable and may be provided with a micrometer head. As in the case of interior meas- urements the caliper must be revolved in two planes about the end that is held at the point from which the diameter is to be measured, and the distance between the points of the caliper must be adjust- ed until touch is made at One point only in each plane. The distance between the points of the caliper, as determined by the length between the outer edges of the jaws of the vernier cal- iper, is then the true length of the exterior diameter. - The frames of the large exterior calipers required for gun meas- urements must be made heavy in order that the calipers shall have sufficient stiffness and not be subject to change of form. In use these calipers are therefore supported from above by a spring con- nection with a frame that is secured to the piece being measured, Fig. 28. Standard Comparator. - In order to insure accuracy in all measurements, all measuring scales are compared with a common standard. For this purpose the standard comparator is provided. A heavy metal bar very accurately graduated in inches and dec- 72 3– ; | ſ \ L º ‘CC (30ſ OW Fig. 28. imal divisions rests in a very stiffly constructed cast iron bed. Slid- ing heads on the bed, one of which carries a reading microscope, may be set accurately at any determined distance apart. RIFLING. The purpose of the rifling in a gun is to give to the projectile the motion of rotation around its longer axis necessary to keep the pro- jectile point on in flight. The rifling consists of a number of spiral grooves cut in the surface of the bore. The soft metal of a band on the projectile is forced into the grooves by the pressure of the powder gases, whereby a rotary motion is communicated to the projectile. ... -- Twist. — The twist of the rifling at any point in the bore is the inclination of the tangent to the groove, at that point, to the axis of the bore. Twist is usually expressed in terms of the calibre, as one turn in so many calibers. If the inclination of the groove is constant the rifling is of uniform twist. If the inclination of the groove increases from breech to muzzle the rifling has an increas- ing twist. 73 Let a Fig 29 be the development of one turn of a groove with uniform twist, n the twist in calibers, or the number of calibers in C A 2nr. B Fig. 29. which the groove makes a complete turn, and r the radius of the bore. Then A B = 2nr, B C = 27tr, and we have tan q = 27tr|2nr = 7tn for the value of the tangent of the angle of the rifling. For the increasing groove # is variable, but at any point its tangent is a ſm. Let v denote the velocity of the projectile at any point of the bore, */ Ç. - q, the angle made by the tangent to one of the grooves with an element of the bore, t go the angular velocity of the projectile, r, the radius of the bore. . . . The velocity of the projectile along the groove is the resultant of two components, v and v tan $, at right angles to each other. The actual velocity of rotation of a point on the surface of the projectile is Gor, and this is equal to the component v tan (b. There- fore - . Gor = z fan ºf and G0 = z fan $/r Increasing Twist.— When the twist is uniform the in- clination of the grooves to the axis of the bore is the same through- out the length of the bore, and therefore it is greater at the breech than the inclination of the grooves of an increasing twist that is equal to the uniform twist at the muzzle. The pressure required to cause the projectile to take the grooves is therefore greater in the case of the uniform twist, and the greater resistance offered to the starting of the projectile serves to increase the maximum pres– sure in the gun. The total energy absorbed by the projectile in 74 taking the rifling is greater with an increasing twist than with the uniform twist on account of the increased frictional resistance due to the continual change in the inclination of the grooves. The total energy absorbed is however small compared with that required to give the projectile its velocity of translation. Service Rifling. — An increasing twist is adopted for the guns in our service. In all guns of recent model the twist is one turn in 5o calibers at the breech, and increases to one turn in 25 calibers at a point about 2% calibers from the muzzle. The pur- pose of the uniform twist for a short length at the muzzle is to give steadiness to the projectile as it issues from the bore. A right handed twist is used in all guns in our service. The number of grooves depends on the caliber of the gun. In the siege and seacoast guns the number is six times the caliber of the gun in inches. Thus the 5 inch gun has 30 grooves and the IO inch gun 60. The 3 inch field rifle has 24 grooves. The shape of the grooves is shown in Fig. 30. The widths of land and groove noted in the figure are the same for all guns of 5 inch caliber and greater. The depth of the groove varies from 0.03 º % - . i ſ > $ Fi 8. 3 O of an inch in the 3 inch gun to O.06 in the seacoast rifles, and ooz in the seacoast mortars. A form of groove called the hook section groove, used in Navy rifles is shown in Fig. 31. The view is from the breech end. Fig. 31. The driving edge of the groove makes a sharp angle with the sur- $ 75 face of the bore, and the other edge has a gradual slope to that sur- face. - The depth of the groove in the larger naval guns is o.os of an inch. In the service 30 caliber rifle the depth of the grooves is o.oO4 of an inch. It is desirable in small arms to limit the depth of the grooves to the minimum, in order to lessen the thickness of bar- rel and to permit ready cleaning of the bore. <--~ * * *** // W v The breech mechanism fomprises the breech block, the obturating device, the firing mechanism, ſand the mechanism for the insertion and withdrawal of the block. The breech block closes the bore after the insertion of the charge, and transmits the pressure of the powder gases as a longitudinal stress to the walls of the gun. There are two general methods of closing the breech. In the first method the block is inserted from the rear. The block is pro- vided with screw threads on its outer surface which engage in cor- responding threads in the breech of the gun. In order to facilitate insertion and withdrawal of the block the threads on block and breech are interrupted. The surface of the block is divided into an even number of sec- tors and the threads of the alternate sectors are cut away." Similar- ly the threads in the breech are cut away from those sectors opposite the threaded sectors on the block. The block may then be rapidly inserted nearly to its seat in the gun, and when turned through a comparatively small arc, say I-8 or I-I2 of a circle depending upon the number of sectors into which the block is divided, the threads On the block and in breech are fully engaged, and the block locked. In the second method a wedge shaped block is seated in a slot cut in the breech of the gun at right angles to the bore, and slides in the slot to close or open the breech. Variations of these two methods will be noted in the descriptions of the breech mechanism of some of the guns in service. / BREECH MECHANISM. f. 76 The breech block is usually º in the jacket of the gun or in a base ring screwed into the jacket." Thus supported it trans- mits the powder pressure to the jacket and relieves the tube of lon- gitudinal stress. The seat in the jacket being of greater diameter than could be provided in the tube the bearing surface of the block in its seat is increased, and the Rºth of the block may be dimin- isled. The Slotted SCrew Breech Mechanism. — The slotted screw breech mechanism is better adapted than any other for use in heavy guns. An example of it as used in the heavier guns is shown in Figs. 32 to 34, which represent the breech mechanism Of the I2 inch rifle. The breech block B has six threaded and six slotted sectors. When the breech is closed the threads on block en- gage with the threads in the breech. The breech is opened by turn- ing the crank K mounted on the shaft PV. The movement of the crank is transmitted through the worm gear to the hinge pin HP, and through the compound gear CG to the rotating lug rl formed on the rear of the block. The block is thus rotated one-twelfth of a turn, and its threaded sectors then lie in the slotted sectors of the breech. Further movement of the crank causes the teeth. Of the com- pound gear CG to engage in the teeth of the translating rack tricut in a slotted sector of the block. The block is thereby caused to slide to the rear on to the tray T, the guide rails of the tray engaging in the grooves g g in the block. When the block is sufficiently withdrawn the bottom of the block depresses the rear end of the tray latch L and lifts the forward end of the latch out of the catch A, where it has been held by the pressure of the spring s. The tray is now unlocked from the breech. The upper front toe of the latch L engages in a groove in the breech block, locking the block and tray together. The further action of the compound gear on the last teeth of the translating rack trithen causes the tray to swing to the right about the hinge pin, carrying the block clear of the breech. As the tray swings clear of the breech the locking bolt lb forces for- ward the operating stud os and enters a seat in the latch. The latch is thus locked in its raised position, and secures the breech blóck against being pushed forward off the tray when open. In closing the breech the operations are reversed in order. When 77 1g. 32. 78 the tray comes in contact with the face of the breech the operating stud os forces the locking bolt lb from its seat in the latch. The latch is depressed by the spring s and thus unlocks the block from the tray. - . The two plugs shown in the obturator head of the breech mech- anism, Fig. 34, are in the seats provided for the insertion of press- ure gauges when it is desired to measure the pressure in the gun. Bofors Breech Mechanism.—The mechanism shown in Figs. 35 to 38, known as the Bofors breech mechanism, is most suit- & e s: ºs º ºs ºs ºms º ºs - sº º ºs- ºr sº tº º sº º § s º s § *śR # % 㺠T-RS//Nº. KiN % % sºil. §/sº º * º / % º:S Rº § %jºšš% º §§/ Ağ% §§§ §§ %HE:328 % Z/ Fig. 35. able for guns of medium caliber. It is applied to the 6 inch gun in Our Service. The block b Fig. 35 is ogival in shape and has six threaded and six slotted sectors. With the ogival shape a very small retraction to the rear is necessary before the block may be swung Open. In the 6 inch gun this retraction is I.2 inches, just sufficient to withdraw the obturator o from its seat in the bore. The block is supported when the breech is opened by the block carrier c provided with a central tube which embraces a spindle s formed in the block. - - This mechanism is not applicable to the larger guns because the - Fig. 33. Closed. - ------- FIG. 34. Open. ---------- -* - - -------- * BREECH Mechanism for Heavy Guns. * - - - 2’ - - -, - *. : - - - - - * - * & - - e - 4 • - - º - * # --- * - º * - gº º - * * * - - g FIG. 36. Closed. Fig. 37. Block Unlocked, READY To Swing OPEN. Fig. 38. OPEN. BoFors RAPID FIRE BREEcH MEchanism. 79 greater weight of the breech blocks in these guns requires better sup- port than can be conveniently given by this method. w The mechanism is actuated by means of the lever l, Fig. 36, which is attached to the lower end of the hinge pin. A spool p mounted on the hinge pin has teeth cut on its lower end which engage in the rack r. The rack slides in a horizontal groove cut in the block car- rier c, and the teeth at its left mesh with corresponding teeth on the block. - - When rotation of the block is completed a lug at Fig. 35 on the spool engages in a slot at the rear end of the block and translates the block slightly to the rear. Before this translation is complete the block carrier is unlocked from the gun and swings to the rear with the block, fully uncovering the bore. The loading tray, shown in Fig. 38, the purpose of which is to protect the threads of the breech from injury as the shot is put into the bore, remains per- manently in the breech. When the block is entered and rotated the tray is pushed aside by the threads on the block until it covers the slotted sector. On opening the block it is brought back into the position shown. In the breech mechanism shown in Fig. 34 the loading tray is a separate piece placed in the breech by hand when loading, and re- moved before closing the block. | | The Welin Breech Block. — The Welin breech block, largely used in naval ordnance, has the threaded sectors arranged in steps at different distances from the center of rotation, as shown *- ===, **- =n. Fº x= - 2% s== s ---, N. * == -* T=__*. Tºº- ---, == == E= T= - a-ºº: := - --E. T. T=== == -:== ==- == E= º===="# === - = ==E" ... ==- s= -º-º- -- *- Sess- Fig. 39. in Figs, 39 and 40. By this means the threaded area may cover two- 80 thirds, three-fourths or even a larger portion of the surface of the block. A large increase in threaded area is thus secured over that obtained on a cylindri- , cal block with alternate threaded sectors, iſ and the block may therefore be made smaller. The amount of rotation re- quired in 10cking and unlocking is also largely diminished, one-twelfth of a turn sufficing for the block shown in Fig. 39, and one-sixteenth for the block of Fig. 40. Obturation.-There must be provided at the breech of the gun some device that will prevent the powder gases from passing to the rear into the threads and other parts of the breech mechan- ism. If any passage is open to the gases they are forced through it with great velocity by the high pressure existing in the bore. Their velocity together with their high temperature gives to them great erosive power, and the threads and other parts of the breech mechanism subject to their action are eroded, chanelled, and worn away to such an extent that the breech mechanism is soon ruined and the gun is rendered useless.. }， O· Holwº '/\\78. L. *00 JLS ONI LW/TSNW/?H_L 19. 5.o. RO TATING RING. * X{OO T18 HO13|3}}8 -*XNw&O ONIATOA38 *$) NI>] 3 SW78 * JL3)4O\/(^* X{0OT XIN\/80 €)NIATO/\2>） FACE PLA z Sº 2. à. ** ſy do OH 88 Fig. 51. -roller. The other thread of the roller works in a corres- ponding thread cut in the tray. Rotation of the translating crank causes the block to move to the rear with a movement equal to the sum of the movements due to each of the two threads. When the front of the roller passes to the rear of the stud shown acting on the tray latch, the block is brought to a stop on the tray, and the shock of its arrest is sufficient to release the tray latch from its hold on the lip Of the recess in the gun. The tray then swings aside carrying the block clear of the breech. The tray is similar in general shape to the tray of the more modern mechanism shown in Fig. 32. 12-inch Mortar Breech Mechanism—The I2-inch mortars are provided with the mechanism shown in Fig. 52. It differs from b-HE Fig. 52. the mechanism just described only in the method of rotating the 89 breech block. A steel plate k is fixed to the rear face of the breech block and extending upwards provides journals for the pinions a, b and c of the rotating gear. The pinion c meshes in the rack e fixed to the gun, and when the crank d is turned the block is rotated to open or close. The block is withdrawn on a tray as described above. The translating stud that engages in the translating roller is seen at the bottom of the block. - - The vent shield f, cut shorter than shown in the figure, is pro- vided with a stud at its lower end that engages with the safety bar of the firing mechanism already described. The stud at its upper end works in the groove g cut in the gun, withdrawing the safety bar as the breech is fully closed. Automatic and Semi-automatic Breech Mechanisms—In guns provided with automatic breech mechanism the energy of re- coil or the pressure of the powder gases is utilized to open the breech, withdraw the fired shell, insert a new cartridge and close the breech. After the firing of the first round the only operation necessary for firing the succeeding rounds is pulling the trigger. The automatic mechanism is at present applied only to guns of small caliber that use the small arm cartridge or fire a projectile weighing not more than a pound. The semi-automatic mechanism is applied to guns of medium caliber, up to 6 inches, and efforts are being made to adapt it to the larger guns. The breech is opened by mechanism that is operated during the recoil or counter recoil of the piece, and if fixed ammu- nition is used the fired shell is ejected. At the same time power is stored in a spring to be later used in closing the breech. In some mechanisms the insertion of the succeeding round by hand Operates the breech closing mechanism. In others the pulling of a lever after the insertion of the round actuates this mechanism. The automatic and semi-automatic mechanisms will be later described with the guns to which they are fitted. - - ". . . ; ; ; ; ;-- . . . . . . ; p* : t - - WX. W ºr -, - s Y ºr, ºf Nº * ſAft wº ! ti \\ \º -N \} º Nºs 90 `s) CHAPTER VI. PROJECTILES. Projectiles are classed as shot, shell, and case shot. The shell is a hollow shot designed to be filled with a bursting charge that by means of a fuze may be exploded at a selected time. The case shot consists of a number of shot held together by an enclosing envelope which º be ruptured by the shock of diº or by a bursting charge in flight. The envelopes of canister and grape shot are rup- tured by shock in the gum, The envelope of shrapnel is ruptured by a bursting charge. - Old Forms of Projectiles.—In the old smooth bore cannon round cast iron shot and shell of diameter nearly equal to the calibre of the gun were used. The grape, canister, and shrapnel for these guns are shown in the illustrations. The shrapnel was invented about 1803 by Colonel Shrapnel of the British Army. In its first form it contained a number of lead balls with loose powder in the in- terstices. The walls of the shell were made thick to resist deforma- tion by the movement of the contained balls. In its later forms the spaces between the balls were filled with melted sulpur, and a cham- ber for the bursting charge was provided as shown. By this ar- rangement the walls were no longer subject to the impact from the loose balls, and therefore could be made thinner, thus providing room for a greater number of bullets. The confining of the bursting N GRAPE. - CANISTER. 91 # E --- | The canister for the 75 m|m Vickers Maxim gun is shown in figure 7. a' The case, c, made of malleable iron is solid at *—T. the bottom and open at the top. It is weakened T S c by two series of cuts, s, each series consisting of three oblique cuts, each of which extends over an arc of I2O degrees. The case contains 244 iron C balls 5% of an inch in diameter and weighing 30 to the pound. The balls are confined in the case by the tin cup, a, riveted in. Three holes, h, drill- ed through the bottom of the case admit the pow- der gases to assist in rupturing the case. The groove, g, is filled with grease for the purpose of preventing the entrance of moisture into the metallic cartridge case, which is attached to the —-b projectile by being crimped at several points into the groove r. The copper band, 6, ---g forms a stop for the head of the cartridge T’ case, and serves as a gas check in the gun. |- & ; : …) Ivº "/* ~\ , Fig. 7. º ſº ==H-4- * ) g. 7 Shrapnel—The modern shrapnel is a pro- #y, ~x y jectile designed to carry a number of bullets to a distance from º ~ gun and there to discharge them with increased energy over an ex- ºfſ # S sº tended area. It is particularly efficacious against troops in masses | 2. and is not used against material. The shrapnel is the principal field/ _^ - Gº! 96 artillery projectile. It is also provided for mountain and siege artil- lery, and for use in the small caliber guns in sea coast fortifications in repelling land attacks. In the earlier models the case of the shrapnel was so constructed as to break into a number of fragments on explosion of the burst- ing charge, with the idea of thus practically increasing the number of bullets carried. With the same end in view the spaces between the balls were filled with the parts of cast metal diaphragms that separated the layers of balls and broke up into additional fragments at the bursting of the projectile. The bursting charge was placed Sometimes in the head and sometimes in the base of the projectile. It was found with these shrapnel that a very large percentage of the numerous fragments had not sufficient energy to inflict serious in- jury. The shrapnel is therefore at present constructed of a stout - case which except for the blowing out of the head, remains intact at the explosion of the loursting charge, and from which the balls are expelled in a forward direction and with increased velocity by the bursting charge in - -- h lº the base. By these means while the number of fragments is less, a greater number poss- «ss the required energy and the effective range of these is-created: \º-&^** Fig. 8 represents the shrapnel for the 3- inch field gun. The case, c, is a steel tube drawn in one piece with a solid base. A steel diaphragm, d, rests on a shoulder near the base, forming a chamber for the bursting fTº charge in the base of the projectile, and a support for a central steel tube which extends through the head, h. A small quantity of gun cotton in the bottom of the tube is ignit- -L-d ed by the flame from the fuze, and in turn ºals. § © e ſº - ; ignites the bursting charge. The balls, of § & e • ; 1ead hardened with antimony, are 262 in num- § & e e º § loer. Each ball is 49-Ioo of an inch in diam- eter and weighs approximately 167 grains, or Fig. 8. 42 to the pound. After the balls are inserted 97 a matrix of mono-nitro-naphthalene is poured in to the case filling the interstices between the balls in the lower half of the case. When cool this substance is a waxy solid. It gives off a dense black smoke in burning. The purpose of its introduction is to render the burst of the shrapnel visible from the gun so that the gun commander may determine whether his projectiles are attain- ing the desired range. Resin is used as the matrix in the forward lmalf of the case. . - The matrix forms a solid mass with the balls and prevents their deformation by the pressure that they would exert upon each other, on the shock of discharge in the gun, if they were loose in the case. Resin gives better support to the balls than naphthalene and there- fore no more of the naphthalene is used than is necessary to produce the desired amount of smoke. On being expelled from the case the matrix burns and breaks up leaving the balls free. . To prevent rotation of the contained mass in the case the interior of the case is fluted lengthwise so that its cross section is as shown in figure 9; and to reduce the friction to a minimum, particularly in the chamber for the bursting charge, the interior of the case is coated with a smooth asphalt lacquer. The head, h, of steel is given a cellu- 1ar form to make it as light as possible. Fig. 9. The weight of the projectile complete is fixed at I51bs., and weight is saved as far as possible in all parts of the case in order that the greatest number of balls may be carried. The head is screwed into the body and fixed by two brass pins, p. The combination time and percussion fuze, f, is screwed into the head. It is protected against injury or tampering by the spun brass cap, b, soldered on to the head of the projectile. The projectile is fixed in the cartridge case as explained for the canister. Shrapnel forms 80 per cent of the ammunition supply of the field gun. * - - A The Bursting of Shrapnel.--When the shrapnel bursts the balls are expelled forward with increased velocity, and as they have 98 at the same time the movement of rotation of the projectile they are dispersed more or less to the right and left. Their paths form a cone, called the cone of dispersion, about the prolongation of the trajectory. The section of this cone by the ground is an irregular oval with its longer axis in the plane of fire. The dimensions of the area will vary, as is evident from figure Io, with the angle of fall, the Fig. IO. height of burst, and the relation between the velocities of translation and rotation at the moment of burst. It is assumed that when a shrapnel ball has an energy of 58 foot pounds it has sufficient force to disable a man, and with 287 foot pounds of energy it will disable a horse. These energies correspond in the service shrapnel bullet to velocities of about 400 and 880 foot seconds. An increased velocity of from 250 to 300 feet is imparted to the balls by the bursting charge. Knowing the velocity of the projectile and the weight of the balls the space within which the balls will be effective may be determined for any range. PoſNT OF BURST.—The best point of burst for a shrapnel is as- sumed to be that point from which the burst of the shrapnel will pro- duce one hit per square yard of vertical surface at the target. The distance in front of the target at which the burst occurs is called the interval of burst. On account of the variation at different ranges in the velocities of translation and of rotation the interval of burst which will produce one hit per square yard of vertical surface at the target varies with the range, decreasing as the range increases. Practically it is found best to consider the height of burst rather than the interval of burst, since the battery commander can more readily estimate the height than the interval. Suitable cross hairs in the field of the battery commander's telescope facilitate this esti- mation. . In our service a height of 3-IOOO of the range, called 3 mils, is adopted as the most favorable mean height of burst. The point of 99 burst at this height gives, over a large part of the range, very ap- proximately the correct interval of burst. For short ranges this height of burst is excessive, and for long ranges it is insufficient. The following table shows for the 3-inch shrapnel the results obtained at different ranges from bursts at the correct interval of burst, and also at a height of burst of 3 mils. The front of target that should be covered depends upon the number of balls in the shrapnel. For the 3-inch shrapnel the front to be covered with one hit per square yard is 18.5 yards. One hit per sq. yj. Height of burst, 3 mils Range - . Interval Front Covered Interval Front Covered Yards Yards Yards Yards Yards 1000 81.4 18.5 118.2 27.0 2000 73.0 18.5 83.4 21.2 2500 68.98 18.5 73.5 19. 55 3000 65.84 18.5 66.6 18.76 3500 63.28 18.5 60.9 18.84 4000 61.07 18.5 56.4 17. 12 4500 58.97 18.5 51.3 16.13 It will be observed that between 2000 and 4500 yards the height of burst of 3 mils gives approximately the desired density of fire at the target. At ranges less than 2000 yards the front covered is largely increased and the density of fire therefore diminshed. The figures refer to a single shrapnel bursting at the mean point of burst. In a group of shrapnel the bursts above and below the mean point would largely make up the discrepancies in distribution and density. • FUZE.—The fuze used in the shrapnel is the combination time and percussion fuze of which a full description will be found in the chapter On fuzes. The fuze is arranged in such a manner that if the projectile is not burst in flight it will be burst soon after impact, a short time being allowed by the delay element in the fuze during which the projectile may rise on a graze and its burst be accom- plished in the air. . The fuze is also constructed to permit of using the shrapnel as 100 canister. When the fuze is set at zero of the time scale, the projectile will burst within 25 feet of the muzzle of the gun. * Shot and Shell.–Solid shot are no longer used in modern can- non except for target practice, at least in our service. Certain hol- low projectiles with thick walls designed principally for the perfora- tion of armor are denominated shot to distinguish them from shell, which name is given to thinner-walled projectiles that have not as great a penetrative power but carry larger bursting charges, and have consequently greater destructive effect after penetration. Shell were formerly made of cast iron, being cast in one piece and subsequently bored for the fuze, figure II. 2–s __ Fig. II. With the adoption of high explosives for bursting charges, greater strength in the walls of shell became desirable in order to insure against accidental explosion of the projectile while in the gun. With the exception of some of the projectiles for guns of minor cali- ber in which black powder is used for the bursting charge, all pro- jectiles are now made of forged steel. & Figure I2 represents a steel shell for the 5-inch siege rifle. The steel projectiles for mountain, field and siege artillery are similarly constructed. _º-A t—º - Th— ºr 101 The base of the shell is closed by a steel base plug, p, which is screwed in after the explosive charge has been packed in the pro- jectile. The plug is bored and tapped for the base fuze, f, which when inserted is flush with the rear surface of the projectile. The wrench holes in base plug and in head of fuze are filled with lead in order to make a continuous bearing .surface for the copper cup, c. The cup is applied to the base of the shell to prevent the powder gases in the gun from penetrating to the interior of the projectile by way of the joints of the screw threads. The edge of the cup fits into the circular undercut groove, g, and the joint there is sealed and the cup held in place by lead wire hammered in. Armor Piercing Projectiles—Armor piercing projectiles are of the same general construction as the steel shell just described. Their distinguishing feature is a soft metal cap embracing the point of the projectile for the purpose of increasing the power of the pro- jectile in the perforation of hard armor. The head and point of an armor piercing projectile are ex- tremely hard, the hardness being attained in the process of manufac- ture by any one of several secret tempering processes. The metal of the projectile before being subjected to the secret process has a tensile strength of about 85000 pounds per square inch, which is undoubtedly increased by the tempering. The cap on the othey hand has a tensile strength of but 60000 pounds with a large per- centage of elongation, and reduction of area, as may be seen in the table on page 5. The metal of the cap is therefore very soft com- pared with the metal in the head of the projectile. A IO-inch armor piercing shot is shown in figure I3 and a IO- inch shell in figure I4. The shot has thicker walls and head, and a less capacity for the bursting charge. The outer diameters of the two projectiles are the same, and the weight of each when ready for firing is the same, 604 pounds. To maintain uniformity of weight the shot is made about 4% inches shorter than the shell. The cap is fixed to the head of the projectile by means of the circular groove, a, cut around the head of the projectile. The cap before affixing is of the shape shown half in section and half in ele- vation in the figure between the projectiles. A shallow recess, b, is Fig. I3. IO-IN. ARMOR PIERCING SEIOT. __º_ * * * Fig. I4. IO-IN. ARMOR PIERCING SHELL 103 An ſ Action of the Cap.–The soft steel cap increases the power of | filled with graphite to lubricate the projectile as it passes through the armor. To fasten the cap, the projectile with the cap resting on its point is placed under a hydraulic press. The press forces a die over the cap, making its exterior cylindrical and pressing the excess metal into the groove in the projectile. - In naval projectiles the caps are sometimes fastened on by passing two wires through holes drilled in the cap and notches cut in the projectile. penetration of the projectile in hard faced armor, at normal impact and up to an angle of 30 degrees from the normal, about I5 per cent with respect to the velocity of the projectile, and more than 20 per cent with respect to the thickness of plate. It confers practically-ng advantage against soft or homogeneets armor.T *:: Among the several theories advanced as to the action of the cap, the following appears the most satisfactory. When an uncapped projectile strikes the extremely hard face of a modern armor plate, the whole energy of the projectile is applied at the point, and the high resistance of the face of the plate puts upon the very small area at the point of the projectile a stress greater than the metal can resist, however highly tempered it may be. The point is therefore broken or crushed and the head of the projectile flat- tened, fig. 15. The flattening of the head brings loss of penetrative ºlº-Iſº wn ºn tº Fig. I5. Power, and the energy of the projectile is expended largely in shat- tering the projectile itself. The head of the projectile adheres to the plate and is practically welded to it. 104 The effect on a plate of thickness equal to the caliber of the projectile may be the partial or complete punching out of a cylin- drical piece, figure I6. But even if the plate is completely perforated, the pro- jectile does not get through as a whole; and behind the plate are found only fragments of the projectile and of the metal forced from the plate. When a projectile provided with a cap strikes a hard faced plate, the pressure due to the resistance of the plate is not confined simply to the point of the projectile, but is dis- tributed uniformly over a comparatively large cross section. In ad- dition the point of the projectile is firmly supported on all sides by the metal of the cap. As a consequence the point is not deformed, and passing easily through the cap it finds the hard face of the plate dished and severely strained and more or less crumbled by the im- pact of the cap. The unexpended energy of the projectile forces the point through the weakened face and through the softer metal of the back. Fig. 17. The face of the plate is crumbled, and a conical hole made through the softer metal, through which the projectile passes prac- tically intact and in condition for effective bursting, fig. I7. 105 ...” ,” —” ~ The form of the cap has not apparently a great effect on the re-> sults. Many different shapes are used by different manufacturers, some of which are shown in figure I8. | Fig. 18. With soft plates the initial resistance offered to the entrance of the point is easily overcome by the hard metal of the point, and con- sequently no benefit is derived from the cap against these plates. ~The pronounced superiority of the capped projectile is also only obtained with high-striking velocities, and therefore advantage, is not derived from the cap at long ranges. The limiting velocity is given at I650 feet which corresponds to a fisºsº yds. with the IO-inch-rifle, and 6000 yds. with the 12-inch rifle-> - º y It has been found that the addition of the cap to the projectile and the consequent moving of the center of gravity of the projectile toward the point favorably influences the trajectory, increasing both the accuracy and range. All projectiles for sea coast guns above 3 inches in caliber will probably be provided with caps. Deck Piercing and Torpedo Shell—These projectiles are provided for the 12-inch mortars. The torpedo shell are longer and of greater interior capacity than the deck piercing shell, and carry large bursting charges of high explosives. The charge for the deck piercing shell is 64 pounds, and for the torpedo shell I34 pounds. Latest Form of Base of Shell.—A form of base with which good results have been obtained is shown in figure 19. The metal of 106 the shell is cut away beginning at a short dis- tance behind the band, leaving only a narrow ring to support the band. In the perforation of armor the band and the supporting ring are sheared off, thus relieving the projectile of the resistance due to the greater diameter of the band. Shell Tracers.-Experiments are now be- ing conducted toward the development of a pro- † jectile that will indicate its line of flight by the Fig. I9. emission of flame, or by the emission of some substance that will be visible from the gun; the purpose of the pro- jectile being to enable the gun commander to follow the flight of a projectile from his gun, and thus determine whether the gun is properly directed. . - The tracer. for use at night consists of a short metal cylinder filled with a slow burning substance that emits a bright flame during the flight of the projectile through the air. It may be screwed into a seat prepared in the base of any projectile. Ignition of the com- pound occurs in the gun. For day tracing a special shell is prepared. The cavity of the shell is partly filled with a mixture of lampblack and water, the mixture having the consistency of thick paint. A small orifice is made through the base of the projectile on one side. The powder gases enter this orifice under the pressure in the gun, and filling the cavity in the shell force from the orifice during flight a spray of black liquid. In recent experiments the flight of a 6-inch day tracing shell was followed for over 7200 yards. - Volumes of ogival Projectiles—The volume of a solid shot with ogival head is given by the following formula, obtained by means of the calculus. rds V. = —(Z – B) 4. In which V, is the volume of the shot, d the diameter, usual- ly taken as equal to the caliber of the gun, Z, the 1ength of the shot in calibers. 3. 107 7rd 3 The factor — Z is the volume of a cylinder whose diameter 3 - - 77 is dand whose length is Ld, the length of the shot. — B is the 4. volume of a similar cylinder whose length is Bd, or B calibers. The volume of this cylinder is therefore the volume of the cylinder cir- cumscribing the head minus the volume of the head. L-B is called the reduced length of the projectile in calibers, as it is the length of a cylinder of equal volume. B is a function of the radius of the Ogive expressed in calibers. Its value is given by the equation. Van — I 6772 — 277 — I B = 2m2 (2n − 1) sin-1 1/4m – 1 27? 3 in which n is the radius of the ogive in calibers. When m = 2, the usual radius of head in Seacoast projectiles, B = 0.58919. - For cored shot the reduced length is less than for solid shot by the length of the cylinder whose volume is that of the interior cavity. Representing by B' the length of this cylinder in calibers, the solid volume of the cored shot, or volume of the metal, is given by the equation º 71.6/3 v=º |z– (84. Bo! 4 Weights of Projectiles.—Representing the reduced length by l, and dividing the expression of the volume of one projectile by a similar expression for another we have - V./ Vſ – dº// a/3// Since the weights are proportional to the volumes: The weights of ogival projectiles are proportional to the products of the cubes of their diameters by their reduced lengths. - The weights of ogival projectiles of the same caliber are propor- tionate to their reduced lengths. Thickness of Walls.-The maximum stress sustained in the gun by the walls of a cored projectile, at any section of the projec- tile, is due to the pressure to which the walls are subjected in trans- 108 mitting to that part of the projectile in front of the section the maximum acceleration attained in the gun. The maximum accelera- tion is due to the maximum pressure in the gun; and this pressure being known the acceleration is determined by dividing the pressure by the mass of the projectile. oc = P/M = P2/w oc being the acceleration, P the total maximum pressure on the base of the projectile, and w the weight of the projectile. Substi- tuting the values of the known quantities oc may be determined. oc being known, if we substitute for w the weight of that part of the projectile in front of the given section and solve the equation for P, the value obtained, which we will call p, will be the pressure Sustained by the walls at the section. The area of the section is ſt (R2 - ??). The pressure per unit of area is therefore p divided by r(R2 – 72). - This pressure must not exceed the elastic limit of the metal for compression, divided by a suitable factor of safety; nor must it cause excessive flexure in the walls. If it does the walls must be made thicker. Thickening the walls will increase the weight in front of the section and therefore a new value of w must be obtained for a second determination. In shrapnel it is desirable to make the walls as thin as possible in order to increase the number of bullets that may be carried. The longitudinal pressure of the contained bullets is borne by the thicker base of the projectile, and the walls sustain only the pressure due to the centrifugal force and that proceeding from the weight of the head and fuze. Their thickness will therefore be deter- mined by the requirement that they must resist rupture by the pres– sure exerted by the gases from the bursting charge when the head of the projectile is blown off. The pressure required to blow off the head is equal to the resistance offered to shearing by the screw threads and shear pins of the head. A much greater thickness of wall than is needed in the gun is required to enable a projectile to withstand the shock of impact on the face of an armor plate. The retardation in this case is much 109 greater than the acceleration in the gun and consequently the stresses on the walls are correspondingly greater. As there is no means of determining the retardation at impact, the proper thick- ness of walls of armor piercing projectiles cannot be calculated, but must be determined by experiment. - We may, however, by assuming that the plate offers a constant resistance to the penetration of the projectile, determine the thick- ness of wall necessary in the projectile to enable it to pass through the plate and have any required velocity on emerging. Thus, to determine the thickness of wall of an armor piercing shell that is required, with a striking velocity v, to penetrate an armor plate of given thickness and to have on emerging a remaining velocity v. : - Let S be the constant resistance offered by the plate, l the thickness of the plate in feet, oc the constant retardation of the projectile during penetration. The work performed by the resistance over the path l is equal to the energy abstracted from the projectile while traversing this path. Therefore M M S! = — (v2 – z12) S = − (z2 – v12) 2 2/ The retardation due to the resistance is equal to the resistance divided by the mass. Therefore S' v2 – z12 OC E — E — M 2/ The pressure sustained by any section of the projectile during penetration is equal to the mass of that portion of the projectile be- hind the section multiplied by the retardation. Denoting by w the weight of that part of the projectile behind any given section, we have for the pressure sustained per unit of area at the section w! OC w/ (z2 – z12) f g z (R2 – 22) 2!gn (R2 – r2) R and r must be given such values, that is, the thickness of the walls must be such, that p will not exceed the elastic limit of the metal for compression, or that the flexure of the walls, considering the shell as a hollow column, will not be sufficient to cause rupture. Sectional Density of Projectiles.—It has been found by ex- Periment, as explained in exterior ballistics, that the retardation in the velocity of a fired projectile, due to the resistance of the air, is expressed by an equation that, for any fixed atmospheric conditions and standard form of projectile, may be put in the form & 2 A = A — f(z) ZU R representing the retardation, A a constant, d the diameter of the projectile, w its weight, and f(v) some function of its velocity. For a given velocity it is apparent that the retardation will increase directly with the square of the diameter of the projectile and in- versely with its weight; or, more concisely, the retardation will in- crease directly with the fraction d2|w. The reciprocal of this fraction, or w/d?, will therefore be the measure of the capacity of the projectile to resist retardation, that is to overcome the resistance of the air. The fraction w/d? is called the sectional density of the pro- jectile. w | }4 7t d? is the weight of the projectile per unit area of cross section, and w/d” is taken as the measure of this weight, 7t|4 being constant. - The sectional density is of importance in considering the mo- tion of the projectile both in the air and in the gun. * EFFECT ON THE TRAJECTORY.—The greater the sectional den- sity of the projectile, the less the value of its reciprocal, the factor dº/w in the above equation, and consequently the less is the value of the retardation of the projectile. - Of two projectiles fired with the same initial velocity and eleva- tion, the projectile with the greater sectional density will therefore lose its velocity more slowly and will attain a greater range. For any given range it will be subjected for a less time to the action of gravity and other deviating causes, and will therefore have a flatter trajectory and greater accuracy. The advantages of increased sectional density are therefore in- creased range, greater accuracy and a flatter trajectory. The sectional density may be increased by increasing the weight of the projectile or by decreasing its diameter. The weight of a 111 projectile for any gun may be increased by increasing its length. This has been done with modern projectiles for large guns until the length is from 3% to 4 calibers. In small arms the weight is in- creased by the use of lead in the bullet. Increase in sectional den- sity by decrease in diameter is found in the modern small arms of reduced caliber, the weight and diameter of the projectile having been reduced in such proportions as to increase its sectional density. EFFECT ON THE GUN.—An increase in the weight of the pro- jectile requires an increased pressure in the bore of the gun if the initial velocity is to be maintained. The maximum pressure for any gun being fixed, it has been possible to increase the weight and sec- tional density of projectiles only by the use of improved powders, which while they exert no greater maximum pressures exert higher pressures along the bore of the gun. The mean pressure on the pro- jectile is therefore greatly increased, and to withstand the increased pressure the chase of the gun is made stronger. i) { MANUFACTURE OF PROJECTILEs. Cast Projectiles.—A wooden pattern of the shape of the pro- jectile is first made, the dimensions of the pattern being slightly greater than the dimensions desired in the projectile, in order to al- low for contraction of the metal in cooling. The pattern is in one Or more parts, depending upon its size. The pattern shown in fig. 2O is in two parts separated at the line b. The parts are slightly coned from this line to facilitate withdrawal from the mold. For hollow projectiles a core box is also made similar in its interior dimensions to the cavity in the shell. The core e, fig. 20, made of Core sand mixed with adhesives, is formed in the core box around a hollow metal spindle wound with tow. The heat of the casting burns the tow, and the gases from the core pass out through the hollow spindle. Figure 2 I shows a mold prepared for casting a shell. The outer box called the flaskis in two sections parting at the line wy. In the lower part the sand is moulded around the pattern, which is also divided into two parts on the same line. In the upper part of the 112 its proper position by means of the frame a bolted to the flask. flask the remainder of the mold is made, and the core attached in * 1lar 1111– 11'C11 ásraeſ，*** • • • ׫º seºsºz， №vº！№vº.º.º.º.， §. *******） 、、、。¿?§§§¿$£§§§§§§§ ！ !! ！！ ！！ ！！ ！！ ！！ ！！ ^^~A（Sae\，{**}）^*•，，，，，，，，，.，，.Ru … ſººs•~ ~ ~~ſººſ §，，，，,%，，，，ſaetae§§§）（?：（?）, Sºſaevº， №， №sſº:№x'ſ S2 5×3，2×2×2×25×5×5 kg::!Sºº！！！！!！!！！！！！！！3！!2！！！！º:S¿№ ºs YR， SOE23;'，'<>> $:$2${}\\} aSt w №wºſº）， $（2:2:$）; āş. * *g *** * 25% § The pressure of ~5~ſºxae'， ººººº £<7，5%;3Æ№ saeaeae ;$22，2%· 3:33，*-º-）. --→ * „º，?！！?！！?！< • • • • •Šī£<.** s *** * º tº:- it st yº/Rºx ºf fº - º * * * *** ~ №vaezº-№vº， £§§@₪$};&ī£5 S★:52， №:43， 23， 。，，，。2;$%^&?}， ſae， !！!！!!S g the c the mold. The patterns old brought together and Fig. 21. in the mold has a c 11ect. k 111 1112. 111 111 & wae，*N。、3。、、。 ∞ √° *， -*.47 *= *a. --•º º-，|- æS:W¿??¿ $3&\;8*ZĀKSNĒ，`№， №83!！2！！53 *>）（?！*,ºzºnº，;$3&\2;43;::№ºººººº，，，（22）： $$$$$$Z×ſº， №.2，-|-Sºº！?!?!<∞§§¶√∞，∞, ∞&ae:2，3，3，-2，2‰）， ies co 11mpuri tS € 1.1SC1 2.SS1S S through the gate b, generally in a at the metal the escape of the gases and brings the to the center and top. The mold is filled with the metal to t * e 111 e ists , where the in th ion so th b 2 h ass 1C drawn and the parts of the m 1 direct d metal 1Qu1 e molten metal enter th ing. 1 Solid shot are cast head down in order that the dense metal may be in the head of the shot. Shells are cast base down, that the wi bolted. gate b and the riser c are also formed Th tangentia ----------- - - - -, purities the top of the riser The 2.1 e motion wh the 1 111 COO 113 base of the shell may be sound and free from cavities that would allow the powder gases to pass into the interior and ignite the burst- ing charge. Chilled Projectiles.—For use against wrought iron armor the heads of cast projectiles were hardened in casting by the process of chilling. A comparatively thin iron mold, the shape of the head, was fixed in the sand around the head of the projectile. This served to rapidly conduct the heat away from the head of the projectile, causing it to cool rapidly and giving it great hardness. These pro- jectiles are no longer used. Forged Projectiles.—The steel for a forged projectile is cut from a cast ingot, and is then bored, forged, and turned to finished dimensions. Armor piercing projectiles are in addition treated with Some secret process of tempering to give them the hardness and toughness necessary for the perforation of armor. º Requirements in Manufacture.—The qualities of the metal of the projectile are prescribed as follows: for cast iron, tensile strength 27000 lbs. per square inch; for steel, in what are called com- mon shell, that is, those of the smaller calibers, tensile strength 85OOO lbs. For armor piercing projectiles the tensile strength or elastic limit is not specified, further than by the requirement that the projectiles in a lot shall not vary in tensile strength by more than 20000 lbs. The strength of these shells is determined by actual firing against armor. The cap must be of steel whose elastic limit tº •. ".$." . does not exceed 60000 lbs., with an elongation at rupture of 30 per º cent and a reduction in area of 45 per cent. The base plugs of all projectiles are made of forged steel. Inspection of Projectiles.—The dimensions of the projectiles are tested by means of calipers, and profile and ring gauges. The slight variations, called tolerances, allowed from the standard dimensions are specified for each dimension, and the gauges for any projectile are constructed for the maximum and minimum of the particular dimension. Thus for the diameter of the band there are two ring gauges, one a maximum, the other a minimum, and sim- ilarly for other diameters. Maximum and minimum plug gauges 114 are applied to the threads of the fuze hole. A ring gauge is shown in figure 22. A profile gauge or templet is shown at a in figure 23. Fig. 22. Fig. 23. Eccentricity in the cavity of the projectile is determined by roll- ing the projectile along two rails, a figure 24, placed on a flat sur- face. Irregular movement of the projectile denotes eccentricity, which may be measured by means of the calipers, d, shown in the figure. For the detection of holes or cracks through the walls of hol- low projectiles all such projectiles are subjected to an interior hydraulic pressure. A pressure of 500 lbs. is applied for one minute to steel projectiles, and a pressure of 300 lbs. for two minutes to those of cast iron. Fig. 24. ~~~~~ * gº.--. ſ .33: " " --~~ 115 To determine whether the treatment received by the armor piercing shot in the tempering process has left in the shot initial strains that might cause rupture in store or in firing, these shot are cooled to a temperature of 40 degrees F. and then suddenly heated by being plunged into boiling water. When thoroughly heated by the water, the projectile is suddenly cooled by being half inserted, with its axis horizontal, in a bath of water at 40 degrees F, After a brief interval it is turned 180 degrees for a like immersion of the other half. Three days must elapse after the treatment of the pro- jectile before this test is applied. The necessity of the test is in- dicated by the not infrequent bursting of the projectiles in the shops after treatment. This test is not applied to armor piercing shell. The thinner walls of these projectiles are more uniformly affected by the tempering process. w The interior walls of hollow projectiles are coated with a lacquer of turpentine and asphalt for the purpose of making them smooth, and of reducing the friction between the walls and the bursting charge. Ballistic Tests. -Each class of projectile is subjected to a ballistic test under conditions assimilating the conditions of ser- vice. For the purpose of the test two or more projectiles are Selected from each lot presented. The projectiles tested are filled with Sand in place of a bursting charge, and after the test must be in condition for effective bursting. Armor piercing shot are fired against hard faced Krupp armor plate, from I to 1 1-3 calibers thick, secured to timber backing. The striking velocities of the shot from 8, IO and 12-inch rifles, against plates one caliber thick are near to 1750 feet, which corresponds to ranges of about 3OOO, 4OOO and 5000 yards, respectively, from the three guns. The shot is required to perforate the plate un- broken and then be in condition for effective bursting. Armor piercing shells must meet similar conditions, the thick- ness of the plate being one half the caliber of the shot. I2-inch deck piercing shell must perforate a 4% inch nickel steel protective deck plate at an angle of impact of 60 degrees. I2-inch torpedo shell are fired into a sand butt from a gun in which the chamber pressure must be 37000 lbs. 116 Common steel shell for sea coast guns of small caliber are tested with service velocities against tempered steel plates from 3 to 5 inches thick, depending on the caliber and service velocity of the projectile. * The shell for field and mountain guns are fired into sand, with a pressure in the gun IO per cent greater than the service pressure and with at least the service velocity. . The Painting of Projectiles.—Projectiles are so painted as to indicate the metal of which they are formed and the character of the bursting charge. The greater part of the body is black. A broad colored band around the projectile over the center of gravity indicates by the color whether the projectile is of iron, cast or chilled, or of steel, cast or forged. The color of the base indicates whether the projectile is charged with powder or with high explosive. In assembled ammunition the base color is painted in a band just above the band of the pro- jectile. ...----------~~~~~~" - - - - - ~~~~~~~--------...-,---. . -sex-ar-e-º" * * * ----> ---- ...~~~~ **" *~~... . --~~~" --~~~~ _--~~~ . ...~~~" ORDNANCE AND GUNNERY. PART III. Armor, - Recoil and Recoil Brakes, Artillery of the United States Land Service. PREPARED FOR THE Cadets of the United States Military Academy. BY ORMOND M. LISSAK, Major, Ordnance Department Instructor of Ordnance and Gannery. * º WEST POINT, N. Y. i U. S. MILITARY ACADEMY PRESS : 1907 | C O N T E N T S. Chapter VII. —Armor............ .................................. '• • * * * * * * * page 1 Harvey and Krupp Armor, 2. Manufacture of Armor, 2. Ballistic Tests, 6. Armor Protection of Ships, 7. Cast-iron Armor, 10. Gun Shields, 11. Chapter VIII. —Recoil and Recoil Brakes................. page 13 Velocity of Free Recoil, 13. Determination of the Circumstances of Free Recoil, 14. Retarded Recoil, 18. Recoil Brakes, 19. Hydraulic Brakes with Variable Orifice, 20. Recoil System of Seacoast Car- riages, 29. Wheeled Carriages, Recoil, 32. Chapter IX. —Artillery of the United States Land Service............................................................ . . . . . . . . . . . . . . . . . page 39 Mobile Artillery, 38. Advantages of Recent Carriages, 41. The Mountain Gun, 42. Field Artillery, 45. The 3-inch Field Gun, 46. Field Howitzers and Mortars, 55. Siege Artillery, 56. The 4.7-inch Siege Gun, 57. The 6-inch Siege Howitzer, 61. Siege Artillery in Present Service, 66. Seacoast Artillery, 68. Pedestal Mounts, 72. Balanced Pillar Mount, 74. Barbette Carriages for the Larger Guns, 76. Disappearing Carriages, 78. Seacoast Mortars, 86. Subcaliber Tubes, 90. Drill Cartridges, Projectiles, and Powder Charges, 92. i f 4. CHAPTER VII. ARMOR. History.—The use of armor for the protection of ships of war began in France in 1853, and soon became general. The first armor was of wrought iron. This metal Opposed a sufficient resistance to the round cast-iron projectiles of that time and to the elongated cast-iron shot of a later date. As the power of guns increased and chilled projectiles came into use wrought iron armor became ineffective. It was replaced about 1880 by compound armor, which consisted of a wrought- iron back and a hard steel face. Compound armor was made either by running molten steel on the previously prepared wrought-iron back, or by welding a plate of steel to another of Wrought iron by running molten steel between them, both plates being previously brought to a welding heat. The hard steel face opposed a great resistance to penetration of the shot and caused the shot to expend its energy in shattering itself. At the same time it distributed the stress over an increased section of the iron back, and the toughness of the wrought iron served to hold the plate together. The chief defect of the compound plate was due to the difficulty of obtaining intimate union between the two metals, and lay in the tendency of the steel face to flake off over considerable areas. The basic principle of this armor, the hard face and the tough back, is still maintained in the construction of the most modern armor. At the same time that the compound plate was used by Great Britain and other powers the all steel plate was being used by France, the effectiveness of the two plates being about equal. . NOTE. –This chapter is largely derived from the chapter on armor by Lieutenant Commander Cleland Davis, U. S. Navy, in Fullam and Hart's Text Book of Ordnance and Gunnery, 1905. 2 In 1889 the homogeneous nickel-steel plate, markedly Superior to the steel plate in toughness and resisting power, was introduced. The Harvey treatment of the nickel-steel plate, developed in the United States in 1890, still further increased the resisting power of armor, and in 1895 the Krupp process followed with further improvement. Harvey and Krupp Armor.—The principle employed in the manufacture of armor by these two processes is the same. In both, the face of the plate is made extremely hard by supercarbonization and subsequent chilling. The superiority of the Krupp plate appears to be due to the composition of the steel. The Harvey plate is made of a manganese nickel steel, while in the Krupp plate chromium is also present, and in greater quantity than the manganese. The composi- tion of the two plates, in percentages, is given as follows: C. Mn. Si. P. S. Ni. Cr. Harvey .30 .80 .10 .04 .02 3.25. 0.00 Krupp .35 .30 .10 .04 .02 3.50 1.90 The nickel, and to a certain extent the manganese, give great strength and toughness to the metal, while the chromium makes the metal more susceptible to the treat- ment that gives the desired qualities to the finished plate. First, it permits the attainment of a very tough fibrous con- dition throughout the body of the plate that makes it less liable to crack; second, it gives the metal an affinity for car- bon which enables supercarbonization to a greater depth; third, it increases the susceptibility of the metal to tempering, which gives a greater depth of chill. These are the qualities that mark the superiority of Krupp armor. - Even when carbonization of the plates is effected in the same manner, carbon will be absorbed to a greater depth in the Krupp than in the Harvey armor, giving a greater depth of hardened face and an increased resistance to penetration of about 20 per cent. - - Manufacture of Armor.—The steel, of proper composition, is made in the open hearth furnace and cast into an ingot of the shape shown in Fig. 1. The head of the ingot affords a T | | | | | | | | | | | f/ Af 144" . - K-–30"--> | | | | | | | ! K-— — — — — — — — — — — 130"—— — — — — — — — — +---> | | | | Sk Pig. 1. means for the attachment of the chains of the cranes em- ployed in handling it. A long heavy beam is used to counter- balance the weight of the plate when slung in the chains. When stripped from the mold and cleaned, the ingot is heated in a furnace and then forged, as shown in Fig. 2, under an immense hydraulic press capable of exerting a total pressure of about 15000 tons. The forging reduces the thick- ness of the plate and increases its length and breadth. The plate is then rough machined approximately to finished dimensions. CARBONIZING.—The carbonization of the face of the plate is effected by one of two methods; the cementation process, or the gas carbonizing process. The cementation process con- sists in covering the surface of the plate with carbonaceous material, usually a mixture of wood and animal charcoal, heating the plate to a temperature of about 1950 degrees, and 4 maintaining it at this temperature for a sufficient time to accomplish the required degree of carbonization. A covering of sand protects the face of the plate and the carbonizing material from the flames of the furnace, and excludes the air. From four to ten days, depending on the thickness of the plate, are required to bring the plate to the desired tempera- ture, and a further period of from four to ten days to effect the carbonization of the face. Under the action of the heat the carbon is absorbed into the face of the plate, and pene- trates into the interior, the quantity of the absorbed carbon diminishing from the surface inward. The gas carbonizing process consists in passing coal gas along the face of the plate heated in a furnace to about 2000 degrees. The heat decomposes the gas which deposits car- bon on the face of the plate, and the carbon is absorbed as in the cementation process. - REFORGING AND BENDING.-After being cleaned of the scale that is formed on it in the process of carbonization the plate is reforged to its final thickness. It is then annealed and bent to the desired shape in a hydraulic press. The opera- tion of bending an armor plate in a 9,000 ton press is shown in Fig. 3. - TEMPERING.-For tempering, the plate is uniformly heated to a high temperature and quickly cooled or chilled by cold water sprayed upon it under a pressure of about 23 pounds to the square inch. - In Krupp plates as first made the tempering produced cracks over the whole hard surface of the plate, some of them a quarter of an inch wide and extending some distance into the plate. The cracks were characteristic of the plate and were not considered abnormal, the resistance of the plate even with the cracks being greater than that of plates made by other processes. With improvement in the process of manufacture smoother plates were produced, and in many of the latest plates the surface appears continuous to the naked eye. When etched with acid however the face is found to be covered with a net work of fine lines and presents an appear- ance similar to that of crackled glass. Fig. 2. 15,000 Tom Hydraulic Forging Press Fig. 3. 9,000 Ton Press, Bending Armor. & 5 Armor Bolts.—The armor plates are fastened to the sides of ships by means of nickel-steel bolts. These are of such strength that they are not broken by the impact of pro- jectiles that badly crack the plate. The bolts pass through the sides of the ship and are screwed into the soft back of the armor plate. To insure a good fit of the plate, and at the same time to lengthen the armor bolt so that its deformation per unit of length under the stresses of impact may not be excessive, wood backing is used between the armor plate and the ship’s side. The wood backing is being reduced in thickness and the tendency is to discard it altogether. Figs. 4 and 5 show types of bolts for armor with and without wood backing. OAKUNM. Rºll ,” W W º | PLATES Fig. 4. The threads on the bolts are all plus threads so that the bolt is of uniform strength. A calking of marline or oakum surrounds the bolt to prevent leakage through the bolt hole. A steel washer is under the head of the bolt. A rubber Washer has also been used under the steel washer to diminish the sudden- ness of any strain on the bolt head. Armor bolts vary in diameter from 6 1.5 inches for plates 5 inches thick or less to 2.4 inches for plates 9 inches thick and upward. In number they are provided one for every five square feet of surface as far as the framing of the ship will permit. Ballistic Test of Armor.—The U. S. Navy specifications require as a test, before acceptance of Krupp and Harvey armor, three impacts of capped shells against a specimen plate, with velocities as given in the following table: Caliber Capped Plate Striking Of gun, projectile, thickness, velocity, inches. poundS. in Ches. f. S. 105 5 1416 6 105 6 1608 6 105 7 1791 7 165 6 1416 7 165 7 1578 7 165 8 1732 8 260 7 1412 8 260 8 1552 8 260 --. 9 1685 10 510 9 1458 10 510 10 1569 10 510 11 1676 12 870 II 1412 12 870 12 I50L The first impact in the center of the plate must not develop a through crack to an edge of the plate, and no part of the projectile shall get entirely through the plate and backing. On the second and third impacts no part of the projectile shall get entirely through the plate and backing. The im- pacts shall not be nearer than 3% calibers to each other or to an edge of the plate. Comparing the requirements for plates attacked by the 8, 10 and 12 inch guns with the requirements of the ballistic tests of armor piercing projectiles for the land service, page 115 Part II, it will be seen that the armor plates one caliber thick are tested with velocities about 200 feet less than that at which the projectiles from land guns are required to perforate similar plates. - Characteristic perforations in hardened and unhardened armor are shown in Figs. 6 and 7. The face of the hardened º sº-, 22.5 ºzº. +7°22&ſºzºs ZZZZZ º & Azºr. ". %22zº Ç ºzºº * 7:2:::::: Fig. 7. armor, Fig. 6, breaks and crumbles under impact, while the metal of the unhardened plate, Fig. 7, being softer and more tenacious, flows under the pressure of the projectile in the direction of least resistance and forms a combing in front of the plate. When the projectile reaches the back of the hard- ened armor the metal of the back, being prevented from flowing by the hard face, breaks out in one or more pieces leaving a broad based conical hole through the back, and producing but slight bulging of the rear surface of the plate. As the metal of the unhardened plate is of the same con- stitution throughout, the perforation does not exhibit the marked differences shown in the hardened plate. The metal Of the back part of the plate flows to the rear producing a greater bulging of the rear surface. - Armor Protection of Ships.-The armor carried by ships of War is of various thicknesses depending upon the size and purpose of the ship and on the position of the armor on or in the ship. The thickest armor is used to protect the water line and the vital parts of battleships. The present practice 8 in the United States is to protect the whole length of the water line with a belt of armor 8 feet wide extending 4% feet above the water line and 3% feet below it. This belt, see Fig. 8, has its maximum thickness over that part of the ship that contains the machinery and the maga- zines. The thickness diminishes from the mid-ship section and is least at the bow and stern. * The gun turrets are protected in front by the thickest armor. Armor of less thickness covers the casemates, barbettes, and sides of the turrets, the thickness depending upon the importance of the part protected. An armored deck of a thickness to prevent penetration by the fragments of exploded shell extends the whole length of the ship. This deck, the berth deck Figs. 8 and 9, is flat Over the machinery and boiler spaces and slopes downward at the sides and at the bow and stern to the bottom of the armor belt. On the heaviest ships the armored deck has a thickness of two inches over the flat part and four inches on the slopes. The gun deck, next above the armored deck, is sometimes an armored splintered deck one inch thick. Across the main body of the ship, bow and stern, extends heavy athwartship armor, which, with the armored barbettes and turrets, provides protection to the body of the ship from fire from the front or rear. Thus with the side armor the main body of the ship becomes an armored box, within which the crew, the machinery, the magazines, and the guns are protected. With the improvements that have taken place in armor within the last fifteen years there has been a gradual reduc- tion in the thickness of armor carried by ships of the various classes. • The battleship Oregon built in 1893 has a water line belt 18 inches in thickness, while the battleship Connecticut, com- missioned in September 1906, has but 11 inches of armor at her water line. FFFs-— MAIN DECK GUN DECK 33" 7" . Cº 3&ss ſ' jºss-º- * *. Supper caseMATE SibE ARMOR. 8" TURRET 12" TURRET 12" BARBETTE. 71. 7" IO" GUN DECK. = < *-s ºf ºf >, >, >, Ez. z= z = z = z:x. E. II" 9" 4" Fig. 8.—Distribution of Armor, United States Battleship Connecticut. 10 The arrangement of the ar- mor on the battleship Con- - *- necticut, is shown in Figs. 8 ºr and 9. - UPPER DEC K. The following definitions —º will assist toward a ready MAIN DECK. .. understanding of the figures. 72 | TURRET.-A revolving af- tf GUN DECK. mored structure in which one *l coal. % | or two guns are mounted. The | |Bºth 25% ; f guns revolve with the turret --→g % coal. ?" and are completely enclosed 9" ro with the exception of the #ico.. FiR E ROOM chase of the gun which pro- ăl s jects through a port hole in COAL. the front plate of the turret. § 0 - BARBETTE.-A fixed circu- 9| |0| OECSCIOS lar structure, armored, which - Fig. 9. protects the mechanism for the ammunition supply of the gun mounted above it, and the mechanism of the turret con- taining the gun. - - CASEMATE.-An isolated gun position for a broadside gun with fixed armor protection. The casemate completely en- closes the gun with the exception of the chase which projects through a port hole. CENTRAL CITADEL.-Armor enclosing a series of broadside guns. There may or may not be splinter bulkheads between the guns. With the bulkheads completely enclosing th guns the citadel becomes a series of casemates. - Chilled Cast Iron Armor.— This armor on account of its thickness and great weight is used only on land. It is manufactured by Gruson of Germany. It is cast in large blocks whose outer faces are made very hard by chilling. The blocks are then built into turrets, usually of rounded shape. On account of the great weight and hardness of the metal and the rounded shape of the turrets, this armor affords better protection than any other armor. ºplºſus q!!！ Juno.IN. [8]søpø. I tio unſ) qouſ-9 '0 L （5 IJI 11 Gun Shields.-Guns of six inch caliber and less mounted in barbette in sea coast fortifications are provided with shields permanently attached to their carriages. The shields are made of Krupp plate 4% inches thick. The requirements of the ballistic test for these shields are as follows: The shield, firmly supported by a backing of oak timbers, is subjected to three shots from a 5 inch gun. The striking velocity of the shot is 1500 feet and the impact normal. On the first impact, near the center of the shield, no portion of the projectile shall get through the shield, nor shall any through crack develop to an edge of the shield. The other two impacts are so located that no point of impact shall be less than three calibers of the projectile from another point of impact or from an edge of the shield. At the second and third impacts no projectile or fragment of projectile shall go entirely through the shield. The supports that hold the shield to the carriage are of steel with a tensile strength of 110,000lbs., elastic limit 75,000lbs., elongation at rupture 15 per cent., contraction of area 25 per cent. The fastening bolts must have a tensile strength of 80,000 lbs. and an elongation at rupture of 27 per cent. The shields are curved around the front of the carriage and are inclined upward and to the rear at an angle of 45 degrees. The chase of the gun protrudes through a hole in the shield and other holes are provided for sighting purposes. Fig. 10 shows the arrangement of the shield on a 6-inch barbette carriage. Shields will probably be provided for all barbette carriages. It is still a matter of discussion as to whether advantage is derived by the use of gun shields, for while they serve to keep out the smaller projectiles they also serve to determine the bursting of larger projectiles whose destructive power may be sufficient to disable the gun and wholly destroy the gun detachment. Without the shields these projectiles would in many instances pass by, doing little or no harm. Field Gun Shields.-Shields of hardened steel plate, two- 12 tenths of an inch thick are attached to the gun carriage and caisson for the 3-inch field gun. These shields are tested by firings, at a range of 100 yards, with the 30 caliber rifle using steel jacketted bullets with 2300 feet muzzle velocity. The plate must not be perforated, cracked, broken, or materially deformed. The front of the caisson chest is made of the same material as the shields and has the same thickness. The door of the chest, which opens upward to an angle of 30 degrees, is made of hardened steel plate Hº of an inch thick. 7 jº CHAPTER VIII. RECOIL AND RECOIL. BRAKES. The stresses to which a gun carriage is subjected are due to the action of the powder gases on the piece. Gun carriages are constructed either to hold the piece without recoil or to limit the recoil to a certain convenient length. In the first case the maximum stress on the carriage is readily deduced from the maximum pressure in the gun. In the second case it becomes necessary to determine all the circumstances of recoil in order that the force acting at each instant may be known, and the parts of the carriage designed to withstand this force and to absorb the recoil in the desired length. Velocity of Free Recoil.-Suppose the gun to be so mounted that it may recoil horizontally and without resistance. On explosion of the charge the parts of the system acted upon by the powder gases are the gun, the projectile, and the powder charge itself, the latter including at any instant both the un- burned and the gaseous portions. While the projectile is in the bore, if we neglect the resistance of the air, none of the energy of the powder gases is expended outside the system. The center of gravity of the system is therefore fixed and the sum of the quantities of motion in the different parts is zero. The movement of the powder gases will be principally in the direction of the projectile. We may therefore write Mvp = mv -- uv, . (1) in which M, m, and u are the masses of the gun, projectile, and products of combustion, respectively; and v, v, and v. the velocities of the same parts. The velocity of the projectile at any point in the bore of the gun may be determined from the formulas of interior ballistics, equations (112) to (115) page 88, Part I. The velocity of the products of combustion is unknown. It varies from Zero near 14 the breech to v at the base of the projectile. We may with- Out material error consider it as equal to half the velocity of the projectile. - Writing v/2 for ve in equation (1), replacing masses by weights, and solving for v, we obtain 1 / 22. v, -**** * (2) W, w, and 6 being the weights of the gun, projectile, and charge. - * At the muzzle of the gun v becomes the initial velocity V, and for the velocity of free recoil at that instant 1/ ×× o' = *** V (3) This value v', is not the maximum velocity of free recoil, though it is the maximum value reached while the velocities of the gun and of the projectile are connected. At the de- parture of the projectile the bore of the gun is still filled with gases under tension, which continue to exert pressure on the breech, and increase the velocity of recoil. The value v', Ob- tained by the above equation is about 7/10 of the maximum velocity of free recoil. It has been determined by experiment with the Sebert veloc- imeter, that the maximum velocity of free recoil may be ob- tained from equation (3) by substituting for the quantity 3% G V the quantity 4700 G3. The equation then becomes 70 V --W700 G3 (4) V, being the maximum velocity of free recoil. The coefficient 4700 applies to smokeless powders. The co- efficient for black powders was 3000. |V = Determination of the Circumstances of Free Recoil.— In the above equations the velocity of free recoil is expressed as a function of the velocity of the projectile, and we have in the ballistic formulas the velocity of the projectile expressed as a function of the travel of the projectile. We may therefore 15 now determine the velocity of free recoil as a function of the travel of the projectile. But in the determination of all the circumstances of recoil it is necessary to know the relations between the velocity, time, and length of recoil; and in order to arrive at these relations by means of equation (2), we must obtain an expression for the velocity of the projectile as a function of the time. - With the velocity of the projectile expressed as a function of the time, equation (2) will then express the velocity of free recoil as a function of the time, and with the velocity of recoil SO expressed we may obtain the length of recoil from the equation a = ſ. v. dt - (5) a representing the length of free recoil. We thus obtain the complete relations between the velocity, time, and length of free recoil. Welocity of the Projectile as a Function of the Time.—The velocity of the projectile as a function of the time is obtained in the following manner. Representing the travel of the pro- jectile by w, we have v=du/at, from which t-ſhau (6) That is, t is the area under the curve whose ordinates are 1/v and whose abscissas are values of u. Therefore if we construct such a curve the area under the curve from the origin to any ordinate will be the time corre- sponding to the velocity whose reciprocal is represented by the ordinate. 16 Construct the curve v, Fig. 1, from the ballistic formulas, _-T 2) %— Tig. 1. the abscissas representing travel, the ordinates, velocity of the projectile. Take the value of v as expressed by any ordinate and lay off its reciprocal on the same ordinate, to any convenient scale. The curve 1/v in the figure, is obtained in this way. Its ordinates are values of l/v, its abscissas are values of u. The areas under the curve are therefore values of t, equa- tion (6). - -- For very small values of v the ordinates 1/v will be very large and will not fall within the limits of an ordinary draw- ing. We cannot determine then from the drawing, the area under the first part of the curve. But we can obtain a suffi- ciently close approximation to this area in the following manner. We may assume, without material error, that.the velocity of the projectile as a function of the time is expressed by the equation of a parabola v= V20t. (7) Multiplying by dt and integrating, we have, since ſv dt =w, w= ſ V2 pt dt =# (2p)* tº . (8) At the instant at which the shot leaves the bore, v in equation (7) becomes the initial velocity V, and denoting the corresponding time by tº we obtain from that equation 17 V= V2nt’ or W2p= V/ Wi", Substituting this value of (2p)* in equation (8), t in that equation becoming t' and M the total travel of the projectile U, we obtain 3 U. 2 V t' is then the total area under the curve 1/v, Fig. 1, and subtracting from tº the area that can be measured we obtain the area under that part of the curve near the origin that is not plotted. Having now from the v curve the values of v = f(w) and from the areas under the 1/v curve the values of t =f(w) we may, by combination, determine the desired values of v = f(t). Using as abscissas the areas under the curve 1/v, which are the values of t, and as ordinates the corresponding ordinates of the curve v, which are the velocities, we obtain 7) the curve of the velocity of the projectile as a function of the time, Fig. 2. Since the velocity of free recoil as given by equation (2) is equal to the velocity of the pro- jectile multiplied by a constant, the curve in Fig. 2 becomes at Once the curve of velocity of free recoil, if we consider the scale of the ordinates as multiplied by the coefficient of v 1 in equation (2). #" Fig. 2. Maximum Welocity of Free Recoil.—The curve shown in Fig. 2 gives the velocity of free recoil only while the projectile is in the bore, and as previously explained the velocity of recoil has not reached its maximum when the projectile leaves the piece. The value of the maximum velocity of recoil is given by equation (4). With this value as an ordinate, Fig. 3, draw a line parallel to the axis of t and continue the curve of Velocity already drawn until it is tangent to this line. It is reasonable to infer that the rate of change in the curvature of the curve of recoil will continue uniform from the point 18 corresponding to the muzzle of the gun to the point of maxi- 2) - _* Elig. 3. mum velocity, and the curve so continued will with sufficient exactness express the circumstances of motion. A slight error made in the selection of the point of tangency will be without practical effect on the determinations to be later made from this curve. The abscissa of the point of tangency is the time corresponding to the maximum velocity of free recoil. - - As, by assumption, there is no resistance to recoil, the maximum velocity attained will never be reduced, and the curve will extend indefinitely parallel to the axis of t. The tangent to the curve at any point is a value of dvº/dt and therefore represents the acceleration at the instant of time represented by the abscissa of the point. The tangent has a maximum value at the point of inflexion of the curve. This point is therefore the point of maximum acceleration. The maximum acceleration being due to the maximum pow- der pressure in the gun the abscissa of the point of inflexion is the time of the maximum pressure. Since, equation (5), a = ſwidt, the area under the curve vi, Fig. 3, from the origin to any ordinate is the length of free recoil corresponding to the velocity represented by the ordi- nate. Retarded Recoil.–In the discussion thus far we have neg- lected all resistances and have considered the movement of 19 the gun in recoil as unopposed. When the gun is mounted on a carriage the recoil brakes, of whatever character, begin to act as soon as recoil begins and consequently the velocity of recoil is less at each instant than the velocity shown by the curves just determined. The manner of obtaining the velocity of retarded recoil will be explained later. - Recoil Brakes.—To absorb the energy of recoil and to bring the gun to rest in a convenient length, all gun car- riages which permit movement of the gun in recoil are provided with recoil brakes. These are of two general classes, friction brakes and fluid brakes. Friction brakes were formerly used on Seacoast carriages but are now confined exclusively to wheeled car- riages. Fluid brakes are either hydraulic or pneumatic. Pneumatic brakes, depending for their resistance on the compression of air, have been used in England to some ex- tent on seacoast carriages. On account of the difficulty of preventing loss of pressure in the brakes through leakage of the air these brakes are not satisfactory. Hydraulic Brakes.—A hydraulic recoil brake consists of a cylinder filled with liquid, and a piston. Relative movement is given to the cylinder and piston by the recoil, and pro- vision is made for the passage of the liquid from one side of the head of the piston to the other by apertures cut in the piston or in the walls of the cylinder. The power of the brake lies in the pressure produced in the cylinder by the re- sistance offered by the liquid to motion through the apertures. If the area of the apertures is constant it is evident that the resistance to flow will be greater as the velocity of the piston, or the velocity of recoil, is greater. Therefore the preSSure in the cylinder, which measures the resistance offered, will vary with the different values of the velocity of recoil. If however the apertures are constructed in such a manner that the area of aperture increases when the velocity of the piston increases and diminishes when that velocity diminishes, the variation in the area of aperture may be so 20 regulated that the pressure in the cylinder will be constant or will vary in such a manner as to keep the total resistance to recoil constant. º Both of these methods have been used in the construction of recoil brakes for gun carriages. The brakes with con- stant orifices and variable pressures were used on the old carriages for 15-inch smooth bore guns. For a fixed length of recoil a constant resistance will have a lower maazimum value than a variable resistance, and con- sequently will produce a less strain on the gun carriage. For this reason and for other advantages that will appear in the discussion which follows, the brake with variable orifices, and constant or variable pressure as circumstances may require, is at present used to the exclusion of all others on gun carriages. - Hydraulic Brake with Wariable Orifice. —The mode of action of the hydraulic brake with variable orifices will be understood from Fig. 4 which represents a longitudinal sec- tion through a recoil cylinder of the form used in our sea coast carriages, The direction of recoil is to the right. r r O – C’ Fig. 4. Fig. 5 represents a cross section through the cylinder. To the walls of the cylinder c are fastened two bars o called throttling bars, of varying cross section as shown. The piston rod, r of the stationary piston, p is fixed to the chassis. The cylinder c forms part of the top car- riage and moves with it in recoil. Fig 5. Through the piston head are cut two slots or apertures s, through which the liquid is forced from one 21 side of the piston to the other as the cylinder moves in recoil. Each slot has the dimensions of the maximum section of the throttling bar with just enough clearance to permit Opera- tion. The orifice open for the flow of liquid at any position of the piston is therefore equal to the area of the slots minus the area of cross section of the throttling bars at that point; and the profile of the throttling bars is so determined that the resistance to the flow of the liquid, or the pressure in the cylinder, is made constant or variable as desired. -- ******** __--~~~~T *º-ºººººº-ºº:::::-----> * Total Resistance to Recoil.—The total resistance to recoil is composed of the resistance opposed by the brake, the re- sistance due to friction, the resistance—either plus or minus— due to the inclination of the chassis rails, and the resistance due to the counter recoil springs if there are such included in the recoil system. The resistance of the springs varies with the degree of compression. Therefore to maintain a constant total resistance when springs are included in the system the resistance of the brake must also vary, the other resistances being constant. Let W be the weight of the moving parts, the mass of the moving parts, the coefficient of friction, the angle of inclimation of the chassis rails, the resistance of the springs at any time t, the total resistance of the hydraulic brake, or the total pressure in the cylinder, at the time t, R the total resistance to motion, o, the velocity of retarded recoil at the time t, V, the maximum velocity of retarded recoil. The resistance due to friction will be f Wcos a; that due to the inclination of the rails will be Wsin a. The total resist- ance at the time t is therefore R = W (sin C. -- f coso) + S + P (9) Dividing the total resistance by the mass, we have, for the retardation, º — do / dt = R/M (10) --~~~~ **. –2. J.--- **** / 22 ;}. R*- t.* ^ \ aſ º () \\ | The retardation R/M being constant, we may substitute it . . for g in the equation that expresses the law of constant forces o” = 2gh Assuming the Origin of movement as at the maximum ve- locity of recoil, V, and designating by l’ the length of recoil from this point to the end, the above equation becomes V.2 = 21' R/M \\ , º * l' = V2 M/2R >{\ , (11) OI’ | * ł, & ' - - N + º- zwº- & , CA. S. l' is the length in with tºº/* à flºo # will OV6I?— come a velocity of recoil V. Walues of the Total and Partial Resistances and Welocities of Recoil.-In the construction of a gun carriage the length of recoil is usually fixed by the design of the carriage. We will therefore assume a length l as the total length of recoil. We must now determine the total constant resistance that will restrict the recoil to this length, and then determine the por- tion of this resistance that is to be contributed by the brake. In so doing we will arrive at the values of the velocities of recoil at all points in the path. Total Constant Resistance.—The curve v, in Fig. 6, which 1) - 7??, v, - - C f 1)b, - 1/r S k O - t Rig. 6. as far as the point m is the curve v, in Fig. 3 drawn to a dif- ferent scale, represents the velocity of free recoil as a function t º \ - º *—- : *~€” f - • '.....--~". , t: , ; * : Y \tº \ } < _º_*-*--, & Cº, N. *— ... ' {i f : ºf f * , ; " ;- , ‘....' ', '--> M-A-º . - ?" - ~...~ * * | | \\ 23 of the time. We have seen that the tangent to the curve at any point represents the acceleration at that point. We may represent the negative velocities due to a constant resistance by the ordinates of some straight line oc, whose abscissas are the corresponding times. The tangent of the constant angle toc is therefore equal to —dv/dt, the retard- ation due to the force. The line oc is for convenience drawn above the axis of t. As its ordinates represent the negative velocities due to the resistance the line properly belongs below the axis. Now if we subtract from the velocities of free recoil, repre- sented by the ordinates of the curve vi, the velocities due to the retarding force, the Ordinates of oc, the Ordinates of the resulting curve v, will be the velocities of retarded recoil. The area under the curve of retarded recoil will be the total length of recoil, see equation (5). We have assumed a total length of recoil, l, and if the area under the curve of retarded recoil, as obtained above, does not give this length, we must change the angle toc and construct a new curve. After a few trials the proper direc- tion of oc will be determined and the area under the curve of retarded recoil, v, Fig. 6, will be the length l. Then the retardation represented by the line oc is given, See equation (10), by the equation —tan toc = — do / dt = R/M (12) from which R, the total constant resistance that will limit the recoil to the length l, is determined. The length of retarded recoil corresponding to any velocity of retarded recoil represented by an ordinate of the curve v, is the area under the curve from the origin to the given Ordinate. Minor Constant Resistance.—The total resistance R is com- posed, equation (9), of the constant part W (sin C.--fcos C.) =k and the two variable parts S and P. The retardation due to the constant part is equal to k/M, and is represented in Fig. 6 by a line ok drawn so that the tangent of the angle tok is equal to k/M. 24 Resistance of the Spring.—The resistance S of the spring has a value expressed by - S-64-6, (13) in which G is the force required to compress the spring, when free, over the first unit of length; and G' is the residual pressure in the spring when the gun is in battery. a, is any length of recoil determined from the curve v, as explained above. The resistance of the spring at any point may there- fore be determined from the above equation. Representing by v' the velocity in the mass M due to the spring alone, the retardation due to the spring is —dv'/dt = G'/M-- Gay/M. But da;=v'dt. Therefore dt = day/w', and —do'/dt= —v'do'./da-G'ZM+ Gaſ/M, and integrating - –v”/2= G'ay/M-H Gaž/2M the constant of integration being 0 since when a is 0, v' is 0. The values of v' determined from this equation are the true Ordinates of the curve os in Fig. 6. They are laid off in that figure from the line ok so that in the figure the ordinates of os are the sums of the true ordinates of ok and 0s. The Ordi- nates of os are therefore the velocities taken out of the sys- tem by resistances other than the hydraulic brake. Sub- stracting these ordinates from the ordinates of the curve of free recoil vi, the resulting curve vi, is the curve of Velocities to be absorbed by the brake alone. For any velocity vu the length of recoil is given by the area under the curve v, limited by the same Ordinate. Resistance of the Hydraulic Brake, -Pressure in the Cyl- inder.—The pressure in the brake cylinder at any point of the recoil may now be determined from equation (9) P= R– W (sin o-H f cos C.) — S, (14) if we substitute for R its constant value from equation (12), 25 & S its value at the given point from equation (13), and for e remaining term its constant value. To find the maximrtim pressure in the cylinder take-Érem C we-ºr-the-maximum-ordinate, find the corresponding length of recoil from-the-eeeeeeººtng area under the curve v, and with this length determine the value of S from equa- tion (13 for substitution in the last equation. cºaº ſº - sº a ºn tº EC Bºlºl. Kl-O I Relation Between the Pressure, Area of Orifice, Velocity, and Length of Recoil.–In this discussion we will designate by the term aperture the cut through the piston, and by the term orifice that portion of the aperture open to the flow of the liquid; and we will consider for simplicity that there is but one aperture and one orifice. Let A be the effective area of the piston in square feet, that is, the area of the piston minus the area of the piston rod and aperture. The square foot is taken as the unit of area, because in the velocities involved in the discussion the foot is the unit of length. Let a be the area of the orifige at any time t. W the maximum velocity of recoil controlled by the brake. v, the velocity of recoil controlled-by-the-brake at any time t. o, the velocity of the liquid through the orifice at the time t. - - - * ºf the weight of a cubic foot of the liquid. P the total pressure on the piston at the time t. The cylinder being full of liquid the volume that passes through the orifice is the volume displaced by the piston. We therefore have at any instant o; A = via Or, for the velocity of flow, vi-vº A/a. (15) From Torricelli’s law for the flow of liquids through Orifices we know that the pressure required to produce this 26 velocity of flow is the pressure due to a column of liquid whose height h is given by the equation o”–2 gh. (16) Substituting for v the value of vi from equation (15) and solving for h we obtain y h = v.*A*/2 ga”. The weight of a cubic foot of the liquid being Y, the weight of the column whose area of cross section is unity will be Yh, and the weight of the column whose area of section is equal to that of the piston will Ayh. Ayh is therefore the pressure on the piston, and substituting in this expression the value of h from the above equation we have, for the total pressure on the piston, for any velocity v. P= YA%),”/2 ga”. (17) This equation is general and expresses the relation that exists between P, A and a for any given velocity of recoil. Constant Pressure.—If P is constant we will have in a given cylinder, for any other values of v, and a, as V, and ao, respectively the maximum velocity of recoil and the maxi- mum area of orifice P= YA*W*/2ga,”, (18) and by combining equations (17) and (18) we obtain for any given cylinder *} v./W =a/a, (19) from which we see that to maintain a constant pressure in the cylinder the area of the orifice must vary directly with the velocity of recoil. . Assuming the maximum velocity of recoil as the origin of movement the following relation may be established, l' rep- resenting the total length of recoil after the maximum velocity has been reached, A z º. : lº º a-a-V 1– 7 Y (20) t- Q , 4- v,” %. { w.L 2 " . % ... ...!--~~ \ ** K -7 *A. (!-0 * {\ , ; V. 2. ^ i. «L - *- V/U 27 that is, with constant pressure in the cylinder the area of orifice varies as the ordinates of a parabola. Area of Orifice.—With the relations established in equations (14), (17) and (18), which are here repeated, and the curve v, in Fig. 6, we are now prepared to determine the variable area of orifice in the piston. P= R— W (sin o.--fcos a) — S, (14) P=0 A*v.”/2 ga”, (17) SP=x A*W*/2 ga.”. (18) The dimensions of the recoil cylinder will be fixed within narrow limits by the design of the carriage, and by the requirement that the pressure per unit of area must not be so great as to render difficult the effective packing of the stuffing boxes through which the piston rod passes. We will therefore assume that the diameters of the cylinder and piston rod are given, and as the relation between the total area of piston and the effective area may be readily estab- lished we will assume that the effective area A of the piston is known. Brake with Constant Pressure.—When there are no springs or other variable resistance in the recoil system, S becomes 0 in the value of P, equation (14), and a constant resistance will be required in the brake. The curve vº. in Fig. 6 will be derived from w; and ok only. - Equation (18) applies in this case. Substitute for P the constant pressure, determined from equation (14) as already explained. The value V, is the maximum ordinate of the curve ty, Fig. 6. A is known. The maximum area of orifice may be now determined from the equation and the area or Orifice at all other points more simply by means of equation (19), using the values of v. taken from the curve. Horizontal Chassis.-If the chassis rails are horizontal and the top carriage is mounted on rollers, so that we may neglect the friction, the term (W sin a + f cos C.) in the value of P, - 2. A : . . . . *. . . * , ; ; ; / ( | *... -- * \ ‘. . J’/\, f } ſ * x i v. 28 equation (14), also becomes Zero, and Preduces to R. Sub- stituting for R in equation (11): the value of P as expressed in (Kiš and solving for a we obtain, since v, and v, are now identical, - ...) a” = &A'z W. The area of Orifice is in this case independent of the velocity of recoil, and is dependent only on the length of recoil. Therefore for a given area of orifice the length of recoil will be the same no matter what the initial velocity of the projec- tile, the charge of powder, or the angle of fire may be. Under these conditions the brake requires no adjustment for varying conditions of fire, and in this respect it possesses further advantage over the brake with constant orifices and variable pressure. Brake with Wariable Pressure.—Equation (17) applies. The value of P for any point in the cylinder is obtained from equation (14), the proper value of S for the point hav- ing been first determined from (12). The value of v is taken from the curve tº in Fig. 6 at the ordinate which makes the area under the curve v, equal to the length of recoil from the origin to the given point. The values of P and v. thus deter- mined are substituted in equation (17). The resulting value of a is the area of orifice at the given point. - Profile of the Throttling Bar.—Suppose there are n simi- lar apertures cut in the piston. The area of each orifice at any point in the cylinder will then be a Zn, a being determined as above. Let b, Fig. 7, be the width and d the depth of each aperture. The throttling bar has the same depth, and a vari- able width y. Then for the area of each ori- fice at the given point in the Hig. 7. cylinder, we have 2- * : \_*- \_, ." 29 a/n = d (b – y). For the brake with constant pressure the profile of the throttling bar from the point of maximum velocity to the end will be a parabola. Its equation obtained by substituting the value of a from the above equation in equation (20) and re- ducing, is - * — h “0. Ø ty = b #Ni-f Neglected Resistances.—In the foregoing discussion we have neglected the resistances due to the friction of the liquid and the contraction of the liquid vein. It has been found by experiment that the error due to the neglect of these resistances may be corrected by assigning to vi, the velocity of the flow through the orifices, equation (15), a value greater than the actual value as expressed in equation (16). The value to be substituted is determined by experi- ment for each class of carriage and takes the form vs= avi + b, a and b being constants. The result of the substitution is a large increase in the value of P. Recoil System of Seacoast Carriages.— The arrangement of the parts of the recoil system on our seacoast disappearing carriages, and on barbette carriages for guns 8 inches or more in caliber, is shown in Fig. 8. The direction of recoil is to the right. The two cylinders c are integral parts of the top carriage; the top carriage, including the cylinders, forming a single steel casting in the sides of which above the cylinders are trunnion seats, for the gun trunnions in a barbette carriage, and for the gun lever trunnions in a disappearing carriage. The piston rods of the recoil cylinders are fixed to the chas- sis in front and supported in the rear. They enter the cylin- ders through stuffing boxes. On discharge of the piece the top carriage moves to the rear with the gun, forcing the liquid in the cylinders through the orifices in the stationary pistons. To equalize the pressure in the two cylinders their in- teriors are connected at the front by the pipe a and at the | s .* * * s y <- 31 rear by the two pipes d and f. Each half of the pipes d and f has unobstructed communication with the other half of the same pipe through a valve box v. A cross pipe b connects the pipe a with the valve box. A path is afforded through the pipes a, b, and d and f for the flow of liquid from one side of the piston to the other, which path must be con- sidered in determining the area of Orifice. The area of orifice, and consequently the length of recoil, is calculated for standard conditions of loading. Any varia- tion in these conditions will vary the length of recoil, and thus, in disappearing carriages, vary the height of the breech of the gun above the loading platform. Standard conditions of loading do not always exist, and it is therefore desirable to have means for varying the resistance in the cylinders in order that the prescribed length of recoil may be obtained under any conditions, as for instance when reduced charges are being used. For the purpose of varying the area of Orifice, and there- fore the resistance in the cylinders, adjustable valves called throttling valves are provided at v1, and v2. The flow from the pipe b into the pipe d communicating with the body of the cylinder, is regulated by the valve v1, and the area open to the flow is affected to increase or diminish the pressure in the cylinder as desired. The pipe d and its valve v1 are for the control of the recoil. To control the counter recoil and to bring the gun and top carriage to rest without shock as they come into battery under the action of gravity, the counter recoil buffer is provided. The rear cylinder head is provided with a cylin- drical recess into which the enlargement n of the piston rod, just in rear of the piston, enters as the carriage approaches its position of rest. The lug n is slightly conical so that the escape of the liquid from the recess is gradually obstructed. The pipe f with its valve v2 assists in the regulation of this part of the counter recoil. The cylinders are filled, through holes provided in the top, with a mineral oil called hydroline. The freezing point of the oil is below 0° F. Its specific gravity is about 0.85 The oil may be drawn off through a hole e in the valve box, Ordinarily stopped with a screw plug. 32 The throttling bars are fastened to the cylinders by screw bolts through the cylinder walls, as shown in Fig. 8. Modification of System.—It will be noticed that any move- ment of either of the throttling valves, that control the recoil and counter recoil, affects the area of Orifice. Therefore the regulation of the counter recoil affects also the recoil. It has been found desirable to separate these two systems so as to have independent control of both recoil and counter recoil; and in a 6-inch disappearing carriage now being tested an additional recoil cylinder is provided in the counter- weight. The control of the recoil is effected wholly by this large cylinder, and the counter recoil is controlled by smaller cylinders whose pistons are acted on by the top carriage in the last part of its movement into battery. Other advantages of this arrangement will appear later in the description of the carriage. Wheeled Carriages, Recoil.—To arrive at the effect of the recoil on a wheeled carriage we must consider the effects of all the forces that act upon the carriage. These forces in- clude the weight of the system composed of the carriage and gun, and the various forces developed by the transmission of the powder pressure to the points of support of the carriage. In Fig. 9 is represented the trail of a wheeled carriage, with ld *6/d tº O F- k-9, % sº: ; C Ks- P-> - º 4. W \,\ } \ }) ; \ Fig. 9. -—S 33 the wheel and spade. For the purpose of discussion we will assume that the carriage is a rigid body, that the wheels are locked, and that the pressure developed in the gun, or the pressure in the recoil cylinder when the gun recoils on the carriage, is transmitted to the carriage at the point 0. The points of application and the directions of the forces acting on the carriage and of the reactions at the points of support are represented in the figure. q) is any angle of elevation, P the transmitted pressure. Let M be the mass of the system composed of the gun and carriage, W its weight, F'- H" + F^, the total friction. The center of gravity of the system is represented at c. The forces acting on the carriage are symmetrically dis- posed with respect to the axial plane, and therefore their resultant acts in that plane. - A system of forces acting in a plane is completely known when its components in the direction of two rectangular axes in the plane and the moments about any axis perpendicular to the plane, are determined. We will assume the rectangular axes as horizontal and vertical. The effect of the forces acting on the carriage will be, under the most general consideration, a movement of the carriage to the rear, and at the same time, since the resist- ance to motion is greatest at the point of support of the trail, there will occur a movement of rotation of the carriage about the point of support. Applying to the carriage, in the manner shown in Fig. 9, all the forces that act upon it, we may consider the carriage as a free body and may then determine the values that the forces must have in order to produce in the free body the actual movement of the carriage in recoil. The movement of a free rigid body acted on by forces may be considered as composed of a movement of translation of 34 the center of gravity and a movement of rotation of the body about the center of gravity. The movements of translation and of rotation may be considered separately. We have for the equations of motion of the center of gravity D+ T – W-P sin (p d°y & ... -- M T dº? " 4-J 4-d / The sum of the moments of the applied forces with refer- ence to an axis through the center of gravity is the same whether the center of gravity is in motion or at rest, and is equal to the product of the acceleration of rotation into the moment of inertia of the body about the axis. Therefore, representing with small letters the lever arms of the forces with respect to an axis through the center of gravity, we have the equation Pp-H Ff--Da-HSs – Ti d20 (23) Mk12 T dº ki representing the principal radius of gyration of the body. Now to introduce into the three general equations of mo- tion, (21), (22) and (23), the condition that the movement of the free body shall be the same as the movement of the car- riage in recoil, we may write g = l sin 0, since this condition holds in the actual movement of the car- riage. Differentiating y twice and dividing by dt” we obtain d”y d”0 , s: ,, a dº diº -loose #2 —l sin 6 diº d6/dt is the angular velocity of the carriage about the point of the trail. lae/dt is therefore the linear velocity of the center of gravity about the same point. Representing this linear velocity by v we obtain from the above equation after multiplying the last term by l,'l, d”y d”0. v’s; diº =l is cos 0–4 sin 9. (24) • -s tºº. *..., (N | ~£ Iº 35 \ This equation expresses that the vertical acceleration of the center of gravity rotating about the point of the trail is equal to the vertical component of the linear acceleration ld”0/dt” about that point, see Fig. 9, minus the vertical component of the acceleration along the radius l. Substituting the value of d”y/dt” from equation (24) in equation (22) we introduce into that equation the actual con- dition of motion. We then have the three equations P cos (p-F—S d’a. MTT dſ : (25) D-- T. W_P si G20 y? . ..” M ** =l cos 0%. – , sin 0, (26). Pp--Ff+Dd--Ss–Tt d”0 (27), 3 Mk12 di” tº ~3 We may determine any three of the quantities in these equations if we establish, or assume, values for the other Quantities; and in this way we may determine the effects . that follow from variations in the values of any of the quan- ; , tities that enter the equations. § The above equations are applicable only while y = l sin 0, • ‘s that is, as long as the point of the trail remains on the ground. Tº *2Problem.—Determine, for the 3-inch field carriage, the relations that must exist between the pressure in the recoil cylinder and the weight and dimensions of the carriage in Order that there may not be any movement of the carriage When the firing is at zero elevation. In the three equations just deduced, q, the angle of elevation, becomes 0; and since there is to be no movement of the car- riage the terms involving the accelerations and the linear velocity become 0. Without movement there will be no friction and F will also be 0. The three equations then reduce to P–S=0, D–– T– W = 0, X- Pp-H Dd-- Ss – Ti =0, \ \ ~, ^\\ Y *~~ * > ...N-,-, ... \ 7:-- 36 which express the relations that must exist between the pressure, the weight, and the dimensions of the carriage under the condition of stability imposed. As the center of gravity of the system moves to the rear when the gun recoils on the carriage, the most unfavorable position of the center of gravity must be used in the equations. This will be the rearmost position. Design of a Field Carriage to Fulfil the above Conditions. —The weight of the system composed of the gun and gun carriage must be such that when the weight of the limber filled with ammunition is added, the weight behind each horse of the team shall not exceed 650 pounds. The length of the trail will be limited by considerations of draft and of the turning angle of the limbered carriage. The height of the carriage must be such that the gun may be readily served and not too easily overturned. The area of the spade must be such that the pressure against it will not exceed 80 pounds per Square inch, as it is found that in average ground the Spade will not satisfactorily prevent movement of the car- riage when the pressure against the Spade exceeds this limit. By carefully weighing these and other considerations, and assuming successive values for the various quantities in the equations of condition established in the problem preceding, satisfactory dimensions for the carriage as a whole are finally determined. Similar equations are established for each of the individual parts of the carriage in exactly the same manner as explained for the carriage as a whole. The stresses to which each part is subjected, and the necessary strength and best form of the part to perform its functions are thus determined. The pressure P determined from the above equations is the greatest pressure that may be transmitted to the carriage under the condition of stability imposed. The 3-inch gun recoils on its carriage and the recoil is controlled by a hy- draulic brake and counter recoil springs. If we neglect the friction of the moving parts, P becomes at once the maximum constant pressure that may be permitted in the recoil con- trolling system. We will then determine, as explained under 37 hydraulic brakes, the length of the recoil when opposed by this pressure, and the length so determined will be the mini- mum length of recoil that may be permitted on the carriage. 3-inch Field Carriage Recoil System.—A longitudinal sec- tion through the gun recoil system of the 3-inch field carriage is shown in Fig. 10, drawn to a distorted scale in Order to show the parts more clearly. m in A GUN III: —º m--|||||||||l |f|H- | *—s _2? Lārī. _b ] r F ŠE. | -I-T-9 º J R |||||||| - |||||||||v |V-NJ--p L- Fig. 10. A cylindrical cradle d, of cross-section as shown in Fig. 11, is pintled by the pintle p in a part of the carriage called the - rocker, not shown. The grooves a of the pintle are engaged by clips provided on the rocker. The rocker embraces the axle of the carriage and has a movement in elevation which is transmitted to the gun by the cradle. The gun is provided with clips k which engage the upper flanges of the cradle; and when fired, the gun slides to the rear on the upper surface of the cradle. The lug l, Fig. 10, is an integral part of the gun. The counter recoil buffer at is attached to the lug by a bolt t, and the recoil cylinder C is attach- ed to the same bolt by means of Fig. 11. the screw v. Integral with the walls of the cylinder are the \ 38 three throttling bars 0. The piston head s is provided with three corresponding apertures, Fig. 11. The hollow piston rod r, made in two parts rigidly joined together, is attached to the front end of the cradle by the bolt b. The rod terminates at its rear end in the piston head S. The collar f on front end of cylinder receives the thrust of the counter recoil springs m transmitted through the annular spring support 17, which also serves to center the cyl- inder in recoil. The gun in recoiling draws with it, by means of the lug l, the recoil cylinder c and the counter recoil buffer M. The piston, attached to the cradle, does not move. When the forward end e of the curve of the throttling bar reaches the piston head S, the apertures in the piston are completely closed against the flow of the liquid, and recoil ceases. The counter recoil buffer M has now been drawn all the way out of the piston rod. - - . Under the action of the springs m, which have been com- pressed by the recoil, the gun returns to battery. The first part of the counter recoil, during which the counter recoil buffer is out of the hollow piston rod, is unobstructed. When the buffer enters the piston rod the escape of oil from inside the rod is permitted only through the narrow clearance between the rod and the buffer. The resistance thus offered gradually diminishes the velocity of counter recoil, and brings the gun to rest without shock as it comes into battery. The buffer is cylindrical for the greater part of its length, with a clearance in the piston rod of 0.025 of an inch on the diameter. The diameter of the buffer gradually enlarges over a length of three inches at the rear until the clearance is but 1/1000 of an inch on the diameter. The pressure on the piston due to the recoil is transmitted through the cradle to the pintle p, and thence to the carriage. The length of recoil is 45 inches. Recoil System of other Carriages.—The recoil-controlling parts of the carriages for siege guns, and of the barbette car- riages for seacoast guns six inches or less in caliber, embody the same principles as the system described above. CHAPTER IX. Artillery of the United States Land Service. Service artillery may be broadly divided into two classes: mobile artillery and artillery of position. Mobile artillery consists of the guns designed to accompany or to follow armies into the field, and comprises mountain, field, and siege artillery. Artillery of position consists of the guns permanently mounted in fortifications. As the fortifications of the United States are all located on the seacoasts, the guns that form their armament are usually designated Seacoast guns. Mobile Artillery.—The mobile artillery of the United States as at present designed will consist of the following gun S: Gun. Caliber. Projectile. Mountain gun 2.95 inch 18 lbs. Light field gun 2.38 inch 73 lbs. Field gun 3.0 inch 15 lbs. Field howitzer 3.8 inch 30 lbs. Heavy field gun 3.8 inch 30 lbs. Heavy field howitzer 4.7 inch 60 lbs. Siege gun , - 4.7 inch 60 lbs. Siege howitzer 6.0 inch 120 lbs. The selection of these calibers is based on the following principles. The field gun, the principal weapon of an army in the field, must have sufficient mobility to enable it to ac- company the rapidly moving columns of the army. Long experience indicates that to attain the desired degree of mobility the weight behind each horse of the team should not exceed 650 pounds. A six horse team is used with the field gun. The total weight of the gun, carriage, limber, and equipment, with a suitable quantity of ammunition is there- fore limited to 3900 pounds. Limited by this requirement 39 40 the power of the gun should be as great as it can be made. The shrapnel being the most important projectile of the field gun the caliber of the gun should be such as to give the shrapnel the greatest efficiency. Consideration of these requirements has led to the adoption of the 3-inch caliber for the field gun of our service. A gun of greater power will, on those occasions when it can be brought into action, be more effective than the 3-inch gun. The heavy 3.8-inch field gun, firing a 30-pound projectile and possessing sufficient mobility to enable it to accompany the slower moving columns of the army, is therefore provided. The weight behind the six horse team is limited to 4800 pounds. With this weight the gun is capable of rapid move- ment for short distances. - - The caliber of the siege gun is limited by the requirement that the weight of the gun shall not exceed the draft power of an 8-horse team. The draft power of this team, for the siege gun, is taken as 8000 pounds. Allowing for bad roads and rough usage and for the occasional necessity of covering considerable distances at high speed, the draft power of a horse for artillery purposes is taken as considerably less than the draft power of the horse used in ordinary commerce. . The guns above named are intended for the attack of targets that can be reached by direct fire, that is, by fire at angles of elevation not exceeding 15 degrees. For the attack of targets that are protected against direct fire and for use in positions so sheltered that direct fire can not be utilized, curved fire, that is, fire at elevations exceeding 15 degrees, is necessary. There is therefore provided, corresponding to each caliber of gun, a howitzer of an equal degree of mobility. The howitzer is a short gun designed and mounted to fire at comparatively large angles of elevation. . In order to reduce to the minimum the number of calibers of the mobile artillery and thus simplify as far as possible the supply of ammunition in the field, the calibers of the guns and howitzers have been so selected that, while both guns and howitzers fulfil the requirements as to weight and 41 power for each degree of mobility, the caliber of each howitzer is the same as that of the gun of the next lower degree of mobility. That is, the howitzer corresponding in mobility to one of the guns is of the same caliber as the next heavier gun, and uses the same ammunition. As there may be occasions when profitable use can be made of a gun throwing a lighter projectile than that of the 3-inch field gun, the light field gun, 2.38-inch caliber, is pro- vided. The weight of the projectile is 7% pounds, this weight being considered the lowest limit for an efficient shrapnel. Advantages of Recent Carriages.—The chief difference be- tween the latest and earlier designs of carriages for mobile artillery lies in the provision made in the later carriages for recoil of the gun on the carriage. By this means a part of the force produced by the discharge is absorbed in controll- ing the recoil of the gun on the carriage, leaving only a part available to produce motion of the carriage; and by the addition to the end of the trail of a spade which is sunk in the ground, the carriage is enabled to withstand the transmit- ted force without motion to the rear. When the spade is once fixed firmly in the earth further firing of the gun does not produce recoil of the carriage. Rapidity offire is thereby greatly increased, and the soldier is relieved from the fa- tiguing labor of running the carriage back into battery after each round. Rapidity of fire is also increased by the use of fixed am- munition, and by the provision for a slight movement in azimuth of the gun on the carriage. The movement in azimuth permits a change in the pointing of the gun of three or four degrees to either side without disturbing the car- riage after the spade is set in the ground. In addition, the gun sights on all modern constructions are fixed to some non-recoiling part of the carriage so that they are not affected by the recoil. The operation of sighting may therefore go on continuously, independently of the loading and firing. 42 Our service field and siege carriages, with the exception of the 6-inch siege howitzer carriage, are so designed that the wheels will not be lifted from the ground under firings at zero elevation. The Mountain Gun.—For mountain service the System composed of gun and carriage must be capable of rapid dis- mantling into parts, no one of which will form too heavy a load for a pack mule. The weight of the load, including the saddle and equipment of the mule, should not exceed 350 pounds. The system must be capable of rapid reassembling for action. - The mountain gun used in our service, originally made by Vickers Sons and Maxim of England, has a caliber of 75 mil- limeters, or 2.95 inches, and fires projectiles weighing 12% and 18 pounds. The caliber of this piece will probably soon be changed to 3 inches so that it may use the same ammuni- tion as the 3-inch field gun. - The gun is made from a single forging, and weighs com- plete with breech mechanism 236 pounds. Fixed ammuni- tion is used in it. The breech mechanism, Fig. 1, is of the interrupted screw type. The block has two threaded sectors separated by flat surfaces. It is provided with percussion firing mechanism so arranged that the gun cannot be fired until the breech block is fully closed and locked. The trigger to which the firing lanyard is attached is seen to the left in the figure outside the breech. In case of a misfire the mech- anism may be recocked without opening the breech. The Carriage.—A low wheeled carriage is provided for the mountain gun. The wheels are 36 inches in diameter and have a track of 32 inches. The principal parts of the carriage are the cradle, the trail and elevating gear, the wheels and N axle. *- THE CRADLE.—The cradle is a bronze casting, with a cen:- - - tral cylindrical bore and a smaller cylinder on each side. ~~~~ The central cylinder embraces the gun to within a few inches of the muzzle and forms a support in which the gun slides in recoil. The side cylinders are hydraulic buffers the piston rods of which are secured to lugs On the gun by interrupted inch Mountain Gun. .95– |- | . || || \,\！ |× Transport of Trail. 43 screws so that the gun may be readily separated from the cradle. Grooves of varying width and depth cut in the in- terior walls of the buffer cylinders allow passage of oil from one side of the piston to the other in recoil. Constant pres- sure is maintained in the cylinder throughout the length of recoil, 14 inches. Spiral springs surrounding the piston rods . return the gun to battery. - The cradle is secured to the trail by a bolt, seen above the axle in Fig. 1, which passes through two lugs formed on the underside of the cradle, the Outer ends of the bolt fitting into two bearings or sockets provided at the forward upper end of the trail. The cradle moves in elevation about this bolt. Light lifting bars are provided for use in dismantling and assembling the gun and carriage. They are passed through the two eye bolts on the top of the cradle, and through one On the gun. Front and rear sights are attached to the cradle. The rear tangent sight is detachable. THE TRAIL.-The trail consists of two outside plates or flasks of steel joined together by a shoe and three transoms. The shoe is provided with a spade on the underside to assist in checking recoil, and with a socket on the upper side, in which a handspike may be fitted, or the shafts attached when travelling on wheels. At the front end of the trail are the bearings for the cradle bolt and further to the rear are bear- ings for the axle. The bearings are Open at the top, Fig. 3, the openings having a width less than the diameter of the bearing. The cradle bolt and axle tree are cylindrical, with flats cut on them so that they can only Fig. 3. enter their bearings at a certain angle. When in position in the bearings they are turned through 90 degrees and thus secured. The crank secured to the axle at the right is for the purpose of turning the axle, in disman- tling the carriage, to bring the flats of the axle in line with the openings of the bearings. When assembled the axle is locked in position by a spring latch bolt in the crank handle which engages in a slot provided in the trail. - 44 rº-rºº THE ELEVATING GEAR.—The | elevating gear is permanently attached to the trail. Motion of the hand wheel is communi- cated to the gun through bevel gears, b Fig. 4, a worm w, and a toothed quadrant q at- tached at its rear end to the cradle. An arm formed on the forward end of the quad- rant embraces the cradle bolt and revolves around it. A cross bar c on each side near the upper end of the arm keeps the quadrant in a central position, and two spiral Springs fast- ened to the front transom and acting on the arm maintain practically a uniform weight on the elevating gear while the gun is being elevated or depressed. The gun may move in elevation from minus 10 degrees to plus 27 degrees. Bºig. 4. Ammunition.—Fixed ammunition is used. The charge is about 8 ounces of smokeless powder. The 110-grain percus- sion primer is used in the cartridge case and a front igniter of about 3% ounce of black rifle powder. Three kinds of pro- jectiles are provided: canister, shrapnel, and high explosive shell. The canister and shrapnel weigh 12% lbs., the high explosive shell 18 lbs. The canister contains 244 cast-iron balls each 3% of an inch in diameter. The shrapnel contains 234 balls. The bursting charge for the shell is 2.07 lbs. of high explosive. The muzzle velocity of the 12%-lb. projectile is 850 feet. The maximum pressure in the bore is 18000 lbs. The gun has an effective range of about 4000 yards. Transportation.—For purposes of transportation the gun and carriage, with tools, implements, and equipments, are divided into four loads, the principal items of which are the gun, the cradle, the trail, the wheels and axle. These loads, 45 without the pack equipment, weigh approximately 250 lbs. each. The pack saddle and equipment weighs 90 lbs., so that the total weight carried by the mule is about 340 lbs. The trail, which forms the most inconvenient load, is shown in Fig. 2, loaded on the pack animal. The ammunition is carried in nine loads of 10 or 12 rounds each, according as the projectiles weigh 18 or 12% lbs. A box holding 5 or 6 rounds, is slung on hooks on each side of the pack saddle by loops formed in wire straps about the box. The boxes open at the end so that the ammunition may be removed from them without disturbing the pack. Field Artillery.—The field artillery as at present designed, will consist of the 2.38-inch gun, the 3-inch gun, the 3.8-inch gun, and the 3.8-inch and 4.7-inch howitzers. It is also the intention to modify the carriage of the mountain gun so that the piece may be fired at high angles of elevation, and be used as a light field howitzer. The caliber of the gun will then be changed to 3 inches so that the projectiles of the 3-inch field gun may be used in it. There is also at present in service a 3.6-inch field mortar. Fixed ammunition is used in all field pieces except the mortar. - The following table contains data relating to the guns and carriages of the field artillery: - Mor- Guns. Howitzers. º Caliber, inches................................ | 2.38 || 3 || 3.8 3.8 4.7 3.6 Date of model.............. * e º e º e º 'º º ſº e º ſº tº e º 'º ſº º | 1905 1905 | 1905 1906 1890 Charge, lbs.................................... ().77 | 1.87 3 | — — 1 Projectile, lbs... .............................. 7.5 15 30 30 60 20 Bursting charge, lbs....................... — 0.82 2.1 2.1 3 || 4 Oz. Cartridge, complete, lbs................. 9.5 | 18.75 |2,38 — — Shrapnel balls, number.................. 118 252 *ś 526 1063 | — Muzzle velocity, f. s....................... 1700 || 1700 || 1700 900 900 | 660 Maximum pressure, lbs................... 33000 33000 33000 23000 T35000 |17000 Weight, limbered, lbs..................... 2400 3900 || 4800 3900 || 4800 K- AT MAXIMUM EL EVATION. : ; N Elevation, degrees.......................... 15 15 15 || 45 45 45 Time of flight, sec.......................... 19.4 21.9 21 |#36.3 || 37.4 21.2 Remaining velocity, f. s.......... * - - - - - - - 664 | 737 769 || 707 || 752 || 515 Range, yards.................................. 5800 6100 .6900 $300 | 6850 || 3360 * -a < ** *. 46 The velocities and pressures are fixed at the low figures given in the table in order that the guns and carriages may be kept within the limits as to weight. With velocities of 400 feet the service shrapnel balls are effective against men, and with velocities of 880 feet, against animals. As the velocity of the balls is increased by from 250 to 300 feet at the bursting of the shrapnel, it will be seen from the table that shrapnel fire from the field pieces is effective at all ranges. The designs of the field guns of different caliber, with their mounts, differ practically only in the size of the parts. A description of one will therefore answer for all. The 3-inch Field Gun.—The 3-inch field gun is the princi- pal weapon of the field artillery. The gun, of nickel steel, is built up in the manner described on page 59, Part II. A hoop called the clip is shrunk on near the muzzle. On the under side of this hoop, and of the locking hoop and jacket, % s º | b § §§§ º #.S º § s n º Fig. 5. 47 are formed clips, k Fig. 8, which embrace the guide rails of the cradle of the carriage. The gun slides in recoil on the upper surface of the cradle. A downwardly extending lug, l Figs. 7 and 8, at the rear of the jacket serves for the attach- ment of the recoil cylinder, which moves with the gun in recoil. - THE BREECH MECHANISM.—The breech mechanism, model 1904, is shown in Fig. 5, in the locked position. The mechan- ism is of the slotted screw type. The breech block b is cylindrical with four threaded and four slotted sectors. It is mounted on a hollow spindle s formed on the carrier c, to which it is held by the lug n which engages in a slot cut in the enlarged base of the spindle. On a semi-circular boss formed on the rear face of the block is cut a toothed rack, outlined at 2 Fig. 8. The teeth of a bevel pinion formed on the inner end of the operating lever g mesh in the teeth of the rack. The lever is pivoted on a pin which passes through two lugs formed on the rear face of the carrier. On grasping the handle of the lever the pressure against a latch t in the handle unlocks the lever from the face of the breech. Swinging the lever to the rear rotates the block until it is stopped by a lug inside the carrier and locked in position by the spring stud a. Further movement of the lever causes both block and carrier to rotate together about the hinge pin h. When the movement is nearly complete the surface o of the carrier bears against the arm of the extractor lever y which causes the extractor a to move sharply to the rear and eject the empty cartridge case. 48 THE FIRING MECHANISM.–The firing mechanism, Fig. 6, º ºrm, ſº f &4a |, ºw º * - O º IFig. 6. is contained in the firing lock case f which is inserted into the hollow spindle from the rear, the interrupted lugs d on the lock case engaging behind corresponding interrupted lugs c on the carrier. Assembled in the lock case are the firing pin p, the spiral firing spring, the firing pin sleeve w, and the trigger fork v, the latter fitting over the squared end of the trigger shaft h which is journaled in an arm of the lock case extending downward and to the right outside the carrier, f Fig. 8. At the lower end of the trigger shaft h, Fig. 8, are two levers at right angles to each other, one marked trigger pro- vided with an eye for the hook of the lanyard, the other acted upon by an upwardly extending lug on the end of the firing lever shaft. A narrow section of the forward end of the lock case, Fig. 6, is cut out for the flat sear spring r. A notch in the sear engages the shoulder formed on the firing pin. The sleeve w at its rear end bears upon the last coil of the firing pin spring. When the trigger shaft h is turned by a pull on the lanyard, or by means of the firing lever, the trigger fork v forces the sleeve w to the front, compressing the firing spring. The forward end of the sleeve pushes the sear Spring aside from its engagement on shoulder of firing pin, and the compressed spring then drives the firing pin forcibly : 49 forward until arrested by the shoulder striking the inner Sur- face of the spindle. When the pull on the lanyard has ceased, the firing spring, still compressed, exerts a pressure against the rear end of the sleeve w, thence on the fork v, and on the head o of the firing pin; and the construction of these parts is such that the spring can regain its extended length only when the parts are in the position shown in the figure. The firing pin is therefore immediately withdrawn, on the cessation of the lanyard pull, until caught again by the Sear. - The system of cocking and firing the piece by one move- ment is called the continuous pull system. The firing spring is compressed only at the moment of firing, whereas in the mechanism that is cocked in opening the breech the firing spring is compressed whenever the breech is opened and may remain compressed for a long time. SAFETY DEVICES.–Safety against discharge before the breech is fully closed is secured as follows. The axis of the Spindle S on the carrier, Fig. 5, lies #6 of an inch below and # of an inch to the right of the axis of the gun. The breech block which revolves on this spindle is therefore eccentric with the bore. The firing mechanism is eccentric with the block, the axis of the firing mechanism being fixed in the axis of the bore. When the block is locked the hole in its front end through which the firing pin protrudes in firing is also in the axis of the bore, but as the block is rotated in opening, the hole rotates out of the axis of the bore and the flat surface at its rear end comes in front of the firing pin, and prevents move- ment of the firing pin until the breech is locked. The headed spring pin u, Fig. 8, enters a hole in the car- ner and retains the firing mechanism in its position in the carrier. By withdrawing this pin and rotating the firing lock case fºupward through 45 degrees the interrupted lugs, d Fig. 6, on the firing lock case disengage from behind the interrupted lugs c on the carrier, and the firing mechanism may be withdrawn from the gun. The breech block is then readily removed. The breech mechan- 50 ism may thus, without the use of tools, be readily dismantled for repair, and the gun quickly disabled in the event of im- minent capture. Four holes are drilled rearwardly through the breech block, b Fig. 5, to permit the escape of gas without injury to the screw threads of the mechanism in case the primer in the cartridge is punctured by the blow of the firing pin. THE 3-INCH GUN, MoDEL 1905.-The 3-inch gun model 1905 is 50 lbs. lighter than the 1902 and 1904 models, the outside diameters being slightly diminished. The twist of the rifling, which in the earlier models increases from 1 turn in 50 calibers at the breech to 1 in 25 at the muzzle, increases from zero at the breech to 1 in 25 at # inches from the muzzle, from which point it is uniform to the muzzle. The purpose of the change in twist is to dimin- ish the resistance encountered by the projectile in the first part of its movement and thereby diminish the maximum pressure. The short length of uniform twist at the muzzle steadies the projectile as it issues from the bore. The Carriage.—The principal parts of the carriage are the cradle, the rocker, the trail, the wheels and axle. THE CRADLE.—The cradle, C Figs. 7 and 8, is a long steel cylinder, which contains the recoil controlling parts. These parts are fully described in the chapter on recoil, and illustrated in Figs. 10 and 11 of that chapter. The gun slides in recoil on the upper surface of the cradle, the clips of the gun, k Fig. 8, engaging the flanged edges. A pintle plate fastened to the bottom of the cradle is provided with the pintle p, Fig. 8, and the grooved arc a, which serve to connect the cradle to the rocker. THE ROCKER.—The rocker r embraces the axle between the flasks of the trail by the bearings at its ends. The cradle pintle fits in a seat provided in the rocker above the axle, and the clips on the rocker engage in the grooved are a of the cradle. This construction permits movement of the cradle and gun in azimuth on the rocker, while the rocker itself revolves about the axle and thus gives movement in elevation ----- l- （---- dz 3 >)92 Cz |- | „) º+→. *L • ~&ŹŃ， ∞2323&NS-ZÁT * ~, ， …«*|- №ſký �\&]} f= Fi g NSI Ó) FIRING LEveR ~8） ） Š €”— 52 to the cradle and the gun. The movement in azimuth, 4 degrees either way, is produced by a screw on the shaft of the hand wheel t, Fig. 7. The shaft is fixed in bearings on the rocker arms and the screw works in a nut pivoted in a bracket fastened under the cradle. The double elevating screw, actuated by either of the crank shafts e fixed in bear-- ings in the trail, rotates the rocker and cradle about the axle. The bevel pinion on the end of each shaft e rotates the bevel pinion b in its bearings. The pinion b is splined to the outer screw m and causes the Outer screw to turn in the fixed nut q which is supported below the pinion b by a transom. The Outer screw ºn has a left handed thread on the exterior and a right handed thread in the interior. When turned it travels up or down in the nut q, and at the same time causes the inner screw m to move into or out of the Outer screw, the inner screw being prevented from turning by its connection with the rocker arms, r Fig. 7. The movement of the inner screw for each turn of the pinion b is thus equal to the sum of the pitches of the Outer and inner screws. THE TRAIL.—The trail, composed of two flanged steel flasks connected by transoms, terminates at its lower end in a fixed spade provided with a float or wings which prevent excessive burying of the spade in the ground. The lower edge of the spade is of hardened steel riveted on so that it may be readily replaced when worn out. The lunette, a stout eye bolt fixed in the end of the trail, engages over the pintle of the limber when the carriages are connected for travelling. Seats for two cannoneers who serve the piece in action are attached to the trail one on either side near the breech of the piece; and two other seats on the axle, facing toward the muzzle, are occupied in travelling by two can- noneers, one of whom manipulates the lever of the wheel brakes. THE WHEELS AND AXLE.—The axle of forged steel is hollow. The axle arms are given a set so as to bring the Iowest spoke of each wheel vertical. 53 The wheels are a modified –pr form of the Archibald pattern, 56inches in diameter with 3-inch Itº . tires. The hub, Fig. 9, consists of a steel hub box h and hub ring rassembled by bolts through the flanges, between which the . | + * spokes of the wheel are tightly - l clamped. The hub box is lined - with a bronze liner forced in. A Fig. 9. steel cap c is screwed on the outer end of the hub box. Riveted to the cap is a self closing oil valve, by means of which the wheels are oiled without removal from the axle. The hollow axle forms a reservoir for the oil. - The wheels are secured to the axle by the wheel fastening, a bronze split ring, hinged for assembling around the axle. The ring revolves freely in a groove in the axle. Interrupted lugs on its exterior engage behind corresponding interrupted lugs, l Fig. 9, in the inner end of hub box, and hold the Wheel on the axle. A hasp connects the hub and the wheel fasten- ing so that they cannot revolve independently and disengage the lugs. THE SHIELD.—The cannoneers serving the piece are pro- tected by a shield of hardened steel # of an inch thick. It is in three parts. One part, the apron, depends from the axle and is swung up forward under the cannoneers’ seats when travelling. The main shield, rigidly attached to the frame of the carriage, extends upwards from the axle to 24 inches below the tops of the wheels. The top shield is hinged to the main shield. When raised its upper edge is 62 inches from the ground, a height sufficient to afford protection from long range and high angle fire to cannoneers on the trail seats. In travelling the top shield is folded over so that should the carriage turn over on the march the shield is par- tially protected from injury. Each shield before being attached to the carriage is tested at a range of 100 yards with a bullet from the service rifle. The plate must not be perfo- rated, cracked, broken, or materially deformed in the test. → pº 54 SIGHTS.—The piece is provided with three different means of sighting. Two fixed sights on the upper element of the gun, Fig. 7, determine a line of sight parallel to the axis, for use in giving general direction to the piece. For more accu- rate sighting a tangent rear sight and a front sight with crossed wires are provided. They are seated in brackets attached to the cradle. A telescopic panoramic sight is seated on the stem of the tangent sight. This sight is used principally for indirect aiming, that is, pointing the gun by means of a line of sight considerably divergent from the line of fire. By means of the panoramic sight any object in view from the gun may be used as an aiming point. The sights are fully described in the chapter on sights. The Limber.—The limber is practically wholly of metal, the neck yoke and pole, and spokes and felloes of the wheels, being the only wooden parts. The body of the limber is a steel frame, composed of three rails riveted to lugs formed on the axle, and braced by steel tie rods. The middle rail is in the form of a split cylinder, one half passing below the axle and the other above. The halves unite in front forming a socket for the pole which is held firmly in place by a clamp. Similarly in the rear the middle rail forms a seat for the pin- tle hook. The pintle hook is swiveled in its seat, so that if at any time the gun carriage turns over the pintle will turn without overturning the limber as well. The ammunition chest, of sheet steel, is fastened to the rails. The front of the chest and the door which forms the rear are strengthened by vertical corrugations. The door Opens downward and is then supported by chains. The metallic ammunition is supported in the chest by three diaphragms each perforated with 39 holes. The middle and rear diaphragms are connected by flanged brass tubes cut away on top to reduce the weight. The tubes support the front ends of the cartridge cases and enable blank ammuni- tion and empty cases to be carried. Seats made of sheet steel are provided for three cannoneers on the limber chest, and a steel foot plate rests on the rails in front of the chest. - tr 3-inch Field Gun, Model 1902. | * - - - W - - - _ 3-inch Field Gun, Limbered. ‘. : Fig. 10. 3-inch Field Caisson. Fig. 11. 3-inch Field Battery Wagon. Tiso". I º R-in rºl, Tial, 1 Sł, , , , , \\"o or, , , , , is ... ' ' - | : * s - & * * - d 55 The wheels of the limber and the wheels of all other car- riages that form part of a field battery are interchangeable with the wheels of the gun carriages. The Caisson and other Wagons.—The construction of the caisson, Fig. 10, does not differ materially from that of the limber. The ammunition chest is larger and carries 70 rounds of ammunition. The front of the chest is of armor plate # of an inch thick; and the door at the rear, which opens upward to an angle of about 30 degrees above the hori- zontal, is of armor plate Hº of an inch thick. A ſº-inch plate also depends from the axle as in the gun carriage. The can- noneers serving the caisson are thus afforded protection for a height of 63 inches from the ground. Attached to the caisson by a hinged bracket at the rear is an automatic fuze setter, by means of which the cannoneer at the caisson may quickly set the fuze of the projectile to the time of burning corresponding to any range Ordered by the battery commander. Three caissons with their limbers accompany each gun into the field. The wagons of a battery include also the forge limber, which as its name indicates, carries a blacksmith’s forge and set of tools; and the battery wagon, Fig. 11, which carries carpenter’s and saddler’s tools and supplies; materials for cleaning and preservation; Spare parts of gun, of carriage, and of harness; tools and implements; miscellaneous sup- plies and two spare wheels. A wagon called the store wagon, Fig. 12, is for use in carrying such stores, spare parts, and materials as cannot be carried in the battery wagon. Experiments are now being conducted toward the develop- ment of an automobile battery Wagon. Field Howitzers and Mortars.-The 3.8-inch and 4.7-inch field howitzers have not yet been constructed. The principles of construction of the guns and carriages will be understood from the description of the 6-inch howitzer and carriage which follows later. i. Jº 56 There is at present in service a 3.6-inch field mortar shown ZZZZZZZZZZ amº S. %: ū N º- ºrſº Vox 3-ºx: ºs- LZ sº - * - - • * * sº -: :Sassºr-º- - - mºs - - m - - * *-* * - * - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ** -, as º § P. º sº §§ º ºº : in Fig. 13. The piece is a short gun intended for vertical fire against troops protected by intrenchments or other shelter. The Freyre obturator described on page 82, Part II, is used in the breech mechanism to save weight. The gun weighs 245 lbs. and its mount 300 lbs. more, so that the gun with its mount may be readily moved in the field. The mount is a single steel casting. The gun is held at any desired eleva- tion by means of a clamp which acts on a steel arc attached to the under side of the gun. When in use the carriage rests on a wooden platform, and recoil is checked by a heavy rope attached to stakes driven into the ground in front. Siege Artillery.—The new siege artillery comprises the 4.7-inch gun and the 6-inch howitzer. The older siege pieces now in service are the 5-inch gun, the 7-inch howitzer and the 7-inch mortar. - The following table contains data relating to the guns and carriages of the siege artillery: 2^ ... ? - Mor- | Guns. Howitzers. tar. Caliber, inches................................ 4.7 5 6 7 7 Date of model.............. • * * * * * * * * * * * * * * * e º ſº 1904 || 1898 || 1905 1898 1892 Charge, lbs....................................| 6 || 5.37 4.6 4.0 Projectile, lbs... .............................. 60 wº ; º 3. 0 ( % #: - Bursting charge, lbs....................... 3. • *y 1. ... Tºſ. & | 12.5 Cartridge complete, lbs................. 73% 1373 3,3 p 7.4 It ºf Shrapnel balls, number.....,............ 1063 — 2150 — — Muzzle velocity, f. S....................... 1700 1830 900 1100 800 - . . . Maximum pressure, lbs.................. ..]33000 |35000 15000 |28000 |28000 ºf Z ( : Weight, limbered, lbs..................... 8000 | 8800 7900 — — AT MAXIMUM ELEVATION. Elevation, degrees .......................... 15 31 45 35 45 Time of flight, seconds................ ... 21.8 38.2 37.5 34.3 || 32.9 Remaining velocity, f. S.................. 971 638 764 || 749 || 641 Range, yards.................................. 7600 10000 7000 || 7700 || 5200 The 4.7-inch Siege Gun.—The gun is similar in construc- tion and in breech mechanism to the 3-inch field gun. Fixed ammunition is used in it. THE CARRIAGE.-The carriage is, in general, similar in con- struction to the 3-inch field carriage. The greater weight of the gun and the increased force of recoil render necessary certain changes in the parts. In the 3-inch carriage the recoil cylinder and counter recoil springs are assembled together in a single cylinder in the cradle. The cradle of the 4.7-inch carriage, Figs. 14, 15 and 16, consists of three steel cyl- inders bound together by broad steel bands, the middle band provided with trunnions. The middle cylinder contains the mechanism for the hydraulic control of recoil. Each of the Outer cylinders contain three concentric columns of coiled Springs for returning the gun to battery. The front end of each of the Outer two spring columns is connected to the rear end of the next inner column by a steel tube, flanged outwardly at the front end and inwardly at the rear end. A headed rod passes through the center of the inner coil and is fixed to a yoke that is fastened to the lug at the breech of the gun, see Fig. 15. The head of the rod acts on the inner coil only, and the pressure is transmitted through the flanged tubes or stirrups to the outer coils. In this way the springs work in tandem and have a long stroke with short assembled length. The length of recoil is 66 inches. 58 The gun is supported, and slides in recoil, on rails r fixed on top of the spring cylinders. The distance apart of the rails broadens the bearing of the gun and gives it steadiness both in action and in transportation. An extension piece, bolted to the front end of the cradle and readily detachable, continues the rails to the front clip of the gun. When trav- elling this extension piece is detached and carried in fasten- ings under the trail. * * THE PINTLE YOKE.—The cradle is trunnioned in a part called the pintle yoke, y Fig. 14, which is itself pintled in a seat p, called the pintle bearing, mounted between the for- ward ends of the trail flasks, its rear end embracing the hollow axle a. A traversing bracket b is attached to the bottom of the pintle yoke and extending to the rear under the axle forms a support for the traversing shaft t and for the elevating mechanism. The rear end of the traversing bracket slides on supporting transoms between the flasks of the trail, motion being given to the bracket by means of i ruo!!!soa ºu! Iº Abu JL ºu nº ºžaſs qouſ-1 * F * 9 T · 5 IJI 59 a screw on the traversing shaft which works in a nut suitably attached to the trail. The gun may be moved in azimuth on the carriage 4 degrees either way. The elevating mechanism is carried on the traversing bracket and moves with the gun in azimuth. It is therefore not subjected to any cross strains. The gun may be moved in elevation from minus 5 to plus 15 degrees. •. THE WHEELS AND THE TRAIL.-The wheels are 60 inches in diameter with 5-inch tires. Exhaustive tests recently con- cluded indicate that no practical advantage is gained by the use of wider tires on vehicles of this class and weight. - The trail is of the usual construction, two pressed steel flasks of channel section tied together by transoms and plates. The front ends of the flasks are riveted to cast-steel axle bearings which extend to the front of the axle and sup- port between them the pintle bearing p. The location of the pintle socket in front of the axle permits the use of a shorter trail and reduces the weight at end of trail to be lifted in limbering. * Bearings are provided at about the middle of the trail, in the opening seen in Fig. 15, for a detachable geared drum which is used in giving initial compression to the counter recoil springs in assembling, and in withdrawing the gun to its travelling position. When not in use the drum is kept in the tool-box in the trail. The spade with its horizontal floats is hinged to the trail on top. For travelling it is turned up and rests on top of the trail, see Fig. 16; for firing it is turned down. In either position it is locked in place by a heavy key bolt. A bored lunette plate is riveted to the bottom of the trail, for engagement on the pintle of the limber. The Limber.—The limber, Fig. 17, is merely a wheeled turntable for the support of the end of the trail in travelling. It has the usual arrangements for the attachment of the team. Its wheels are interchangeable with those of the carriage. The turntable, shaped to fit the end of the trail, is mounted on a frame fixed to the axle. It forms a seat for the trail. The 60 4. A ! & º .* - - § G _, 2 T ... --> - ~~~ gº?” ~~~~~ 5.2. __2- - - → *. --~~~ * * n 2 S. 2 }% º & % 2×zº: º afº ºzºº 2:7; rºzzº%3 & yº. SVZ SS %:7,777% S N ess" º Q sº º N N assº" º S N N ~ -- ~ T § à i º à 2Z § & º º f § Z Ż ZT º º ZU * * s * * * r e L”* : § * - * Fo l sº N S Ly Sls, sº I º jº Fig. 17. % º tº º seat is pivoted at the rear end and its front end rests on rollers which travel on a circular path on the limber. A pintle on the seat engages in the lunette in the bottom of the trail. - When travelling, in order to distribute the weight as evenly as possible between the front and rear wheels of the limbered carriage, the gun is disconnected from the piston rod and spring rods, and drawn back 40 inches to the rear, Fig. 16. In this position the recoil lug is secured between two stout braces attached to a heavy trail transom. The breech of the gun is thus supported and rigidly held in travelling, and the elevating and traversing mechanisms are relieved from all strains. The braces referred to are pivoted in the trail, and when not in use are turned down inside the trail. Weights.-The weight of the gun carriage complete is 4440 lbs., and that of the gun and carriage, 7170 lbs. The weight at the end of the trail, gun in firing position, or the weight to be lifted in limbering, is 400 lbs. ; with the gun in travelling position, this is increased to 1150 lbs., which is the part of the weight of the gun carriage sustained by the limber. - Siege Limber Caisson.—For the transportation of ammuni- tion for siege batteries there is provided a vehicle called the / 61 - 49 # \ siege limber caisson. As the name indicates this vehicle is composed of two parts. Each part supports an ammunition chest arranged to carry 28 rounds of 4.7-inch ammunition or 18 rounds of 6-inch ammunition, thus making 56 rounds of 4.7-inch ammunition or 36 rounds of 6-inch ammunition per vehicle. For each siege battery of 4 guns 16 limber caissons are provided. - The 6-inch Siege Howitzer.—This is a short piece, 13 calibers long, mounted on a wheeled carriage so constructed that the piece can be fired at angles of elevation from minus 5 to plus 45 degrees. This wide range of elevation on a Wheeled mount introduces into the carriage requirements not encountered in the construction of the carriages previously described, which provide for a maximum elevation of 15 degrees. - The piece is made from a single forging, Fig. 18. A lug l Fig. 18. extends upward from its breech end for the attachment of the recoil piston rod and the yoke for the rods of the spring cyl- inders. Flanged rails r formed above the piece support it on the cradle of the carriage, on which the piece slides in recoil. The Carriage.—The cradle, Figs. 19 and 20, is provided with recoil and spring cylinders. The gun is placed below the cylinders in order that the center of gravity of the system may be as low as possible. The trunnions of the cradle rest in beds in the top carriage which in turn rests on and is pintled in the part called the pintle bearing. Flanges on the top carriage engage under clips on the pintle bearing. The forward ends of the trail flasks are riveted to the pintle bear- ing, which forms a turntable on which the top carriage, and the parts supported by it, have a movement of three degrees ·* 6 I º ‘ēļJI �\j N **ş /^«^|× „^>><<� ^ `º„KN«> NN zºN }^} ^ < „ „a- @ a-, º�w^ ~）@ 2ST T-5ÑIHw38 3–1 LN13 `S， O`J ? ’NQ9* Nºnº) �„”_—T | X„?”O3 TQ \; >}O \ø |}，„z']，O |\\ | ,，,^}}|30|N|~1,\O 9N!}} dS |•O |- \ \,^|- ſ（\]）$',|-*>| 30NĮ TI „NO TIO OJÈJ|- 63 in azimuth to either side. The traversing is accomplished by means of the hand-wheel t on the left side. The traversing Fig. 20. shaft is supported in a bracket a fixed to the left flask, and its Worm works in a nut o pivoted to the top carriage. THE ROCKER.—The rear part of the rocker is a U-shaped piece that passes under the gun and is attached to the cradle by the hook k, pivoted in the cradle. Arms extend forward from the sides of the U and embrace the cradle trunnions between the cradle and the cheeks of the top carriage, so that the rocker may rotate about the cradle trunnions. The sights are seated on a bar supported on the left vertical arm 64 of the rocker. The upper end of the elevating screw n is attached to the bottom of the rocker, while the lower end of the screw and the elevating gear are supported by trunnions in lugs on the under side of the top carriage. The rocker therefore moves in elevation in the top carriage and gives elevation to the gun-supporting cradle fastened to the rocker by the hook k. The elevating apparatus is operated by a hand-wheel e on either side. THE TRAIL.-The flasks of the trail extend separately to the rear a sufficient distance to permit free movement between them of the gun in recoil at any elevation. They are then joined by transoms and top and bottom plates, and terminate in a detachable spade which is secured to the top of the trail when travelling. Sockets are provided for two handspikes at the end of the trail. Two lifting bars are also fixed to the trail. In order to permit the desired movement of the cradle in elevation the axle is in three parts, the middle part lower than the two axle arms. The three parts are held by shrink- age in cylinders formed in the sides of the pintle bearing. The wheel brakes, used both in firing and in travelling, are manipulated by hand-wheels b in front of the axle. RECOIL CONTROLLING SYSTEM.–The feature of this carriage which chiefly differentiates it from other carriages described is the provision for the automatic shortening of recoil as the elevation of the gun is increased. From minus 5 degrees to 0 elevation the gun has a recoil of 50 inches. As the eleva- tion increases from 0 to 25 degrees the length of recoil dimin- ishes continuously from 50 inches to 28 inches. For eleva- tions between 25 and 45 degrees the length of recoil remains at 28 inches. The variation in length of recoil is necessitated by the approach of the breech to the transoms and to the ground as the piece is elevated. - The piece is set low in the carriage to diminish as far as possible the overturning moment; but the maximum velocity of free recoil of this light piece is so great that stability of the carriage at all angles of elevation could not be obtained without exceeding the limit of weight and making the recoil unduly long. The carriage will be stable for angles of eleva- 65 tion greater than about 10 degrees. The wheels are expected to rise from the ground in firings at angles of elevation less than 10 degrees. - It will be observed that the operating lever of the breech mechanism of the gun is above the axis of the gun instead of below it as in other guns. It is so placed for the purpose of increasing the clearance in recoil and for convenience in Operating. - The automatic regulation of recoil is produced in the follow- ing manner. Four apertures are cut in the piston of the recoil cylinder, and two longitudinal throttling grooves in the walls of the cylinder. The total area of apertures and deepest section of the grooves is the proper maximum area of orifice for the 50-inch length of recoil, while the grooves alone fur- nish the proper continuous area of orifice for a recoil of 28 inches. A disk rotatably mounted on the piston rod against the front of the piston, and provided with apertures similar to those in the piston and similarly placed, is rotated on the piston rod during the recoil of the piece by two lugs project- ing into helical guide slots cut in the walls of the recoil cyl- inder. The rotating disk gradually closes the apertures in the piston, and the twist of the guiding slots is such that the area of orifice is varied as required for limiting to 50 inches the recoil of the gun when fired at 0 elevation. The recoil cylinder is rotatably mounted in the cradle. Teeth cut on its outer surface, Fig. 21, mesh in the teeth of a ring surrounding the right spring cylinder, and the teeth of the ring also mesh, at any elevation between 0 and 25 degrees, in a spiral gear cut on the cylindrical _2% sº block 8 fast to the right cheek of º the top carriage. As the gun is %*== | elevated from 0 to 25 degrees 2 = the spiral teeth of the gear cause 22.22% 2 jº the ring to rotate clockwise and tº the cylinder counter clockwise. Fig. 21. The rotating recoil cylinder car- 66 ries with it the disk in front of the piston, causing the disk to close the piston apertures more and more until at 25 de- grees elevation they are completely closed. The throttling grooves in the walls of the cylinder then provide the proper area of orifice for the 28-inch length of recoil permitted to the gun at elevations between 25 and 45 degrees. LoADING Position.—To load the piece after firing at high angles the hook k which holds the cradle to the rocker is disen- gaged by means of a handle h conveniently placed on top of the cradle, and the cradle and gun are swung by hand to a convenient position for loading. The center of gravity of the tipping parts is in the axis of the trunnions. A pawl, 3, attached to the cradle automatically engages teeth, 4, on the top carriage and retains the gun in the loading position until released by means of the same handle h that was used to disengage the cradle hook. As the sights and elevating screw are attached to the the rocker, their positions are not affected by the position of the piece in loading. The operations of laying the piece may therefore be performed at the same time as the loading. THE LIMBER.—The limber is the same as the limber of the 4.7-inch siege carriage previously described. When limbered the rear end of the cradle is locked to the trail in order to relieve the elevating and traversing mechanisms from strain. The short length of the howitzer renders it inadvisable to move the gun to a more rearward travelling position. WEIGHTs.—The weight of gun and carriage is about 6900 pounds, and the weight of the limber 1000 pounds. The total weight is slightly less than the limit of 8000 pounds con- sidered as a maximum load for a siege team. Siege Artillery in Present Service.—The wheeled siege pieces in present service are the 5-inch gun, shown in Fig. 22, and the 7-inch howitzer, Fig. 23. - - When emplaced in a siege battery the carriage for either piece rests on a wooden platform. Recoil is limited by means of a hydraulic buffer attached to the trail and pintled in front to a heavy pintle fixed to the platform. The howitzer also recoils on the carriage, the recoil of the piece being controlled by hydraulic buffers one on each side in front of the trunnions. Springs, strung on rods in rear of the trunnions, return the gun to the firing position. The springs are either coiled or Belleville springs, the latter being saucer shaped disks of Steel strung face to face and back to back. The pieces are mounted at a height of about six feet above the ground to enable the guns to be fired over a parapet of sufficient height to shelter the gunners. 68 For travelling, the guns are shifted to the rear into trunnion beds provided on the trail. - The 7-inch siege mortar and carriage are shown in Fig. 24. ** ºl (||} - - -G Jºs, K -j {{| – ( ; ) 3\ O ZO O O O O tº gº 2–5–3–2–3 s ºf z/AE4.2% - g” \f" £" £" O' ‘Jº Pig. 24. The carriage rests on three traverse circle segments f bolted to the platform. It is held to the platform by the overhang- ing flanges of the segments g. Elevation is given to the gun by means of the handspike l which, for the purpose, is seated in a slot in the trunnion; and direction is given by means of the handspikes j which are engaged against lugs on the car- riage. The means of controlling the recoil of the piece are similar to those employed with the 7-inch howitzer. - Sea Coast Artillery.—Comprised in the sea coast artillery are guns ranging in caliber from 3 inches to 16 inches, their projectiles ranging in weight from 15 pounds to 2400. The 3-inch guns are used for the defense of the sea fronts of forti- fications against landing parties and for the defense of the submarine mine fields. The guns of medium caliber, from 4 to 6 inches, are best used for the protection of places subject to naval raids, and for the defense of mine fields at distant ranges. Their fire is effective against unarmored or thinly armored ships. The 8 and 10-inch guns are effective against armored cruisers and against the thinly armored parts of battleships. 2 69 The proper target for guns 12 inches or more in caliber is the heavy water line armor of the enemy's battleship. The 12-inch gun is the largest gun at present mounted in our fortifications. One 16-inch gun has been manufactured and satisfactorily tested, but no guns of this caliber are mounted. The latest model of 12-inch gun was designed to give the 1000 pound projectile a muzzle velocity of 2550 feet, which would insure perforation, at a range of 8700 yards, of the 12-inch armor carried by the latest type of battleship. But it has been found that in the production of this high muzzle velocity in a heavy projectile the erosion due to the heat and great volume of the powder gases is so great as to materially shorten the life of the gun. It has been decided therefore as a measure of economy to reduce the muzzle velocities of the larger guns from 2550 feet to 2250, and to build for the defense of such wide waterways as cannot be properly defended by the 12-inch guns with the reduced velocity, 14-inch guns which will give to a 1660-pound projec- tile a muzzle velocity of 2150 feet, sufficient to insure perfora- tion of 12-inch armor at a range of 8700 yards. The wide channels that exist at the entrances to Long Island Sound, Chesapeake Bay, Puget Sound and Manila Bay will require these 14-inch guns for their defense. The table following contains data relating to seacoast guns: Projec- Burst- Muzzle *: For MAXIMUM RANGE. Gun. *::::: Chºe tile. cºe Vºe. Press- * Range Time of Remain- - - e lbs. lbs - f's Ull’e. tion, yóls "| flight. ing veloc- ** * - Mº - f. S. deg. º SeC. ity. f.S. 3-inch 1902 5 15 || 0.35. 2600 || 34000 15 8500 24.1 776 4.72-inch Armstrong | 10.5 45 | 1.96 2600 34000 | 15 || 10000 26.4 718 5-inch 1900 26 58 || 2.75 2900 36000 15 10900 || 27.6 865 6-inch 1903 43 106 5 2900 || 36000 | 15 12400 29.4 926 8-inch 1888 80 318 19 2200 || 38000 | 12 || 11000 23.5 1080 10-inch 1900 245 606 || 36 2250 38000 | 12 || 12300 24.7 | 1148 12-inch 1900 || 375 | 1048 || 64 2250 38000 || 10 | 11600 21.5 1269 14-inch 1906 280 | 1660 58.52. 2150. 38000 || 10 | 11300 20.9 || 1302 16.inch 1895 || 640 || 2400 || 139.23 2300 38000 || 10 | 12800 22.4 || 1373 MORTAR. -- J 12-inch 1890 | 660 1051 64 || 1325 33000 || 45 || 13400 52.7 1055 The bursting charges given in the table are for shell. The bursting charge for a shot is about one-third of the bursting charge for a shell of the same caliber. Af 70 Seacoast Guns.—The Seacoast guns and mortars are con- structed as shown on pages 60 and 61, Part II. As the considerations that limit the weights of the guns of the mobile artillery do not apply to Seacoast guns mounted on fixed platforms, and as with longer guns higher muzzle ve- locities may be obtained without increasing the maximum pressure, the Seacoast guns are much longer, in calibers, than are the field and siege pieces. This may be noted in the table on page 117, Part I. - All seacoast guns up to 4.7 inches in caliber use fixed ammunition. In guns of greater caliber the projectile is inserted first and is followed by the powder charge made up in one or more bags. In general the breech mechanism of the guns using fixed ammunition is of the type described with the 3-inch field gun. Guns five and six inches in caliber are provided with the Bofors mechanism. Larger guns have the cylindrical slotted screw mechanism described on page 76, Part II. - Seacoast Gun Mounts.—The mounts for the seacoast guns, commonly called carriages, are distinguished as barbette or disappearing carriages according as they hold the gun always exposed above the parapet Or withdraw the gun behind the parapet at each round fired. The disappearing carriage has the advantage of excellent protection for the carriage and gun crew, and, for guns of the larger calibers, the added ad- vantage of greatly increased rapidity of fire. The increased rapidity of fire is due to the lowering of the gun to a height convenient for loading, so that the heavy projectiles and charges of powder need not be lifted in loading. On high sites the disappearing carriage is not necessary to secure pro- tection for the gunners, for behind the parapets the gunners can only be reached by high angle fire from the enemy’s ship, and on account of the excessive strain on the decks that would accompany such fire guns aboard ship are not so mounted that they can be fired at high angles. Disappearing carriages, emplaced, are more costly than barbette carriages, but the advantage of the more rapid fire from the disappearing 71 carriage has determined its use in this country for all sea-- coast guns above six inches in caliber, on high sites as well as on low sites. , - - Most of the 6-inch guns and all guns below six inches in caliber are mounted on barbette carriages provided with shields of armor plate for the protection of the gunners. Seacoast guns being permanently emplaced the weights of the gun and the carriage, and simplicity of mechanism in both gun and carriage, are not matters of such importance as they are in the field and siege artillery. We consequently find adapted to the Seacoast guns and carriages every mech- anism that will assist in increasing the rapidity of fire. Fixed ammunition is used in guns up to 4.7inches in caliber and its use will probably be extended to larger calibers. Ex- periments are being made with mechanisms for the auto- matic or semi-automatic Opening and closing of the breech. The mechanisms for elevating the gun and for traversing the carriage are arranged to be operated from either side of the carriage, and in the carriages for the larger guns provision is made for the operation of these mechanisms both by hand and by electric power. Sights are provided on both sides of the gun, and the Operations of aiming and loading may pro- ceed together. . Finally the magazines and shell rooms in the walls of the fortifications are so arranged with regard to the gun em- placement, and so equipped, as to insure a rapid delivery of ammunition to every gun. The seacoast gun mounts differ for guns of different caliber. A description of one mount of each distinct type will follow and will serve to show the principles that govern in similar Constructions. GENERAL CHARACTERISTICS.–In general, the mount consists of a fixed base bolted to the concrete platform of the emplace- ment, and of a gun-supporting superstructure resting on the base and capable of revolution about some part of it. The Superstructure supports, in addition to the gun, all the recoil Controlling parts and the necessary mechanisms for elevat- ing, traversing, and retracting the gun. - 72 Fastened to the fixed base or to the platform around the base is an azimuth circle graduated to half degrees, and on the movable part of the carriage is fixed a pointer, with vernier, that indicates the azimuth angle made by the gun with a meridian plane through its center of motion. The gun, supported by means of its trunnions on the Super- structure of the carriage or contained in a cradle which is itself so supported, has movement in elevation about the axis of the trunnions. The elevating mechanisms, or the sights, are provided with graduated scales which usually indicate the range corresponding to each position of the gun. . Protecting guards are provided wherever necessary for the protection of the gunners against accident, or for the protec- tion of the mechanisms of the carriage against the entrance of dust or Water. ~ Pedestal Mounts.-Seacoast guns up to six inches in cal- iber are mounted in barbette on carriages similar in construc- - tion to the carriages shown in Figs. 25 and 26. - * * A conical pedestal of cast- steel, p Fig. 25, is bolted to the $22. concrete platform. A pivot N yoke y free to revolve is seated in the pedestal. In the upwardly extending arms of the pivot yoke are seats for the trunnions of the cradle c. The gun is sup- ported and slides in recoil in the cradle. The weight of all the revolving parts is supported by a roller bearing r on a central boss in the base of the pedestal. In the lower rear portion of the cradle are formed a central recoil cylinder and two spring cylinders, Fig. 26, similar to the corresponding cylinders de- scribed in the 4.7-inch siege carriage, but much shorter. As the Seacoast gun mounts are firmly bolted to platforms and as they may be made as strong as desired without limit as to % § al % §4 § zºzº Fig. 25. º unae) qouſ-9 uo] quino.IN. [13] sºpº， *{) «» 73 weight, these mounts will stand much higher stresses, with- out movement or rupture, than can be imposed on a wheeled carriage. We therefore find that shorter recoil is allowed to the seacoast guns than to the lighter field and siege guns. Thus the recoil of the 5-inch gun on the pedestal mount is but 13 inches, and of the 6-inch gun 15 inches, while the 4.7-inch siege gun recoils 66 inches on its carriage, and the 3-inch field gun 45 inches. Bolted to the arms of the pivot yoke, on each side, are brackets to which are attached platforms for the gunners. The platforms move with the gun in azimuth and carry the gunners undisturbed in the operations of pointing and of manipulating the breech mechanism. The carriage may be traversed from either side. The shafts of the traversing hand-wheels extend downward toward the pedestal and actuate a horizontal shaft held in bearings On the pintle yoke. A worm on this shaft acts on a circular worm-wheel surrounding the top of the pedestal, t Fig. 25. Elevation is given by the upper hand-wheel, on the left side only. The elevating gear is supported by a bracket bolted to the platform bracket and works on an elevating rack attached to the cradle, the center of the rack being in the axis of the trunnions. The traversing rack, or worm-wheel, surrounding the upper part of the pedestal is held to the pedestal by an adjustable friction band; and a worm-wheel in the elevating gear, con- tained in the gear casing fixed to the elevating bracket, Fig. 26, is held between two adjustable friction disks. These friction devices are so adjusted as to enable the gun to be traversed or elevated without slipping of the mechanism, and yet to permit slipping in case undue strain is brought on the teeth of the worm-wheels. , A shoulder guard is attached to the cradle on each side of the gun to protect the gunners from injury during movement of the piece in recoil. Open sights and a telescopic sight are seated in brackets on the cradle on each side of the gun. Dry batteries in two boxes held in brackets secured to the platform brackets 74 supply electric power for firing the piece and for lighting the electric lamps of the sights. The shield, of hardened armor plate 4% inches thick, is fastened by two spring supports to the sides of the pivot yoke. The bolt holes for the shield support are seen in Fig. 26. The shield is pierced with a port for the gun and with two sight holes, and is inclined at an angle of 40 degrees with the horizon, see Fig. 10, Chapter VII. The Balanced Pillar Mount.—A variation of the mount just described is found in the balanced pillar mount, also called the masking parapet mount. This mount is constructed for guns up to five inches in caliber. The purpose of this mount is to afford a means of withdrawing the gun, when not in use, behind the parapet and Out of the view of the enemy. The gun is withdrawn behind the parapet only after the fir- a ing is completed, and not after each round. Guns mounted ºft on the disappearing carriages later described are withdrawn from view after each round fired. The construction of the balanced pillar mount will be un- derstood from Fig. 27. The pintle yoke, with all the parts supported by it, rests on the top of a long steel cylinder which has movement up and down in an outer cylinder. The base of the pintle yoke is circular. It embraces a heavy pintle formed on the top of the cylinder and rests on conical rollers which move on a path provided on the cylinder top. Clips attached to the base of the pivot yoke engage under the flanges of the roller path and hold the top carriage to the . cylinder. - Imbedded in the concrete of the platform is the outer cast- iron cylinder in which the inner cylinder slides up and down. The weight of the inner cylinder and supported parts is balanced by lead and iron counterweights strung on a central rod which is connected to brackets on the inside of the inner cylinder by three chains. The pulleys over which the chains pass are supported on posts that pass through holes in the counterweight and rest in sockets formed in the bottom of the cylinder. For lifting and lowering the inner cylinder 75 'qunow deſſä peoutſieg uo unº useooges %ZZZZ q0 UIſ-g * ?, Z 76 with the gun and top carriage, a vertical toothed rack is fixed to the exterior of the inner cylinder. A pinion is seated in bearings provided at the top of the Outer cylinder and engages in the rack. The pinion is turned by means of two detach- able levers mounted on the ends of the pinion shaft. By means of a friction clamp the pinion is made to hold the elevated carriage against any sudden downward shock. The construction permits a vertical movement of the gun and carriage of about 3% feet. When firing, the muzzle of the gun projects over the para- pet; and before lowering, the gun is turned parallel to the parapet. - - In a similar mount provided for 3-inch guns the Outer cyl- inder is a\ double cylinder) The counterweight is annular and occupies the space between the two cylinders composing the double outer cylinder. The lifting levers are applied directly to the shaft of one of the chain pulleys, over which pass the chains that connect the counterweight to brackets on the outside of the inner cylinder. The brackets move in slots provided in the interior of the double cylinder. Barbette Carriages for the Larger Guns.—Guns from 8 to 12 inches in caliber are mounted in barbette on carriages similar in construction to that shown in Fig. 28. The car- riages are made principally of cast steel, all the larger parts 77 with the exception of the base ring being of that metal. The cast-iron base ring, A Fig. 29, has formed on it a roller path b Fig. 29. on which rest the live conical rollers E of forged steel. The rollers are flanged at their inner ends and kept at the right dis- tance apart by outside and inside distance-rings B. The cen- tral upwardly extending cylinder c forms a pintle about which the upper carriage revolves. Embracing the pintle and rest- ing on the rollers is an upper circular plate called the racer. Clips attached to the racer, see Fig. 28, and engaging under the flange of the lower roller path hold the parts together under the shock of firing. The two cheeks, C Fig. 28, of the chassis are cast in one piece with the racer and are connected together by transoms and strengthened by inner and Outer ribs. A groove or recess is formed in the upper part of each cheek, see Fig. 30, for the series of rollers seen in Fig. 28 on which the top carriage moves in recoil. The axles of the rollers are fixed in the walls of the grooves * à) at such a height that the tops of º the rollers are just above the top ##" of the chassis. º The top carriage, D Fig. 28 and Fig. 30. a Fig. 30, rests on the rollers and is held to the chassis by means of the clips d Fig. 30. The top carriage is cast in one piece. It consists of two side frames united by a transom a passing under the gun. The side frames contain the trunnion beds c for the gun trunnions, and the two recoil cylinders b. The piston rods of the recoil cylinders are held in lugs formed on the front of the chassis. Elevation from minus 7 to plus 18 degrees is given by means of the hand-wheel seen near the breech of the gun, Fig. 28, or by the hand-wheel just under the top carriage. 78 The carriage is traversed by means of the crank handle in front of the chassis. Through a worm and worm-wheel the crank actuates a sprocket-wheel fixed in bearings on the chassis. A chain that encircles the base ring and that is fast to the base ring at one point passes over the sprocket-wheel. When the sprocket-wheel is turned it pulls on the chain and causes the chassis to revolve. . In later carriages the chain is replaced by a circular toothed rack attached to and surrounding the base ring, and the sprocket-wheel is replaced by a gear train whose end pinion meshes in the rack. There is less lost motion with this con- struction. & The shot is hoisted to the breech by means of a crane attached to the side of the carriage. When the gun is fired, the gun and top carriage recoil to the rear on the rollers. The length of recoil is limited by the length of the recoil cylinder, and on this type of carriage is about five calibers. The recoil is absorbed partly in lifting the gun and top carriage up the inclined chassis rails and partly by friction, but principally by the resistance of the recoil cylinders, as explained in the chapter on recoil. On cessation of the recoil the gun returns to battery under the action of gravity, the inclination of the chassis rails, four degrees, being greater than the angle of friction. N-- Disappearing Carriages.—The importance of the function of the heavy seacoast guns, the difficulty in the way of quick or extensive repairs to their mounts, the great cost of the guns and their carriages, are all considerations that point to the desirability of giving to these guns and carriages the greatest amount of protection practicable. - * The guns are therefore emplaced in the fortifications behind very thick walls of concrete, which are themselves protected in front by thick layers of earth. Additional protection is obtained by mounting the guns on carriages which withdraw the guns from their exposed firing position above the parapet to a position behind the parapet and below its crest, where the gun and every part of the carriage except the sighting 79 platforms and sight standards are protected from a shot that grazes the crest at an angle of seven degrees with the horizontal. - An additional and very important advantage gained by the use of these carriages is the increased rapidity of fire obtained from the guns mounted upon them. The guns in their lowered positions are at a convenient level for loading, and the time and labor that must be expended in lifting the heavy projectiles and powder charges to the breech of a gun of the same caliber mounted in barbette are practically eliminated. 12-inch Disappearing Carriage, Model 1901.-The annular base ring, b Fig. 31, surrounds a well left in the concrete of the emplacement. The racer a rests on live rollers on the base ring and is pintled on a cylinder formed by the inner wall of the base ring. The racer supports the superstructure as in the carriage just described. It is held to the base ring by clips c which engage under a flange on the inside of the pintle. A working platform or floor, of steel plates, is fixed to brackets a fastened to the racer, and moves with the car- riage in azimuth. - The forward ends of “the chassis cheeks are continued upward, and on the inside of the cheeks and of the upward extensions are formed vertical guideways for the crosshead k from which the counterweight w is suspended. GUN LIFTING SYSTEM.–The top carriage, similar in con- Struction to that of the barbette carriage, rests on flanged live rollers which roll on the rails of the chassis. The rollers are connected together by side bars in which the axles of the rollers are fixed. - The gun levers l are trunnioned in the trunnion beds of the top carriage. They support the gun between their upper ends; and between their lower ends, the cross-head k from which the counterweight is suspended. The cross-head is provided with clips that engage the ver- tical guides formed on the inside of the chassis cheeks. Cut on the front faces of the clips of the cross-head are ratchet teeth in which pawls p engage to hold the counterweight up 80 after the gun has recoiled. The pawls are pivoted on the chassis. Levers v pivoted on the ends of a shaft across the front of the chassis serve as means for releasing the pawls when it is desired to put the gun in battery. - The counterweight consists of 102 blocks of lead of varying size, weighing approximately 164,700 pounds. It is piled on the bottom plate m which is suspended by four stout rods from the cross-head. The preponderance of the counter- weight may be adjusted, within limits, by the addition or removal of small weights at the top. - ELEVATING SYSTEM.—The gun elevating system consists of the band n dowelled to the gun and provided with trunnions that are engaged by the forked ends of the elevating arm h. The elevating arm has at its lower end a double ended pin which rotates in bearings in the elevating slides. The ele- vating slide has a movement up and down on an inclined guideway machined on the rear face of the rear transom. Movement is given to the slide by means of a large axial screw on which the slide moves as a nut prevented from turn- ing. The screw is turned by gearing on the shaft e actuated by hand-wheels outside the carriage. In order to counter- balance the weight of the elevating arm and band, and to. equalize the efforts required in elevating and depressing the gun, a wire rope passes from the elevating slide over pulleys, and supports a counterbalancing weight g. The gun moves in elevation from minus 5 degrees to plus 10 degrees. TRAVERSING SYSTEM.–Crank-handles on the traversing shaft t actuate, through gearing, a vertical shaft carrying at its lower end a pinion 0 which works in a circular rack on the inside of the base ring. In a convenient position on the racer near the azimuth pointer is placed the lever of a trav- ersing brake, not shown, which works against the base ring. By its means traversing is retarded as the carriage ap- proaches any desired azimuth. RETRACTING SYSTEM.–Means are provided to bring the gun down from its firing position when for any reason it has been elevated into battery and not fired. Detachable crank-han- dles mounted on the ends of the shaft r turn two winding mÐ qou!-ŻI º Iº º 5 įJI ‘UIO!!!SOq $uſ IĮĮĶI ‘93，3||JJ8O 3uȚIgºddesſ CT UIO UĐ đ• ; |× ~~ ~~~~ ~~~~- ---_____.* 82 drums on the shaft M inside the chassis. A wire rope y leads from each drum around a pulley at the rear end of the chassis to the top of the gun lever, a loop in the end of the rope engaging over the hook of the lever. SIGHTING SYSTEM.–Elevated platforms are provided on each side of the carriage. The telescopic sight, see Fig. 32, is mounted above the left platform on a hollow standard that rises from the floor of the racer. A vertical rod passing through the standard is connected at the top to a pivoted arm carrying the sight, and at the bottom the rod is so geared to the elevating shaft that the same movement in elevation is given to the sight arm as is given to the gun. Within reach of the gunner at the sight are two crank-handles, at the upper ends of vertical shafts, by means of which the gunner has electric control of the elevating, traversing, and retracting mechanisms. - Trials are being made of the panoramic sight fitted to dis- appearing carriages. The vertical tube of the sight is made very long and the sight is attached to the side of the carriage in such a position that the eye piece is convenient to the gunner standing on the racer platform, while the head piece of the sight is above the parapet. . OPERATION.—The operation of the carriage for firing is as follows. The gun is loaded in its retracted position, Fig. 32, being held in that position by the pawls p engaged in the notches on the cross-head k. After the gun is loaded the tripping levers v are raised, releasing the pawls from the notches in the cross-head. The counterweight falls and the top carriage moves forward on its rollers, the last part of its motion being controlled by the counter-recoil buffers in the recoil cylinders, so that the top carriage comes to rest without shock on the chassis. By the movement of the gun levers the gun is lifted to its elevated position above the parapet. When the piece is fired the movements are reversed in direction. The recoil forces the gun to the rear, the top car- riage rolls back on the chassis rails and the counterweight rises vertically under the restraint of the guides engaged by the cross-head. - - - ruo!!!soaſ ſuſpeoT ‘ože ſaeo ºutubºddesſ (I uo unae, qouſ-ZI ---- «* 8 · 5 IJI · 83 In the movement either way the upper end of the gun lever describes an arc of an ellipse. The path of the muzzle of the gun, indicated in Fig. 31, is affected by the constraint of the elevating arm. The ellipse is the most favorable figure to follow in the movement of a gun on a disappearing carriage. From the firing position the movement of the gun is at first almost horizontally backward, and the movement downward occurs principally in the latter part of the path. Therefore the carriage that moves the gun in an elliptical path can be brought nearer to the parapet and thus receive better pro- tection than any other carriage. The recoil is controlled principally by the recoil cylinders, and the shock at the cessation of motion is mitigated by two buffers f which receive the ends of the gun levers. The buffers are composed of steel plates alternating with sheets of balata - Balatá is a substance that resembles hardened rubber. It has not as great elasticity as rubber but does not deteriorate as rapidly under exposure to the weather. Modification of the Recoil System.–In the chapter on recoil it was pointed out that there is a disadvantage in hav- ing the control of the counter recoil in the same hydraulic cylinders that control the recoil. The adjustment of the counter-recoil system affects the adjustment of the recoil System. - It will also be observed in the carriage just described that in the latter part of the movement in recoil the gun is moving almost vertically downward. Consequently the movement of the top carriage to the rear is very slight during this part of the recoil, and the slight movement affords little oppor- tunity for the close control by the recoil cylinders of the final movement of the gun. But it is in the last part of the recoil that complete control of the movement of the gun is most desirable, in order that the gun may be brought to rest with- Out shock to the carriage. While the movement of the top carriage is least rapid at the latter end of recoil the counterweight has then its most 84 rapid movement. Therefore a recoil cylinder fixed so as to move with the counterweight will afford the best control of the final movement of the gun. The top carriage has its most rapid movement at the latter part of the movement of the gun into battery, while the coun- terweight has its least rapid movement at that time. The control of the counter recoil is therefore best effected through the top carriage. • By retaining therefore, to act on the top carriage, recoil cylinders adapted for the control of the counter recoil only, and by adding to the counterweight a cylinder adapted for control of the recoil, we will obtain the advantage of com- pletely separating the two systems, thus making them capa- ble of independent adjustment, and the advantage of obtain- ing from each system the greatest control of the movement to which it is applied. 6-inch Experimental Disappearing Carriage, Model 1905. —The modification of the recoil system as above indicated has been applied to a 6-inch experimental carriage. The recoil cylinder is held in the center of the counter- weight, Fig. 33. The lower end of the piston rod is fixed in the lower member d of a frame whose sides f are bolted to the bottom of the racer a, as shown in the left and rear views. The counter recoil is checked by the short cylinderss mounted on each chassis rail in front of the top carriage. The pistons of the counter-recoil cylinders are not provided with apertures for the flow of the liquid from one side of the piston to the other, but the flow of the liquid takes place through the pipes p which are led from both cylinders to a valve v, by which the area of orifice is controlled and through which the pressure in the two cylinders is equalized. The pressure in the counter-recoil cylinders does not exceed 500 pounds per square inch, while the pressure in the recoil cylin- der is 1800 pounds. As the top carriage comes into battery the front of the car- riage strikes the rear end 0 of the piston rod and forces the piston through the cylinder against the liquid resistance and d Left View. Elig. 33. Rear View. against the action of springs g mounted on each side of the cylin- der. The springs act on central rods connected to the forward end of the piston, and as the top carriage moves from battery the springs move the piston to the rear in position to be acted on by the top carriage as it comes back into battery. 86 There are other points of difference between this carriage and the carriage last described. The retraction of the gun from the firing position is accom- plished without the use of wire ropes by the vertical racks 6, shown in the left and rear views, attached to bars that connect the cross head k and the bottom section m of the counterweight. The end pinions 5 of two trains of gears, one on each side, mesh in the rack; the gear trains being actuated by the cranks on the shaft r. The retracting mechanism is partially shown in the smaller views. The parts are similarly. numbered in all the figures. The mechanism is thrown out of gear when not in use. The rollers of the top carriage are geared to the top car- riage so that they are compelled to move with the top carriage and there can be no slipping of the top carriage on the rollers. In present service carriages this slipping sometimes occurs as the gun recoils, so that on counter recoil the rollers reach their position in battery before the top carriage, and prevent the top carriage from coming fully into battery. - The sight standard is moved to the front of the chassis in order to get better protection for the gunner, for the sight, and for the elevating and traversing mechanisms under control of the gunner. Through the upper hand-wheel e and the shafts and gears also marked e the gunner has control of the elevat- ing mechanism; and through another hand-wheel at his right hand, covered by the wheel ein the figure, and the shafts and gears marked the controls the traversing mechanism. Seacoast Mortars.-The thick armored sides of ships of war protect the ships to a greater or less extent, against the direct fire from high powered guns. The great weight of armor that would be required for complete deck protection is prohibitive. The decks of war ships are therefore thin and practically unarmored, the heaviest protective deck on any battleship being not more than two inches thick over the flat part. The decks therefore offer an attractive target. As the elevation above sea level of the sites of the guns in most fortifications is not sufficient to permit direct fire against the decks, there are provided for use against this target the 12-inch seacoast mortars, short guns so mounted that they can be fired at high angles only. The heavy projectiles fired 87 from these guns carry large bursting charges of high explo- sive. Descending almost vertically on the deck of a ship they easily overcome the slight resistance offered, and penetrating to the interior of the ship burst there with enormous destruc- tive effect. . The mortar carriages permit firing only at angles of eleva- tion between 45 and 70 degrees. With a fixed charge of powder a limited range only would be covered by fire between these angles. Charges of several different weights are there- fore used in the mortars. With eaeh charge a certain zone in range may be covered by the fire, and the charges are so fixed that the range zones overlap. Any point within the limits of range may thus be reached by the projectile. The least range with the smallest charge provided is about a mile and a half. Mortar batteries are therefore usually erected at not less than this distance from the channels or anchorages that are under their protection. The 12-inch Mortar Carriage, Model 1896.-The construc- tton of the 12-inch mortar carriage, model 1896, will be under- stood from Fig. 34. The mortar is supported by the upper ends of the two arms of a saddle d which is hinged on a heavy bolt to the front of the racer. The arms of the saddle are connected by a thick web. Extending across under the Web is a rocking cap-piece, c, against which five columns of coiled Springs act, supporting the gun in its position in bat- tery and returning it to battery after recoil. Fig. 34. 88 The lower ends of the springs rest in an iron box trun- nioned in two brackets bolted to the bottom of the racer. The box oscillates as required during the movement of the saddle in recoil and counter recoil. Holes in the bottom of the box and in the cap-piece and saddle web permit the ends of the rods on which the springs are strung to pass through during the movement. The recoil cylinders h are trunnioned in bearings fixed to the top of the racer. Bolted to the top of each cylinder is a frame f which serves as a guide for the cross head o at the upper end of the piston rod. The cross head embraces the stout pin r which extends outward from the trunnion of the mortar and communicates the motion of the piece in recoil to the piston rod. - The provision for the flow of liquid in the recoil cylinder 22:7; from one side of the piston to the other differs in this carriage from that described in other carriages. A small cylinder, A Fig. 35, is formed outside the recoil cyl- inder proper, H. Holes a bored through the dividing wall form passages through TS which the oil may pass from the front of the piston to the rear. The piston head in its movement closes the holes succes- sively. Thus as the velocity of recoil de- creases the area Open to the flow of the liquid is reduced. The area of aperture is also regulated by screw throttling plugs b that are seated in the Outer wall of the small cylinder. These plugs have stems of different diameters, and are used to partially or wholly close any of the pas- sages in the proper regulation of the re- coil. The recoil cylinders on each side of zº the carriage are connected by the equaliz- Yo ing pipe p. The counter recoil is checked and the gun brought into battery without shock by the counter-recoil buffer S, an annular projection formed on the cylinder head 2 º §º Z 2 % i à ‘89 surrounding the piston rod. The buffer enters, with a small clearance, an annular cavity in the head of the piston, and the liquid in the cavity escapes slowly through the clearance. As an added precaution against shock when the gun returns to battery, buffer stops composed of alternate layers of balata and steel plates are held between the cross-head guides of the frame f, Fig. 34, under the cap. * The gun is elevated by the mechanism shown mounted on the saddle, Fig. 34, and traversed by means of the crank shaft and mechanism supported in a vertical stand on the racer. A pinion p on the end of a vertical shaft engages in a circular rack bolted to the inner surface of the base ring. For determining elevation, a quadrant, similar to the gunner's quadrant described in the chapter on sights, is per- manently attached to a seat prepared on the right rimbase of the mortar. - - The 12-inch Mortar Carriage, Model 1891. –The 12-inch mortar carriage, model 1891, on which many 12-inch mortars are mounted in Our fortifications, is shown in Figs. 36 and 37. Fig. 36. The spring cylinders E are formed in the vertical cheeks sº 90 PH- sº i % % # % º % º % º% -\ Fig. 37. § º r - % % º % § º s N § § % à }% % § ſm...K bolted to the racer. Inside the cheeks º are inclined guideways for sliding zº cross-heads G. The cross-heads re- p Té ceive the trunnions of the gun. The pistons h of the recoil cylinders pro- ject downward from the cross-heads and enter the recoil cylinders H at- tached to the lower parts of the --G Spring cylinders. The cross-head G has at its upper end an arm g’, Fig. % £5 37, which projects outwardly into the £º ---E' Spring cylinder and carries at its == Outer end the adjusting screw K 5: which rests on top of the column of == Springs. The Springs are compressed == when the gun recoils, and return the AE gun to battery on the cessation of == recoil. By means of the adjusting #. screw K the height of the trunnion 3–5 s carriages G may be adjusted to bring == * to the proper height for E=Nºbs loading. Eriº The hand-wheel g, Fig. 36, works P: the shot hoist a, by means of which = ſ Q-> the shot is lifted to the breech of the gun for loading. Subcaliber Tubes.— For the pur- *-*. =r + pose of enabling troops to become ãº- familiar with the operation of the guns and carriages by actual firing yet without the expense attendant upon the use of the regular ammuni- tion, there are provided for use in- side the various service guns smaller guns or gun barrels called subcaliber tubes. These are seated in the bores of the larger guns in such position i:º }% that the breech of the subcaliber tube is just in front of the 91 breech block of the gun when closed. The subcaliber tube is loaded with fixed ammunition arranged to be fired by the fir- ing mechanism of the larger gun. Three calibers of subcali- ber tubes are provided: one of 0.30-inch caliber, using the small arm cartridge, for guns that use fixed ammunition; one of 1.475-inch caliber, using 1-pounder ammunition, for use in all guns 5 inches or more in caliber; and one of 75 mm. (2.95 inches) caliber, using 18-pounder ammunition, for use in the 12-inch mortar. For those guns that use fixed ammunition the 30-caliber Subcaliber tube, a 30-caliber rifle barrel, is fixed in a metal mounting that has the shape and dimensions of the complete cartridge used in the piece. Fig. 38 shows the subcaliber tube for the 3-inch rifle. S; ZZZZZZZZ”2 § Z. Ž3S ST & J NY º ZZZ §. & Fig. 38. The 30-caliber small arm cartridge is inserted in the barrel b and is fired by the percussion firing mechanism of the piece. It is extracted, far enough to be grasped by the hand, by the extractor, two bowed springs which are under compression when the small arm cartridge is forced to its seat by the breech block of the gun. A special primer is used in the small arm cartridge, strong enough to withstand without puncture the heavy blow of the firing pin of the gun. The head of the subcaliber cartridge is permitted longitu- dinal movement in the body in order to allow for expansion of the 30-caliber barrel in firing. . The 1-pounder tube is provided with different fittings to adapt it to the particular gun in which it is to be used. It is fitted in the gun in the manner shown in Fig. 39, which rep- resents the 75 mm. subcaliber tube in the 12-inch mortar. The 75 mm. tube is a gun similar to the mountain gun, without its breech mechanism. The cartridges for the moun- tain gun are used in it. º 92 The wheel-shaped fittings, called adapters, are screwed on % N Z// \ WWWW \ | ſ | | ſ -— | H -- l | | l | | | * l ſ —T- ! | | / & \\\\\\\\\\\ z C’ z ...” % Fig. 39. the gun. The front adapter fits against the centering slope in the bore for the band of the projectile. The Outer rim of the rear adapter is cut through at the top and the rim is ex- panded against the sides of the bore by the wedge w, which is forced between the parts of the rim by means of the screw seated in one of them. The tube is prevented from turning in the adapters by the clamp screw c. - The firing mechanism of the guns in which the two larger subcaliber tubes are used is not of the percussion type. The cannon cartridges uséd in these two tubes are therefore pro- vided with the 110%rain igniting primer, described in the chapter on primers, in place of the usual percussion primer. The igniting primer in the cartridge is ignited by the flame from the ordinary primer seated in the rear end of the breech mechanism of the ºy Drill Cartridges, Projectiles, and Powder Charges.—For ordinary use at drill, without firing, dummy cartridges are provided for guns that use fixed ammunition, and dummy projectiles and powder charges for other guns. The dum- mies have the dimensions and weights of the parts they represent. . The drill cartridge for guns using fixed ammunition are hollow bronze castings, Fig. 40, of the shape of the service V } 93 shrapnel. For the instruction of cannoneers in fuze setting 1- - - - - - - - - - - - - - - - - - - - YZZZZ º SN Fig. 40. there is fitted at the head of the cartridge a movable ring graduated in the same manner as the time scale on the com- bination time and percussion fuze. Drill projectiles, for guns separately loaded, are of the con- struction shown in Fig. 41. A bronze band b is inset at the bourrelet to prevent wearing of the rifling in the gun by fre- quent insertion of the projectile. The rotating band r is pressed to the rear on a sloping seat by Springs S. When the projectile is rammed with force into the gun the band is likely to stick in its seat and thus to resist efforts to withdraw the projectile. The method of attachment of the band is for the purpose of affording a means of readily overcoming this º resistance. The extractor, a hook on the end of a pole, is engaged over the inner lip l. A pull on the pole will, if the band is stuck, first move the remainder of the projectile to the rear until it strikes and dislodges the band. The dummy powder charge, Fig. 42, circular in section, is 94 | º Fig. 42. made up of a core of metal surrounded by disks of wood, the whole covered with canvas. The parts are assembled by means of a central bolt. An inner lip l formed in the hollow .. metal base piece is engaged by the hook of the extractor. x #: .# i &- f: • Cº-cCCºcº . • * + . A. * # lº §§le. *-o-c ! { ~ : , , º: * > . ~~. Afs *...*- : *.*_! {-*-8. ; Fu cult # *-*.*.*, *...K. . Exteri; Baſſistics: - *-wº- ...----~~~~ * ata and Additions. 1. Fº Cº. v. tºº C- Page 11, equation (5), make dt =#-dv/R cos 0. line 7, write d0/cos2 0 for dB/cos 0. Page 15, equation (21), write A1 for A2. line 11, write 2600 for 2700. line 17, after indeterminate insert: and the values of the functions cannot be determined as above. Page 22, first formula, put in minus sign between 4vo and (Z-Zo),"100. Page 27, line 15, write 24 =10° 2'.6. line 26, after Z write (4 v1-400) being negative. line 27, change – .3 to +.3, and 0.15360 to 0.15366. change last zero to 6 in log B' and log tan Go following. Page 28, line 25, make characteristic of log tan q, negative. Page 29, equation (44), put Square brackets about the numerical term. Page 30, line 1, write 34 for 49.6, and 28 for 29.3. line 10, make A =0.03024=a'0. line 17, make log C. 0.76750. Page 34, last line, make 4X=–69.2 feet. Page 37, line 6, write 5' 9" for 5' 8". line 17, write (0.09778) for (0.99778). 4th line from bottom, change Z to 2. Page 40, line 3, before the second word insert Under this law. Erase is after B' and write becomes. last line, make log T 1.09155. Page 43, line 2, write 394.68 for 393.77. line 4, write .47 for .38, and 3677.7 for 3663.4. line 5, write .47 for .38, and 19.788 for 19.752. line 6, write .47 for .38, and 346.29 for 345.66. line 7, write .47 for .38, and 32.2 for 31/.4. line 9, write X=31225 ft. = 10408 yds. T=57.66 sec. vo–1009 f.s. Page 44, line 11, write 386.46 for 386.40. line 12, Write 948.24 for 948.11. line 22, write 1.76 for 1.89, 7.59 for 7.72, and 13.8 for 14'.7. line 25, write 1.76 for 1.89, 6.20 for 6.07, and 16.8 for 18'.6. Page 45, line 10, write 38.5 for 65. line 11, write 29 for 31. Page 14, change Q." to Q and Q,” to Q, in the two equations. Page 15, erase the second marks from Q, Qi and Q2 throughout the page. Page 16, equation (24), write 2 before cos 2q2. Page 34, line 3, write The first method for These methods. line 4, write is for are. line 5, add The second method may be used in all problems of direct fire. C | 47 ...” gº -4° 2 * {...,’ Fºº { …~": ; ". …” {< º" # * ExTERIOR BALLISTIC FORMULAS, Y DIRECT FIRE. V-825 f. s. (p<20° -o-fº ". Z= X/C ö cal T=CT'/coso sin 2 q = A (ſ ** * * * 0) = M. COS GOS6 tan Go = B' fan q) q/ Aſ = I, tan • (A–a)/A a' = A =sin 2 g/C tan 6 =tan (p (A —a')/A - vo-ao" C tan (p C{}RRECTION FOR ALTITUDE. log (log f) = log yo-H [5.01765–10] DANGER SPACE AND DANGER RANGE. (A — a) z=2 y cos” (p/0% ao' ad" = 2 yo/O’ DRIFT. Sea Coast Guns. Drift (yds.) = [7.79239–10] C*D'/cos" (p Field Guns. Drift (yds.) = [7.92428–10] C*I)'/cos” (p WINI) EFFECT-BANGE. A V = }}'', cos (p V’ = V + A V sin A q = W, sin q/ V' (p = q + A q) \ X (ft.) = X’ – (X+ W. T.) WIND EFFECT-DEVIATION FOR 8, 10, 12-INCH PROJECTILES. Deviation (yds.) = [7,16633] sin a W (f.s.) (ºxº dº) CURWEI) FIRE. -- - V3825 f.s. p <25° . , Abi w_ Z Y y O'-f b cd2 Z= X/O sin 2 (p=[5.80618] AC/ V* tan co–B' tan q vo- [7.09691–19) was cos (p V/cos to T= [2.90309] CT'/ V cos (p HIGH ANGLE FIRE. - (p-25° Use Table IV and coefficient of reduction, or Table I or II and correct for altitude. (SURWATURE OF EARTH. Curvature (ft.) = [3.33289–10] X? (yds.) W.Y....…" . . . . .< * * ...” * fº ºr w º * *} : .# º, "...º.º. * Y { - w f ,’ 2. “ f ; : f 2. T. v, ** 3 . - - ; 2 ºf '--> *...* §.f. * , ‘Y o cla tº ^ O L. ºr lo y V Y - : { - & - r - 4. - - *— ar. * * -> - *.* ºt &l- 3. "…C. * *, ... " ~} .# § 3. J ...- s: º ~. :* -6.2. 3. 4. ".*. 3. Yºr | a +&c S. tº "4:30. & sºft 2 (. { | Cºci f \! * * * t .” < , …? . . . . -*.--- -!. f {22. fºr tº s -. ***, *.* ſ R. . . . . .” - ~P Ø-C*-6- 4/y ~ } !. | *... |- C& {} 0| & i - * } * ^2. **~~~ Ü 2. } * . • * - • . * - ". r * t - w * - s - * * - 1. * % - * * - - *.. - *- - * * “x * - - * - * , ~ .* - -. - - 2: . * - NON _2^ S C-" Exterior Ballistics. Exterior Ballistics treats of the motion of a projectile after it has left the piece. In the discussions the dimensions of the gun are considered neg. legible in comparison with the trajectory. The Trajectory, b d f Fig. I, is the curve described by the cen- ter of gravity of the projectile in its movement. F1 G. 1. The Range, b f, is the distance from the muzzle of the gun to the target. - The Line of Sight, a b f, is the straight line passing through the sights and the point aimed at. & The Line of Departure, b c, is the prolongation of the axis of the bore at the instant the projectile leaves the gun. - The Plane of Fire, or Plane of Departure, is the vertical plane through the line of departure. - - The Angle of Sight, 8, also called the Angle of Position, is the angle made by the line of sight with the horizontal. The Angle of Departure, b, is the angle made by the line of de- parture with the line of sight. The Quadrant Angle of Depatrure, b + 8, is the angle made by the line of departure with the horizontal. 4 The Angle of Elevation, q', is the angle between the line of sight and the axis of the piece when the gun is aimed. - The Jump is the angle j through which the axis of the piece moves while the projectile is passing through the bore. The move- ment of the axis is due to the elasticity of the parts of the carriage, to the play in the trunnion beds and between parts of the carriage, and in some cases to the action of the elevating device as the gun re- coils. The jump must be determined by experiment for the individ- ual piece in its particular mounting. It usually increases the angle of elevation so that the angle of departure is greater than that angle. The Point or Fall, f, or Point of Impact, is the point at which the projectile strikes. The Angle of Fall, Go, is the angle made by the tangent to the trajectory with the line of sight at the point of fall. The Striking Angle, 60', is the angle made by the tangent to the trajectory with the horizontal at the point of fall. Initial Velocity is the velocity of the projectile at the muzzle. Remaining Velocity is the velocity of the projectile at any point of the trajectory. Drift, k f/, is the departure of the projectile from the plane of fire, due to the resistance of the air and the rotation of the projectile. Direct Fire is with high velocities, and angles of elevation not exceeding 20 degrees. Curved Fire is with low velocities, and angles of elevation not exceeding 20 degrees. High Angle Fire is with angles of elevation exceeding 20 de- grees. The Motion of an Oblong Projectile. — The projectile, as it issues from the muzzle of the gun, has impressed upon it a motion of translation and a motion of rotation about its longer axis. The guns of our service are rifled with a right handed twist, and the rotation of the projectile is therefore from left to right when regarded from the rear. After leaving the piece the projectile is a free body acted upon by two extraneous forces, gravity and the resistance of the air. When the projectile first issues from the piece, its longer axis is tangent to the trajectory. The resistance of the air acts along this 5 tangent, and is at first directly opposed to the motion of translation of the projectile. The longer axis of the projectile being a stable axis of rotation tends to remain parallel to itself during the passage of the projectile through the air, but the tangent to the trajectory changes its inclina- tion, owing to the action of gravity. The resistance of the air act- ing always in the direction of the tangent, thus becomes inclined to the longer axis of the projectile, and in modern projectiles its result- ant intersects the longer axis at a point in front of the center of gravity. In Fig. 2, G being the center of gravity, and R the resultant resistance of the air, this resultant acts with a lever arm l, and FI G. 2. tends to rotate the projectile about a shorter axis through G, per- pendicular to the plane of fire. The resultant effect of the resistance of the air on the rotating projectile, is a precessional movement of the point of the projectile to the right of the plane of fire. After the initial displacement of the point to the right, the direction of the resultant resistance changes slightly to the left with respect to the axis of the projectile, and pro- duces a corresponding change in the direction of the precession, which diverts the point of the projectile slightly downward. If the flight of the projectile were continued long enough the point would describe a curve around the tangent to the trajectory; but actually, the flight of the projectile is never long enough to per- mit more than a small part of this motion to occur. The precession of the point is greater as the initial energy of rotation is less. It is therefore necessary to give to the projectile sufficient energy of rotation to make the divergence of the point Small. Otherwise the precessional effect may be sufficient to cause the projectile to tumble. - When the point of the projectile leaves the plane of fire, the side of the projectile is presented obliquely to the action of the re- 6 sistance of the air, and a pressure is produced by which the projectile is forced bodily to the right out of the plane of fire. It is to this movement that the greater part of the deviation of the projectile is due. .." - DRIFT.-The departure of the projectile from the plane of fire, due to the causes above considered, is called drift. Form of Trajectory. — It may be shown analytically that the drift of the projectile increases more rapidly than the range. The trajectory is therefore a curve of double curvature, convex to the plane of fire. The trajectory ordinarily considered is the projection of the actual curve upon the vertical plane of fire. This projection so near- ly agrees with the actual trajectory that the results obtained are prac- tically correct; and the advantage of considering it, instead of the actual curve, is that we need consider only that component of the re- sistance of the air which acts along the longer axis of the projectile and which is directly opposed to the motion of translation. Determination of the Resistance of the Air.— The relation between the velocity of a projectile and the re- sistance opposed to its motion by the air has been the subject of numerous experiments. - - In the usual method of determining this relation the velocity of the projectile is measured at two points in the trajectory. The points are selected at such a distance apart that the path of the pro- jectile between them may be considered a right line, and the action of gravity may be neglected. The resistance of the air is then re- garded as the only force acting to retard the projectile, and is con- sidered as constant over the path between the two points. The loss of energy in the projectile, due to the loss of velocity, is the measure of the effect of the resistance of the air; and is equal to the product of the resistance into the path. The resistance thus obtained is the mean resistance, and corresponds to the mean of the two measured velocities. - - EARLY ExPERIMENTS.–The first experiments were those of Robins in 1742. For the measurement of velocities he used the bal- listic pendulum. His conclusions were, that up to a velocity of I IOO 7 foot seconds the resistance is proportional to the square of the veloc- ity; beyond I IOO f. s. the resistance is nearly three times as great as if calculated by the law of the lower velocities. Hutton in 1790, with the improved ballistic pendulum, made numerous experiments with large projectiles. His conclusions were, that the resistance increases more rapidly than the square of the velocity for low velocities, and for higher velocities that it varies nearly as the square. - General Didion made a series of experiments at Metz in 1840 with spherical projectiles of varying weights. His conclusions were, that the resistance varied as an expression of the general form a (v*-Höv"), a and b being constants. This formula held for low velocities only. Experiments were again made at Metz in 1857. Electro-ballis- tic instruments were now used for the measurement of velocities. The conclusions from these experiments were, that the resistance varies as the cube of the velocity. Experiments by Prof. Helie at Gavre, in 1861 gave practically the same results. The experiments above described were made principally with spherical projectiles. The difference in the nature of the resistance experienced by oblong and spherical projectiles, together with the difference in the velocities, then and later, may account for the wide difference in the results obtained from these and from later experi- ments. - LATER ExPERIMENTS.–The Rev. Francis Bashforth made ex- haustive experiments in England, in 1865 and again in 1880, using comparatively modern projectiles and accurate ballistic instruments. His conclusions were, that for velocities between 900 and IIoo f. s. the resistance varied as the sixth power of the velocity; between IIoo and I350 f. s., as the cube of the velocity; and above I350 f. s., as the square of the velocity. - The most recent experiments are those made by Krupp in 1881 with modern guns, projectiles and velocities. The results of these experiments were used by General Mayevski in the deduction of the formulas for the resistance of the air which are now generally used. CONCLUSIONS FROM THE EXPERIMENTS.—The experiments have shown that the resistance of the air varies with the form of the pro- jectile, with its area of cross section, with the velocity of the projec- 8 tile, and with the density of the air. Considering the form of the projectile the resistance is affected principally by the shape of the head, and by the configuration at the junction of the head and body. The ogival head encounters less resistance than any other form of head. The resistance was found to increase directly with the area of cross section of the projectile, and directly with the density of the a11". Mayevski’s Formulas for Resistance of the Air. In expressing the relation between the resistance of the air and the velocity of the projectile, General Mayevski placed the re- tardation, as determined in Krupp’s experiments, equal to an expres- sion which, together with an unknown power of the velocity, in- volves quantities whose values are dependent on the weight, form, and cross section of the projectile, and on the density of the air. Calling p the resistance of the air, w the weight of the projectile in pounds, g the acceleration of gravity. the retardation is ſo g/w Make * - R = 0 g/w = v"A/C (I) in which A is a constant and n some power of the velocity, both to be determined from the experiments. THE BALLISTIC COEEFICIENT, C.—The quantity C in the equa- tion was given a value Ól zº Ó C dº in which 6, is the standard density of the air, " — 6 the density at the time of the experiment, c the coefficient of form, d the diameter of the projectile in inches. w the weight of the projectile in pounds. By the introduction of this coefficient into the value of the retarda- tion, the effect of variations in weight, form, and cross section of the projectile, and in the density of the air, may be considered. 9 The coefficient of form c was taken as unity for the standard projectiles. For projectiles of a form that offers greater resistance the value of c will be greater than unity. Examination of equa- tion (I) shows that as c increases, and C decreases, the retardation is increased; a result also obtained by increase in d or 6, that is in the cross section of the projectile or in the density of the air; while by an increase in w, C is increased and the retardation is diminished. The coefficient C is therefore the measure of the bal- listic efficiency of the projectile. The value of c for all projectiles in our service is usually taken as unity. The density of the air is a function of the temperature and of the atmospheric pressure. The values of 6,26 for differ- ent atmospheric pressures and temperatures, are found in Table VI of the ballistic tables. New tables for the determination of this factor have recently been published. Mayevski determined, from Krupp’s experiments, values for n and A for different velocities as follows: velocities - Il - log A f. S. above 2600 1.55 3.6090480 2600 to 1800 1.7 3.0961978 1800 to 1370 2 - 4.1.192596 1370 to 1230 3 8.9809023 1230 to 970 5 14.8018712 970 to 790 3 8.7734430 below 790 2 5.6698914 N Trajectory in Air. Ballistic Formulas.-In the deduction of the ballistic formulas the trajectory is considered as a plane curve. The line of sight is taken as horizontal. The angle of ele- 10 vation is taken as the angle of departure, and the striking angle becomes the angle of fall. ... " - - - The trajectory so considered is called The Horizontal Trajec- tory. - - Considering the motion of translation only, and that the re- sistance of the air is directly opposed to this motion, let, Fig. 3, F I G 3. R be the retardation due to the resistance of the air, its value being given by equation (I); * - V, the initial velocity; - - 7", the velocity at any point of the trajectory whose co-ordinates are 4 and y; - v, the component of z, in the direction of a ; ‘b, the angle made by the tangent to the trajectory, at the origin, with the horizontal; or the angle of departure. - 6, the value of q for any other point of the trajectory; G9, the angle of fall; + and y, the co-ordinates of any point on the trajectory, in feet; X, the whole range in feet. EQUATIONS OF MOTION.—The only forces acting on the projec- tile after it leaves the piece are the resistance of the air and gravity. The resistance of the air is directly opposed to the motion of the projectile, and continually retards it. Gravity retards the projectile in the ascending portion of the trajectory, while it accelerates it in the descending portion. - Considering the ascending portion of the trajectory, the veloc. ity in the direction of a is -- - - - - - - - - - - - - -- z cost = z = dx/dt dx = widt , (2) The velocity in the direction y is v sin() = vitant} = dy/dt dy = vitant dt (3) _^ 11 The retardation in the direction of y is —d(vitant?) Adt = g-HR sing (4) Since gravity has no component in a horizontal direction, the re- tardation in the direction of A is - sº- dvydt = R cost? dt = — dvºcosé (5) Substituting this value of di in (2), (3) and (4), and performing the differentiation indicated in (4), d tanë being d6/ cosé, we obtain. dx = — widviſ ſº cost - - (6) dy = — vitanſdv/R cosp (7) d6 = g cost dy/Rv, (8) - ... "Sº, _---- ~ * **T _> \ } 2–~~~<}^{* \ % // *" The fouñéquations (5) to (8) are the differential equations of mo- tion of the projectile, and if they could be integrated directly they would give the values of ar, y, t and 6 for any point of the tra- jectory. But as they are expressed in terms of R, v, and 6, three independent variables, the direct integration is impossible. Tº The value of R is given by Mayevski's formulas, R = Az”/ C, n representing the exponent of v for any particular velocity. Sub- stituting this value of R in (6), the equation may, by reduction, be put in the form dx = —C cost-16dviſ/Az’,”- (9) The second member would be an exact integral were it not for the factor cosm-16. In direct fire cosé differs but little from unity, and it might be taken as unity without appreciable error. A closer approximation, however, as shown by Siacci, results from making cost-16 = cos”-*q, \ Making this substitution, equation (9) may be brought by reduction to the form. - - C d(z)isecº) dx = — — — (Io) . * , A (z, secº)”- Make v, secº = w cost/cost = u V. secº = V cosºp/cosq = W 12 Integrating (10) between the limits b and 6, remembering that for the value $, v, becomes V, we obtain - - C I I g ** F. g-º-º-º-; ººº-ºº-ºººº- (II). (m-2) A un-º Vn-2 - * And similarly equations (5) and (8) may be brought to the forms - C I I !. f = e-º-º: - (12) ' (n-1 )A cosºp 2/72-1 Vn-1 . g C I I tand – tanff = — | –—— I anº a11 n Acos' b |: # (13) To simplify equations (II) to (I3), make * 1 - S - - - • * (*) = 0–2. Tº at 9 yº ºf -* S(V) = 1 - ~ ; ~ (n-3) fººt 9 \ . \ - {--> 1 (14) ~ T (M) = / …" T ) (n-1) Aw"T" + Q 3. º º Rºº, —*— // ... --- I (u)=– º – + Q The reason for the addition of the constants will appear. Making these substitutions, equations (II) to (13) become * E c{s(u) — S(V) | (15) C ... * * -:}ro-roo! - (16) . cost . tanff = tang — 2 2 COS | (u) — I (V) | (a) ' * 13 Making in the last equation tanff = dy/dx, and making 1 / / (tº) du ~~ A (u) = — A.J. T.T - (147) this equation (17) may be brought to the form q, C 4(a) – 4(V) , (V) º º —— m. t -- - ; I A” a11 2 cosº | S(u) — S(V) ; - Equations (15) to (18), with the equations ‘. - 6 ZZ = 7) cos" (19) 1- cos @ and - /-. Tº adº (20) are the fundamental equations of Exterior Ballistics, and constitute the method of Siacci, an eminent Italian ballistician. The essence of the method lies in the use of u, called by Siacci the psu.edo velocity, for v the actual velocity. - In all problems of direct fire, since the difference between q and 6 is not great, u may be used for v, with sufficient accuracy. In problems in curved and high angle fire, and in direct fire when greater accuracy is desired, we pass from the value of it to the value of v by means of equation (19). It will be seen from this equation that, since tº cos b = w cos 0, u is the component of v parallel to the line of departure. - The ballistic coefficient C in (20) differs from that assumed by Mayevski, by the presence of the two factors f and ſº. f is called the altitude factor, and brings into consideration the diminution in the density of the air as the altitude of the trajectory increases. The value of f is greater than unity, and depends upon the mean altitude of the trajectory, which is taken as two-thirds of the maximum altitude. ſł is an integrating factor, and corrects for the error due to cer- tain assumptions made in deducing the primary equations, when these equations are applied to a trajectory whose curvature is consid- 14 erable, ſº is approximately unity in all problems of direct fire. The product ſic is called the coefficient of reduction. When, in the statements of ballistic problems, the data required to determine 6/6, ſº or c is not given, the value unity is assumed for the factor, f is also assumed as unity unless a correction for altitude is desired. When all these factors are unity the ballistic coefficient becomes -- ~~ C= w/dº The Functions. – The functional expressions in equations (15) to (18) are called: I (w) the inclination function, T (u) the time function, S (u) the space function, and A (at) the altitude function. Their values are given by the equations (14) and (14). The values of these functions for values of it from 3600 to IOO foot seconds were calculated by Captain James M. Ingalls, Ist Ar- tillery, U. S. Army, (now Colonel, retired), and form Table I of the Ballistic Tables. - Since V is a particular value of u, the values of the functions of V are included in the table as values of the functions of u. The quantities Q, Q', and 9", in the values of the functions, (14), are arbitrary constants; and the purpose of including them is to provide a means for avoiding abrupt changes in the tables at those points where in Mayevski's formulas the values of A and m change. CALCULATION of THE FUNCTIONS.—The method of employing the constants in forming the tables is best shown by an example. The value of the S function is, (I4), + (X I S (u) = ——— (u) (m-2) A un-2 For values of v greater than 2600 f. S., we have from May- evski's formulas, n = 1.55. Therefore for a velocity greater than 2600 f. s. - - Z/ 0.4 in I S (u) = — — —H Q^{= — Z/ 9.45 X - o.45 A. o.45 Al ( + Q ) In order to avoid the use of large numbers, Table I of the lat- est ballistic tables, published in 1900, is so constructed that the S, A, 15. and A. and T functions reduce to zero for u = 3600. I (u) reduces to zero for u = Co. We have then for S (u), when u = 3600 I ------0.45 -(3600 + QX ...A." + QX) S (u) = 0 = — and therefore * Q& = — (3600)* For any other value of u down to 2600 ...--— . - I - —” _- y S (u) = (3600 — u"*) = Aſ — A ſu” ( (21) / O.45 As. - S – For velocities between 2600 and 1800 f. s., n = 1.7, and I * S (u) - - -- (u" –H QA) - o.3 As sº, mºst have such a value as to make the value of S (u) for 24 * the same as the value determined from equation (21) with this value of u. Therefore 5 - *-*. º —0.4 (26oo + QN) = Aſ — A7 26oo o.3 A. - from which the value of Q.” can be determined. The same process is followed at each change in the values Qf it ; - .. vaº- zº / w - sº e :----., - ºr-- ~~1- ? *** When n = 2 equation (II) becomes indeterminatº but mak- £ 2'-." cº × 2 ing n = 2 in equation (IO) and integrating we obtain ſ A k . C. cºvvct'ſ * = — – (loge u — loge V) Cº-º-º-º-º/A.R. 2 C-gº & sº A * * ** : ; * * \ *_{ ~~~~ S (14) becomes in this case \s loge tº INTERPOLATION IN TABLE I:-This is effected by the ordinary rules of proportional parts. The difference between successive val- S(u) = 16 ues of u varies from unity in one part of the table to 2, 5, and Io in other parts. This difference must be carefully noted in interpo- lating. - * Formulas for the Whole Range. — Designate the whole range, fig. 3, by X, the corresponding time of flight by T, the angle of fall (considered positive for convenience) by Go and use the subscript o to designate the values of u and v at the point of fall. At this point y = 0 and 6 = – 69; and after combining equa- tions (17) and (18) to eliminate I (V) from (17), equations (15) to (19) become, respectively x – c s (u) – S (V) (22) C r=#: 7" (uo) —rº (23) cos @ º-º-º-ºº! Go Cº) – —— Z! * 2 * .…]”-sº-sº * sin 2 b = S(,) TS (V). T ( 25 tºo = zo, cos Go / cos ‘b (26) At the summit of the trajectory 6 = 0. Using the subscript o to designate the summit, equations (17) and (19) become, after reduction / (u,) = sin 2 b/C + 1 (V) (27) tºo = zºo/cos ‘b (28) Combining (27) and (25) we have * T S (uo) – S (V) 9 17 Therefore (24) and (25) become C 2 cos ‘b tan (c) - | I (uo) *- Z (uo) | (30) sin 2 b = C{ / (uo) — I (V) } (31) The Ballistic Elements. – The quantities C, u, V, b, 6, 69, 7' and X in the previous equations are called the ballistic elements. When referring to the end of the range they are written as capitals, or with the subscript w. For any other point of the tra- jectory they are written as small letters, with suitable subscript if desired. The subscript o always refers to the summit of the trajec- tory. The equations, by reason of Siacci's assumption for the value of cos”6, express the relations existing between these elements in direct fire only. When three or more of the elements are given the others may be determined. The Rigidity of the Trajectory. — According to the principle of the rigidity of the trajectory, which is mathemati- cally demonstrated, the relations existing between the trajectory and the chord representing the range, in direct fire, are sensibly the same whether the chord be horizontal or inclined to the horizon, provided that the quadrant angle of departure and the angle of position are small, or that the difference between them is small. That is to say that, considering b + 6 and 8 as small, in fig. I, if the trajectory and its chord bf were revolved about the point b until bf were horizontal, the relation of the trajectory to bf would not change. Therefore when the quadrant angle of departure, b + 8, is small We may consider bf, or any other chord of the trajectory, as a hor- izontal range; and we may apply to the trajectory which it subtends, the formulas deduced for a horizontal range. If however, the quadrant angle of departure is large, the prin- ciple of the rigidity of the trajectory applies only when the angle of position is also large, that is when b + 6 does not differ much from °. The principle therefore applies only to a part of the trajectory near the origin. This part may be treated as a horizontal range 18 whose angle of departure, later referred to as ‘ba, is the difference between the quadrant angle of departure of the horizontal trajectory and the angle of position. When the difference between b + 6 and 8 is small, b must be small. It is therefore evident that, in direct fire, the principle of the rigidity of the trajectory applies whenever the angle of departure is small. This principle enables us to use the elements calculated for a horizontal range, when firing at objects situated above or below the level of the gun. ,- d .* ~ ; , , , , , – “... < * , 'y º r i. Use of the Formulas.--The method of using the formulas . may best be shown by considering a problem. - - Problem 1–What is the time of flight of a 3-inch projectile Aweighing I 5 1bs., for a range of 2000 yards; muzzle velocity, I 7oo f. S. P - The given data are C = 15/9, W = 17oo, and X = 6ooo, the range being always taken in feet. T is required. These formulas apply: cosº, A (uo) — A (V ** = c/ (uo) – A ( |-ſº (25) C - - 7 = º 7" (to) – 7( º! (23) S’ (tºo) - S (V) x = c(S(u,) – S (V) (22) Take the T, S, A, and I functions of V from Table I. Determine S (uo) from (22). - - Find tºo, from Table I, and take from the Table 7" (tºo) and A (tºo). - Find b from (25). - Find T, required, from (23), Ams. T = 4.48 seconds. Secondary Functions. – The most important problems in gunnery may be solved by means of equations (22) to (31) and 19 ballistic Table I, but some of the solutions are indirect and tentative and therefore very laborious. The processes of solution have been greatly abbreviated, and the labor greatly reduced by the introduc- tion of secondary functions, whose values, for all the requirements of modern gunnery, have been calculated by Colonel Ingalls and collected in Table II of the ballistic tables. From equation (15) we have S (u) = x/C+ S (V) and substituting the values of S(u) and S (V), see (I4), ** I - Wº I * =sº sºmsºmºsºmsºmºmºmºmºsºmsº + – — (n-2) A zºn-2 C (n-2) A Vn-2 From this equation it is apparent that the value of the pseudo Velocity u, at any point, is a function of 4:/C and ſº only, and is independent of the height of the point in the trajectory. Make 2 = x/ C - Z = X/C It is apparent from (16), (17) and (18) that t, 6, and y are also functions of 2 and of V. The secondary functions, whose values are here given, are all functions of Z and V, and are tabulated with Z and V as arguments. - * (*) - 4 (tº , (v) S (u) – S (V) AE = Z (w) _A (u) – A (W) S (w) — S( V) A / = A + B = Z (u) — I (V) Tº = T(w) – 7 (V) B1 = B/A The subscripts are dropped in these expressions since they only 20 serve to indicate particular values of u, while the table contains the values of A, B, &c, for all the values of u. The table also contains, in the column u, the values of u for all values of Z and V. Equations (23), (24) and (25) may now be put, by reduction, into the following exceedingly simple forms. 7' = C 7'1/ cos q, (33) sin 2 q = A C (34) tan G0 =B / tan q = B C/2 cosº, (35) Equations (17) and (18) may also be put in the forms tan ºp anº — —I- (A — aſ) (36) y = * (A ––a ) (37) In these equations a and aſ are the values of A and A' corre- sponding to 2 = x/ C for the particular point of the trajectory con- sidered, while A and A / are the values corresponding to Z = X/C for the whole range. - - - At the summit tan 6 reduces to zero; and we obtain from equation (36) writing a'o for a' at the summit a'o = A (38) Equation (37) then becomes wº-ººººoº-o) - (38') From the third equation (32) we have for the Summit, b0=a'o-ao. With this relation and 20=aco/C, and making a”0=b920/a'o equation (38) may be reduced to the form ,-- vo-a"00 tan (p (39) go representing the maacimum ordinate. is.” To obtain a," for use in this equation, we find in Table II, in the A / column, the value of A as determined for the whole range. 21 With this value as A' and the given value of V we find a " in the A // column. Write Z — X/C - (40) z = u cosb/cost (41) 6, w c=ſ+ cal” (42) and Drift (yds)= | [379-39) C*/D//cos' (seacoast guns) (43) [3.92428] C*D// cos’q (field guns) which is Mayevski’s formula for drift, abbreviated for tabulation by N. Colonel Ingalls. The values of D/ are found in Table II. \,, We have in the equations (33) to (43) the principal formulas required for the solution of nearly all the problems of direct fire. - While the formulas apply strictly to direct fire only, where the values of q and 6 are such as to permit the use of Siacci's value of cost-16 without appreciable error, they give sufficiently accu- ~ & rate results for curved fire, and they are used for curved fire as well. For high angle fire they are modified. is, \\ The formulas are found assembled on page VIII of the book of sº ballistic tables under the heading “Formulas to be used with Table II,” so that this book contains all that is needed for the solution of most of the problems in gunnery. * -- The formula S(u) - Z + S(z) … . . . . . * . * &- which is another form of A = C(S(u) — S(V)} is included with the others. This formula, which requires the use of Table I, is sometimes convenient to use. To understand the additional formulas under this heading, it is Only necessary to know that 8 represents the angle of position; that ‘by, as explained under the rigidity of the trajectory, is the angle of departure for a horizontal range represented by a chord of another 22 * trajectory whose quadrant angle of departure is b, the chord being considered as practically equal to the value of a for its extreme point. The quantity a in these formulas is the particular value of A for the value of 4. Interpolation in Table II. — Exact formulas for in- terpolation in Table II are deduced and explained in a separate pam- phlet. These formulas, which are here written, will be used in place of the interpolation formulas given on page VIII of the ballistic tables, as the latter formulas are approximate only. Double Interpolation Formulas—Ballistic Table II. f= non-tabular value of any function corresponding to the non- tabular values V and Z. ſ = tabular value of function corresponding to tabular values V. and Z, always neart less than V and Z. // sº difference between velocities given in caption of table. Az, and A2% = tabular differences for f. /\z\, = tabular difference next following /\v, in same table. Tº indicates that function decreases as V increases, and increases as Z increases. Use the following formulas for the functions A, A ', B, 7”, log C/, and D / throughout the table. They also apply for some values of the functions A" and log B' when V-2500. .k. (—V) *–42. V–V, 23-4, * *(A Av.) (j-Z) = f; + IOO &o /. Z/6 T OO 2 Z/1 7'0 Z–Z, & f. -- A2, – f V = V, -- I OO X ſh 0 Z— 2, - Z\v, + (Azn – Z\v) IOO V— V. J-U--- Av. Z = Z., + |W – W. X I oo A2% º (Az', ºms Az) /. * 23 Use the following formulas for the functions A" and log B when V 3500, and for some values beyond that point. Gºv. Z— Z, , , V-V, 2–2, V-V. f}. Z) = f; + —A2, + Az', + O (Azºl-Az'.) - IOO - IOO Z– Z, f— [f. -- — A2, V = W. + IOO X /. -- 0 l. Z— Z, Az', -H (Av. - Avo) - IOO V— V, f– [ſ, H- →-Av. 2 Z = Z. -- V— l’. X I oo A2, + (Av. - Avo) / --- - / Use the following formulas for the function u. Z— Z. W– V. Z— Z, V– V. - £(+ V) – £– —/\2, + ——/\z, - Q (/\z, - /\z') e-z-y, IOO /? Ioo ſh Z— Z, - f— J. -*. A2, -*. - IOO IV = V. -- - X ſh Z— Z. Avo - (/\z, - Avi) * IOO - V – V. - f. -- h Av, ) – f 2 = Z., + — # = x .sº A2, + (Az', - Azh) 24 Inspect the tables to determine how the function varies with V and Z, and select the proper group of formulas. Exercise great care in the use of the plus and minus signs. As the numbers in the difference columns of the table are writ- ten as whole numbers, we must, when using the interpolation for- mulas, treat the tabular values of the functions as whole numbers, and afterwards put the decimal point where it belongs. Regarding the interpolation formulas, we will note that the pro- portional parts of the differences A2, and Av, are always ap- plied to the tabular value of the function, fo, with a sign indicated by the manner of variation of the function with Z and V respective- ly; positive if the function is increasing, negative for a decreasing function. The sign of the last term of the fiformulas is positive if the signs of the preceding terms are similar, and negative if they are dissimilar. - In the formulas for V and Z the fractional coefficients of h and W– V, Z—Z, 100 are equal respectively to — — and too These coefficients will always indicate by their values whether we are working with the proper tabular values. Numerator and denominator of the fraction should always be positive, and the value of the fraction less than unity. The deduction of the interpolation formulas was made by means of Fig. 4, and as this figure gives a conception of the construction of the table, and a clear idea of the method to be pursued in all double interpolation, it is here included. 25 * h N- gº Vo (V-Vo) V V2 ; : | | : | ; : fo | C. | *Av ! --— — t A. \4 At/0 (Z-Z.) | \ .d | > : `Azo \\ \ &| - º, : \ Y. - 100 & º! k \ \ ; : l * * * * * * * * * * * * * * |--|--|-- : |\, \ |\ : | X | | | | \ \ l \ \, N i \ ºf \ - Nº || | \ i n N \ i \ : \| : 770, --> i . `Av. ! ~T lºg ass, sº - - -ºr e º 'ºm' as sº a “ - - - - - - º ºs*X F G. 4. We may represent the value of a function of two independent variables by a right line drawn perpendicular to the plane which contains the axes of the variables. f. in the figure represents such a value, and the other heavy corner lines of the figure represent suc- ceeding tabular values in both directions. f is the interpolated value of the function for the non-tabular values V and Z. It is obtained from the four corner values (tabular) by applying the rule of pro- portional parts, first, to the two upper values, next to the two lower values, and finally to the two results of these operations. Interpolation in Table VI is performed in this way. It must be remembered that in the formulas the large letters represent values of the quantities for the whole range, or complete horizontal trajectory; while the small letters represent values of the same quantities for particular points of the trajectory. In the tables all these values are gathered in columns headed with the large letters, which are thus used in a general sense. In what follows, either in general discussions or when demonstrating the use of the tables, the large letters will be used. 26 The Solution Of Problems.-With the ballistic formulas and the tables, the solution of the problems of gunnery be- comes very simple. We will remember that all the functions in table II are functions of V and of Z = X/C, the arguments of the table. Therefore, given any two of the three quantities, V, Z, and a value of a function, the third may be determined from the table, and also the corresponding value of any other function in the table. For in- stance, suppose V and A/ are given and the corresponding values of . A”, log B' and 7" are required. With V and A/we obtain Z from the table, and with V and Z we obtain A", log B and 7”. Inspecting the formulas on page VIII of the tables, we select those that contain the given quantities, and such other formulas as, with Table II, will enable us to pass to the formula containing the required quantity. To show the advantages derived from the use of Table II with the abbreviated formulas, let us consider the problem whose solution by means of Table I has been indicated on page 18. - Problemn I.--What is the time of flight of a 3-inch projectile weighing I5 lbs., for a range of 2000 yds. ; muzzle velocity, 1700 feet 2 C = 1.5/9, W = 17oo, and X = 6ooo are given. T is re- quired. These formulas apply : T = C7'ſ sec q, (33) sin 2 q = AC (34) Z = X/C (40) Determine Z from (40). With Z and l’ take A and 7” from table II. Determine ‘b from (34). Determine 7" from (33). Ans. Z'E 4.48 seconds. Compare this with the process indicated on page 18. To show the most convenient method of performing the work, the solution of a problem is here given in full. - Problem 2–A 575 lb. projectile is fired from a 10-inch gun at a target 8000 yds. distant; muzzle velocity, 2540 f. s. What is the angle of elevation required, and what are the other elements of the trajectory P 27 • - No data being given for the determination of 61/6 or f. the . value C= w/dº is taken for the ballistic coefficient. Z= X/ C. log w 2.75967 2 log d 2.ooooo log C o. 75967 sº log X 4.38oz I log Z 3.62O54 Z = 4173.9 To find the angle of departure, use sin 2 b = A.C. From table II, with V = 2540 and Z = 4174 A = (o.o.3054) + .74 × 107 – 4 × 243 – 3 × 10 = oogo.33 The inclusion of the number in parenthesis is to indicate that, in applying the corrections, this number is treated as a whole number. 1og A 2.48187 log C o. 75967 1og sin 2 @ T.24154 2 b = Io" 2, (2 q, – 5” I / To find the time of flight. use Z = C7' sec (b. From Table II, with V and Z, T = (2.145) + .74 × 68 – 4 × 89 – 3 × 3 = 2.1588 log Z’’ o.3342 I log C o.75967 I.O.9388 log cos $ 7.99833 - log T' I.og S55 T = I 2.46 seconds To find the angle of fall, t1Se tan o'- A / tan b. ' “. . From Table II, with V and Z J. C AW, - A Vo) 64.4% /** 3. log B / = (o. 1513) + .74 × 38 – 4 × 12 + .3 = o, 1536% a 1og B o. 1536so log tan $ 2.9434o log tan Go T.og7odo 69 = 7° 8, 28 To find the striking velocity, use z = u cost secó. 6 in this case becomes co. From Table II with V and Z w = 1481 — .74 × 20 +. 4 × 66 = 1492.6 log tº 3. I7394 log cos ºf T.998.33 3. I 7227 log cos & 1.99663 1og z. 3. I7564 z = 1498 ſ. s. It is evident from these values of u and v that no material error is made by considering, for this shot, that u = w. - To find the marimum ordinate, use yo = a,” C tan (b. As already explained, see (39), we find the value of a," in this equation by means of the value A obtained from the equation sin 24 = AC. At the summit, see (38), a..' = A = sin 24/C This value of A is therefore the value of A' for the summit. Using this value of 4 in the A' column of table II, with the given value of V, we obtain from the A” column the value of a,". The value of A obtained above is O.O3O33. From table II with V= 2540 and A’ = o.o.303. Z - Z. 303 - (3oo – 4 × 24) , 7 I IOO 18 — 4 ac// = I2 oo + .7 I X 59 = I 241.9 1og doſ' 3.09.409 1og C o.75967 log tan $ 2.9434o log yo 2,797 I6 yo = 626.8 feet Problem 3–Compute the drift for the shot in problem 2. Use Mayevski's formula, D (yds.) = [3.792.39] C* D//cos' b. V = 2540 Z = 4174 b = 5° 1' log C= o 75967 29. * N From Table II Dº = 81 + .74 × 5 – 4 × 6 = 82.3 1og D’ 1.91540 2 log C I.5.1934 const. 10g 3.79239 I. 227 I 3 3 log cos ºf I. 99.499 log /2 I. 23214 D = 17 yards, Correction for Altitude.-The altitude factor / in the ballistic coefficient, see (42), takes into account the diminution in the density of the air as the projectile rises, and it corrects with sufficient exactness for the error that arises from the use of the standard density with which Table II is computed. When accuracy is desired the altitude factor is calculated and applied to the ballistic coefficient, in all firings at angle greater than IO degrees. Under the assumption of the mean height of the trajectory as two-thirds of the maximum ordinate, the value of the altitude fac- tor is given by the equation —t (44) log (log f) = logy, + sº The summit ordinate is, (39), yo = a,” C tan ºb As Centers the value of y, we must assume, for an approxi- mation, the value of C obtained by considering the altitude factor as unity. Call this value C. With C compute yo as explained in problem 2, and determine f from (44). Then applying the value of /, thus determined, to the assumed value C, a new value of C. C., is obtained with which a second approximation is made. The result- ing value of f is applied to the value C, first assumed, and the pro- cess is repeated until there is no material change between the cor- rected values of C, resulting from the last two operations. The final corrected value is then used as C. Problem 4.—Correct for altitude the ballistic coefficient of problem 2 using the angle of elevation there deduced; the atmos- 30 pheric conditions being, thermometer 49:6 degrees, barometer inches. We have from (42) 6, w Ó c d” C = f Consider small c unity. Pronn Table VI. We find for 6/6 the value I.OO3 1og 61/6 o.o.o 130 From problem 2 log w/d’ o.75967 log C. o. 76097 (1st approximation) log sin 2 b T. 241.54 1og A 2.48o A : O24 = a,’ g 4öO57 =02. 4. - With this value of A compute y o as explained in problem 2, and find log Wo 2.7974 I constant log 5.01765 log (logy) is sisoo log fooooss log C. o. 76097 log C. 6.76750 (2d approximation) log sin 23 F.24154 • , - log A 2.474O4 A = o.o.29788 Z — Z, 298 - (300 – 4 × 24) I OO 18 — 4 .43 Note that in this operation we have taken a tabular value, O.O3OO, for A, larger than the given value, O.O297, because the tabu- lar value when corrected for the variation in V becomes less than the given value. 31 V al,” = 1200 + .43 × 59 = I225.4 log a,” 3.08828 - 1og C. o. 7675o log tan ºb 3.91349 log ſy, 2.79918 …” const. 10g 5..or 765 log (1ogf) 3.81683 s log ºf •oss log C. o. 76097 log Co. o. 76753 As this value of C is practically the same as the value Cº, 2nd approximation, no further correction is necessary, and this value should be used for C to obtain a more accurate solution of prob- lem 2. - Thus, to get the true angle of elevation in that problem : log r 4.38oz I log C o 76753 log Z 3.. 61268 . Z = 4099 4 = (.92950) + .99 × 104 – 4 × 235 – 4 × 8 = 0,029sss log A 2.47068 log C o 76753 - log sin 2 b T. 2382 I - 2 q = 9° 58' q = 4° 59' differing by two minutes from the value of ‘b found with the ap- Proximate value of C used in the solution of problem 2, 32 The Effect of Wind.—In considering the wind we assume that the air moves horizontally, and that the effect on the velocity of the projectile is due to the component of the wind in the plane of fire only. We also assume, as practically correct, that the time of flight of the projectile is not influenced by the wind. Let W be the velocity of the wind in f. s. W, the component of W in the plane of fire. G: the angle, reckoned from the target, between the direction - of the wind and the plane of fire. Then W, - W cos oc. Call W, positive for a wind opposed to the projectile, and nega- tive for a wind with it. THE EFFECT ON RANGE.-We will assume that the effect of the wind component, W, is simply to increase or diminish the resist- ance encountered by the projectile; and that therefore this resistance, instead of being due to the velocity v, is due to the velocity (v-- W.). Represent by AX the correction to be applied to the range in a calm to produce the true range, this correction being the variation in range, with its sign changed, caused by the wind. We may put equations (23) and (22), when b is small and cosºp nearly unity, in the following forms; using the upper signs when the direction of W, is toward the gun, and the lower signs when it is toward the target : ... -- T (v -- W,) = 7/C + 7 (V-E W.) A x = C ( S (w = W.) – S (W = W.) ) - (x + TW) in which 7" (w = W.) and S (z = W.) are the 7" and S functions in table I. - - Compute the range X and the time of flight T without consider- ing the wind. Then from the first of the foregoing formulas find v + W., and from the second, the desired value of A 2. Another Method.—Let ob, fig. 5, represent the initial direction of the projectile and its velocity V. Let b c represent the velocity Referring to Fig. 6, let b represent the position of the gun, Fig. 6. and bá the range X in calm air. In the head wind the range is reduced to be. cd is therefore the variation in range due to the wind. While the projectile travels from b to c the air particle travels from b to a, the distance W. T. ac, or X', is therefore the distance that separates the projectile and the air particle at the end of the time T, that is, it is the relative |range of the projectile with respect to the air particle. The |relative initial velocity of the projectile is as shown in Fig. 5 |its velocity in a calm, V, increased by the component, A V, of |the air’s volocity in the direction of motion. V’= V+ A V is therefore the initial velocity necessary to produce the relative range, and simliarly q' = q – Aq) is the necessary angle of departure. - It is apparent from the figure that cd=bd–bo-bd– (ac—ab) !or col= X— (X'— W,T) land calling cd with its sign changed AX, we ha #. *-*-ºil– ve. \ \ • * ºr . . . . - 3. 4. . . ." º 33 F | G. 5. W, of the wind component in the plane of fire, reversed in direction. # While the projectile moves from o to b the air particle b moves to the left a distance equal to b c. The direction of movement of the projectile relative to this particle of air is therefore o c, which is also the relative velocity, P’’, of the projectile. ‘b’ is the relative inclina- tion, and A$ the relative change in inclination. Draw c d perpen- dicular to ob, and call ba', A V. Then, using the upper signs only, A W = W, cosºp º V/ = V + Z\ V (nearly) V' sin Aq = W, sing, q ( = q T /\b Compute the range X' with the values V' and ºb' using the formulas of table II. X' will be the range relative to the moving air. While the projectile is traversing this relative range the air particle moves over a distance W. Z'. The actual range traversed by the projectile is therefore Y' -E Wr 7 and the variation in range due to the wind is * - _X – (X, T W., 7') Changing the sign and re-arranging, we get Ax = x - (x + W.7") ſ in which X and 7 are computed from Vand b without considering the wind. The upper signs in the above equations apply when the wind 34 blows toward the gun, the lower signs when it blows toward the target. - APPLICATION OF METHODS.--There methods of obtaining the ºftºl Only when th9 angle of variation in range dye to wind ºne us# - roblem 5–Wh à One o'clock wind, blow- - twº trºpéthé fºrd ing 30 miles an hour, &n the range of the shot in problem I ? Velocity in miles per hour × 44/30 = velocity in foot seconds. W= 30 × 44/32 = 44 f s. oc = 30°, W = W coso. - 38.1 f. s. • * From problem I : log C = o 22 185, X = 6ooo, V = I 7oo, " – 4.48, q = 2" 42 /. - Therefore W, Z = 17o. 7, and X + W, Z = 617.0. 7. First Method. V -- We – 1738. I, From table I, S (1738. I) = 6220. 2 - .81 × 43.8 = 6184.7 7 (1738. 1) = 2 .508 – .81 × .oz.5 = 2.4878 log 7 o. 651 28 - - log C o. 22.185 log 7./C o,42943 - -- 7./ C = 2 - 688o 7 (1738. 1) 2.4878 T' (z) + W.) 5, 1758 From table I, z -- We – I I I 2 + 5. 189 – 5. 176 ... o I 8 X 2 F. I I I 3.4 and S (1 1 13.4) = 986.O. o – 14/20 × 20.6 = 9845.6 S (11 13.4) 9845-6 S (1738. I) 6184.7 log 3660.9 3.56359 log C O. 22.185 log 6 IOI.5 3.78544 A + W., T 617o.7 A X = – Ser8 feet. (97.2. 35 Second Method.—Find A V = 38. oé V/ = 1738. I A b = 3 /.6 - - $ 1 = 2" 38/.4 From sin 24 = AC A = o.o.5521 . From Table II Z = 3671.5 From Z = X// C A / = 6119. I A + W., T = 6176.7 A X = – 51.6 feet Note the difference in the results of the two methods. Neither method is very satisfactory. . . THE EFFECT OF WIND ON DEVIATION.—The component of the wind perpendicular to the plane of fire, W sin oc, is alone considered as producing deviation. The deviation due to the wind can only be determined by experiment for each kind of projectile. The following formula is given in the Coast Artillery Drill Regulations for the deviation of 8, 10 and 12 inch projectiles, due to a wind blowing normal to the plane of fire at a rate of Io miles per hour. 7. 2 Deviation (yards) = (; 3 + R/ a) in which - T is the time of flight. R the range in yards. .The deviation due to any other wind is proportional to the velocity of the normal component. Problem 6.—Compute the deviation of the shot in problem 2 for a two o'clock wind blowing 20 miles an hour. - W- 20 m. p. h. oc = 60° W, = W sin oc = 17.32 T = 12.46 A’ = 8ooo I 2.46 2 I 7.32 *-*-*- Deviation = 8ooo = 16 yards, left. IO 3.3 + I OOOO 36 The Danger Space—The danger space is the horizontal dis- tance over which an object of a given height will be struck. It is the horizontal length of those portions of the trajectory for which the ordinates are equal to and less than the given height. Usually the danger space at the end of the range is alone considered. The elements of the trajectory are assumed to be known. Substituting C2 for a in equation (37) we obtain by reduction (A - a) 2 = 2 y cos ºb/C (45) in which A is the value of the function for the whole range X, and a the particular value of the same function for the abscissa a cor- responding to the ordinate y. The elements of the whole range be- ing known, and y given, there remain two quantities, a and 2, to be determined from the equation. This is done by applying the rule of double position. - METHOD of Double Position.—From table II, with the given value of V, and an assumed value of Z, which we will designate Z, take out the corresponding value of a (from the A column). Sub- stitute these values, Z, and a, with the known value of A in the first member of (45). The difference between the resulting value of the first member and its true value, as given by the known second mem. ber, is the error e, which we will call positive when the first mem- ber is greater. Assume now another value of Z, Z, such that when substituted with the co-ordinate value of a in the first member of (45) it will make the error e, preferably less than €. Then the true value of Z may be obtained from the proportion. **** - - — - - e, – 2, #4- Z. :: es : Z. — Z As Z - Z is the correction that must be applied to Z, to pro- duce Z, we may enunciate this proportion as follows: “--- Rule of Double Position.—The algebraic difference of the errors is to the difference of the assumed values of Z, as the smaller error is to the correction to be applied to the corresponding assumed value of Z. To arrive at correct values in this way the errors must be small, and should be on both sides of the true value. From the value of Z= x/ C thus determined we obtain the value of a corresponding to the given abscissa 'y. --- f y | --~~~~ Let abo, Fig. 7, be the known trajectory for the range X, Fig. 7. and let y represent the height of the object for which the danger space is to be determined. The danger space for this height is evidently so much of the range that lies beyond the Ordinate y. It is equal to the whole range minus the abscissa a corresponding to the ordinate y. Calling the danger space AX we obtain AX= X—ac. . The problem of determining the danger space therefore consists in finding the value of a corresponding to the given value of y, and subtracting from the given range. Continue this process until there are found two values of Z, Za and Z, that give the smallest errors, e, and e, lying On both sides of the true value. The correction to be applied to Za to produce the true value of Z is then given by the following equation in which the signs of the errors e, and e, are not considered, their numerical values only being used: – “a *- AZa= €a-Héb (Z, -Z.) Then * Z= Za-HA2a. To make this demonstration general we will consider that We are determining the value of either one of two functions, f and f', whose product c is given. ff’-c. We may write either for f' for Z in the above equations, which then be- COOOle – “a * Af.- : *z, (f-f.) f=f.--Af. 37 ºt We then have for the danger space - AX = X – x. s Problem 7.—What is the danger space for an infantryman in the 1,000-yard trajectory of the new magazine rifle 2 This assumes that the rifle is fired from the ground. The height of a man is assumed at .# = 5.75 ft. F ſy. The value of C as given in the handbook of the rifle is O.39408. log C= 1.59558 V- 2300 ſ. s. X = 3000 We must first find A and b, in (45), as in problem 2. * . A = o.o.977.84 - q = 1° 6/ Then in (45) 2 log cosº T.99984 log 2 y I.o.6070 I.o.60.54 2 log C T. 19116 log (A * a) 2’ 1.86938 - (A -- a) 2 = 74, O3 We therefore have, (45), (A - a) 2 = (olºg; 78 – a 2 = 74.03 \ By inspection of table II for V = 2300 we see that the value of 2 = 7200 with the corresponding value of A, o. o8873, will give a close approximation. For Z = 72OO we obtain (o og 778 — o,08873) 72 oo F 65. 16 e = 65. 16 – 74. og = — 8.87 For Z = 7IOo we obtain - (o, og778 – o.o.S661) 7 Ioo = 79.3 . e, F 79.31 – 74. Og = + 5. 28 º Then se- . I4. I5 : Ioo :: 5. 28 : 37.3 … and - 2 = 7 Ioo + 37.3 = 7 137.3 log Z 3.85353 log C F. 59558 log & 3.449 II .x = 2813 feet. - AX = 30oo – 28 13 = 187 ft. = 62 yds. For V-2300 we will also find that the value 2=799.62 with • the corresponding value of a will satisfy the equation (A-a).2=74.03. This value of 2 gives ac-315 feet, which is, at once, the danger Space at the inner end of the trajectory. See Fig. 7. - . . . . ** \ 38 The Danger Range.—When the danger space is continuous and coincides with the range it is called the danger range. To determine the danger range we compute the horizontal tra- jectory whose maximum ordinate 4', is given. Combining equations (34) and (39) and making cos ‘b unity, since ‘b for all danger ranges is very small, we obtain - From this we determine a, by trial by the method just explained, using the Aſ and A/ columns of table II. Since at the summit a,' = A, see (38), with this value of a, we go to the A column of table II for the given value of V and find the corresponding value of Z, from which the required X is obtained. - - Problem 8. What is the danger range; for a cavalryman, Of the new magazine rifle, fired from the ground P The height of a cavalryman is assumed as 8 feet. yo = 8. W = 23 oo log C = I. 59558 log 2 yo I. 204 12 2 log C I. 19.1 16 log a,' a.” 2. or 296 ao' ao' F I O2 . O3 By inspection of table II for V = 23Oo we find that the pro- duct of A' and A' for Z = 31 oo will give a close approximation. for Z = 31 oo A/ A// = o . OS 73 × 1755 = 1 oo. 56 - e1 = I oo.56 — Iog.o.3 F – 2.47 for Z = 32 oo Al A// = o.o.6oo X 1818 = Io9.08 e, H Io9.08 – I of .og := + 6.05 Then for A/ 8.52 : 27 :: 2.47 : 7.8 - and A’ = a/ = or oS73 + .oOo.78 = o.o.5808 or it may sometimes be more convenient to find the value of Z and then the value of adſ, thus: for Z 8.52 : Ioo :: 2.47 : 29 Z = 31 oo + 29 = 31 29 and - a/ = o os 73 + .29 × 2.7 ... o.o.5808 - - 39 Using this value of a,' in the A column, we obtain 5808 – 5666 Z = 55oo + -— X Ioo = 5586 • . 165 6., log Z 3. 747 Io log C I.59558 log X 3.34268 A = 220 1 ft. = 734 yds. This means that at any point of the trajectory for this range a cavalryman would be struck. Curved Fire.-Problems involving angles of departure less than 25 degrees, and initial velocities less than 825 f. S. are solved by means of the first part of table II, pages 14 to 16, Ballistic Tables. The formulas to be used are collected on page VIII of the tables under the heading, “Formulas to be used with the first part of table II.” For velocities less than 825 f. s. the resistance of the air is as- Sumed to vary as the square of the velocity, or as it is called, accord- ing to the Quadratic Law of Resistance. Under this law the formulas for direct fire are capable of modification into the forms on page VIII that we are now considering. It may be shown that, under the quadratic law of resistance, the function A, for the same value of Z = X/C, that is, for the same range and projectile, will vary for different values of V in the ratio Vº/ V*. If therefore we obtain the values of A with the value V, and all the necessary values of Z, we can pass by means of the above ratio to the value of A for any other velocity. The value V = 8oo was used in calculating the part of table II that refers to velocities less than 825 f. s. - The value of sin 2 b, see (34), calculated for V = 800 becomes for any other velocity - – 8oo \ 2 A C S111 2 p. - A C Jy F [5.80618] º the form in which it appears among the formulas we are considering. 40 \. Under the quadratic law the other functions vary according to different ra º Of 4. angly V, as shºw, by the formulas in which they appear. e function B.iº independent of the muzzle velocity, and therefore V does not appear in the formula for tan G). Problem 9. A shot weighing 120 lbs. is fired from the 6-inch howitzer at a range of 2800 yds., muzzle velocity 750 f. s. What is the angle of elevation and the time of flight 2 - V : 750 A = 84oo C = w/dº log C- o. 52288 Use the formulas: - sin 2 b = [5.806 18] A C / W. “ 7 – [2.903.og] C7''/ V cosºp log Y 3.92428 log C o. 52288 log Z 3.4o 140 Z == 252 O A = o 13602 + . 2 X 590 - F o. 1372 o log A T. 13735 log C o.52.288 const. log 5.806 18 5.46641 2 log | 5.75o 12 log sin 24 T. 7 I 629 2% = 31° 21' + q = 15° 4 1/ 7 / = 3.31.5 + .2 × 141 -- 3.34.32 log 7’’ o. 524 16 log C o. 52288 const. log 2.90309 3.950 I 3 log V' 2.87.506 log cosº I. 98352 2.85858 2.85858 log T b.og 155 7 = 12.35 sec. l 41 High Angle Fire.-Problems in high angle fire are solved by means of table IV. This table was computed under the quadratic law of resistance and is practically a range table for velocities less than 825 feet, for a projectile whose ballistic coefficient is unity. To make it applicable to other projectiles the tabular numbers involve the value of the ballistic coefficient with the values of the different elements. Therefore with C known, and applied as indicated in the headings of the columns, we may, with any other known element of the trajectory in addition to the elevation, obtain from the different columns the values of the remaining elements. Thus C, b, and V being known, find V/v/C and take out of the tables for the particular value of $, the values of X/0, 7/1/C, etc., corresponding to V/v/C as obtained. X, T, etc., may then be obtained. If h is not a tabular value, solve the problem for the tabular values of q on either side of the given value, and interpolate between the results. To correct for altitude use the formulas log (log f) given at the head of each table. The value of the maximum ordinate is also there given in the terms of the range. While the quadratic law of resistance applies to velocities less than 825 f. S., table IV may be used for the higher velocities now Obtained from our mortars by the introduction of the coefficient of reduction c into the ballistic coefficient. Compensation may thus be made for the errors arising from the use of the table for higher velocities. The values of c for the 800 lb. mortar projectile have been calculated from actual firings for different ranges and angles of elevation. - Values of the coefficient of reduction, c. for the 800 lb. pro- pectile for 12-inch mortar. - - - RANGE IN YARDS. | 4ooo sooo | 60oo 7ooo sooo 90oo Ioooo | - 45° I. I9 I. OI I.OI I. OO I. OI I. O5 I. I.4 46 I. 20 | I. O3 || I. O2 I. OO I. OI I. O7 || I. I5 47 I. 21 | I. O6 | I. O3 | I.OI I.O2 I. O9 I. I6 48 I. 23 I.O8 I. O4 I. O2 | I.O3 | I. II | I. I8 49 I. 25 I. II | I. O5 I. O3 I. O4 || I. I3 | I. 20 50 | I. 27 | I. I3 I. O7 I. O5 I. O6 i.15 | 1.22 5I I. 30 I. I5 I. O8 I. O7 I. O9 I. I6 I. 24. 52 I.33 I. I7 || I. IO I. IO I. I2 I. I8 I. 26 53 I. 36 I. I9 I. I2 I. I3 I. I6 I. 20 | I 29 54 I.39 I. 2I I. I5 I. I6 | I.20 I. 23 I. 32 55 I. 42 I. 24 I. I8 I. I9 I.24 | I. 27 | I. 35 56 |. I.45 I. 27 | I. 22 || I. 22 | I. 28 I. 32 - 57 I 48 I. 3O | I. 26 I. 26 | I. 32 | 1.37 58 I. 5 I I. 33 I. 3O | I. 3O | I. 37 I.43 \, ', 59 I. 54 || I. 36 | I. 35 | I. 34 I. 42 | I. 50 | * 4 6O I. 56 I.40 | 1.40 I. 39 I.47 | I.58 PROBLEMS IN HIGH ANGLE FIRE —When C, b, and V or X are given, to determine the remaining elements. - Problem I.O. — Given d = 12", w = Iooo 1bs., b = 58°, V = 1 150 f. s., and c = 1, 1 I. Correct once for altitude and find A , 7, 69 and zo. *** y •º ºve-sº C = w/cd” 1og C = o. 796 V/l/ C = 459. From table IV, for b = 58°, we find with this value of V/w/C A / C = 4572 + .98 × 164 = 4732.7" log X/C 3.675 II 1og C o. 79632 log & 4.47 143 const. 10g 5.34888 log (log f) T. 1.2255 log f o. 1326o log 0 o. 79632 log C. o. 92892 43 ! } We will hereafter use this value as C. - - , (o We find V/v/C= sºft# - From the table, with this value, as an argument, we obtain : X/ C = 3603 +% X 159 = 36%l 7/1/C = 19.6 + 4. . 4 + 1933% vo/v/C+= 343 + #/. * = a&# A *- a = 61° * +! … = or 32.” - From these "...]” derive (a 1.7-S DY. Ø f £70 2% = 3rºes, ft. = .#yds. T = 57.55 sec. zºo, F * f. s. When C, V, and X are given, to find ºb. Problem II. An 800 lb. projectile is fired from the I2 inch mortar at a range of 7700 yds., muzzle velocity 950 f.s. What is the angle of elevation required? The value C = w/dº is first used to determine X/ C and . V/v C. - log C = o. 74473, X/C = 4.158 V/v/C = 403.04 These values of X/C and V/VC are used only to find an ap- proximate value of $ in table IV. We look through table IV and find, under b = 51°, that the value of X/C corresponding to V// C = 403.o.4 is very nearly 4.158. We will therefore use this table and the next one, b = 52°, for further computations. In each table, that is, for each value of b, we will find the value of V that will produce the given range; and then, considering the given value of V as an interpolated value between those found from the tables, we obtain the corresponding value of ºb. First correct C, by c and f. From the table of values for c we find, with X = 7700 yds. For b = 51° c = I. ob4 For p = 52° C = I. I. I.4 With X = 23 Ioo * 44 (q = 51°) log A 4.36361 (q = 52°) 4.36361 const. log 5.46384 5.45087 log (log f) 2.89977 2.91274 log ºf o.o.7939 o, o&I 8o log C o. 74473 o. 74473 c log c T. 96497 I. 953 II log C. o. 78.909 o, 77.964 With A = 23 roo and these new values of C find A ZºC = 3754. 2 = 3836.8 Corresponding to these values we find, from the respective tables, V/l/ C = 379.77 . = 386. 4's and V = 942. o.4 - F 948 3:? As these values of V are both on the same side of the given value, V+= 950, and less than it, it is apparent that the true value of § 1ies between 52° and 53°, and we must make a further compu- tation with the table q = 53°. We will obtain from this table, by the process just pursued, the value V= 955.83, which is greater than our given value, V+= 95o. - With 950 as an interpolated value between 948.ii and 955.83, to which correspond respectively the values of $ 52° and 53°, we find .1 to 1. , 13% q = 52° + — x 1° = 52° -º-º: 7 ºffº If we had used the wºe of V first found, we would have for p . (> - º - - I. = 52° + l, X I o E sº 6.e.7 - - 2. O THE CoEFFICIENT OF REDUCTION.—A recent addition to table IV, provides a simple method of computing the coefficient of re- duction for any projectile, when b, V and X are determined from actual firings. - - 45 A column containing values of Vºx, obtained by combining the two columns V/v/C and X/C, is added to the table. With # and V*/X as arguments, we may obtain C from the value in the column V/V/C. The value of C thus obtained is the complete 6, w 3 ca’ of the table, and 6,70 from table VI. c is then readily deter- mined. . • .. 2 value, C = f Determine f from the formula at the head Problem 12. Compute the coefficient of reduction for the Looo lb. projectile for 12 inch mortar, when V = rooo f. s. ; % = 59°; A = 778o yds.; thermometer, 6.5 degrees; barometer, *3r inches. - 33 St First obtain V*/A = 42.85 From table IV for b = 59°, with this value of V*/A, we find V/l/ C = 386.6. ." log V 3.ooooo log V/l/C 2.587.26 log V C o.41274 log C o.82 548 To determine f log AT 4.368 Io const, 5.329I4 log (log f) f.o.3896 - log ºf o. Io939 From table VI, log 6, A 6 ſ.98989 log w/d * o.84164 clog C T. 17452 - --- log c o. I I544 C = I. 3O45 Perforation of Armor—The following empirical formulas are used by the Ordnance, Department, U. S. Army for calculating perforation of the earlier Krupp armor. Uncapped projectiles, T r. ... Z0Z) Capped projectiles, - O. 5, t” = [H.84060] * *. go. 75 in which, * l # = thickness perforated, in inches, w = weight of projectile, in pounds, w = Striking velocity, in foot seconds, = diameter of projectile, in inches. The following formula has been proposed by the Ordnance Board for capped projectiles against thin plates: - f o. 7 w°-52, º - - [4. 92665] S11] CC go. 75 in which oc is the angle of impact, that is to say, the angle between the axis of the projectile and the face of the plate. This formula is applicable to tempered nickel steel plates from 3 to 4% inches thick, and for angles of impact varying from normal to 50 degrees. The following formulas are used by the Bureau of Ordnance, U. S. Navy, for calculating the perforation of face hardened armor without backing. They apply to Harvey armor only. No formula satisfactory to the Bureau has yet been developed for the perforation of the most modern Krupp armor: - Uncapped projectiles, - - a * , 34 v = [3,34512] w” Capped projectiles, % . ; . . f v = [3.253.12] ZU 2 in which the letters represent the same quantities as in the formulas above. - - " * 47 The formula for capped projectiles is tentative only. Range Tables.—The elements of the trajectories for different ranges are calculated for each gun in the service, and embodied with other information in a range table. The standard muzzle velocity and standard weight of projectile are used in the construction of the table for each gun. The range is the argument in the table, the suc- cessive entries in the range column differing from each other by 200 yards. The perforation of armor and the logarithm of the ballistic coefficient, corrected for altitude at standard temperature and pres- Sure, are entered at intervals of 1,000 yards. The Construction of range tables will be understood from the following data taken from the first line of the range table for Io- inch rifle: * Muggle Velocity, 2250 f. s. Projectile, capped, 604 lbs. Range, X, * * - - * - Iooo yards Angle of departure, ‘b º * - - o° 34/. I Change in elevation for Io yds. in range, - - - O/.4 Time of flight, 7. - - wº - I. 37 seconds Angle of fall, Go, * - - - * o° 36/ Slope of fall, * - - * º I On 95 Maximum ordinate, yo, - - * - 8 feet Striking velocity, v, & ~ * * - 2 I I6 f. s. Perforation of Krupp armor, impact normal, I3.3 inches - 30° with normal, II. 2 inches Ballistic coefficient, log C, * * * o. 781 12 Curvature of the Earth.-The angle of elevation is affected by the curvature of the earth about 15 seconds of arc for each IOOO yards of range. The amount of curvature, in feet, is approximately two-thirds the square of the range in miles. * * ,* *, , ， , ,-|- - ~*ș -→și s-• |-|- - |--ÂşTš ș*. • !---- ∞- ‘í|-- { · · · ·\,?ç; ' , : * «ſø- |-ș** � -● -* +ę-*· ?� +-} ș -is � -* ·z• • - ~|-} -• * r - • .*, , -، …|- ►ș.*→|- ·, ！ '-- → . ę�*|- *.~ - +** -} +''- , ， ,-4 *• ----į,· } ',|- ·· ·\ a^3* •* . ^ •* *s→ ·ș -●|- ș *. -* . . -.-'） - * ： ? » |- • ! �-* �·|- · *£ . ~} *|- �·|- rº|- ！ ∞ *· ſą • |-→ſą ‘, • |-� *• |- * -• * �*• PRIMERS AND FUZES FOR CANNON By Major ORMOND M. LISSAK, ORDNANCE DEPARTMENT Instructor of Ordnance and Gunnery, U. S. M. A. RIMERS are the means employed to ignite the powder charges in guns. They may be divided, according to the method by which ignition is produced, into three classes : Friction primers, Electric primers, • Percussion primers. Combination primers are those so constructed that they may be fired by any two of the above methods. Primers that close the vent against the escape of the powder gases are called obturating primers. All primers should be simple in construction, safe in hand- ling, certain in action and not liable to deterioration in store. Electric primers in addition should be uniform as to the electric current required for firing. The primer known as to the common friction primer, for- merly used in all cannon, is shown in figure 1. - a! & | | | | | & FIG. 1. . . The body band the branch d are copper tubes. The tube b is filled with rifle powder, and is closed at its lower end by a wax stopper 0. The tube d is filled with the friction composi- 2 PRIMERS AND FUZES FOR CANNON tion, whose ingredients are chlorate of potash, sulphide of antimony, ground glass and sulphur, mixed with a solution of gum arabic. Imbedded in the friction composition is the ser- rated end of the copper wire c, the other end of the wire being formed into a loop for attachment of the hook of the lanyard. The outer end of the tube d is closed over the flattened end of of the wire which is bent over into a hook, as shown, and serves to hold the wire securely in place except when a stout pull is given to the lanyard. The pull on the lanyard straightens out the hook, and draws the serrated wire through the friction composition, igniting it. The fire is communicated to the rifle powder in the tube b, and thence through the vent to the pow- der charge in the gun. e For use in axial vents, in order to prevent the primer being blown to the rear among the men of the gun detachment, a coiled copper wire e is added to the primer, one end of the wire being made fast to the top of the primer body, the other end to the loop for lanyard hook. The coil is extended by the pull of the lanyard, and the primer when blown to the rear remains attached to the lanyard. - Obturating primers.--The primer above described is blown out of the gun by the explosion of the powder charge, leaving the vent open for the escape of gas. The disadvantage is overcome in modern practice by the use of obturating primers. The breech mechanisms of all guns now made are adapted to obturating primers, and the primer just described is no longer used in service cannon. Screw friction primer. —This primer, figure 2, has a brass body i bored as shown. A pellet of friction composition d is &| a! .l f * . | | | | | CZ moulded around the shank of the serrated wire g just above the Serrations. A paper cylinder e encloses the composition to prevent disintegration. The priming charge is composed of the two cylinders of compressed powder, b and c, and the loose rifle powder, which facilitates ignition. The safety block h, Soldered to the wire g, prevents any forward movement of the PRIMERS AND FUZES FOR CANNON 3 ſ O wire through the pellet, which might cause premature firing in transportation or handling. The conical brass gas check f is loose on the wire g. A brass cup a, shellacked in place, closes the mouth of the primer. The primer is screwed into its seat in the gun. When, by the pull of the lanyard, the serrated wire is pulled quickly through the pellet of friction composition, ignition occurs. The gas check f comes to a bearing in the coned seat in rear and prevents the escape of gas through the body of the primer. The primer fits closely in its seat in the gun, and at discharge the thin walls of its mouth are expanded against the walls of the primer seat, preventing the escape of gas around the body of the primer. This primer was formerly in use in all siege and seacoast cannon. It has been Superseded in Seacoast cannon by the combination primer described later, but its use will be con- tinued in the 3.6-inch and 7-inch mortars. To assist in increasing the rapidity of fire of all guns a primer that can be more readily inserted in the gun is required. The desired object has been attained by the addition of firing mechanisms to the breech mechanisms of most guns, the firing mechanisms being so designed as to permit the use of a smooth- sided primer that can be readily pushed into its seat. The head of the primer is firmly held by the firing mechanism, so that the primer cannot be blown out on the discharge of the piece. The firing wire is engaged and pulled by a slotted lever actuated by the pull on the lanyard. Friction primer, latest patterm. —The primer, figure 3, has a body h of brass. The brass firing wire i passes loosely FIG. 3. through the hole in the serrated cylinder g, the end of the wire being flush with the end of the cylinder when the nut on the wire bears against the interior shoulder of the cylinder. The friction composition, pressed into the brass case e, surrounds the cylinder g above the serrations. The vulcanite washer f holds the friction composition in place and prevents it from crumbling 4 PRIMERS AND FUZES FOR CANNON when the pull is applied. The nut d, screwed to a bearing on the case e, holds the assembled parts in place. Three holes through the nut permit the passage of the flame from the fric- tion composition to the priming charge of powder. When the wire i is pulled, ignition of the friction composi- tion is effected. The conical end of the cylinder g is pulled to its seat in the body of the primer, and prevents escape of gas to the rear. Should the primer for any reason fail to fire, the wire i is now free to move forward without carrying the cylin- der g and the friction composition with it, and therefore with- out danger of firing the primer in its reverse movement. In earlier models the teeth were formed on the wire, and it was found that when a primer had failed to fire it might be fired by an accidental reverse movement of the wire forcing the teeth quickly through the composition. All metal parts of this primer are tinned to prevent corrosion. FIG. 4. Figure 4 shows the more cheaply constructed drill primer of this form. Electric primers.-The electric screw primer, figure 5, is used in the 3.6-inch and 7-inch mortars, these guns being * ! º }s , . . y FIG. 5. adapted for Screw primers only. The single copper wire j, in- sulated with silk except at its Outer end, passes through the vulcanite bushing i and the body of the primer to the brass ob- turating plug f into which it is screwed. The plug is insulated from the primer body by the vulcanite washer h, the leather washer g, and the Vulcanite cylinder e. The platinum wire bridge d, 0.002 of an inch in diameter, is soldered to the plug f and to the brass washer c. The latter is put in electrical con- PRIMERS AND FUZES FOR CANNON 5 nection with the walls of the primer by the brass closing screw b. A small quantity of guncotton surrounds the platinum WIY’e. - When the primer is inserted in the gun the base end of the wire j is grasped by the parts of an electric contact piece through which is passed in firing an electric current insulated from the gun. The current passes through the wire j, the platinum bridge and the body of the primer to the walls of the gun, and thence to the ground. The passage of the electric current heats the platinum wire, igniting the guncotton and the priming charge of powder. Another electric primer for use in a different breech mech- anism is described after the 110-grain percussion primer. Combination electric and friction primer.—This primer is used in all seacoast cannon except those fitted for percussion firing. FIG. 6. The primer is shown complete in figure 6. The igniting elements enlarged are shown in figure 7. 2. - The parts of the friction elements of this (*= primer are similar in construction and action to the parts of the friction primer \!. | | shown in figure 3. Prºr-tº- | For electric firing the wire k is & & d a f g h 4 covered with an insulating paper cylin- FIG. 7. der j and enters the primer body through a vulcanite plug i. The wire is in electric contact with the serrated cylinder h, figure 7, but this is insulated from the primer body by the Vulcanite washer g and the pellet of friction composition, a non-conductor of electricity. The electrical elements of the primer are assembled in the metal case f. The head of the forked metal support e is in con- tact with the headed end of the wire k, but not fastened to it. The forked end of the support is held in the vulcanite cup c. The brass contact nut b, screwed into the end of the case f, presses the assembled parts into intimate electrical contact. A platinum wire d is soldered to the head of the support e and to 6 PRIMERS AND FUZES FOR CANNON the contact nut b. An igniting charge of guncotton surrounds the wire. The electric current enters the primer by means of the button on the outer end of the wire k, and passes through the primer and gun as described for the previous primer. It will be observed that the friction elements of the com- bination primer are independent of the electrical elements, and that when one of these primers fails to fire by electricity it may still be fired by friction. If, however, the primer fails in an attempt to fire it by fric- tion, it will not generally be possible to fire it electrically since the cylinder h, which has been pulled into the head of the primer, is out of contact with the part e and the platinum wire bridge. The current will then pass directly from h through the primer body and gun to the ground. - The primer should in this case be at once removed from the vent, and not be again used. The outer button and wire k may be turned without danger of breaking the platinum wire bridge d. When an electric or friction primer fails to fire, it should be removed from the vent, and the wire bent down and around the primer to prevent attempts to use it again. Percussion primers. —The friction and electric primers de- scribed are used in guns in which the projectiles and powder charges are loaded separately, the primer being separately in- serted in the breech block. Percussion primers, and the elec- tric primer described with them, are, on the other hand, inserted in cartridge cases, in which are usually assembled both the projectile and the powder charge. The essential parts of a simple percussion primer such as the cap in a small arm cartridge, are the primer cup, the anvil, and the percussion composition. Formerly the percussion composition of all service primers contained a large percentage of fulminate of mercury. On account of the danger involved in handling mixtures containing the fulminate of mercury, its use as a primer ingredient in ser- vice primers manufactured at the Frankford Arsenal has been abandoned, and a mixture known as the H-48 composition is now employed. This mixture contains the same ingredients as the friction composition, but in different proportions, as follows: Chlorate of potash, 49.6. Ground glass, 16.6. Sulphide of antimony, 25.1. Sulphur, 8.7. º . : * f : f j ! PRIMERS AND FUZES FOR CANNON - 7 To insure the practically instantaneous ignition of smoke- less powder charges, the addition of a small charge of quick- burning black powder is required. This may be inserted in the base of the smokeless powder charge, or may be contained in the primer. It is desirable, on account of the smoke produced by black powder, and the fouling of the bore, that the quantity of black powder used be limited to the smallest amount that will produce prompt and complete ignition of the Smokeless powder. The minimum amounts required for different charges have been determined and, for fixed ammunition, are contained in the percussion and igniting primers. These primers are inserted in the head of the cartridge case, in the position occupied by the primer in the small arm cartridge. Two sizes of percussion primers, the 110-grain and the 20- grain have been adopted for all guns from the 1-pounder to the 6-inch Armstrong inclusive. 110-grain percussion primer.—The body f is of brass, 2.93 inches long. A pocket is formed in the head of the case for the reception of - the metal cup e U__U__U_HIV *** *-*. ſ containing the tº - . EAT/h, percussion com. --→-- | | | position d. Pro- | | | jecting up from & & & d e f the bottom of FIG. 8. the pocket is the anvil c against which the percussion composi- tion is fired. Two vents are drilled through the bottom of the pocket. The priming charge consists of 110 grains of black powder inserted under high pressure into the primer body around a central wire. The withdrawal of the wire after the compression of the powder leaves a longitudinal hole the full length of the primer. Six radial holes are drilled through the walls of the primer and through the compressed powder. The compression of the powder increases the time of burning of the priming charge and causes the primer to burn with a torch-like rather than an explosive effect, making the ignition of the Smokeless powder charge more complete. The holes through the priming charge increase the surface of combustion and the mass of flame, and direct the flames to different parts of the charge of powder, thus facilitating its complete ignition. The paper wad a, shellacked in the mouth of the primer, and the tin-foil covering b, serve to keep out moisture and to protect 8 PRIMERS AND FUZES FOR CANNON the primer from the impact of the powder grains when trans- ported assembled in cartridge cases. This primer is used in cartridge cases for guns from the 6- pounder to the 6-inch Armstrong gun inclusive. The 20-grain percussion primer, shown in figure 9, length 1.1 inches, is used in cartridge cases for 1-pounder subcaliber tubes, 1-pounder machine guns and 1.65- FIG. 9. inch Hotchkiss guns. 110-grain electric primer.—This primer, figure 10, is similar in form to the 110-grain percussion primer F- just described, and has the same priming charge similarly arranged. Ignition is produced electrically through the brass cup g to which one end of the platinum wire e is soldered. The cup is insulated / Y from the body of the primer by the | H cylinder f and bushing d, both of vul- & d 4 f & canite. The brass contact bushing c, to FIG. 10. which the other end of the platinum wire is soldered, completes the electrical connection. 20-grain saluting primer.—This primer, figure 11, costing less to manufacture than the 110-grain primer, is to be used in place of the latter with blank charges only. The primer contains a charge of 20 grains of §: § g § º § loose rifle powder. As black powder only is used in blank charges, a smaller igniting FIG. 11. charge answers. The percussion primers and the electrical primer of the same form, are so manufactured as to have a driving fit in their seats in the cartridge cases to which they are adapted, the diameter of the primer being from one-and-a half to two thousandths of an inch greater than the diameter of the seat. Special presses for the insertion of the primers are provided. The primer must not be hammered into the cartridge case. The primer seats in all cartridge cases using these primers are rough bored to a diameter about 20 per cent. less than the finished size, and then mandrelled to finished dimensions with a steel taper plug, to toughen the metal of the cartridge case around the primer seat. The toughening is necessary to prevent expansion of the primer seats under pres- PRIMERS AND FUZES FOR CANNON 9 sure of the powder gases, and consequent loose fitting of the primers in subsequent firings. Combination electric and percussion primer.—In figure 12 ºmº" is shown a combination electric and = percussion primer used in rapid-fire º # Wſ à % # º % N ºf º #3 4:37 § . º 2. Ø } § (§ guns in the U. S. Navy. Its con- § tº struction can be readily understood – from the figure. The insulation is FIG. 12. shown by the heavy black lines. When fired by percussion the percussion cap is not directly struck by the firing pin, but by the point of a plunger forced inward by the blow. Igniting primers. —The igniting primers are for use in cartridge cases for subcaliber tubes for Seacoast cannon not pro- vided with percussion firing mechanism. They contain no means of ignition within themselves, but require for their ignition an auxiliary friction or electric primer which is inserted in the vent of the piece in the same manner as for service firing. The flame passes from the service primer through the vent in the breech block to the igniting primer in the head of the cartridge case. The flame from the service primer would not be sufficient to ignite properly the smokeless powder charge in the cartridge case, and, therefore, the igniting primer is added. Sºº-Sº Hi-º-º-º: Sº--a-S- --S Y | FIG. 14. & & FIG. 13. The 110-grain and the 20-grain igniting primers, figures 13 and 14, differ from the corresponding percussion primers in the substitution of the obturating cup a and obturating valve b, both of brass, for the percussion cup and anvil. The obturat- ing cup a is provided with a central vent to allow passage for the flame from the auxiliary primer. The obturating valve b is cup-shaped, and has three sections of metal cut away from its top and sides to allow passage of the flame. The valve b has a sliding fit in the cup a, and when the pressure is greater in front of the valve than behind it, the valve is forced to the rear and the Solid top of the valve closes the vent in the outer cup. The valve is shown in section in figure 13, in the position it 10 PRIMERS AND FUZES FOR CANNON assumes after firing ; and in elevation in figure 14, in its posi- tion before firing. , \ - N FUZES : -----> *~. ~. -- S. >s ſº..." } º' Fuzes are the means employed to ignite the bursting … /charges of projectiles at any point in the flight of the projectile, or on impact. - They are of three general classes: Time fuzes, Percussion fuzes, - Combination time and percussion fuzes. All fuzes should be simple in construction, safe in handling, certain in action, and not liable to deterioration in store. In addition the rate of burning of the time train of the fuze must be uniform. - The time fuze alone, that is, without percussion element, is no longer used in modern ordnance. Percussion fuzes, –A percussion fuze is one that is prepared for action by the shock of discharge, and that is caused to act by the shock of impact. - When ready to act, as after the shock of discharge, the fuze is said to be armed. - Percussion fuzes are inserted at the point or in the base o the projectile. In some of the projectiles for guns of minor caliber, including the 6-pounder, the fuze is inserted at the point ; in others at the base. The percussion fuzes for field, siege and seacoast projectiles are base insertion fuzes. The percussion fuze consists essentially of the case or body of brass, which contains and protects the inner parts and affords a means of fixing the fuze in the projectile ; the plunger, carrying the firing pin and provided with devices to render the fuze safe in handling ; the percussion composition, which is fired by the action of the plunger on impact ; and the priming charge of black gunpowder. The percussion composition of all service fuzes manu- factured at Frankford Arsenal is the same. The ingredients are chlorate of potash, sulphide of antimony, sulphur, ground glass and shellac. The thoroughly pulverized ingredients are mixed dry, and alcohol is added to dissolve the shellac. The percussion pellets are formed by pressing the mixture while in a plastic state into the percussion-primer recess. Upon the evaporation of the alcohol, the shellac causes the pellet to adhere to the metal of the recess. PRIMERS AND FUZES FOR CANNON f 11 : A mercuric fulminate percussion composition was formerly used in fuze primers, but on account of the danger incident to handling this compound it has been abandoned as a primer ingredient. - It is still used abroad, and the percussion composition of both the Ehrhardt and Krupp combination time and percussion fuzes contains mercuric fulminate. Point percussion fuze, for minor caliber shell.-These are adapted to the projectiles for 1-pounder, 2-pounder and 6- pounder guns. FIi G€” 3. }* & : FIG. 1. FIG. 2. The body a, Fig. 1, is of brass. Into the recess formed in the head is screwed the cup-shaped primer screw e, Fig. 3, into which are previously assembled first the brass primer shield g and then the primer cup d, Fig. 3, containing the percussion composition f and the charge of black powder c. A tin-foil disk b closes the mouth of the primer cup. The primer screw e has a central hole through the bottom which permits the firing pin to reach the percussion composition. The primer shield g prevents any dislodgment of the composition during trans- portation, or by shock of discharge, and also restrains the firing pin during flight of the projectile. The primer cup d has two chambers separated by a solid partition, through which are bored two circular vents. The lower chamber, 0.03 inch deep, holds the percussion composition, and its wall is undercut to assist in holding the composition in place. The primer cup is 0.03 of an inch longer than the recess in the primer screw e, so that when the latter is screwed down hard the primer cup bears against the head of the fuze body. Contained in the body of the fuze is the plunger, which consists of the firing pin j, the firing-pin sleeve h and the split- 12 PRIMERS AND FUZES FOR CANNON ring Spring k, all of brass. The firing pin has an enlarged rear part joined to the forward part by a conical slope, and provided near the bottom with a groove l of diameter slightly larger than the diameter of the forward part of the pin. A radial hole i through the pin near its forward end, and an axial hole from this point to the rear end of the pin, provide a pas- Sage for the flame from the priming charge. The rear part of the bore through the sleeve, h, is of diameter just sufficient to admit the split ring, which rests against the forward shoulder of the counterbored recess, and holds the firing so that its point is wholly within the sleeve. The front part of the sleeve is counterbored to permit ready entrance of the flame from the priming charge into the passage through the firing pin. The plunger thus assembled is placed in the fuze body, which is closed by the brass closing screw m, provided with a central vent which is in turn closed by the brass disk m. To prevent pressure of the closing screw on the plunger, which might cause expansion of the split ring and the arming of the fuze, the plunger is allowed a longitudinal play in the fuze body of from one to two hundredths of an inch. With the parts of the fuze in this position the point of the firing pin is prevented from coming into contact with the percussion composition, and, therefore, the fuze cannot be fired. If sufficient force is applied rearwardly to the sleeve, h, the split ring, k, will be forced over the enlarged portion of the firing pin until it rests in the groove l, near the bottom ; and the sleeve, moving to the rear, will expose the point of the firing pin. The fuze is then armed, as shown in Fig. 2. To insure arming of the fuze when fired the resistance of the split ring to expansion is made less than the force necessary to give the sleeve the maximum acceleration of the projectile. Therefore when the piece is fired and while the projectile is attaining its maximum acceleration, the pressure of the sleeve will force the ring over the enlarged part of the firing pin into the groove at the rear. The diameter of this groove being greater than the diam- eter of the front part of the firing pin, the ring is now. expanded into the counterbored recess in the sleeve and locks the sleeve and firing pin together, with the point of the firing pin projecting beyond the sleeve. As the plunger of the fuze does not encounter the atmos- pheric resistance which retards the projectile in its flight, it is PRIMERS AND FUZES FOR CANNON 13 - probable that during the flight of the projectile the plunger moves slowly forward until the point of the firing pin rests against the brass primer shield. At impact of the projectile the combined weight of the plunger parts acts to force the point of the firing pin through the primer shield and into the percussion composition, igniting the composition. The flame from the primer charge passes through the for- ward vents, through the passages in the plunger, and through the vent in the closing screw, blowing out the closing disk and igniting the bursting charge in the shell. Base percussion fuze, for minor caliber shell. This fuze, as well as the point percussion fuze, is adapted to the projectiles for 1-pounder, 2-pounder, and 6-pounder guns. - The fuze, Fig. 4, is similar in construction and action to the point percussion fuze. As the primed end of the fuze is toward the interior of the shell the flame from the priming charge passes directly to the bursting charge in the shell. without passing through the body of the fuze. The flame passages through the plunger parts are therefore omitted. The primer cup b containing the percussion composition. and priming charge is closed at its outer end by the brass disk a, which is secured in place by crimping over it a thin wall left on the brass closing cap screw c. FIG. 4. There are two classes of ring-resistance fuzes manufactured, the ‘‘high resistance ’’ and the “low re- sistance,” so called because the arming resistance of the ring is relatively high or low. High-resistance fuzes are safe under all ordinary conditions of handling and transportation, and are transported fixed in the projectiles in which used. Low resistance fuzes cannot, on account of the danger of premature arming, be transported fixed in projectiles. The low-resistance fuzes are transported packed in hermeti- cally sealed boxes, and, to prevent premature arming of the plunger in handling, the firing-pin sleeve and the firing pin are locked together by means of a safety wire passing through them and the body of the fuze. Just before using this wire must be pulled out, after which the fuze may be screwed into { … ºf 14 PRIMERS AND FUZES FOR CANNON the projectile. The only low-resistance fuze at present issued is the 28-second combination fuze, for 7-inch mortar shrapnel. The act of arming a ring-resistance percussion fuze short- ens the plunger and increases materially its longitudinal play in the fuze body. This fact permits a ready and simple means of inspecting for premature arming without dismantling the fuze. If the fuze be held close to the ear and shaken, the marked difference between the play of the plunger in an armed fuze and in an unarmed one can be readily discerned. Percussion fuzes for larger projectiles. Centrifugal fuzes.— The centrifugal fuze of service pattern is the result of a long series of experiments made for the purpose of developing a fuze that would fulfil the requirements of absolute safety in handling and transportation, and certainty of action. In the case of ring-resistance fuzes, or any fuze the action of which depends on the longitudinal stresses developed by the pressure in the gun, the conditions of Safety in handling, and certainty of action are opposing ones. It was impossible to meet successfully both sets of condi- tions in all cases, the stress developed in the direction of the axis by accidental dropping of a fuze being in many cases higher than that developed in the gun. - As has been shown, two classes of ring-resistance fuzes are required, the “low resistance ’’ and the “high resistance”; the low resistance for use in the 7-inch mortar, in which piece the low powder pressures impart a low maximum acceleration to the projectile, and require greater sensitiveness in the fuze. The greater liability to accident in handling fuzed projec- tiles due to the rapidity of fire required from modern guns, has caused the retirement of fuzes—that depend for their action upon the acceleration of the projectile in the gun, except in some of the smaller guns where the acceleration is so great that a very high arming resistance can be given. A fuze which is armed by the centrifugal force developed by the rotation of the projectile, and which is safe until the maximum velocity of rotation is nearly attained, has been developed at the Frankford. Arsenal, and is now applied to all projectiles above the 6-pounder in caliber. . . . . . . . . . . . . . . . . . . . . . . . * T--~.As already stated, ring-resistance fuzes are used in 1-pounder-2-pounder, and 6-pounder shell. The limited fuze dimensions render the construction of a satisfactory centrifugal fuze for these shell mechanically very difficult; but, by reason of the high maximum acceleration of these projectiles in the PRIMERS AND FUZES FOR CANNON 15 un, the sensitiveness of the ring-resistance fuzes need not ~..."...º.º..."...” the Safe handling of the projectiles. - “r------ - The centrifugal fuze, before arming, is shown in Fig. 5. Figure 6 is a view of the plunger after arming. >3-->-——— & ## -4–––6 º & Żºłº _----C º zº e” d _2^ 2–– FIG. 5. The fuze body, or stock, and the primer parts of the cen- trifugal fuze do not differ materially from the corresponding parts of the ring-resistance fuzes. To protect better the prim- ing charge, the closing cap screw b is lengthened and the vented primer-closing screw a is added. The body of the centrifugal plunger is in two parts, nearly semi-cylindrical in shape, which, when the fuze is at rest, are held together by the pressure of a spiral spring g contained in the cylindrical bushing e which is secured to one of the plunger halves. The spring exerts its pressure on the other half of the plunger through the bolt f. Pivoted in a recess in one half of the plunger is the firing pin d, which when the fuze is at rest is held with its point below the front surface of the plunger by the lever action of the link c which is pivoted in the other half. Under the action of the centrifugal force developed by the rapid rotation of the projectile the two halves of the plunger Separate. The separating movement causes the rotation of the firing pin d, the point of which is now held in advance of the front surface of the plunger, Fig. 6, ready, on impact of the projectile, to pierce the brass primer shield and ignite the per- cussion composition. When the fuze is armed the end of the link c rests on the axis of the firing pin, thus affording support to the firing pin when it strikes the percussion primer. The 16 PRIMERS AND FUZES FOR CANNON separation of the plunger parts is limited by the nut i coming to a bearing on a shoulder in the bushing e, so as not to permit the diameter of the expanded plunger to equal the interior diameter of fuze stock, see Fig. 8, below. w A rotating piece, h, Fig. 7, screwed into head of fuze stock, engages in a corresponding slot cut through the bottom of both plunger-halves and insures rotation of the plunger with the shell. The strength of the spring g is so adjusted that the fuze will not arm until its rapidity of revolution is a certain percent- age of that expected in the shell in which it is to be used, and that it will certainly arm when the rapidity of revolution approximates that expected in the shell. Should the parts of the plunger be accidentally separated and the fuze armed by a sudden jolt or jar in transportation or handling, the reaction of the spring will immediately bring the plunger to the unarmed condition. . The fuze just described, called the “F” fuze, is used in siege detonating fuzes for 5 and 7-inch shell charged with high explosive. The fuze shown in Fig. 8, the “S” fuze, is for use with 3.6 and 7-inch mortar shell, powder-charged, and with the detonat- ing fuzes in 8, 10 and 12-inch armor- piercing projectiles. The additional priming charge in end of fuze gives a greater body of flame than is emitted from the “F” fuze. A similar fuze of larger size is used in powder-charged shell of 8- inch caliber and over. A fuze, called the “12 M'' fuze, is provided for use in detonating fuzes for the 12-inch mortar and tor- pedo shell. This fuze is similar in construction to the other centrifugal fuzes, but on account of the low velocity of rotation of mortar pro- jectiles and their low striking velocity a much heavier plunger is needed to FIG. 8. provide the force necessary for arm- ing the fuze, and for puncturing the primer-shield on impact. in-addition, the plunger carries on its front surface a swivelled piece which when the fuze is armed falls into–the aperture PRIMERS AND FUZES FOR CANNON 17 between the plunger-halves, so that when once armed the two parts of the plunger are automatically locked apart in the armed position. This prevents the closing together of the plunger-halves, and the consequent withdrawal of the firing pin, if the shell should strike upon its side, as it does sometimes when fired at low velocities. & * ~ . Combination fuzes. –All combination fuzes used in the service are point insertion and combine the elements of time and percussion arranged to act independently in one fuze body. Combination fuzes contain two plungers and two primers, arming and firing by concussion and percussion respectively. The concussion plunger arms and fires the concussion primer by shock of discharge in the bore of the piece and ignites the time element. The percussion plunger is armed by the shock of discharge and fires its primer on impact. . - There are at present two general classes of combination fuzes in service, differing principally in the details of the time- train elements. In the first class this element consists of a wire-drawn lead tube filled with mealed powder, wound in a Spiral groove around a lead cone. In the second class the time element consists of two superposed trains of mealed powder compressed under heavy pressure into annular grooves in disks of brass. The first class is represented by the following Frankford Arsenal combination fuzes: The 15-second, the 28-second high-resistance, and the 28-second low-resistance combination fuzes. No more fuzes of this class are to be manufactured, and the fuze will become obsolete when the supply now on hand is exhausted. The second class is represented by the Frankford Arsenal 21-second combination fuze. The method of preparing the time train of this fuze insures much greater uniformity of action than was obtained in the fuze with lead-cased time train. This fuze has, therefore, been adopted for use in service Shrapnel. Combination fuze, latest patterm. Fig. 9.-The upper part of the fuze contains the time elements, the lower part the percussion elements. The time elements consist of the con- cussion or time plunger b, the firing pin c, and the time train. The firing pin is fixed in the body of the fuze, and the plunger carries the percussion composition and a small igniting charge of black powder. The plunger is held out of contact with the firing pin by the split resistance ring a. On the shock of dis- ~... 18 PRIMERS AND FUZES FOR CANNON charge, the inertia of the plunger acting through the conical Surface in contact with the split ring expands the ring so that the plunger can pass through it and carry the percussion com- position to the firing pin. & *** ***** 3&ºtº ºº::348% The time train of the fuze is composed of two rings of powder f and h contained in grooves cut in the two time-train rings m and n. The grooves are not cut completely around the rings, but a solid portion is left between the ends of the groove in each ring. Mealed powder is compressed into the grooves under a pressure of 70,000 pounds per square inch, form- ing a train 7 inches long, the combined length of the two grOOves. The flame from the percussion composition passes through the vent digniting the compressed tubular powder pellet e which in turn ignites one end of the upper time train f. When the fuze is set at zero the flame passes immediately from the upper time train through the powder pellet g to one end of the lower time train h; thence through the pellet i and vent j to the powder k in the annular magazine at the base of the fuze. Under each of the time rings is a felt washer, o and p, that closes the joint under the ring against the passage of flame, except through the hole in the washer directly over the vent in the part below. The upper washer 0 is glued to the upper corrugated surface of the lower time ring n and moves with that ring, the lower washer p is glued to the fuze body and is PRIMERS AND FUZES FOR CANNON 19 stationary. The upper time ring m is fixed in position by two pins l halved into the fuze body and the ring, The lower time ring is movable, and any of the graduations on its exterior, see Fig. 10, which correspond to seconds and fifths of seconds of /_G) \ Q?, ?A º \\\\\ FIG. 10. burning, may be brought to the datum line marked on body of fuze below, the ring. The ring is moved, in setting, by means of a wrench applied to the projecting stud w. To set the fuze for any time of burning, say 20 seconds, move the lower time ring n until the mark 20 is over the datum line. On ignition of the primer the flame ignites the upper time train f, which burns clock-wise, looking from base to point of fuze, until the hole through the washer over the zero mark of the lower ring n is encountered. The flame then passes through the vent g to the lower time train n, which burns anti- clockwise until the 20 is reached. This mark being over the Vent i in the body of fuze, the flame now passes to the maga- Zine k. The setting of the fuze consists in fixing the position of the passage from the upper to the lower time train, so as to include a greater or less length of each train between the vent e and the vent i. In each time ring a vent opens from the initial end of the powder train to the exterior. The vent contains a pellet of powder and is covered by a thin brass cup. The vent in lower time ring is seen at a in figure 9. The caps, x, x', of both Vents are shown in figure 10. The blowing out of the cap affords a passage to the open air for the flame from the burning 20 PRIMERS AND FUZES FOR CANNON time train, thus preventing the bursting of the fuze by the pressure of the contained gases. When the fuze is set at safety, indicated by the letter S stamped on the lower time ring, the position shown in figure 10, the solid metal between the ends of the upper time train is over the vent g to the lower train, and the solid metal between the ends of the lower train is over the vent i leading to the magazine. In case of accidental firing by the time plunger, the upper train will be completely consumed without communicat- ing fire to the lower train and to the magazine. The fuze is habitually carried at this setting, which serves also when it is desired to explode the shell by impact only. - For percussion firing the fuze is provided with a centrifugal plunger r, as described in the base percussion fuze. To insure rotation of the plunger with the shell the bottom of the plunger is cut away on the sides to fit in a slot in the rotating ring q, which is held firmly against the walls of the plunger-recess by the pressure of the closing screw u, screwed down hard. A vent s leads from the percussion primer to the annular maga- zine k. A thin brass cap t separates the lower plunger-recess from the powder in the four radial chambers v cut in the bot- tom closing screw. The central vent in the closing screw is closed by a piece of shellacked linen, held in place by a brass washer. These fuzes are issued fixed in the loaded projectiles. For protection in transportation the fuze is covered by a spun brass cap, soldered on to the head of the projectile. The soldering strip is torn off and the cover removed before using the projectile. Combination fuze, old patterm. —The time train b, encased in a lead tube, is wound spirally around the lead cone c. To set the fuze for any time of burning the time train and lead cone are punctured, by means of a tool provided for the purpose, at the point on the scale marked on the cover of fuze correspond- ing to the time of burning desired. The puncture passes com- pletely through the time train and the lead cone behind it, forming a channel from the annular space in which the letter b appears to the powder in the time train. When the projectile is fired the percussion flame ignites the compressed powder ring d, and the flame from this ring ignites the time train at the point at which it has been punctured. The safety pin a re- tains the time pluger in its unarmed position, and must be with- drawn before placing the projectile in the gun. (See Fig. 11.) PRIMERS AND FUZES FOR CANNON 21 FIG. 11. Ehrhardt combination fuze.—This fuze is similar in con- struction to the Frankford Arsenal fuze, latest pattern, de- scribed above and differs only in details. e’ sº & * b & C A322:47, Sººty: § § ºSNI. Nºvº Nººgº | 3. § §§ WºRººs ::/2\º § §§ º * jś §§ §§§:#######$º § &\ºº >}\;\ºº §§§§§§ y -:g FIG. 12. 22 PRIMERS AND FUZES FOR CANNON The arming of the time plunger of the Ehrhardt fuze, Fig. 12, is resisted by the U-shaped spring a, the upper ends of which are sprung out into a counterbored recess in the closing cap, and by the slender brass pin b, which passes through the plunger and both sides of the closing cap. At discharge of the piece the inertia of the plunger shears the pin b and straightens the U-shaped spring a, permitting the plunger to strike the firing pin. In the percussion mechanism the composition is carried in the plunger and the firing pin is fixed in the diaphragm d in body of fuze. The plunger is held away from the firing pin before firing, by the brass restraining pin c. The pin is let in to a hole in the diaphragm d, the head of the pin abutting against a shoulder near the bottom of the hole. The restraining pel- let of powder e is pressed in to fill the recess above the pin. A perforated brass disk and a piece of linen close the hole at its upper end and prevent the powder pellet from being jarred out of place. The burning of this pellet on ignition from the time plunger leaves the restraining pin and percussion plunger free to move forward at impact. A compressed charge of black powder, g, is inserted into the extension of the closing screw f to re-enforce the magazine charge and effectually to carry the flame to the base charge in the shrapnel. - The Krupp combination fuze does not differ essentially from the Ehrhardt fuze. The shear pin through time plunger is omitted, the U-shaped spring being made strong enough to offer sufficient resistance against accidental arming. The per- cussion plunger, carrying the percussion composition, is held away from the firing pin, before firing, by a sleeve and an in- verted U-shaped resistance spring. A spiral spring between plunger and firing pin prevents the creeping forward of the plunger during the flight of the projectile. Detonating fuzes.—These fuzes are for use in shell con- taining high explosives. . PRIMERS AND FUZES FOR CANNON 23 Fig. 14 shows the form of detonating fuze for point inser- tion in field shell. Fig. 13 shows the form of fuze for base in- sertion in siege and Seacoast projectiles. Ns +4'29. p FIG. 14 In order to prevent the unscrewing of the fuze during flight of the projectile, all point insertion fuzes are provided with right-handed screw threads, and base insertion fuzes with left-handed threads. Notes For Artillery Course Trajectory in Vacuo - Gunnery Interpolation by Differences Exterior Ballistics concerns itself with the behavior of projectiles in their motion through the air. In order to avoid unnecessary complexity the following hypotheses are made as a basis: 1. That the projectile travels through still air of uniform density. 2. That the resistance of the air is tangential and, at any instant, is directed along the longitudinal axis of the projectile. 3. That the resistance of the air for a given projectile is a function of the velocity alone. 4. That the surface of the earth is a plane and without motion. 5. That gravity is constant in intensity and direction. The errors incident to these hypotheses are corrected, with some exceptions, by the use of mean values and where prac- ticable by formulas involving known elements, or by actual determination of the range or elevation-corrections due. In the excepted cases, the exact solution of the problem would demand great complexity of equipment in the position-finding Service, due to elements varying with the locality, circumstance and direction of fire; and the corrections are not large enough to justify this, or to produce a material effect. Hence they are ignored, or, where it is advisable to do so, they are represented by mean values. As Part I. is necessarily short, it will be impracticable to demonstrate the principal formulas on account of the amount of analysis invloved, though wherever it is practicable to demonstrate a formula with simplicity, this will be done. In order to fix ideas and to give a clear conception of the limitations of even the best projectile, the unresisted motion of a projectile will be first considered. This is technically known as motion in vacuo, and the trajectory is referred to as the trajectory in vacuo. It is clear that if we suppose the air gradu- ally thinned and reduced to nothing, a projectile will experi- ence less and less retardation and, finally, none at all, except that due to gravity in the ascending branch, which becomes, in 2 the descending branch, an acceleration, beinging the projectile back to the level from which it started with the same velocity that it had at the muzzle. THE TRAJECTORY IN VACUO We shall approach the discussion with the following state- ment of the problem: A projectile is fired with a muzzle velocity V, and a quad- rant angle of departure p; it is subject to no other influence except that of the attraction of gravity, which will be repre- sented by g. (a) Find at the end of any time, t seconds after the gun is fired, what will be the horizontal range x, the height, y, the remaining velocity V, and the inclina- ation of and their relation to one another. (b) Find these elements for the point of fall, and represent them by corresponding capital letters. (c) Find the coordinates, x0, y0, of the summit of the tra- jectory and the time of flight to, to this point. In order to solve this problem we need only algebra and a very elementary knowledge of trigonometry. (a) Let us suppose first that gravity does not act; then at the end of the time t, the projectile will have gone Vt feet along the line of departure, corresponding to a horizontal distance x = Vt cos 2, and a vertical distance y = Vt sin o. Now the action of gravity will not affect x, since it acts in a vertical direction only. It will, according to the law of falling bodies, cause the projectile to drop from the line of departure, vertically a distance J/3 gt”; and it will reduce the vertical velocity by gt. Hence we have, in vacuo, x = Vt cos p (1) y =Vt sin go – }% gt” (2) v cos 0 = V cos go (3) v sin 6 = V sin p – gt (4) From (1) we find t = , and substituting this in (2) - V cos p we have y gx -º- = tan go — —º- (5 X 40 2 V8 cos” to ) From (3), (4), and (1) we find, by eliminating v and t tan 0–tan e--É– (6 Ø V” cos” (e ) 3 We may now find x from (1); y from (5); 6 from (6) and v from (3), in the order named. (b) At the point of fall the projectile has again reached the level of the gun, hence y = 0. Then from, (5), V*sin22 X = (7) g Y = 0 and from (6) by substituting for X its value from (7) tan a = tan p – 2 tan ºp whence a = — p” (8) and since V, cos a = V cos p, we have * v, - V (9) From (1) and (7) we find T = 2 V sin e (10) (c) The summit is reached when the vertical velocity is Zero, then; equation (4) gives us _ V sin p to (11) g Hence from (10) …- to = % T (12) From (2), (11) and (12) we find - yo = # g T2 (13) and from (1), (7) and (13) yo = }4 X tan go From the above we see that, with no retardation, we have Sin 2 p = gx OI’ Sin 2 e. V2 X X x- T cos e = * or v T cos • – X = 0 tan a Co - ºp = 1 tan ºp v., cos w = V cos o or v. = V cos e Sec as - yo = % X tan p = # g Tº. * The angle of fall, ay, is recognized as negative but will be spoken of as positive since we are concerned only with its magnitude or absolute value. The position angle, the value of whose tangent is "/, is designated e and is regarded as an absolute value in this discussion since this prevents any confusion; and simplicity is secured, as in Coast Artillery work it is always applied in the same direction. - 4 In air, forms analogous to these are useful in direct fire. For high angle fire we use by preference the ratios 10 X 10 T Vo yo V2 V V X and the angle w. The reason for this is that in direct fire we have a constant velocity and vary the elevation. In high angle fire we have certain elevation limits and muzzle velocities vary with the zones used; besides, the simplicity of the law of retardation renders it desirable to tabulate these values on pages, one page for each degree of p; and p is the range table argument. THE PRINCIPLE OF THE RIGIDITY OF THE TRAJECTORY This “principle” assumes that, if the angle of departure necessary to reach a certain point at a horizontal range, x, from the gun and on the same level, is known, it will only be neces- . sary, in order to reach another point h feet below the former and at the same horizontal range, x, to subtract from the first angle of departure, the angle e, called the position angle and given by the equation tan e = "/. This is a useful conception, and is always assumed, in direct fire. It is indispensable in Case I, and furnishes a simple and satisfactory method for finding quadrant elevations for use with Case II. - The assumption, while sensibly true, is not exact, and the error in elevation thereby introduced will now be found. (a) Denote by p, the angle of departure that will bring the point of fall to a horizontal range x, with a given muzzle velocity V. (b) Denote by a the angle of departure that will deliver the projectile at a distance h feet below its position in (a), and at the same range x. Equation (5) then becomes, in the two cases, since y is zero for (a) and — h, for (b). X º _8 Sin 240, TV. (a) and tane = -\ = —É-–tan •- ". 2V2 cos” (p X gx Or tane = tan o 8 ——º- - 1. (b € © lv sin 242 ) Hence º tane = tan (p #-1} - sin 20 From this sin 24” – sin 20 = 2 cos” p tan e (14) Now, assuming the principle of the rigidity of the trajectory the sight angle of departure used is p’ = p + e. Substituting º' – e for p in equation (14), we will find after reduction sin 20, - sin 2 • {1 + tan o' tan e} What we assume, then, under the principle of the rigidity of the trajectory is that tan p' tan é is zero or negligible. The amount of error in to thereby introduced can be readily ascertained from this equation and its range equivalent found from a range-table. - It is to be noted that pº is the p of a range table, and that when we correct for height of site and curvature” we introduce e assuming that os = o' = p + e. The assumption that the principle of the rigidity of the trajectory is applicable in direct fire is productive of no error that is worthy of note. Such error as is produced is just about compensated for by the difference in height above sea-level between the gun and the target, corrections for altitude being made for the height of the gun as if the entire trajectory were above that level. This simplifies the correction for altitude also. - The term “rigidity” in this connection refers to the sup- posed rigid shape of the trajectory and its chord (drawn from the muzzle to the target or point of impact); and the assump- tion practically involves the hypothesis, that the figure whose outline is composed of the trajectory and its chord, behaves as if it were cut out of card board and rotated up or down with the muzzle of the gun as a center. EXAMPLES 1. Given V = 2250, X = 24000 feet, compute remaining elements of the trajectory for each thousand yards of range, i.e., for each 3000 feet of range. This includes y, t, v and 0. * See Appendix VII. 6 (b) Compute also x0 and yo. (c) Taking the value of p which was found, enter the 12" Range Table and take from it the range, maximum ordinate and angle of fall. (d) Note the effect of air resistance even on this heavy projectile. (g = 32.16 foot-seconds per second.) 2. From the 12” Fange Table V = 2250 f.s. opposite the range 10000 yards, take out p, T, yn, V., w. Calculate from these the values VT cos p V* sin 240 V cos a 4yo 8y, X gx V cos e X tane gT" These values are unity for the conditions in vacuo. Note the effect of air resistance. ` 3. Assuming the curvature of the earth to produce angular depression of an object on the water surface at the rate of 1 minute for every 4000 yards of range; find e for each 4000 yards of range for a height above sea level of 65 feet. Apply the depression due to height of site and curvature, at each of the ranges 4000, 8000 and 12000 yards, to the values of the angles of departure given in a 12" range table for V =2250 and give the resulting quadrant angles of departure. (b) Find these values of e from Artillery Notes, No. 29, and compare results. - (c) What principle is assumed in making these corrections? Answer to (b) 0°19'.6 for 4000 yards. 0°11'.3 for 8000 yards. 0°09'.2 for 12000 yards. GlüNNERY Gunnery in its widest sense embraces a knowledge of Ex- terior and Interior Ballistics and the use of Artillery Material, the object of the art and science involved being the destruction of or maximum damage to the object at which artillery fire is directed. This chapter will, however, be restricted to the application of ballistic data to practical use at the battery, and to a de- scription of the general character of the auxiliary instruments by which the range of the target is found and the gun directed so as to hit the target. 7 1. For every type of gun, carriage, and projectile, there should be certain firings, called range-firings, conducted with a view to determining the peculiarities of gun, mount, and pro- jectile. This having been done, a range-table may be com- puted, and range- and deflection-corrections found. 2. The elevations given in the range-table referred to must be corrected by subtracting the absolute value of the posi- tion angle, w, due to height of trunnions above mean low water and to curvature, before they are applied to the range-scale of any gun. This applies only to guns using direct fire. The range-table elevations are not altered for mortars. 3. The ranges and the corresponding elevations having been tabulated for each gun (see Artillery Notes, No. 29) may be applied to the range-scale of the gun by the use of a clino- meter, the gun being given (by means of the clinometer) the elevations pertaining to ranges at fixed intervals (say 200 yards) and the range is marked on the range-scale in place of the eleva- tion. The ranges of the range-scale thus represent elevations under range table conditions. Ordinarily these graduations are applied under the super- vision of the Ordnance Department to the metal of the range- scale. If, however, conditions should arise under which it has to be done by the personnel of the battery the graduations should be made on a scale made of durable paper applied to the surface of the range-scale by an adhesive which does not greatly affect the dimensions of the paper, and, after it has set, and the paper is firmly in place the scale should be graduated by use of the clinometer as already described. The scale should then be varnished with shellac to prevent injury or distortion by mois- ture. The same precaution against moisture whould be ob- served in the use of paper scales in any portion of the equipment of a battery. The adjustment of the gun for elevation is now complete until calibration firings show further adjustment in elevation to be necessary, due to peculiarities of the individual gun and mount. The adjustment of the sight for deflections, is readily made by causing the vertical hair of the sight to bisect the same object as that bisected by bore sights, the object being station- ary and at a range not less than half the extreme range-table range of the gun, the sight being set for “no deflection.” 4. The gun and carriage having been thus adjusted to range-table conditions, it is necessary to have some means of 8 finding the range of the target, its rate of travel, and the cor- rections in elevation and deflection necessary in order to allow for the motion of the target and for variations from range-table conditions. The means of adapting range-table data as applied to a battery to conditions actually obtaining at the time of firing constitute collectively THE POSITION FINDING SERVICE This consists of The Position Finder. The Plotting Board. The Range Board. The Deflection Board. The Atmosphere Chart. The Wind Component Indicator. . The Powder Chart. These will be briefly described in the order given. 1. Position Finders Position finders, properly so called, are used for finding, at any instant, the position of a target; that is, its range and azi- muth. They are thus distinguished from range-finders which merely give the range. Range-finders may be used with the Secondary armament but a position-finder is needed with a heavy battery for reasons which will appear. Position-finders are of two principal kinds: Horizontal and vertical (or depression) position-finders. a. A horizontal position-finder consists essentially of two azimuth measuring instruments, one at each end of a base-line, and having a telephone connection and a time interval bell to Synchronize the observations. The position of the target at fixed intervals is thus determined by the two azimuths. A map may thus be made on the plotting board of the path of the target showing its position at uniformly spaced times. b. A depression position-finder consists of only one in- strument and is thus self-contained or under the complete con- trol of a single individual. By it are found the range and azi- muth of the target at fixed intervals of time. The vertical position-finder mechanically solves a triangle in a vertical plane, after correcting for the conditions of tide curvature and atmospheric refraction that actually obtain. It may be said that the vertical position-finder, by means of a triangle within itself and similar to the right triangle 9 formed by the height of the instrument above the water (cor- rected for refraction and curvature), the true horizontal range, and the straight line from the instrument to the water-line of the target, gives at any time, from a scale, the range in yards; and the azimuth is given by the azimuth measuring feature of the instrument. The correction for height of tide above mean low water is accomplished in the insrtument by an adjustment indepen- dent of the range, since the tide is assumed both the same over the field of fire. Curvature of the earth varies with the range, and may be readily computed. Atmospheric refraction is the least stable and least uniformly acting abnormality with which the D.P.F. has to contend. Normal refraction is assumed quite generally to counteract # of curvature; that is to say, by assuming the earth to have only ; of its actual curvature we may, once for all, allow for normal refraction. But if the refraction be abnormal (or different from its assumed value), which is usually the case, this arrangement fails as an exact solution, since the correction for curvature and refraction depends on the range. The general solution of the problem would appear to be the combination of refraction and curvature into one effect represented in the instrument by a similar curve, the curve to be able to adapt itself to actual conditions, as determined by making the instrument read true on all the important datum points in the field of fire. Of course conditions of refraction are frequently temporary and readjustment from time to time will be necessary. . In some forms of instrument the correction is made once for all in the instrument for curvature and normal refraction, and any other state as to refraction is allowed for by adjusting the tide-scale, thereby making the instrument read approxi- mately true on two datum points. The ranges and directions of the target at fixed time-inter- vals are thus obtained from the position-finder (of whatever kind) and sent to the plotting room where the position of the target is continually plotted on a plotting board. 2. The Plotting Board This device consists essentially of a drawing board pro- vided with means of quickly mapping the data sent fromt the position-finder, finding the rate of travel of the target in range 10 and azimuth for use with the range and deflection correction devices, finding by means of an arm (called the gun arm) the range and azimuth to the target from the directing gun (or point) of the battery, and applying to the true range and azi- muth the corrections received from the correction boards. For direct fire the plotting board takes no cognizance of azimuth corrections which are sent to the gun directly from the deflection correction board, which, however, receives the data as to rate of travel in azimuth from the plotting board. 3. The Range Board This board applies through mechanical addition and sub- traction the information received as to atmospheric conditions, tide, travel of target and variation in muzzle velocity, obtaining a range-correction which when applied through the gun arm of the plotting board gives the artificial or corrected range for use in elevating the gun. . 4. The Deflection Board This combines by mechanical addition and subtraction the deflections due to cross-wind, drift and travel of target, giving a corrected azimuth or deflection. 5. The Atmosphere Chart This device gives the reference number due to the reading of the barometer and thermometer, for use with the atmosphere correction feature of the range board. 6. The Wind Component Indicator Information as to the wind velocity and direction having been received from the meteorological station, this device is used to resolve the wind velocity into components across and in the plane of fire, the direction in which the components are to be applied on the range and deflection correction devices being shown by the reference numbers. 7. The Poujder Chart This is a chart showing the muzzle velocity to be expected from a slight change in the weight of the charge and from vari- ations in the temperature of the powder from the standard temperature 70° F., at which the prescribed charge should give the muzzle velocity of the range-table. f 11 Reference Numbers These are used to avoid the confusion incident to the use of positive and negative numbers. II. PRACTICE In our service practice is preceded by trial shots. In the preliminary work of determining conditions the adjustment of all scales should be carefully made and especially the elevation and deflection devices of the individual gun on its mount. All such adjustments having been made and the ammunition being prepared for use in accordance with orders and regulations, the gun should (at the proper time) be loaded, especial pains being taken to properly seat the projectile. This is of great im- portance, as, with the projectile not properly seated, the powder gas escapes past the projectile and considerable loss of velocity OCCU). TS. As no intelligent correction can be made for such an im- proper seating it is essential to avoid such a situation by careful drill in loading. Trial shots are fired giving the gun the proper corrected elevation and deflection for the assumed range which it is ex- pected that the shot will attain. This involves taking into consideration, the knowledge if any, that is on record from prior firings with the same powder as to what muzzle velocity is to be expected from it, as well as the data as to meteorological and other conditions, In making the range-corrections, it is to be remembered that the range used on the range-board as a basis for corrections, is the corrected range. Hence, taking the actual data and the actual supposed range find the range corrections and thence the corrected range; using this corrected range apply the corrections again until they cease to change. The plotting board now shows the corrected range which is to be sent to the gun. The deflection board gives the proper deflec- tion correction. The gun may now be fired and the range and deflection of the shot observed. In this manner fire all the trial shots, cor- recting no further for muzzle velocity, but correcting for other conditions, provided the latter show any change. Then take the mean of the observed ranges as the true range. To find what the muzzle velocity must have been, take the difference between the expected range and the mean observed range and find from the range board what diminution or in- 12 crease must be applied to the velocity as assumed in the trial shots. In case the first of the trial shots falls very wide of the ex- pected range, so as to interfere with observation of the fall of the shots by parties stationed in the neighborhood of the point of fall, a suitable correction in range should be made and at- tributed to velocity. Suitable correction having been made in the recorded value of the range of the first trial shot for the difference in velocity between itself and the succeeding ones, it may, if it was accurately observed, be incorporated with the suc- ceeding ones in obtaining a mean observed range. Otherwise In Ot. - The results of trial shots should be carefully considered and recorded with a description and history of the powder used, for use in estimating muzzle velocity in future trial shots. The mean of the ranges of the trial shots is taken because as shown by the theory of errors it is safest to do so. Having found the muzzle velocity, the corrections for the first shot of the practice msut be found by first finding the cor- rected range from the range-finder range as in the case of the first trial shots. For subsequent shots the corrected range of the shot last fired is already known. III. CALIBRATION Calibration, as the term is now used, is the process of ascer- taining the peculiarities of individual guns of a battery and correcting the range scale so that each gun ranges correctly on an average. This enables the battery commander to treat the battery as if it were a single gun when in action or at practice. In calibrating a battery the powder should be such as to give the proper muzzle velocity. In that case the range scales of each gun should be adjusted so as to agree with the range which the calibration test indicates for range table conditions. For calibration to be of great and certain value, the powder used must be uniform in action, and carefully weighed. It is best when practicable to measure the muzzle velocity for each round, but this is generally impracticable. The elevation should be given by clinometer and verified after loading and before firing. The elevation given should be that due to the range corrected for actual conditions by use of the position finding service. The mean range of each gun should be re- corded, also the mean deflection and the mean error of each. The difference from the expected range of the actual mean 13 range for each gun should be properly applied in yards to the range scale of that gun. It is then calibrated, for that range. For example a gun is laid for an assumed actual range 6000 yards. The tentative corrections based on 6000 yards show a Corrected range less than that; using this find new corrections until the corrected range is established before firing. Lay the gun at the corrected range which we will take for illustration to be 5500 yards. If the ranges are observed to be 5870, 5920, 5840, 5850, 5900, the average is 5876. Thus the gun lacks 124 yards of shooting up to the range. Set the range scale now at 6000 yards, and move the scale or pointer until the pointer is Opposite 5876 on the scale. The gun is now adjusted in range. If each gun is thus adjusted on its mean range the battery will behave as a unit. - Deflections could be tested in the same way, but they are usually small and unless there is material discrepancy between guns the sights need not be changed, the principal object of calibration being the adjustment of the range scale. - As all our direct fire guns elevate by scale and use the sight Only to follow the target in azimuth, the only adjustment of the Sight that is necessary is that by which the axis of collimation of the sight is made to cross the axis of the bore at a mean range. This is done by the use of bore sights in the gun and the sight seat or the deflection scale, as the case may be, is adjusted by the adjusting screws until axis of telescopic sight and line of bore sights bisect the object on which observations are taken. (See also Artillery Notes, No. 22. For mortars see Artillery Notes, Nos. 22 and 11). - IV. ACCURACY The accuracy of a battery is determined by the percentage of hits which it may be expected to make on the service target. It is best determined from the firings used in calibrating. It must be found for each individual gun and then, after the battery is calibrated the errors may be treated individually as errors of the battery. Suppose for instance that of three guns in a battery firing with an assumed actual range 6000 yards, Say, the record is as follows: Gun No. 1 Gun No. 2 Gun No. 3 5780 5946 6024 5820 5910 • 6085 5910 5982 5990 5875 5895 6041 5850 5647 6000 Sum 29235 29680 30140 º . —Iſlead 5847 5936 É 028 - : 14 The pointers should now be set at 6000 on each gun and the scales of pointers moved without changing the elevation so that the pointer is opposite the actual mean range reached in each case. - - - Taking the observed range in each case minus the mean range of the group we have, averaging errors without regard to sign, for guns 1, 2, 3, respectively. –67 +10 — 4 –27 –26 +-57 +63 +46 –38 +28 —41 +13 + 3 +11 - —28 SUIT). 188 134 140 Iſlea Il +37.6 ––26.8 +28.0 These are the mean errors for the individual guns. After cali- bration, their mean is the mean for the battery, thus 37.6 26.8 28.0 3)92.4 * 30.8 or +30.8 is the mean error for the battery. To show how the mean error of a gun or battery may be expected to give a definite idea of the number of hits that may be expected with perfect work on the part of the personnel the following is given: . The mean error is the average of errors without regard to sign. In each case the sum of the negative errors is the same as that of the positive errors, since the mean range is subtracted from the observed range in each case. Positive and negative errors are equally likely to occur. This mean error is the error of average size of the average error having regard to the fre- quency of occurence of each size of error. Large errors are less frequent than small ones and this play as an important part in the theory of errors. Another error, called the probable error, is such that just half the shots fired will in the long run have an error equal to or less than this probable error. This is gotten from the mean error by multiplying by 0.845. The 15 mean error thus enables us to find the probable error. The probable errors for guns 1, 2, and 3 are, respectively, 31.772; 22.646; and 23.660. and for the battery when calibrated 26,026 Now since half the shots will in the long run fall over or short of the mean range by amounts not greater than the prob- able error, and half of them short and half over, it follows that, if we draw two lines perpendicular to the line joining the bat- tery to the mean point of impact of the shots, the two lines be- ing beyond and short of the mean range by a distance equal to the probable error, we will have a zone or belt in which half or 50% of the shots will fall. This is called the 50% zone and the total width of the 50% zone, is thus twice the probable error or 1.69 times the mean error. The width of the 50% zone being taken as the unit of com- parison the theory of errors deduces the widths of zones of other percentages in terms of the 50% zone width as a unit. Thus the factor Z/Z, in the table shows the width, in terms of that of the 50% zone, of the zone whose percentage is given in the column on its left. As an illustration, the 60% zone is 1.25 times as wide as the 50% zone, the 82%% zone is twice as wide. The 99% zone is four times as wide. The zone four times as wide as the 50% zone is called the enveloping zone, since it practically contains all the shots. ^ - To apply this table to the illustration of a battery of three guns which we have used we find the enveloping zones to be for the three guns and the battery, respectively, of widths 254, 181, 189 and 208 yards We are mainly concerned however with hitting the stan- dard target of such guns which is 30 feet high and 60 feet wide. We are also concerned chiefly with its height, as errors in de- flection are small. The slope of fall at this range we will take to be say 1 on 10. Hence the horizontal equivalent of the ver- tical target is 10 × 30 = 300 feet, and, aim being taken at the water line, this will, if the gun is properly laid (the center of impact on point aimed at) give a space of 300 feet or 100 yards beyond the target; allowing half that or 50 yards for ricochet by shots falling short, we find that the danger space consists of two half zones, one beyond and the other short of the target. 16 % |Factor 9% Factor 9% |Factor 9% Factor 9% Factor Z/Z1 Z/Z1 Z/Z1 Z/Z1 | | | Z/Z1 1 || 0.02 21 0.40 41 || 0.80 61 | 1.27 | 81 | 1.94 2 : 0.04 || 22 || 0 . 41 || 42 0.82 62 | 1.30 82 | 1.98 3 || 0 , 06 || 23 0.43 || 43 0.84 63 | 1.33 83 2.03 4 0.07 || 24 0.45 44 || 0.86 64 | 1.36 | 84 2.08 5 0.09 || 25 0.47 45 0.89 || 65 | 1.39 || 85 2.13 6 0.11 26 || 0.49 46 0.91 | 66 | 1.42 | 86 2.18 7 || 0 , 13 27 | 0 , 51 || 47 0.93 || 67 || 1 , 45 87 2.24 8 || 0 , 15 28 0.53 || 48 0.95 | 68 | 1.48 || 88 2.30 9 0.17 | 29 0.55 49. 0.98 || 69 | 1.51 | 89 2.37. 10 || 0 , 18 30 || 0 . 57 50 | 1.00 || 70 || 1 , 54 90 2.44 11 || 0 . 20 31 0.59 || 51 | 1.02 || 71 || 1 , 57 91 2.52 12 || 0 , 22 || 32 0.61 52 | 1.04 || 72 || 1 , 60 92 2.60 13 0.24 || 33 0.63 || 53 | 1.07 || 73 || 1 , 64 || 93 2.69 14 || 0 , 26 34 0.65 54 | 1.09 || 74 | 1.67 94 2.78 15 0.28 35 | 0 . 67 || 55 1-, 12 75 | 1.71 95 2.91 16 || 0.30 || 36 0.70 || 56 || 1 , 14 76 || 1 , 74 96 || 3.04 17 | 0.32 37 || 0 . 72 57 || 1 , 17 77 || 1 , 78 97 3, 22 18 0.34 38 0.74 58 || 1 , 19 78 || 1 , 82 98 || 3, 45 19 || 0.36 39 0.76 59 || 1 , 22 79 | 1.86 99 || 3.82 20 0.38 40 || 0 , 78 60 | 1.25 80 || 1 , 90 100 As out table concerns itself with zones symmetrically dis- posed with respect to the center of impact (or point aimed at) we must not take 100 yards and 50 yards as zone widths but as half zone widths. Taking, then, 200 and 100 yards as the zone widths we may find the percentages and halve the latter. - Thus for a zone 200 yards wide we find for the battery, the 50% zone width of which is 52 yards - Z. –*–3.84 ... per cent = .99 Z1 52 - and for the 100 yard zone width : Z 100 – = −1 = 1.92 . . per cent = .80% Z1 52 p % Hence the percentage for the two half zones, taken together, is .805 . .990 1.795 = * * * * * = .893 2 + 2 2 34% or we may expect 89%% of hits with the calibrated battery; 17 that is 9 hits out of 10 shots. This is the best shooting that can be required of the personnel. It follows that it is important (a) To adjust all scales and instruments to a condition of minimum error. (b) To keep a record of all shots fired, with deductions from results, especially from trial shots. (c) To draw correct conclusions from trial shots. It is safest to take the mean range and the mean deflection, unless sone definite cause of error existed for a particular shot. (d) To avoid correction of fire during the practice when the error of a particular shot is within the mean error of the trial shots, or when the difference between two shots is less than the width of the 50% zone. (e) To exercise the greatest care with the ammunition and verify the weights of projectiles and powder charges and note the condition and temperature of the latter about the time of firing. (f) To so train the loading detail that the projectile is properly seated, always. This is of great importance, not on account of the density of loading but on account ot the escape of gas past the projectile. The bands of the projectiles should be examined beforehand and all burrs and irregulatiries re- moved. Eacercises: 1. To fire at a target 6300 yards distant, with the following data, how is the range-scale to be set? Data: 10-inch rifle, w = 634 lbs. ; range table V, 2250. Antici- pated powder velocity 2250; temperature of powder, 50° Fahr.; temperature of air, 37° Fahr.; barometer, 29.85 inches; wind component (WA) miles per hour, + 4.0 (rear); powder, Inter- national smokeless, lot 6, 1906, 7 days in service magazine. (a) 5, /ö = .974 –.85 (.974 –.943) = .974 –.85 ×.031 = .948 Hence º = – (1 – 0.948) + - 052 (See Table A: Values of 6,76) (b) AV = — 33 f.s for powder temperature 50° F. 18 TABLE B TABLE OF VALUES OF PERCENTAGES CHANGE IN MUZZLE VE- LOCITY OF POWDERS TESTED AT 70° F. AND FIRED WITH A MAGAZINE TEMPERATURE tº F. AV for V = t *V/V 2000 T 2250 2500 0° — . 0323 | – 65 — 73 —81 10° — . 0302 || – 60 – 67 — 75 20° — .0275 | –55 – 62 – 69 30° — .0241 –48 — 54 – 60 40° – .0199 || –40 –45 –50 50° — . 0147 — 29 –33 — 36 60° — . 0082 — 16 — 18 – 20 70° — .0000 0 0. 0 80° .01.01 20 23 25 90° .0228 47 52 58 100° .0387 77 87 96 (Computed from the formula * = .00867 {2^* — 2.03270t } V - - .00867 {2° — 4.73 }). (c) W. = + 4 mi/hr. I. From these using data for construction of range board (see Table D.) or, preferably, range board itself, based on these data, find the following corrections based on actual range 6300 yards. - - • . — 5.2% in C, + 74 yards — 34 f.s. in V, + 151 yards + 4, W., — 10 Total + 215 Corrected range 6515 II. Taking the corrected range as 6500 yards, we find in the same order . + 78, + 154, - 11, = + 221 yards Corrected range 6300 + 221 = 6521 yards. From the range table for this gun (Table C.) it is seen that this corresponds to an elevation (uncorrected for height of site and curvature) of r 19 4°31'.1 + (4°42'.3 – 4°31'.1) × ### = 4°31'.1 + 11’.2 × .605 = 4°37.9 - - The correction for height of site and curvature was that due to a battery 31 feet above mean low water at 6300 yards range or 7.2. Hence the clinometer should be set at 4°37'.9 — 07.2 = 4°30'.7. - Hence set elevation range-scale, at 6521 yards correspond- ing to 4°30'.7 clinometer elevation. - 2. Suppose that as a result of setting and conditions as in 1, the following ranges are recorded, with a particular gun: Shot No. Range Shot No. Range 1. 61.90 6 6320 2 6215 7 6235 3 6185 8 6155 4 6290 9 6230 5 6205 10 6230 (a) What is the mean range? (b) What is the mean error? (c) What is the probable error? & 30 cot (a + e) (d) What is the danger space, –5–- 10 (cota + e) (e) What is the correction to be applied to range-scale? (f) What is the percentage of hits to be expected on the service target, (30 feet high and 60 feet long) includ- ing ricochet hits, if the gun is properly laid? (a) Mean range = * = 6225.5 = Ro (b) Mean error found as follows: Ranges Errors - R – Ro 61.90 —35.5 6215 — 10.5 6.185 — 40.5 6290 +64.5 6205 —20.5 6320 - +94.5 6235. + 9.5 6155 — 70.5 6230 + 4.5 6230 - + 4.5 2R 6225.5; 2e 355.0 R, 6225.5; eo. 35.5 20 That is, mean error is + 35.5 yards. (c) Probable error, r = +35.5 × 0.845 = +30 yards. (d) From range-table a =5°36' and a + e =5°43’; cot(a + e) = 10.0 and danger space = 10 × 10 = 100 yds. (e) Correction to be applied to range scale is 6300 – 6225.5 = +74.5 yards. (f) The width of the 50% zone is Z1 = 2 xro = 2 ×30 = 60 yards The danger space consists of halves of two zones whose widths are 200 and 100 yards. Hence Factors 4, in the two cases are 200 and 100 or 3.33 and Z1 60 60 1.67. Hence percentages are .97 and .74 and the mean of these is 1; =85.5%. Hence this gun if properly laid can be expected to make 17 hits out of 20, in the long run, at this range. This is the best that can be required of the personnel. 3. If the other gun of the battery when laid at the same elevation gave a mean range of 6086 yards and a mean error +101 yards. - (a) What correction should be made in the range elevation scale? (b) What is the probable error of this gun? (c) What is the probable error of the calibrated battery? (d) What percentage of hits may be expected with this battery, as calibrated? (a) 6300–6086 = +214 yards. (b) == 101 X.845 = +85.3 yards. (c) sº. +58 yards. (d) Factors Z/Z, for battery as calibrated are #}} and 4% Hence percentage is ** = 60% or 6 hits out of 10. That is, the second gun shoots so inaccurately that it seriously impairs the shooting of the battery even after calibration; and its inaccuracy should be remedied or the gun or carriage replaced. - 21 4. Find the percentage of hits to be expected of the second gun alone. - - NOTE: When a gun is calibrated by changing the relative positions of pointer and range-scales it is assumed that the car- riage has an angular jump (positive or negative) that is con- stant for all ranges. This is the best that can be done, and is practically satisfactory. Thus the jump in gun No. 1 is negative and equivalent to 75 yards at that range. But at 6300, 11’ =200 yards. Hence the assumed jump is — ſº X11'= –4'.1. - The term jump as here used may include not only the action of the carriage, but aslo virtual jump, “kiting” of pro- jectile, slight unsteadiness in flight, changing inclination of trajectory, etc. Tables A, B, C, and D are, respectively, a table of values of 6,76, the atmosphere factor, a table of percentage changes in V due to change in temperature of powder; a range-table for the 10-inch B. L. Rifle; and a table of data for the construction of range-correcting devices. Table A is reprinted from Artil- lery Note No. 25; Table B and the formula given are deduced from results obtained by the Ordnance Department at Sandy Hook. - INTERPOLATION BY DIFFERENCES Let a quantity vary in accordance with a law, which law need not be known except in a general way from a set of values of the quantity corresponding to a set of values of another quan- tity taken at regular intervals; for example, suppose ew have glven: R T 1000 1.37 4000 5.85 7000 11.11 10000 17.36 13000 25.17 We have values of R at regular intervals, and correspond- ing values of T. It is required from the values furnished to interpolate intermediate values which will follow a law con- sistent with that of the given values. To do this we may resort to the method of differences as follows: - . * Given the quantities a, b, c, d, e, etc., as the values cor- responding to values of a related quantity taken at regular inter- 22 vals, to find any intermediate value, corresponding to an intermediate value of the related quantity, called the argument. 1st. Subtract each value from the next following and designate the remainder öl. 2nd. Arrange these differences in order and treat them in the same way. Designate their remainders 62. - 3rd. Do the same with the new set of differences and obtain 6, etc. - 4th. Continue until the differences are small, or zero, or the terms given afford no more differences. To illustrate: R T ô1 ô2 63 * 64 1000 1.37 4.48 .78 .21 .36 4000 5.85 5.26 .99 .57 7000 11.11 6.25 1.56 ..10000 16.36 7.81 13000 25.17 Here a = 1.37; b = 5.85; c = 11.11; d = 16.36; e = 25.17; b — a = 4.48; c – b = 5.26; d – c = 6.25; e – d = 7.81. - These latter are tabulated as 61 and their differences (tabu- lated as 32) are: - - c – b – (b — a) = c – 2b + a = 0.75 d – c – (c — b) = d – 2C + b = 0.99 e – d – (d — c) = e – 2d + c = 1.56 The 63 column is obtained in the same way from the 62 column, and so on. Considering now only the first horizontal line of the abov table, we write: + - a = 1.37; 61 = 4.48; 62 = 0.78; 63 = 0.21; 64 = 0.36 and this is all that is needed for interpolating. For, reverting to the letters, we find b — a = 6, ... b = a + 61 c — 2b + a = 62 ...". C = ô2 + 2b —a = a +261 + 62 d – 3c -- 3b — a = 98 ... d = a + 361 + 362 + 63 e – 4d -- 6c – 4b + a = 6, ... e = a +46, + 632 + 56s-i-6, and, in general, for a term the nth after first In 61 + n (n − 1) 5, n(n-1) (n −2) 62 Tº = a+= 1.2 1.2.3 23 To illustrate the use of this formula, let us take the numer- ical example used. The term corresponding to R = 1000 is a - 6 & 2 2 & 4 R = 4000 is the 1st after a .”. n = 1 & 6 G. & & 4 “ R = 7000 is the 2d after a ... n = 2 * 6 & 4 6 & “ R = 10000 is the 3d after a . . n = 3 and so on - - Suppose it is required to find the values of T for every thousand of R; we have, R Il R Il 2000 # 7000 2 3000 # 8000 4 4000 1 9000 § 5000 # 10000 3 6000 3. 11000 19. Hence for 2000, - T = 1.37+% (4.48) +} X –$ X} X.78+} X – # X –$ X; X.21 +} X – 3 × —# X –$ X ºr X.36 = 1.37 -- 1.493 — .087 -- .013 — .015 = 1.37 -- 1.493 — .087 -- .013 — .015 = 2.774 For 3000 T = 1.37+% (4.48) +3 × —# ×3 ×.78+3 × —# × –$X; X.21+3 × —# × —# X –$X ºr X.36 = 1.37 -- 2.987 — .087 -- .010 — .010 = 4.27. We may continue thus, making n = 3, 4, §, *, and obtain the values of T for the missing ranges without difficulty. The values taken from the range table are R T R T 2000 2.78 : 8000 13.06 3000 4.27 9000 15.14 5000 7.52 11000 19.75 6000 9.27 12000 22.35 Let the student obtain the values for these ranges by inter- polation and compare. It is frequently necessary to interpolate values from ranges 1000 to 3000 yards apart, and as a rule, second differences are regarded as sufficient. In the following methods of interpolating any number of means between values, all differences of a higher order than the second will be neglected. 24 Our formula becomes n(n º 1)Sg 2 Suppose we desire the interval between successive values to be only 1/m of what it was; then making n = 1/m, 2/m, T. =a+n ô1 + 3/m, . . . . . . (m –1)/m we get the intermediate terms. The terms are in order: ô1 1 1 61 (m e-º-º-º-º: 1) a -- — — ( — — 1 | 6′ = a — — — 6 + IQ + 2m (; ) 2 + II] 2m2 2 a +º, +; (; – 1) . = a + , -ºs. II] 2m \ m g Iſl 2m2 a ++ + š, (º - 1) s. - a + º-ººººº. IQ 3m \ m m 2m2 If we now take the differences between these successive values and call them A1, A2, A3, . . . . . . . . . . A, we have: A1 __31 – m - 1 ô2 IIl 2m? A2 - * II] — 3 ô2 IIl 2m? A3 = * † ==ºp II? -- 5 ô2 I]] 2m? etc. etc. Placing now m = 3, 4, . . . . . . . . . . . . . . 10 we have m = 3 m = 5 A1 -: – ; 3. A1 -- * ô2 6, 6 A2 -: - A2 –4– 3's 32 6 6, A4 -: + # ôs A3 -: ô1 , II] = A4 -#4 3's 62 A1 -* – A As -* + , , 4 5 6 A2 - - g's ô2 25 As - 4 * 6, A4 = * + *s ô2 - 4 m = 10 A1 = *- wºn 3, A6 = * + ** 6, 10 10 As =*— ** 6, A7 = ** aid 3s. 10 10 - A -- *, *. A, -º-; its A4 =*— ** 6, A9 –44 ** 62 10 * 10 - As – “… wºn 3, Al-&# ** 62 10 10 62 2 by B, we have, generally, Representing * by A and II] whatever the value of m, A1 = A – (m – 1) B A2 = A – (m – 3) B = A1 + 2B As = A – (m — 5) B = A2 + 2B A4 = A – (m – 7) B = A3 + 2B As = A – (m – 9) B = A4 + 2B etc., etc., ad libitum, according to the value of m. For m = 3; A = 61/3 and B = + 62/18 For m = 4; A = 31/4 and B = + 32/32 For m = 5; A = 61/5 and B = + 62/50 For m = 10; A = 61/10 and B = + 62/200 Suppose now values are interpolated in the numerical example with m =3. We have, using second differences only, A = } ôi, B = 32/18 Hence for 2000. A – tº - 1493; B – º – 04383 3 - 18 and A = A- (m-1) B = A-2B = 1.493 – .087 = 1.406 A2 = A1+2B = 1.406+.087 = 1.493 A0 = A3+2B = 1.493+.087 = 1.580. . 26 Hence for R =2000, T = 1.37 --A1 = 2.776 = 2.78 R = 3000, T = 2.776 --A = 4.269 = 4.27 R = 4000, T = 4.269 + As = 5.849 = 5.85 Now, owing to the fact that third differences are neglected and that the cumulative effect will appear, we begin again at 4000 yards using a = 5.85; 61 = 5.26; 62 = .99; and hence A = 1.753; B =.055; 2B = 0.110 A1 = A —2B = 1.753 –.110 = 1.643 A2 = A1 +2B = 1.753 A3 = A2 +2B - = 1.863 Hence for R = 5000, T = 5.850+1.643 = 7.493 6000, T = 7.493 + 1.753 = 9.246 7000, T =9.346 - 1.863 = 11.109 The case where m = 10 is an important one, and it will be desirable at the same time to illustrate a case where the succes- sive differences do not increase in order or may be negative— R V (º) 61 62 100 2153 —269 +10 4000 1884 —259 +28 7000 1623 —231 +44 10000 1392 — 187 13000 1205 Find'vo for each 300 yards between 7000 and 10000 yards. Here 61 = —231; 62 = +44 - A = 61/10 = —23.1; B1 = 6°/200 = .22; 2B = .44 Hence A1 = A —9B = —23.10 – 198. = —25.08 As = A +2B = } –24.64 A3 = * - –24.20 A4 = —23.76 A5 = —23.32 A6 = –22.88 A7 > –22.44 As = –22.00 A9 - —21.56 A10 –21.12 27 Hence R V (9 A Range table va, 7000 1623.00 —25.08 1623 7300 1597.92 –24.64 - 7600 1573.28 —24.20 1574 7900 1549.08 —23.76 8200 1525.32 —23.32 1526 8500 1502.00 - –22.88 8800 1479.12 –22.44 1480 9100 1456.68 —22.00 94.00 1434.68 —21.56 1435 97.00 1413.12 —21.12 ... 10000 1392.00 1392 It will be noticed that there is in no case a difference of 1 foot per second. THE USE OF DIFFERENCES IN DETECTING ERRORS OF CALCU- LATION OF OBSERVATION Where it is known that a quantity varies very gradually and regularly, it is safe to assume the third order of differences constant. On the other hand, if a set of observed or calculated values are somewhat irregular, imperfections may be eliminated by assuming 63, uniform. To show the differences in correcting such errors the fol- lowing given in the Rev. Francis Bashforth’s “Motion of Pro- jectiles” is of value: “The following example is given as illustrating the great utility of differencing when systematic experiments or indepen- dent calculations are being made. - Suppose the following logarithms of numbers had been in- dependently calculated, and that the calculator wished to test the accuracy of his work. Numbers º alculated ô1 62 Corrections logarithms 1280 .107210 +3380 –27 1290 .110590 3353 35 +10 1300 .113943 3318 05 –20 1310 .117261 3313 35 +10 1320 . 120574. 3278 24 – 1 1330 .123852. 3254 26 + 2 1340 .127106 3228 23 — 1 1350 .130334 3205 23 1360 .133539 3182 . 1370 .136721 28 Applying these corrections to the column of 62 values we get a new set of values of 62, giving a sensibly constant 63; using the new values of 62, we construct new values of 61, and thence of the logarithms. . Numbers cº 61 ô2 1280 .107210 +3380 –27 1290 .110590 3353 —25 1300 .113943 3328 —25 1310 .117271 3303 —25 1320 .120574 3278 —25 1330 .123852 3253 –24 1340 .127105 3229 —24 1350 .130334 3205 —23 1306 .133539 3.182 1370 . 136721 The difference of type indicates the erroneous figures in the first table and their corrections in the other. A method similar to this is in use in our service in checking calculations in range tables. It is also used in calculations of corrections of elevation due to curvature and height of site above mean low water. After the calculations are completed they are checked by making the second or third differences practically constant according to the interval of calculation. APPLICATIONS I. THE USE OF DIFFERENCES IN CALCULATING RANGE TABLES Much of the labor of computing range tables may be avoid- ed by the use of differences. ^ In the first place, it is possible by calculating log C for zero, 5000 and 10000 yards range and taking these, for purposes of interpolation, to five places to calculate by differences a closely approximate value of log C for every thousand yards range. 2 - The value of C for zero range is Co -*.*. since f = 1 C and p + q =0. 3 Now 6 _d” 1+2+* | . Representing the factor in par- 2w 135 enthesis by 6' we have * = 1 + º-e 135 and g C = f, . 2 w". ce d'à Co = 2 W2 C dé Hence C = f. X Co Y” Now f, and Y’ may be much more briefly calculated if we have a closely approximate value of the true log C. This we may find by differences as above indicated. An illustration will be given using the 8-inch, 10-inch and 12-inch guns, the weights of projectiles being 316, 604 and 1046 pounds, and the muzzle velocities 2200, 2250 and 2250, respec- tively. We have for log Co in the three cases 0.78484; 0.86309; and 0.94419. respectively. Calculations for 5000 give for log C in the three cases 0.76022; 0.84247; 0.92455; and , for 10000 yards, 0.72189; 0.81199; 0.89834. - The computations may be arranged as follows: 8” 10’’ 12” R log C log C log C 0 .78484 .86309 .94.419 5000 .76022 .84.247 .92455 1000 .721.89 .81 190 .89834 61 — .02462 — .02062 —.01964 ô2 — .01371 — .00995 — .00657 A — .004924 — .004124 — .003928 B – .0002742 — .0001990 —.0001314 A —2B — .003827 — .003328 — .003402 + 2B — .000548 -- — .000398 — .000263 — A1 —.003827 — .003448 — .003402 A2 — .004375 — .003726 — .003665 A3 — .004124 — .004124 — .003928 A4 — .005472 — .004522 — .004.191 As — .006020 — .004920 — .004.454 A6 —.006569 — .005318 — .004716 A7 —.007.117 — .005.716 — .004799 As — .007666 —.006114 — .005242 A9 –.008214 — .006512 — .005505 A10 –.008763 — .006910 — .005768 A11 — .009311 — .007308 —.006030 A12 — .009859 —.007706 — .006293 A18 –.010408 — .008104 —.006556 30 Applying these differences we find the following interpo- lated values of log C for ranges at intervals of 1000 yards and beside them are placed the computed values from the range tables showing how closely these values approximate to those of the true log C and hence that they will answer admirably for computing r' and f. - values of log C. 8-inch 10-inch 12-inch R Int True Intº .True Int. True 0000 .784.84 .784.84 .86309 .86309 .94419 .94419 1000 .781.01 .87.118 .85976 .85922 .94079 .94034 2000 .77664 .77602 .856.04 .955.74 .93712 .93690 3000 .77171 .77081 .85191 .85.177 .93320 .93310 4000 .76624 .76555 .84739 .84733 .92900 .92897 5000 .76022 .76022 .84247 .84.247 .92455 .92455 6000 .75365 .75359 .837.15 .83724 .91983 .91986 7000 .74654 .74620 .83144 .83.167 . .91.486 .91492 8000 .73887 .738.23 .82532 .82572 .90961 .90971 9000 .73065 .73006 .81881 .. 81922 .904.11 .90420 10000 .72189 .72189 .81190 .81190 .89834 . .89834 11000 .71258 .71371 .80459 .80345 .89231 |.39208 12000 .70272 } .79689 .79351 .88602 .88536 13000 .69222 .78878 .87946 .87812 It will be noted that the greatest percentage error in the trial C is that for the 10-inch gun at 12000 yards. erence of the logarithms is then .00338. The ratio of the inter- polated value to the true value is in this extreme case 1.008:1, or the difference between the two C's is less than 1%. As an illustration of the use to which these interpolated values may be put, the value of C will be found for the 12-inch gun for 9000 yards range using the interpolated C as a basis of The dif- computation. log X = 4.43136 a.c. log C = 9.09589 log Z = 3.52725 Z = 3367 we find, 1 + º-e = 1.1166, and f = 1.0184 * † 155 all Cl I1 31 • Now f C :- ; Co 7. log f. = .00792 a.c. log r' = 9.95210 log Co = .94419 log C = 0.90421 APPLICATION II. Second differences are sensibly constant in relation to range-corrections in the vicinity of a given range. The necessity for plotting each range-correction opposite a non-tabular corrected range is obviated by the method about to be deduced for finding the correction by the use of a tabular corrected range, opposite which this range correction is plotted. The difference in the two procedures may be illustrated thus: With 9000 yards as the uncorrected range we found fore changes in C of +20%, H–10%, -10% and —20% the follow- ing values AR: +366, +200, -259 and –591, respectively, and these were to be plotted opposite the corrected ranges 8634, 8800, 9259 and 9591, respectively. Using the data for the other ranges we determine similar groups for each of them. Finally, curves are drawn with each correction opposite the corrected range and we might then scale off the values of AR opposite 9000 yards. We should find thus that for a corrected range 9000 yards we would have in round numbers +400, H-210, -245 and –530 yards, respectively. It is desired to find these directly using the corrected range (9000 yards in the case quoted) as a basis of computation. If second differences are constant, we have AR = h -- kR + 1 R2 since there could be no term in R3. Of the three constants h, k and l, h is zero for all correc- tions due to changes in V, C and W1; k is zero for all changes in C and W., and k is 2 AV for changes in V; 1 is negative for V velocities and positive for C and W. Taking the general expression and representing the cor- rected range by R = R – AR we have AR *** = k + 1 R R + 32 * - k + 1 R 'R whence, by division, - - R R k + 1 R R , AR #. ~ R - H.- 1 + ...) (1 + R #) AR’ AR AR AR -(1 ++ . #) (1 x * – k);) For k = 0 we find # - #1 * (*) tº or AR AR’ AR’ AR’\ 2 * - * {1 + 2* + 5(.)) The values of # are tabulated for use in C and W, changes y with º as argument. Table I is for positive values of AR’. Table II is for negative values of AR’. For k = 2 y we find after reduction and neglect of in- significant terms, as before, The use of these formulas will be illustrated by the follow- ing typical cases. • . Given values of AR’, for 9000 yards, for 12" gun V = 2250, the values of AR’ corresponding to the changes in C, W1 and V indicated, find AR in each case, (R' =9000). Solution: I AC/C AR’ AR/R’ AR/R’ From AR +.20 +366 .0407 .0443 } Table I +399 +.10 +200 .0222 .0232 +209 —.10 —259 —.0288 –.0272 } Tabel II —245 —.20 +591 —.0657 –.0585 —527 33 II III. AV + 150 + 50 — 40 — 150 Wi +50 —50 AR’ AR’/R’ AR/R' From AR +220 .0244 .0256 Tabel I +230 —292 –.0324 — .0305 Table II — 275 TD / AR 2 º' 2*Y Aſ I AR AR R’ V AR AR’ + 877 .1949 .1333 .06.16 1.062 +931 + 305 .0678 –.0444 .0234 1.023 +312 — 318 –.0707 ––.0444 – .0263 .974 – 310 — 1002 — .2227 —H. 1333 – .0894 .911 —913 With these values of AR we may use the corrected range 9000 yards; and the corrections to be plotted opposite that cor- rected range may be interpolated from these values to intervals of 2% in C, 10 miles per hour in W, and 10 f.s. in V. AR’/R’ .00 .01 .02 *03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .0 000 010 020 .0 102 113 123 .0 208 TABLE I. AR/R' 0 1 2 3 4 5 6 031 041 051 061 134 144 155 165 241 152 263 274 354 365 377 388 471 483 495 506 593 606 619 631 722 735 749 762 856 870 884 898 997 *011 +026 +040 143 158 174 189 297 313 329 345 458 474. 491 507 626 643 661 678 802 820 838 856 986 *005 *024 *043 177 197 217 237 219 331 447 568 696 828 958 113 265 425 591 766 948 138 130 342 459 581 709 842 982 128 281 442 608 784 967 157 319 435 556 683 815 954 098 250 409 574 748 929 119 Positive Changes in C and W. 7 8 9 071 082 092 176 186 197 285 296 308 400 411 423 519 531 543 644 656 669 775 788 801 912 916 940 *054 *069 +083 204 219 234 361 377 393 524 540 557 696 713 731 874 892 910 *062 +081 + 100 257 277 297 34 TABLE II. Negative Changes in C and W. AR/R’. AR'/R' 0 1 2 3 4 5 6 7 8 9 —.00 – .0 000 010 020 030 040 050 060 069 079 088 —10. –.0 108 098 117 127 136 145 155 164 174 183 –.02 – 0 192 202 211 220 229 238 247 256 265 274 –.03 –.0 283 292 301 310 319 328 337 345 354 362 —.04 —.0 371 379 388 396 405 413 422 430 439 447 —.05 –.0 456 464 473 481 489 498 506 51.4 523 531 —.06 –.0 539 547 555 563 571 579 587 595 603 611 –.07 –.0 619 627 635 643 651 659 667 675 683 691 –08 –.0 698 706 713 721 729 736 744 751 759 767 –09 –.0 744 782 790 797 805 812 820 827 835 842 —.10 –.0 850 857 865 872 879 887 894. 902 909 918 — .11 — .0 925 933 940 947 955 962 969 977 984 991 – .12 — .0 998 +006 +013 *020 *028 +036 #043 *050 *058 +065 — .13 — . 1 072 080 087 O94 102 109 117 124 131 134 — .14 — . 1 145 153 160 167 175 182 190 197 204 212 —.15 — . 1 219 227 234 242 249 257 264 271 279 z186 CURVATURE OF THE EARTH If a point move away on the surface of a sphere from another point, it will, after moving 90° on a great circle, have acquired an angular depression of 45° from the tangent to the circle at the stationary point. When it has reached a point 180° from the stationary point it will be diametrically opposite the stationary point and a line joining the two points will be at right angles to the tangent to the circle at the stationary point; and, in general a simple figure will show that the de- pression due to curvature is half the curvature of the circle between the two points. There is thus a depression angle of % of a minute for a curvature of the earth of 1'. Now 1' is equal to a knot or 1.152 miles. That is to say 2027 yards. We thus have a depression due to curvature of 1' for 4054 yards, or, 0.2467 per 1000 yards. That is 0.987 or, practically, 1’ per 4000 yards. The curvature in feet may be found if de- sired from this relation. . . ; - It becomes * K-33333–10 R-2154 (ii). - * 1000/ r 35 in which Kis the curvature in feet and R the range in yards. The curvature in feet for ranges at 1000 yards interval are given in the following table. CURVATURE IN FEET. *. Curva- : - Curva- Range | ` ture | * Range ture A yds. feet feet yds. feet | feet 1000 || 0.22 || 0.64 || 12000 || 31.02 || 5.38 2000 || 0.86 | 1.08 || 13000 || 36.40 || 5.82 3000 1.94 | 1.51 || 14000 || 42.22 || 6.25 4000 3.45 | 1.94 || 15000 || 48.47 7.66 5000 5.39 2.36 || 16000 || 55.14 || 7.11 6000 7.75 2.80 || 17000 | 62.25 | 7.54 7000 | 10.55 3.24 || 18000 || 69.79 || 7.97 8000 || 13.79 3.66 || 19000 || 77.76 8.40 9000 || 17.45 9.04 || 20000 | 86.16 || 8.83 10000 || 21.54 4.52 || 21000 | 94.99 || 9.26 11000 | 26.06 || 4.96 || 22000 || 104.25 36 THE METHOD OF DIFFERENCES Let a, b, c, d, etc., represent the successive terms of a series formed according to any fixed law, then if each term be subtracted from the succeeding one, the several remainders will form a new series called the first order of differences. If we subtract each term of this series from the succeeding one, we shall have another series called the second order of differences, and so on, as exhibited in the annexed table. a, b, C, d, €, b-a, c-b, d-c, e-d, etc, 1st. c-2b-Ha, d-2C+b, e-2d + c, etc., 2nd. d-c-H3b-a, e-3d-H3c-b, etc., 3rd. e-4d +6c-6c-4b +a, etc., 4th. If, now, we designate the first term of the first, second, third, etc., orders of difference, by di, d2, da, d4, etc., we shall have, di = b -a, whence b =a+di, d2 = c –2b +a, whence c = a +2di +do da = d – c.3 +3b —a, whence d = a +3di +3dg-H da 4 = e –4d +6c –4d +a, whence e = a +4di +6.d3+4d3+da, etc., etc. - And if we designate the term of the series which has n terms before it, by T, we shall find by a continuation of the above process, T=a+nd,+*(*="2d, ""T"-ºld, 1.2 1.2.3 .n(n − 1)(n −2) (n −3) 1 + 1.2.3.4 d4+ etc., (1) This formula enables us to find the (n+1)th term of a series when we know the first terms of the successive orders of difference. To find an expression for the sum of n terms of the series a, b, c, etc., let us take the series 0, a, a +b, a +b+c, a +b+c+d, &c (2) The first order of differences is evidently •. a, b, c, d, &c., (3) Now, it is obvious that the sum of n terms of the series (3), is equal to the (n+1)th term of the series (2). But the first term of the first order of differences in series (2) is a; the first term of the second order of differences is the same as di in equation (1). The first term of the third order of difference is equal to d2, and so on. 37 Hence, making these changes in formula (1), and denoting the sum of n terms by S, we have, * n (n − 1) (n − 1)(n −2) (n − 1) (n −2) (n −3) s-na-Fººd, Ha*-ī;-ºd, ºn 1.2.3.4 da When all of the terms of any order of differences become equal, the terms of all succeeding orders of differences are 0, and formulas (1) and (4) give exact results. When there are no orders of differences, whose terms become equal, then formu- las do not give exact results, but approximations more or less according to the number of terms used. Eacamples 1. Find the sum of n terms of the series 1.2, 2.3, 3.4, 4.5, &c. Series, 1. 2, 2.3, 3.4, 4.5, 5.6, &c. 1st order of differences, 4, 6, 8, 10, etc. 2nd order of differences, 2, 2, 2, &c. 3rd order of differences, 0, 0. Hence, we have, a = 2, d1 =4, d2 = 2, da, d4, &c., equal to 0. Substituting these values for a, di, do, &c., in formula (4) we find, n (n − 1) ×4+n\"T")." –2) 1.23 1.2.3 S _n(n – 1)(n +2). 3 2. Find the sum of n terms of the series 1.2.3, 2.3.4, 3.4.5., 4.5. 6, &c. 1st order of differences, 18, 36, 60, 90, 126, etc. S = 2n + X 2; whence, 2nd order of differences, 18, 24, 30, 36, e&c. 3rd order of differences, 6, 6, 6, &c. 4th order of differences, 0 0, &c. We find a = 6, d1 = 18, d2 = 18, d3 = 6, da = 0, e&. Substituting in equation (4), and reducing, we find, s_n(n+1)(n+2)(n −) 4 tº 3. Find the sum of n terms of the series 1, 1+2, 1+2+3, 1 +2+3+4, &c. Series, 1, 3, 6, 10, 15, 21 1st order of differencss, 2, 3, 4, 5, 6. 2nd order of differences, 1, 1, 1, 1. 3rd order of differences, 0, 0, 0. 38. a = 1, di =2, d2 = 1, da= 0, d4 = 0, &c. - n(n-1).9 n(n-1)(n −2) na-H3n2+2n h 9 S = 2 €In Ce nºt-º-º-º-º: g 1) (n +2 , red 5 S n(n+1)(n+2). Or, reducing 1.2.3 ) 4. Find the term of n terms of the series 12, 2, 3, 4, 5, &c We find, a = 1, di =3, d2 =2, da =0, d. =0, &c, &c. Substituting these values in formula (4), and reducing, S _n(n+1)(2b +1) | 1.2.3 tº 5. Find the sum of n terms of the series, 1 (m+1), 2(m+2), 3(m+3), 4(m+4), &c. We find, a = m +1, di = m +3, d2 = 2, d3 =0, ;&c. n(n − 1) ſ.v. as Ln(n − 1)(n −2) J isº (m-3)+n-º-ºx2. we find, whence, S = n(m+1)+ s_n(n+1)(1+2n+3m) 1.2.3 Of Piling Balls See plate OT, The last three formulas deduced, are of practical applica- tion in determining the number of balls of different shaped piles. First, the Triangular Pile. A triangular pile is formed of successive triangular layers, such that the number of shot in each side of the layers, decreases continuously by 1 to the single shot at the top. The number of balls in a complete triangular pile is evidently equal to the sum of the series 1, 1+2, 1+2+3, 1+2+3+4, &c. to 1 +2+ ... + n, n denoting the number of balls on one side of the base. But from example 3rd, last article, we find the sum of n terms of the series, - s_n(n+1)(n+2) (1) 1.2.3, Second, in the Square Pile. The square pile is formed, as shown in the figure. The number of balls in the top layer is 1; the number in the second layer is denoted by 22; in the next by 38, and so on. Hence, the number of balls in a pile of n layers, is equal to the sum of the the series 12, 22, 32, &c., n2, which we see, from example 4th of the last article, is 39 s_n(n+1)(2n+1) (2) 1.2.3 Third, in the Oblong Pile. The complete oblong pile has (m -- 1) balls in the upper layer, 2Gm +2) in the next layer, 3(m+3) in the third, and so on; hence, the number of balls in the complete pile, is given by the formula deduced in example 5th of the preceding article, s_n(n+1)(1+2n+3m) - (3) 1.2.3 If any of these piles is incomplete, compute the number of balls that it would contain if complete, and the number that would be required to complete it; the excess of the former over the latter, will be the number of balls in the pole. The formulas (1), (2), and (3) may be written, triangular, S =#. nºgº (n + 1 + 1) (1) (n +n+1) (2) 1. 3. Square, S *; 1) "#" (nºm)+(n+m rectangular, S =#. 2 Now since nºgº is the number of balls in the triangular face of each pile, and the next factor, the number of balls in the longest line of the base, plus the number in the side of the base opposite, plus the parallel top row, we have the following RULE Add to the number of balls in the longest line of the base the number in the parallel side opposite, and also the number in the top parallel row; then multiply this sum by one-third the number of the triangular face; the product will be the number of balls in the pile. - TABLE A values of 8/8 for temperature and pressure of atmosphere 78% saturated with moisture. (From Artillery Note No. 25). —17° Ther. Barometer Ther. Barometer F. 28” 29” 30” 31” F. 28” 29” 3'0 –20° 0.890 0.861 0.831 0.806 41° | 1.017 0.982 0.951 0.919 –19° 0.892 0.863 0.833 0.808 42° | 1.019 0.984 0.593 0.921 —18° 0.894 0.864 0.835 0.809 43° | 1.021 0.987 0.955 0.923 0.896 0.866 0.837 0.811 || 44° | 1.023 0.989 0.957 0.925 —16° 0.898 0.868 0.839 0.813 45° 1.026 0.991 0.959 0.927 —15° 0.901 0.870 0.841 0.815 || 46° | 1.028 0.993 0.961 0.929 —14° 0.903 0.872 0.843 0.816 || 47° | 1.030 0.995 0.963 0.931 —13° 0.905 0.874 0.845 0.818 48° | 1.033 0.997 0.964 0.933 —12° 0.907 0.876 0.847 0.820 || 49° | 1.035 0.999 0.996 0.935 –11° 0.910 0.878 0.848 0.822 50° | 1.037 1.002 0.968 0.937 —10° 0.912 0.880 0.850 0.824 51* | 1.040 1.004 0.970 0.939 — 9° 0.914 0.881 0.852 0.826 52° | 1.042 1.006 0.972 0.931 — 8° 0.916 0.883 0.854 0.827 53° 1.044 1.008 0.974 0.943 — 7° 0.918 0.885 0.856 0.829 || 54° 1.046 1.010 0.976 0.945 — 6° 0.920 0.887 0.858 0.831 55° | 1.048 1.012 0.978 0.947 — 5° 0.922 0.899 0.860 0.933 56° | 1.050 1.014 0.980 0.949 — 4° 0.924 0.891 0.862 0.835 57° 1.053 1.016 0.982 0.951 — 3° 0.926 0.893 0.864 0.836 || 58° | 1.055 1.018 0.984 0.952 – 28 0.928 0.895 0.866 0.838 59° | 1.057 1.020 0.986 0.954 — 1° 0.930 0.987 0.868 0.840 || 60° | 1.059 1.022 0.988 0.956 0° 0.932 0.899 0.870 0.842 || 61° | 1,062 1.025 0.990 0.958. 1° 0.934 0.901 0.971 0.844 || 62° | 1.064 1.027 0.992 0.960 2°| 0.936 0.903 0.873 0.845 || 63°| 1.066 1.029 0.994 0.962 3° 0.938 0.905 0.876 0.847 64° | 1.068 1.031 0.996 0.964 4° 0.940 0.900 0.878 0.849 65° | 1.071 1.033 0.998 0.966 5° 0.942 0.909 0.880 0.851 66° | 1.073 1.035 0.101 0.968 6° 0.944 0.911 0.881 0.853 || 67° | 1.075 1.037 1.003 0.970 7°| 0.946 0.913 0.838 0.885 68° | 1.078 1.040 1.005 0.973 8° 0.948 0.915 0.885 0.856 69° | 1.080 1.042 1.007 1.975 9° 0.950 0.917, 0.887 0.858 70° | 1.082 1.044 1,009 0.977 10° 0.952 0.919 0.899 0.860 71° | 1.085 1.046 1.011 0.979 11° 0.954 0.921 0.890 0.862 72° | 1.087 1.048 1.013 0.981 12° 0.956 0.923 0.892 0.864 73° 1.089 1.050 1.015 0.983 13° 0.958 0.925 0.894 O. 866 74° | 1.09.2 1.053 1.017 0.985 14° 0.960 0.927 0.897 0.867 75° | 1.094 1.055 1.019 0.987 15° 0.962 0.929 0.899 0.869 76° 1.096 1.057 1.022 0.989 16° 0.964 0.931 0.901 0.871 77° | 10.99 1.059 1.025 0.992 17° 0.966 0.933 0.903 0.873 78° | 1.011 1.062 10.27 0.994 18° 0.968 0.935 0.905 0.875 79° | 1.104 1.064 1.029 0.996 19° 0.971 0.937 0.907 0.887 80° | 1. 106 1.066 1.031 O. 998 20° 0.973 0.939 0.909 0.879 81° | 1. 109 1.068 1,033 1.000 21° 0.975 0.941 0.911 0.881 82° | 1.111 1.071 1.035 1.002 2° 0.977 0.943 0.912 0.883 83° | 1.114 1,074 1.038 1,005 23° 0.979 0.945 0.914 0.885 84° | 1.116 1.076 1.041 1.007 24° 0.981 0.916 0.947 0.887 || 85° | 1. 119 1,079 1.033 1,009 25° 0.983 0.949 0.918 0.888 86° | 1. 121 1.081 1,045 1,011 26° 0.985 0.951 0.920 0.890 87° | 1.124 1,083 1.047 1.013 27° 0.987 0.953 0.922 0.892 88° | 1.126 1.086 1.049 1,016 28° 0.990 0.955 0.924 0.894 89° | 1.129 1.089 1.053 1.018 29° 0.992 0.958 0.926 0.896 90° | 1. 131 1.092 j . 055 1 .020 30° 0,994 0.960 0.928 0.898 91° | 1 , 134 1.094 1.057 1,022 31° || 0.996 - 0.962 0.930 - 0.899 || 92° | 1.136. 1 .096 1.059 1.025 32° 0.998 0.964 0.932 0.902 93° | 1. 139 1.009 1.062 1,027 33° | 1.000 0.966 0.943 0.903 94° | 1. 142 1. 102 1.064 1.029 34° | 1.003 0.968 0.936 0.906 95° | 1. 144 1. 105 1.066 1.031 35° | 1,005 0.970 0.938 0.907 96° | 1. 147 1. 107 1,068 1,033 36° | 1.007 O.972 0.940 0,909 97° | 1. 149 1. 110 4.071 1,035 37° | 1.009 0.974 0.943 0.911 98° | 1 , 152 1. 112 1.074 1.037 38° | 1.011 0.976 O. 945 0.913 99° | 1.155 1. 115 1.076 1. 040 39° | 1,013 0.978 0.947 0.915 || 100° | 1. 157 1, 117 1,079 1,042 40° | 1.015 (). 980 O. 949 (). 917 TABLE C RANGE TABLE FOR 10-INCHRIFLE Perforation of | * ... • ‘’’, ºr . . . . . . . . . . . . Krupp armor, Deflection for— | capped projec-| ! | ... 30° | Nor- || with mal. |normel. ==f|§ | i | f a ; . |s...Tº . . . . . . . ; ºf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.37 || 0 35 | 97.3 || 8 || 2136 || 14.5 13.3 0.019 0.10 (0.85922 1.65 || 0 43 80.5 | 12 2114 | | | .023 .012 1.94 || 0 51 67.9 | 16 || 2002 | | | .027 . .014 || r : |Inches. Inches. | Degrees. |Degrees. Log. G. © 1 o — ºr- e º f S f ; 5 1 1600 || 0 23 || 0 59 58.4 || 21 2070 | | | .031 .016 1 1 26 2048 || | | .035 | .018 #1 | 1.1 4| 32 2026 || 13.4 | 12.2 .040 || 0.21 11 || 1 24 40.8 || 39 || 2005 || | | .044 || 023 0.85745 | 1 : 0 || 47 | 1983 || | | . .049 | .025 | : -: i i i ; i -s 1 56 1961 | | | .054 || 027 | | 76 1918 || 12.3 || 11.3 .064 .032 0.85177 .3 || 88 1897 || | | .069 .034 .3 || 100 1875 || || || .074 | .037 2600 | 2800 3000 || 1 3200 || 1 3400 : i i 3 4 2 2 8 i 7 6 .5 114 | 1854 || || | .079 .039 || 3 | 1812 || 11.2 | 10.2 | .091 || .044 |0.84733 159 || 1791 || | | .097 .046 || || 177 1770 | | | .103 | .049 | 3600 || 2 3800 | 4000 | . 4200 | 4400 i 2 : i : 9 6 i 6 1 9 : 1 4 3 196 || 1749 | 109 | .051 | 216 1728 . . . 115 .054 . 238 || 1708 || 10.1 | 9.2 . 121 .056 |0.84247 261 | 1688 | . 127 | .059 285 | 1668 - . 134 .061 4600 4800 5000 5200 5400 i 1 8 i : i : 6 9 . 5 9 1 4 : 2 3 8 311 | 1648 - . 141 .064 | 338 1628 .148 .067 | 6 | 1608 || 9. 1 || 8.4 | . 155 | . 070 (0.8374 396 || 1588 . 162 | .073 • 428 1569 . 170 | .076 5600 5800 6000 6200 6400 : 0 -9 : . i 5 3 i 1 2 1 1 i 3 6 6 162 1550 . 178 .079 497 | 1531 || . . . . 186 .082 534 || 1512 | 8.2 | 7.5 | .194 | .085 0.83167 573 || 1493 .203 | .088 614 || 1475 .212 .091 6600 6800 7000 7200 7400 i ; 6 : : : 1 1 5 0 : 3 8 : i 910 || 1369 . 271 , 111 968 || 1352 .282 . 115 1029 || 1335 | 6.6 6.1 .294 . 118 (0.81922 1093 || 1319 .306 . 122 1160 || 1303 . 318 . 125 1 5 4 6 : 5 3 º 10 19 16.41 || 10 46 16.90 | 11 14 7 i : 1 5 9 3 i : 1231 1288 .331 | . 129 1306 || 1273 .344 | . 133 1384 1258 * * .357 | . 137 0.81190 1466 | 1244 .371 | . 141 --- 1552 | 1230 .385 | . 145 17.40 11 43 17.91 | 12 12 .43 | 12 42 18.96 || 13 13 19.50 | 13 45 9600 9800 10000 10200 10400 5 9 5 5 52. : 3 5 : : : 1 8 : i 1642 | 1217 .400 . 149 1737 | 1204 .415 . 153 1836 | 1.192 - * .430 , 157 |0, 80345 1940 || 1180 .446 . 161 2049 || 1169 .463 . 166 657 | 1.457 .221 | .094 703 || 1439 . 230 | . 098 751 || 1421 || 7.4 || 6.8 .240 | . 101 . 801 || 1403 .250 | . 104 0.82572 854 || 1386 .260 | .108 7600 7800 8000 8200 8400 i 7 i : i 1 3 6 3 : 1 9 : : 8600 8800 9000 9200 94.00 i -1 20.05 || 14 18 20.61 | 1.4 52 21. 18 15 27 21. 76 16 03 22.36 | 16 40 10600 9 27 10800 9 45 11000 || 10 04 11200 10 24. 11400 || 10 44, i : : i i 5 4 5 () 2164 1159 .480 . 170 2285 || 1 149 , 498 . 174 241.1 | 1140 || 5 | 0 || 4, 6 . 516 179 0.79351 22.97 || 17 19 23, 59 17 59 24, 22 | 18.40 ! 3. 11600 || 1 1 05 11800 | 11 27. 12000 || 11 49. 1 1 : 2 1 s 1 1 1 1† () Projectile (cappe) 604 lbs. NSTRUCTION OF RANGE BOARD - . . . . . . 2100 2110 2120 2130 2140 2150 2160 2170 2180 2190 2200 2220 2210 2230 2240 2250 2260 2270 2280 2290 2300 2310 2330 2340 2350 2360 2370 2380 2390 2400 1000 1.37 124 116 108 99 91 83 74 66 58 49 42 32 24 16 8 0 9 17 25 33 42 48 58 66 74 83 92 101 109 118 127 2000 2.81 242 227 211 194 178 162. 145 129 113 97 81 64 48 32 16 0 17 33 49 65 82 98 114 130 157 164 181 198 215 232 250 3000 4.34 355 332 308 284 261 237 213 190 166 143 119 95 71 48 24 0 24 48 72 95 121 145 168, 192 217 242 266 291 317 342 367 31 63 94 124 157 88 219 251 283 315 347 879 411 444 477 37 76 114 151 190 228 266 306 344 383 422 461 600 539 578 44 88 132 176 220 255 309 355 400 445 490 535 580 625 670 5000 7.69 560 523 486 449 412 375 338 301 264 226 189 151 114 6000 9.53 652 609 566 523 480 437 393 351 308 263 220 176 132 7 6 3. : 50 99 149 199 248 298 348 399 450 500 551 601 652 702 763 55 109 164 218 273 328 383 438 494 549 604 659 715 770 827 59 118 177 236 295 354 413 472 532 591 650 709 769 828 888 7000 11.50 736 688 639 590 542 493 443. 396 347 297 248 149 : 99 8000 13.63 812 759 705 651 597 543 488 435 381 327 273 218 164. 109 9000 15.93 808 822 764 705 646 588 528 471. 412 354 295 236 177 188 59 1 () 9 5 ! 63 126 188 251 313 376 438 501 564 626 889 751 813 876 938 66 130 197 263 328 394 459 525 590 655 720 785 850 915 980 68 136 204 272 340 408 476 543 610 677 744 811 878 944 1010 10000 18.43 940 877 815 752 689 627 563 502 440 877 314 251 188 126 63 11000 21.18 991 924 859 792 726. 660 594 528 462 396 330 264 198 132 66 12000 24.22 1034 964 895 826 757 688 619 550 481 412 343 274 205 136 68 : Range correction for Change in ballistic coefficient (per cent) - Wind compinent (mile” per hojr) # -- # É # s . . . . . . . . . . . . . . . - =– head wind + = rear wind +2 +4 +6 +8 +10 +12 +14 +16 —50 –40 –30 —20 –10 0 +10 +20 +30 +40 +50 1 1 1 2 2 3 3 4 3 3 2 2 1 0 1 0 1 2 2 2 0 i 0 ; 1000 324 7 6 5 4 3 2 1 1 1000 151 26 22 19 15 12 9 5 3 . . . . . -- . 5 3 0 3 5 7 to 11 14 15 18 11 9 7 5 3 1 4 6 7 9 3000 95 57 59 42 34 27 20 12 6 6 12 17 23 27 33 36 42 25 20 16 11 6 a 9 13 17 22 i 8 16 24 33 42 14 27 40 56 70 ; 22 43 63 87 108 11 21 31 41 49 59 66 76 47 37 29 20 10 7 33 49 64, 77 92 105 119 79 62 47 32 16 24 47 70 92 112 13 152 172 122 96 72 48 24 1000 65 100 86 73 60 47 35 22 11 5000 48, 155 133 112 92 72 54 35 17 6000 37 220 189 159 131 103 77 51 25 :- 3 3 32 64 94 127 158 45 90 134 178 222 61 123 184 242 302 33 64 95 124, 153 180 207 234 177 139 104 68 35 43 83 123 161 200 235 270 306 244 193 144. 93 48 54 104 154 202 252 297 341 387 325 254 193 125 64 7000 29 294 254 214 176 139 104 69 34 8000 23 377 327 276 227 180 124 89 44 9000 17 468 407 344 283 225 167 111 55 81 163 245 322 401 106 212 319 421 524 136 272 407 543 678 65 127 188 248 800 365 420 477 422 338 252 165 83 77 152 225 298 371 439 507 575 538 431 322 213 106. 90 179 266 352 437 520 601 681 686 540 404 270 135 11000 11 667 581 494 407 324 240 160 79 12000 10 776 673 574 474 376 280 185 92 6 5 ! 0 6 §§§ §§ §§§ &&&&& ¿ ģ} 、、。、、。 ;&#。 §§§§ 。£ 3. ſae！§§ §§§ §§§§ §§§§ §§ ≡ №ſº · ::X } Ā， #:;;，,， ， ; é º ¿? §§ ¿ §§§ III|||| "3" 9015 0276 __-_---—--------——---—------------~~~ TTTTTTTT af º, º # * * * * tº: - ſº & rºº tº . , , , , ; * * * **** º º - tº . . . . . . : * * * * * * * * * | i º º : ſ * * *.*.*, * & tº * , , :*: sº * † * .. 3 & ºr * sº-º § “ * † º ºg .# * * f º º * * * * * º º tº 3 & s = * : * * * * * # * * * * * * * * * * * * * * º, , , * * * A : * * * * * º & º * * * - º sº sº gº is .# tº . " * : * * , ºr . : " ' "." a tº r * * - * ... sº ... ". - . . . . . . " sº is e. f * - - . . . . . . . . . . x * : * : *- : * ~ * | * , * * * *... . º - * : * * f. a . . . . * ºf . y º- * & - - - - * *, * * * *. , * . . º