HN B. Me r u . " "iL- 0, L. 
 
 NE\V YOR.. : 
 NOSTRAKD, PUBLISHE 
 
THE VAN NOSTRAND SCIENCE SERIES. 
 
 18mo, Green Boards. Price 50 Cents Each. 
 Amply Illustrated when the Subject Demands. 
 
 PQ 
 
 
 
 O 
 
 h I 
 
 J 
 O 
 
 W 
 
 EPLACES AND 
 rongr, C. E. kd 
 i Chimneys, by 
 
 Zerah Colbv 
 NING WAL 
 
 IN BRIDGE 
 
 f W. F. Butler. 
 Greenleaf. 2d 
 
 DRUCTIOE OF 
 mr Jacob, A.B. 
 ORMS OF RE- 
 te, C. E. 
 ENGINE. By 
 Vith Additions 
 
 L., to which is 
 ICIAL FUELS 
 By John Wor- 
 
 ilated from the 
 s of American 
 dition. 
 
 Allan. 
 
 . By Prof. W. 
 
 IS. By J. J. 
 To which is 
 STS. By Ed. 
 
 r. J. Atkinson, 
 le, C. E. 
 ?G CERTAIN 
 Prof. Geo. L. 
 
 1 ~ r p r of. W. II. 
 
 Prof. w7H.To rfieTd. 
 No. 19. STRENGTH OF BEAMS UNDER TRANSVERSE 
 
 LOADS. By Prof. W. Allan. 
 No. 20. BRIDGE AND TUNNEL CENTRES. By John B. 
 
 McMasters, C. E. 
 No. 21. SAFETY VALVES. By Richard H. Buel, C. E. 
 
THE VAN NOSTRAND SCIENCE SERIES. 
 
 No. 22. HIGH MASONRY DAMS. By John B McMaster. 
 
 No. 23.-THE FATIGUE OF METALS UNDER REPEATED 
 STRAINS, with various Tables of Results of Ex 
 periments. From the German of Prof. Ludwig 
 Spangenberg. With a Preface by S. H. Shreve 
 
 No. 24.-A PRACTICAL TREATISE ON THE TEETH OF 
 WHEELS, with the Theory of the Use of Robin 
 son s Odontograph. By Prof. S. W. Robinson. 
 
 No. 25. THEORY AND CALCULATIONS OF CONTINU 
 OUS BRIDGES. By Mansfield Merriman, C. E. 
 
 No. 26.-PRACTICAL TREATISE ON THE PROPERTIES 
 OF CONTINUO US BRIDGES. By Charles Bender. 
 
 No 27. ON BOILER INCRUSTATION AND CORROSION- 
 By F. J. Rowan. 
 
 No. 28. ON TRANSMISSION OF POWER BY WIRE 
 ROPES. By Albert W. Stahl. 
 
 No. 29. -INJECTORS ; THEIR THEORY AND USE. Trans 
 lated from the French of M. Leon Pouchet. 
 
 No. 30. TERRESTRIAL MAGNETISM AND THE MAGNET 
 ISM OF IRON SHIPS. By Prof. Fairman 
 Rogers. 
 
 No. 31.-THE SANITARY CONDITION OF DWELLING 
 HOUSES IN TOWN AND COUNTRY. By George 
 E. Waring. Jr. 
 
 No. 32.-CABLE MAKING FOR SUSPENSION BRIDGES, 
 as exemplified in the construction of the East 
 River Bridge. By Wilhelm Hildenbrand, C. E. 
 
 No. 33. MECHANICS OF VENTILATION. By George W. 
 Rafter, C. E. 
 
 No. 34. FOUNDATIONS. By Prof. Jules Gaudard, C. E. 
 
 Translated from the French. 
 No. 35. THE ANEROID BAROMETER: Its Construction 
 
 and Use. Compiled by Prof. G. W. Plympton. 
 
 3d Edition. 
 
 No. 36. MATTER AND MOTION By J. Clerk Maxwell 
 No. 37. GEOGRAPHICAL SURVEYING: Its Uses, Methods 
 
 and Results. By Frank De Yeaux Carpenter. 
 No. 38.-MAXIMUM STRESSES IN FRAMED BRIDGES. 
 
 By Prof. Wm. Cain. 
 
 No. 39. A HANDBOOK OF THE ELECTRO-MAGNETIC 
 , TELEGRAPH. By A. E. Loring, a Practical Tel 
 egrapher. 2d Edition. 
 No. 40. TRANSMISSION OF POWER BY COMPRESSED 
 
 AIR. By Robert Zahner, M E. 
 
 No. 41.-STRENGTH OF MATERIALS. By William Kent. 
 No. 42. VOUSSOIR ARCHES, applied to Stone Bridges, Tun 
 
 nels, Culverts and Domes. By Prof. Wm^Cain. 
 No. 43. WAVE AND VORTEX MOTION. By Dr. Thomas 
 
 Craig, of Johns Hopkins University. 
 
HIGH MASONRY 
 
 DAMS. 
 
 JOHN B. McMASTEE, 0. E. 
 
 AUTHOR OF "BRIDGE AND TUNNEL CENTRES. 
 
 NEW YORK: 
 
 D. VAN NOSTRAND, PUBLISHER, 
 23 MURRAY AND 27 WARREN STREET. 
 
 1876. 
 
PREFACE. 
 
 In the preparation of the following treatise 
 three points have been constantly in view, to 
 avoid as far as possible all purely theoretical 
 discussion, to discover the most economical 
 forms of profiles consistent with perfect 
 strength, and to consider none that have not, 
 after repeated practical application demon 
 strated their excellence, even under the severest 
 tests. To treat the subject, however, in a logical 
 way, it has been found best to begin with a 
 theoretical determination of the strongest and 
 at the same time least expensive form of profile, 
 and afterwards to modify this to meet the re 
 quirements that arise in actual construction. 
 
 The theoretical type is, as I have attempted 
 to show, that composed of a vertical face on 
 the inner, and a concave surface on the outer 
 side. Of this, there are, of course, an almost 
 unlimited number of possible modifications. 
 But when we impose the condition of economy, 
 the number of really useful ones dwindle down 
 to less than half a dozen. Those treated of in 
 the present work number four. The first (il 
 lustrated in Fig. 9), is, beyond all doubt, the 
 
very best. It has indeed, been often urged 
 against this type of profile, that it is difficult 
 to determine with accuracy the equations of 
 the logarithmic curves forming the bounding 
 faces, as also to cut the facing stones to such 
 a curve. As to the first objection, no equations 
 can surely be simpler than those we have 
 given, while the second is a difficulty most 
 easily removed, not by argument, but by de 
 termination. 
 
 The three other types are also profiles of 
 equal resistance, and are treated of so fully in 
 the work as to call for no remark here. It will 
 also be observed that I have touched very 
 lightly on the sliding of dams on their founda 
 tion, or of any portion of them along a hori 
 zontal joint. This has been done, because, 
 though I have examined as fully as possible the 
 causes that have led to the destruction of dams 
 of all style of profile and of all heights, both 
 abroad and in this country, I have been able 
 to find extremely few that may justly be said 
 to have yielded by sliding. It has almost in 
 variably been by revolving about an axis near 
 the outer face, caused by taking too great a 
 limit of vertical pressure, and thus throwing 
 the line of resistance, when full, too far out 
 ward from the centre of thickness. 
 
 JOHN B. McMASTER. 
 YORK, February, 1876. 
 
HIGH MASONEY DAMS, 
 
 THE subject proposed for consideration 
 in the following work is that of the pro 
 file of masonry dams of such height, 
 breadth and general dimensions as would 
 be required for reservoir purposes, or for 
 impounding the waters of rivers and 
 large streams for mill or irrigation use. 
 We would observe, however, at the out 
 set, that as this matter has already been 
 treated with such fullness by several 
 writers, and especially by MM. Delocre 
 and Sazilly to whose excellent "me 
 moirs " we are greatly indebted we can 
 hope to add little that is really new, but 
 shall endeavor, by drawing from many 
 sources, to supply our own deficiency, to 
 diminish the errors of others, and thus 
 obtain results very much more accurate 
 
6 
 
 than could be derived if we relied solely 
 on ourselves. 
 
 Before, however, we take up the con 
 sideration of the matter of the form of 
 profile that shall combine the greatest 
 strength with the least amount of mate 
 rial, there are a number of important 
 points to be considered somewhat in de 
 tail. Thus, it is necessary, in the first 
 place, that we should know the forces to 
 which dams are subjected, their kind? 
 whether constant or variable, the meth 
 ods of determining their direction and 
 calculating their intensity, and the ef 
 fects they are likely to produce, and 
 these matters being known, we may pass 
 to the consideration of the conditions of 
 stability, first when the dam has only its 
 own weight to support, and, secondly, 
 when it has to withstand both its own 
 weight and the pressure of the water. 
 We may then deduce a theoretical profile 
 of equal resistance, and, finally, adopt 
 one so modified by the requirements of 
 practice and suggestion of experience, 
 that it shall serve as a profile type, ful- 
 
filling to the utmost the requirements of 
 great strength and stability, beauty of 
 outline and economy of material. 
 
 Now, it becomes evident, after a mo 
 ment s consideration, that there are but 
 two forces that may at any time be re 
 garded as acting with vigor on a dam, 
 and these are, the weight of the mason 
 ry, cement and other material composing 
 the structure, and the pressure or thrust 
 of the water whose flow it checks. The 
 first becomes, to all intents and purposes, 
 a constant quantity as soon as the dam 
 is finished, and continues so for ever 
 after, acting vertically downwards 
 through the centre of gravity of the 
 mass. But, on the other hand, the latter 
 force is one of great variability. For, 
 as its intensity at any moment depends 
 on the depth or head of water behind 
 the dam, increasing as the water deepens 
 and 1 decreasing as the water falls, and 
 the head of water, especially in reservoirs 
 used for mill or irrigation purposes, be 
 ing subject to frequent rise and fall, it 
 follows that this thrust must be consid- 
 
8 
 
 ered as a variable quantity and treated 
 accordingly. It is, moreover, to be ob 
 served, that this thrust acts horizontally, 
 and unlike the weight, is not distributed 
 uniformly over the entire face of the dam, 
 being almost, if not quite, zero at the 
 point where the water cuts the masonry, 
 and growing greater and greater as we 
 descend towards the foot of the dam. 
 The weight, it is true, also increases as 
 we go from the top to the bottom, yet, 
 if we suppose the dam to be at any point . 
 ten feet thick, the pressure on any hori 
 zontal section taken at that point will be 
 everywhere the same, and this is by no 
 means the case if we take an area ten 
 feet square on the water face of the 
 dam, and against which the fluid presses. 
 In order that the dam may not yield 
 under the first force, and be thrown 
 down by the greatness of its own weight, 
 it is necessary, should the structure be 
 of such height, or the material of such 
 heaviness, that the pressure per unit of 
 surface at any horizontal section is in 
 excess of the "limit of pressure" for 
 
masonry, that the surface of the section 
 be increased so that the pressure being 
 distributed over a more extended area 
 the load at each unit of surface shall be 
 less. The second force, or thrust of the 
 water, is resisted at any point by the 
 weight of the masonry above that point, 
 and by the friction of the stones, which 
 is of course dependent on the weight. 
 Some resistance is indeed afforded by the 
 bonding power of the hydraulic mortar 
 used in setting the stones, but this is so 
 ^small that precautions of safety require 
 that it shall in all calculations be disre 
 garded entirely. 
 
 But these two forces, the weight act 
 ing vertically downwards and the thrust 
 of the water acting horizontally, counter 
 act each other to a certain extent, and 
 give rise to a third power or resultant, 
 the position of which, as regards the base, 
 will determine the stability of the dam. 
 To illustrate, let A B C D (Fig. 1) repre 
 sent the profile of a dam composed of 
 horizontal courses of masonry bedded 
 on each other, and K the centre of gravi- 
 
10 
 
 FIG, I. 
 
 ty of the mass, lying above the line E F. 
 Represent by K G the direction and in 
 tensity of the weight of AF, and by 
 KP the direction and intensity of the 
 thrust of the water from D to F. Then, 
 constructing in the usual way the paral 
 lelogram P K Gr R, we shall have for the 
 resultant of KP and KG, the line KR. 
 Now, supposing the dam to be perfectly 
 secure as to its weight, the force P of the 
 water can demolish the wall only, when, 
 exceeding the weight and friction K G, 
 it shoves the mass AF along the joint 
 E F, or causes it to rotate about an axis 
 through E. Which of these motions, 
 
11 
 
 the slipping or rotating, shall take place 
 depends entirely on the magnitude and 
 direction of the resultant K R. If the 
 pressure of the water is so large com 
 pared with the weight that the angle 
 RK G, which the resultant makes with 
 the vertical, is larger than the angle of 
 friction (32 for masonry on masonry), 
 the mass A F will then slide along the 
 line EF; while if the position of the re 
 sultant is such that it passes without the 
 base B C, then rotation will take place 
 about the axis of E. Of these two mo 
 tions, the latter is in practice the most 
 likely to occur, inasmuch as in nine cases 
 out of ten when rotation does take place 
 it does so about some point as E , nearer 
 the resultant than E, because the press 
 ure concentrated at E, breaks off the 
 stone, and thus throws the axis of rota 
 tion nearer the resultant. 
 
 The condition of stability then, in dams 
 that do not transmit laterally to the 
 sides of the valley, the pressure they sus 
 tain (and this is the ease in all large 
 dams) is, that they must resist this press- 
 
12 
 
 ure at every point by their own weight. 
 If the material employed were of con 
 siderable resisting power, as well as the 
 soil of the foundation, and if there were 
 between them an unlimited degree of 
 adhesion, the only condition of stability 
 to be fulfilled would be, as we have just 
 seen, to give the wall such a profile that 
 the resultant of the thrust of the water 
 and the weight of the dam shall pass 
 within the polygon of the base. But 
 this condition is not found sufficient in 
 practice ; the material and the soil of 
 the foundation will, in fact, support only 
 a limited pressure (depending on their 
 nature), and they have not between 
 them an unlimited degree of adhesion. 
 Hence, the two following indispensable 
 conditions : 
 
 1 The several courses of masonry in 
 the wall must be incapable of slipping 
 the one over the other, and the wall in 
 capable of sliding on its base. 
 
 2 In no point of the structure may 
 the material employed, or the soil of the 
 foundation be required to bear too great 
 
13 
 
 a pressure. To begin with the first con 
 dition. 
 
 STABILITY AS TO SLIPPING. 
 
 We shall take up first the condition 
 of stability as to the slipping of the va 
 rious courses of masonry, and then pass 
 to that of the entire dam. The first 
 thing to be now determined, is the hori 
 zontal thrust of the water. Suppose 
 A B C D (Fig. 2) to represent the face of 
 a dam pressed by water, and let h=A J 
 denote the height; a= J C the projection 
 of the slope of the dam on the horizon 
 tal plane; and, finally, let 1= A B denote 
 the length of the dam, and b= A G is 
 breadth across the top. Then will the 
 vertical pressure of the water on the face 
 A B C D be expressed by 
 
 al- y\ alliy ... 1 
 
 and the horizontal thrust by the* expres 
 sion 
 
14 
 
 in each of which y denotes the density 
 of the water. These equations are ob 
 tained as follows : 
 
 Let E P, in Fig. 2, represent the nor 
 mal pressure of the water on the surface 
 A C, which we will call F, and resolve it 
 
15 
 
 into two components, one vertical E P , 
 and one horizontal E P", and call them 
 respectively P and P". Then expressing 
 the angle P E P" made by the horizontal 
 component P" and the normal E P, by a 
 
 we shall have from the triangle E P P" 
 pp// 
 
 -^5 = sin P E P" or sin a. 
 L Jr 
 
 But PP"=EP =P , hence 
 
 P ") 
 
 p-=sin a or P = P sin a. 
 
 In the same way we find 
 p// 
 
 = cos a or P"=:P cos a. 
 
 Now, let a projection A B C D, of the 
 surface A B D, be made on a plane at 
 right angles to P", and call the area of 
 the projected surface F . Then will F 
 = F cos AC A , or since the angle of 
 inclination A C A of the surface to its 
 projection is equal to the angle P E P" 
 = a, between the normal to A C, and 
 the perpendicular to A 7 C, we shall 
 
 F 
 have F =F cos a or cos a=^r But cos 
 
 JD TD// 
 
 a is by equation 3 equal to ^-, and 
 therefore, 
 
16 
 "p// TJ^ TT 
 
 From the principles of mechanics, we 
 know that the pressure P of water on 
 any given area is the product of the 
 area,^the height h of the water, and its 
 density y, so that in the present instance 
 F being the area of the surface A B C D, 
 we shall have for the value of P the ex 
 pression P=F h y, and this substituted 
 in equation 4 gives 
 
 P"=FAy or F hy . . . 5. 
 
 Therefore is the pressure with which 
 water presses against a surface in a given 
 direction equal to the weight of a column 
 of water, which has for its base the pro 
 jection of the surface pressed, and for 
 height the depth of the centre of gravi 
 ty of the surface below the top of the 
 water. We see, moreover, from the 
 above, that since the projection at right 
 angles to the vertical is the horizontal, 
 and the projection at right angles to the 
 horizontal is the vertical projection, the 
 
11 
 
 vertical component of the pressure of 
 water against a surface may be found if 
 the horizontal projection, or its trace, be 
 considered as the surface pressed, and, 
 on the other hand, the horizontal com 
 ponent may be found if the vertical pro 
 jection of the surface, or its trace, be 
 considered as the surface pressed. 
 
 Applying these two principles to the 
 case of Fig. 2, and replacing F in equa 
 tion 5, by its value 111, we shall have for 
 the horizontal thrust of the water on the 
 face A B C D of the dam the equation 
 P" ^h*ly, and in the same way the 
 vertical component will be found to be 
 equal to P -J a h I y. Now, b being the 
 breadth of the dam, and a r the projec 
 tion of the slope G K, and y the density 
 of the masonry composing the dam, it is 
 evident that the area of K C E G will 
 
 / T a + a f \ 
 be tcH -- - I fi ; the cubic contents 
 
 lily . The whole vertical pressure on 
 the base will therefore be equal to this 
 
18 
 
 weight plus the vertical pressure of the 
 water, or 
 
 We have seen, however, that the force 
 which tends to counteract the push of 
 the water, and on which the stability as 
 to slipping must therefore depend, is 
 equal to this weight of the dam increas 
 ed by the friction of the stones. De 
 noting this co-efficient of friction by/, 
 we shall then have for the force to push 
 the dam forward the expression 
 
 and in the case where the horizontal 
 thrust of the water is to effect the dis 
 placement 
 
 or dividing each member through by 
 ^ h ly^ we shall have 
 
19 
 
 In order therefore that the dam may not 
 be pushed away by the water, we must 
 have one of the two following conditions 
 fulfilled ; either 
 
 For safety, we may further assume 
 that the base of the dam is quite per 
 meable, in which case there is (on the 
 principle that a pressure in one direction 
 produces an equal pressure in the oppo 
 site direction) a pressure from below up 
 wards equal to (2 b + a + a f ) Ihy, equal 
 the weight of the dam, and as this is, of 
 course, to be subtracted from the above, 
 we have finally, 
 
 (2 b + a + a ) - - l-a 9. 
 
 These equations are applicable not 
 only to the sliding of the entire dam on 
 
20 
 
 its foundation, but also to any particular 
 layer of stone at any point in the dam. 
 The value of the co-efficient of friction 
 f will of course be very different in cases 
 where we consider the stability of differ 
 ent parts of the wall, from that in cases 
 where we consider the dam to slide on 
 an earthty foundation. In the former 
 case, it is that of masonry on masonry, 
 in the latter, that of masonry on earth, 
 and in general clay. In fact, it may be 
 restricted almost solely to clay, because 
 in a sandy, porous or yielding soil, it 
 is better, on principles of economy, not to 
 build a dam, but a dyke. For masonry 
 on masonry, or, indeed, bricks on bricks, 
 we may with safety take the co-efficient of 
 friction as equal to .67 ; for masonry on 
 dry clay .51; but for masonry on wetted 
 clay the co-efficient falls to .33. 
 
 A few examples may, perhaps, serve 
 to illustrate the above remarks. We 
 shall confine ourselves first to the case 
 of rotation about one of the joints, as 
 that is really the most likely one to arise 
 in practice : 
 
21 
 
 Let Fig. la represent the profile of a 
 FlG.U. A D 
 
 dam, constructed say of brickwork 
 weighing 112 pounds per cubic foot. 
 Let the thickness on top be 10 feet, and 
 that at the base 20 feet, required to find 
 the perpendicular height, the dam must 
 have in order that, when the water stands 
 at the brim, the wall shall be just on the 
 point of turning about the point B under 
 the pressure of the water. Denote by h 
 the height of the dam, or the quantity 
 we are in search of, = C D. Now, by 
 
22 
 
 equation 2, the thrust of the water on 
 
 one lineal foot of surface is X62.5 Ibs., 
 
 A 2 
 and the moment of this thrust is X62.5 
 
 h A 3 
 
 Ibs. X - or X 62.5 Ibs. The pressure 
 o 6 
 
 of one foot of the dam, or what is the 
 
 AD + BC z 
 same thing, its weight is -- h X 
 
 104-20 
 112 Ibs., or h X 112 lbs.= 1680 h 
 
 Ibs., and the moment of this pressure 
 with reference to the point B is 1680 h 
 XBE. Before we can obtain this mo 
 ment, then, we must find the value of 
 B E, and this is found as follows : 
 
 It is evident from a moment s inspec 
 tion of Fig. la, that the area of 
 ABCDxG<7 = area ABC FxBH + 
 
 area of ABFxIB, or 
 denoting A D by a\ B C by ~b\ D C by c; 
 and G g by d, we have since B H = 
 
 2ba 2 (ba) 
 
 --,andIB=. -L. 
 
 2 b a c(ba) 
 ~~ 
 dividing by c 
 
23 
 
 . 11 a (baY 
 
 - 
 
 
 Substituting for the above quantities 
 their values, we have : 
 
 c?=^-W = 1 J iI , 
 
 The moment of the dam therefore is 
 1680 h X -%*. 
 
 |pX 62.5 Ibs. = 1680 A X J R 
 
 62.5 A 3 _184800A 
 
 6 9 
 
 ^ 2 A/197.12 
 A = 44.3982. 
 
 Again, preserving the same dimensions, 
 let it be required to find the " modulus 
 of stability" of a masonry dam of the 
 profile, shown in Fig. 1, the stone weigh 
 ing 200 pounds per cubic foot. Draw 
 from the middle of the top A D to the 
 middle of the base B C the line R Y, and 
 take its length as 45 feet, and the depth 
 of the water behind the dam, 44 feet. 
 
24 
 
 Now, by geometrical principles, which it 
 is not worth while to repeat here, we 
 have : 
 
 10 + 10 _ 245 
 15 = T5~ 
 
 g being the centre of gravity of the 
 wall. Again, in the two similar triangles 
 E Y S and g V T, we have : 
 
 RV: VS;:V<7: VT. 
 
 The value of N g we have just found. 
 VS is evidently equal toYC SO, or 
 10 5 = 5. In the triangle RYS, we 
 also have R S =R V -VS 2 , or RS 3 = 
 (45) 2 -(5) 2 ; hence RS=44.38. Substi 
 tuting these values in the above propor 
 tion, we shall have : 
 
 45 : 5;:- : VT 
 15 
 
 The weight or pressure of the wall 
 acting through the centre of gravity g 
 of the dam is, as we have already seen, 
 
 20+10 
 - X 1 X 44.38 X 200=133140 Ibs., 
 
25 
 
 and that of the water 44X1X^X62.5 
 = 60500 Ibs. If now we denote by P 
 the " centre of pressure " of the water, 
 that is to say, that point where a single 
 pressure will counterbalance the thrust 
 of the water against the entire face D C 
 of the dam, then P=C P=^=14.6 feet. 
 The quantity we are in search of, the 
 modulus of stability of the wall is the 
 ratio of T B to T O. The value of T B 
 we have already, and may obtain that of 
 T O from the proportion that the press 
 ure of the dam is to the height of the 
 centre of pressure (P) of the water 
 above the base of the dam as the press 
 ure of the water is to the entire pressure 
 of the water acting on its centre of press 
 ure P. Thus : 
 
 133140 : 14.6! .60500 : x 
 aj=6.6=TV. 
 
 Dividing this last found quantity by T B, 
 we have : 
 
 rp "TT f /i 
 
 i^-^= --^ = .53*1 = modulus of stability. 
 1 J3 llf 
 
 In a well built structure, this quantity 
 
26 
 
 should never be less than .5, hence, as in 
 the present case, the modulus is somewhat 
 above this value, we are justified in re 
 garding the dam as a perfectly stable 
 structure, when the water is not over 44 
 feet in depth. 
 
 In these considerations, we have taken 
 no account of the resistance offered by 
 the adhesion of the mortar. Should 
 this be taken into account and it is al 
 ways best that it should not then equa 
 tion 9 will require to be modified some 
 what as follows : Let H equal the dis 
 tance of the centre of gravity of a layer 
 of stones below the top of the dam. 
 The shove of the water tending to throw 
 down this portion of the dam is, as we 
 
 . _ . , 
 have just seen, - , in which expression 
 
 2i 
 
 6 is merely a short notation for ly. The 
 forces resisting this shove are the friction 
 of the two layers sliding on each other, 
 and the adhesion of the masonry. The 
 first is proportional to the weight of the 
 masonry above the stratum in question, 
 and the second or adhesion of the mason- 
 
27 
 
 ry is proportional to the thickness of 
 the dam at this point. Representing as 
 before the co-efficient of friction by/*, by 
 c the cohesion of the mortar per unit of 
 surface, by s the area of the upper sur 
 face of the course next below, and by 5 
 the thickness of the dam at this section, 
 we shall have for the resistance R to 
 sliding : 
 
 and therefore, in order to insure stabili 
 ty, we must have : 
 
 2 
 
 or clearing of fractions, and then divid 
 ing by 
 
 H 2 
 
 10 
 
 Neglecting the adhesive power of the 
 mortar, the above becomes : 
 
 ad 
 
 The second case of slipping, or that of 
 the dam on its foundation will rarely, if 
 
28 
 
 ever arise, when the dam is founded on 
 a rock, for in that case the value of the 
 co-efficient of friction will be the same 
 for the horizontal section of the founda 
 tion as for any section of the masonry- 
 It is, however, very likely to arise when 
 ever circumstances will not enable us to 
 lay the foundation on bed rock. In such 
 cases the soil will almost always be of 
 an argillaceous nature, for, should it 
 prove to be of a gravelly, sandy or very 
 permeable character, the employment of 
 some common form of dyke will be much 
 preferable to the construction of a dam. 
 We may, therefore, reasonably assume 
 that in all cases where the foundation 
 course does not rest on a rock surface, it 
 will be laid on argillaceous soil, and as 
 this will readily give, under the action of 
 water, a slippery slimy surface, we must 
 assume a co-efficient of friction very 
 much less than that used for masonry on 
 masonry. With this point kept clearly 
 in view, the conditions of stability will 
 be given by the above equations. Yet 
 there are one or two other considerations 
 
29 
 
 that must not be overlooked. Thus, as 
 the stability will depend in large meas 
 ure on the lateral resistance of the soil, 
 it is not sufficient to be sure that this 
 resistance is large enough to prevent the 
 sliding of the wall, but is also necessary 
 to be assured that at any point of the 
 front of the foundation wall, the normal 
 pressure does not exce ed the limit R/ of 
 which the soil or the wall is susceptible. 
 Again, in order to prevent any slipping 
 likely to arise from the lateral compres 
 sion of the earth, it is not necessary to 
 interpose any packing between the face 
 of the wall and that of the ditch, and, 
 finally, that in all cases it never comes 
 amiss to " step " the rock or the earth on 
 which the foundation course rests, a mat 
 ter to be considered more in detail here 
 after. 
 
 SECOND CONDITION OF STABILITY. 
 
 To return now to the second condition 
 of stability, namely, that in no point of 
 the structure may the material employ 
 ed, or the soil of the foundation, be re- 
 
30 
 
 quired to bear too great a pressure. For 
 this purpose let A B C D (Fig. 3) repre- 
 D C 
 
 Rc.3. 
 
 sent the profile of a dam. Then from 
 the principles we have already establish 
 ed, it follows that any section of this, 
 equal in length to a lineal unit, may be 
 considered as subject to the action of 
 two forces, which are, respectively, the 
 vertical component P of the resultant of 
 the weight of the structure above that 
 unit, and the horizontal pressure or 
 
31 
 
 thrust of the water, and the horizontal 
 component F of the thrust of the water. 
 In the section A B C D, these two forces 
 act through the centre of gravity G, and 
 produce a resultant of their own which 
 cuts the A B at E. This latter resultant 
 R may therefore be regarded as applied 
 directly to the point E, and resolved into 
 two components, one vertical and equal 
 to the force P, and one horizontal and 
 equal to the force F. The horizontal 
 force tends to slide the wall along the 
 base AB. This we have considered. 
 The vertical spread^ itself over the base 
 from the extremity B, which is nearest 
 the point of application of the resultant, 
 according to the well known decreasing 
 law. Now, in all works on mechanics, 
 we have given a formula which applies 
 to a homogenous rectangle, pressed by a 
 force acting upon one of the symmetri 
 cal axis, and this is : 
 
 ( x ) 
 
 and , 1 
 
32 
 
 Where N is the entire load or pressure, 
 and D, the entire area of the surface 
 pressed. In the case we are considering, 
 the quantity 1ST in equations cc and /?, is, 
 of course, represented by P the vertical 
 component. iQ, by I, if by this letter we 
 designate the breadth of the base A B, 
 and if we denote the distance E B by u, 
 then will the quantity n in equations oc 
 
 and ft be represented by - . 
 
 Substituting these quantities, we shall 
 
 have : 
 
 * 
 
 , 3l6u\ P 
 
 -- 
 
 and 
 
 : 12. 
 
 3 u 
 
 Equation cc is applicable in all cases 
 where ^<J, and therefore equation 11 is 
 
33 
 
 I _ 2 u 
 applicable when - <; that is when 
 
 i 
 
 Equation /3 is applicable to all cases 
 when ^>J, and consequently equation 
 
 I _ 2 u 
 
 12 to all cases when - > J, or, what 
 l 
 
 is the same thing when ^^< J L We have 
 seen that the condition of stability re 
 quires that some limit, R , should be 
 placed on the pressure each superficial 
 unit is expected to bear. The pressure 
 at the point B, must therefore be less or 
 never greater than R , and we shall have 
 according as u is greater or less than J 1 9 
 
 and 
 
 2P 
 
 -_-or<R ..... 14. 
 
 3 u 
 
 And this condition is to be fulfilled for 
 each section made in the profile, neglect 
 ing the force of cohesion of the mortar 
 which is unfavorable to resistance. 
 
 These expressions are susceptible of 
 yet further modification, if we introduce 
 
34 
 
 into the calculation the maximum height 
 A that may be given to a wall with ver 
 tical faces, so that the pressure upon^the 
 base shall not exceed the limit B/ of 
 safety. Indeed, if we represent the den 
 sity of the masonry, or the weight per 
 cubic yard by d, we shall have R = tf A, 
 and the above equation become : 
 
 r- 7 =or<A . . 15. 
 o I 
 
 and 
 
 - = or<d / A=-5r=or<\ . . 16. 
 3 u ud 
 
 The conditions expressed in these 
 equations would be quite sufficient if the 
 water was always up to the top of the 
 dam, but as this is by no means always 
 the case, the wall must be capable, even 
 when the dam is quite empty, of sup 
 porting its own weight without being 
 subject at any point to a pressure per 
 unit of surface exceeding the limit d A. 
 
 In this case the resultant of all the 
 
35 
 
 forces acting on the wall is reduced to 
 the weight P , and denoting by K A, 
 the distance from the resultant passing 
 through the centre of gravity of Fig. 
 (3) to the nearest extremity A of the 
 base, by u, the pressure at A, will be given 
 according to circumstances by equations 
 11 or 12, and the stability of the wall 
 will require that one of the relations ex 
 pressed in equations 15 or 16 be satisfied 
 when P is substituted for P. 
 
 The next step, therefore, is to determ 
 ine the proper 
 
 PEOFILE FOE A DAM HAVING ONLY ITS 
 OWN WEIGHT TO CAEEY. 
 
 In order to study under all conditions, 
 the question we are now about to con 
 sider, it is perhaps well to inquire, in the 
 first place, what form it is most conven 
 ient to give a dam having only its own 
 weight to carry, in order that each point 
 of the masonry shall not be subjected to 
 a pressure larger than the limit of safety, 
 and then to determine the alterations 
 which economy require to be made in 
 
36 
 
 this assumed profile. It is evident, to 
 begin with, that when the height of the 
 dam is such that it does not go over the 
 limit A (i. e. the greatest height we can 
 give to a vertical wall, without the press 
 ure on the base becoming larger than 
 H , we shall be quite justified in giving 
 the dam vertical facings, and that, in 
 such case, the load for each unit of sur 
 face at the lower part will be somewhat 
 less than d A, or at least, never greater. 
 Again, we know that whenever the press 
 ure on a horizontal surface of masonry 
 is larger than the limit of safety, we may 
 correct this, by enlarging the area of the 
 surface pressed, and so lessen the load 
 on each superficial unit. And these are 
 the two fundamental principles of dam 
 construction, and may be summed up in 
 brief as follows : If we are construct 
 ing a dam of a height equal to or less 
 than A, and having only its own weight to 
 support, it is a safe practice to give it 
 vertical facings from top to bottom. If, 
 however, we are constructing a dam of 
 a height greater than A, yet having only 
 
its own weight to support, we must make 
 the faces vertical for a distance from the 
 top equal to A, and from this point to 
 the base slope them outward. 
 
 A dam constructed on this latter prin 
 ciple would give a profile similar to that 
 in Fig. 4. From the summit A B to the 
 section C D, the pressure per superficial 
 
 B 
 
 F.c/K 
 
 F.c.6. 
 
 TH 
 
 unit is nowhere greater than tf A, and 
 
38 
 
 therefore from A to C the face is verti 
 cal, but below C D, the load exceeds the 
 limit and increasing at each section to the 
 base, and hence from C to Y the face is 
 sloping. And just here we are met by the 
 great question in dam construction that 
 of profile. Should the bulging portion 
 C Y Y D, be bounded by right lines as 
 in Fig. 4, should it be stepped, should it 
 be curved, and if so, should the bound 
 ing curves be logarithmic curves, simple 
 or compound ? these are questions we 
 propose to consider. 
 
 It is an easy matter to determine the 
 force to be given to the facing, so that 
 the condition that the load per unit of 
 horizontal surface shall never go over 
 the limit tf A, shall be satisfied. To do 
 this, we may choose arbitrarily one face 
 and then determine the other, but if we 
 desire to use the minimum of material 
 consistent with perfect safety, then the 
 wall must be symmetrical as to its axis. 
 In such a case as that illustrated in Fig. 
 4 that of a high masonry dam, whose 
 height is greater than A the slopes 
 
39 
 
 D N Y and C M Y, ought to satisfy the 
 requirement that, if in any section, as 
 M 1ST, the load per surface unit is equal 
 to any given quantity, the pressure will 
 be the same for any other section as 
 m ri , infinitely near to it. This will be 
 fulfilled, if the increase given to the base 
 is proportional to the increase of press 
 ure, or as the profile is to be made sym 
 metrical to the axis O S, if the increase 
 of the half surface L 1ST or L M is pro 
 portional to the increase of load on that 
 half surface. If we denote by P the 
 pressure on L 1ST, arising from the weight 
 of the structure above, and a the surface 
 of this section, then, it is evident, the 
 above condition will be expressed by 
 
 K.da. . . . 17. 
 
 In which K is a constant quantity, 
 and denotes the limit of pressure on the 
 unit of surface or d A. Again, by #, de 
 note the dimensions of the dam in the 
 direction perpendicular to the section we 
 are concerned with, and by x the length 
 of the half section LN, or, to express 
 
40 
 
 it mathematically, the abscissa of the 
 curve or line sought (i.e. DN Y ), and 
 finally, by y, the distance of MN from 
 a horizontal line taken as the axis of x. 
 Then the surface a will equal to bx, and 
 consequently an increase of surface as 
 da in equation 17, will be expressed by 
 
 da dbx 
 and moreover 
 
 <JP=d bxdy 
 
 These values substituted in equation 
 17 give for the differential equation of 
 the curve, 
 
 d bx.dy= K.b.dx. . 18. 
 whence 
 
 7 K dx 
 
 dy = w^ 
 
 But K equals the limit of pressure per 
 unit or # A, and this value replaced for 
 K, we shall have 
 
 7 # A dx , dx 
 
 d y=-r-ir OTd y = *-z 
 
 Integrating this between the proper lim 
 its, we shall have 
 
 -2/ = A log ... 19 
 
41 
 
 Now, from this equation we see that, 
 the curve being referred to rectangular 
 axes, one of the co-ordinates is equal to 
 the logarithm of the other, and, hence, 
 the curve must be a logarithmic curve. 
 Here then we have one property of the 
 curve D N Y. To find in the next place 
 the origin of its co-ordinates, we may 
 make in the foregoing equations # = A, 
 in which case we shall have : 
 
 and y =0 2(X 
 
 From this last relation it is quite ap 
 parent that the origin of co-ordinates is 
 to be taken at a point where the value 
 of x is equal to that of A, and in this 
 point the tangent to the curve makes an 
 angle of 45 with the axis of x. Re 
 turning now to equation 19, let us replace 
 y and x by their respective values, 
 given in equation 20, when we shall have : 
 
 y=\ log. x 
 
 A i 
 
 or passing from the system of Napier to 
 the common system of logarithms, 
 
42 
 
 #=2.302658509 A, log. -T- . . 21. 
 
 6 A 
 
 This curve, when constructed, will give 
 the form of the facing of a wall of in 
 definite height for which the pressure 
 per unit of surface equals the limit of 
 pressure K. It is not to be forgotten in 
 in making use of equation 21, that the 
 direction in which ?/ s are usually esti 
 mated has been reversed ; in other 
 words, y when positive is to be estimat 
 ed downwards, and when negative up 
 wards, or in the direction of L O. Fig. 
 5 represents this curve constructed, by 
 assuming the pressure limit or K as 
 132,000 Ibs., and the density of the 
 masonry as double that of water. 
 
 In such a profile, as Fig. 4 has, the 
 sloping faces below C D being bounded 
 by right lines, we may obtain the neces 
 sary breadth of the base Y Y , as soon 
 as we have determined the height and 
 
 * We may also pass from the Naperian to the common 
 system, by multiplying the Naperian logarithm by the 
 modulus of the common system, which is 0.434294. Its 
 logarithm is 9.63TT84. 
 
43 
 
 Fic.5. 
 
 the breadth at top. Denote by 1) the 
 breadth at top A B ; by h the distance 
 AC=A, and by h the distance from C 
 to the base Y Y ; by $ the density of 
 the masonry, and by x the quantity we 
 are seeking for, or the base YY . Then 
 we shall have : 
 
 The quantity h in this equation, which 
 is merely another expression for the 
 quantity A,, has been determined by a 
 
44 
 
 number of investigators, but the most 
 reliable results are those obtained by the 
 French engineers,* who, in the construc 
 tion of their great masonry dams, such 
 as Furens, have taken the limit of press 
 ure K at 60,000 kilogrammes, or about 
 132,000 Ibs. per square metre, and K 
 being equal to tf A, and tf being equal 
 to 2,000 kilogrommes, A becomes equal 
 to 30 metres. As we shall hereafter see, 
 however, the limit of pressure varies for 
 the outer and inner face of the dam. 
 
 If, again, the profile adopted be such 
 as is illustrated in Fig. 3, that is to say, 
 if the faces of the dam slope continu 
 ously from the top to the bottom, then 
 the thickness or breadth of the base will 
 evidently be obtained by dividing the 
 product of the height of the wall and 
 its thickness on top by the difference be 
 tween 2 A and the height. For d r A or 
 the limit of pressure is equal to the area 
 of the profile, multiplied by the density 
 of the masonry divided by the thickness 
 of the base. In the figure, the area is 
 
 * MM. Delocre, Sazilly and De Graeff. 
 
45 
 
 plainly equal to half the sum of the two 
 parallel sides by the altitude, and denot 
 ing this latter by H, we shall, therefore,, 
 have : 
 
 The conditions which govern the con 
 struction of such a dam, and the height 
 to which it is safe to build it, become 
 from this equation quite apparent, should 
 
 we make H= 2 A, then x would equal ,. 
 
 and the base of the wall would spread 
 out to infinity. Should we, upon the 
 other hand, make H greater than 2 A, 
 then A would become negative, and 
 hence it follows that the greatest height 
 we can give to a masonry dam with 
 straight sides equally inclined from the 
 summit and not go over the limit of re 
 sistance for masonry, is equal to twice 
 that of a wall with vertical sides. Yet, 
 within this limit, such a profile for a 
 masonry dam of any height, occasions a 
 gross waste of material. This becomes 
 strikingly apparent, if we compare the 
 
46 
 
 breadth of base of a dam constructed 
 with inclined faces from top to bottom, 
 with that of a dam of the same height, 
 but having a profile such as that of Fig. 
 4. Suppose each dam to be 30 metres 
 high and 5 metres thick on top; required 
 the thickness at the base. For the first 
 ease, using equation 23, we have : 
 
 30X5 
 
 For the second form of profile, we us e 
 equation 22, and have, since the quantity 
 h equals A, the same value, or x = 5 
 metres. 
 
 If we raise the dam by 10 metres, then 
 equation 23 
 
 B=*>^ = 10 metres. 
 
 60 40 
 
 and by equation 23 
 
 30X5 + 10 
 or since 
 
 O i^/&-j-/&i 
 
 : *, ,\ 7 77^==? metres. 
 
47 
 
 If, once more, we add ten metres to 
 the height, then equation 23 
 
 x=25 metres, 
 and eq. 22 cc=10 metres. 
 
 The saving thus affected when the 
 dams are of great height becomes simply 
 enormous. The difference, however, be 
 tween the profile when the dam below 
 C D (Fig. 4) is bounded by right lines, 
 and when bounded by logarithmic curves, 
 such as shown in Fig. 6, is not so marked 
 as in the cases just considered, yet is 
 considerable. To take but one case in 
 illustration, a dam of a profile such as 
 Fig. 6 illustrates, with the faces below 
 CD bounded by curves, would require 
 (equation 21) a breadth of base equal to 
 9.739 metres, the height and thickness 
 at top being as before, 50 and 5 metres 
 respectively, while, as we have just seen, 
 if the faces below C D were right lines, 
 the base would be 10 metres. 
 
 Such, in brief , is the relative merit of 
 these three forms of profile, for a dam 
 having nearly its own weight to support. 
 
48 
 
 In practice, however, such a dam can, of 
 course, never exist, and it thus becomes 
 necessary to take into consideration the 
 second condition, or that of a dam sup 
 porting a charge of water. 
 
 PROFILE FOR A DAM RESISTING THE 
 PRESSURE OF WATER. 
 
 And here, again, we are to throw aside, 
 at first, all practical considerations, and 
 determine a theoretical profile of equal 
 resistance, one in every part of which 
 the pressure shall not be greater than the 
 limit R . For this purpose we return to 
 the two equations, deduced some time 
 back, which express the conditions of 
 stability for a dam resisting the thrust 
 of water, and neglecting the signs > 
 and < and the values corresponding to 
 them, take only those corresponding to 
 the sign of =. We then have the two 
 following equations : 
 
 and *=* 
 
49 , 
 
 If we now replace the quantities w, I 
 and P, by their respective values, ex 
 pressed in functions of the height of the 
 dam, we may readily deduce two equa 
 tions which, on examination, will show 
 two things. 
 
 1. That the profile offering the least 
 thickness, consistent with the conditions 
 of stability, is one in which the side 
 turned towards the water, has a vertical 
 face, and the side turned from the water, 
 or the outer face of the wall, a concave 
 face. 
 
 2. That as the height increases, the 
 thickness increases less rapidly, so that 
 in a wall constructed with a vertical face 
 on the water side and a curved face on 
 the other side, and so planned that it 
 shall satisfy the conditions of stability 
 as to its base will present an excess of 
 strength for the surplus of height. 
 
 Fig. 7 is the profile of a dam of this 
 description. It will be observed, more 
 over, that in this form of profile the 
 thickness of the wall at the top is zero 
 This, of course, in practice is never ad- 
 
50 
 
 missible, inasmuch as it presupposes the 
 water to be at all times in a perfectly 
 quiescent state, and thus makes no al 
 lowance for the very considerable force 
 of the waves raised by the wind. It is, 
 therefore, necessary, whatever the profile, 
 to give the dam quite a thickness at the 
 summit, in general, about fifteen feet, is 
 
51 
 
 a good width, as it thus enables us to 
 construct a footpath and roadway on the 
 top of the dam, which is quite a conven 
 ience. 
 
 Before we consider any other modifi 
 cations, it may be well to determine as 
 nearly as possible the co-ordinates of the 
 concave curve forming the outer face. 
 For this purpose we will take the verti 
 cal face A B as the axis of #, and for the 
 axis of y, a perpendicular to this pass 
 ing through the point A, and call it AD. 
 Anywhere on the curve we will take a 
 point C, and denote its co-ordinates B C 
 =y and C e=x ; then the relation exist 
 ing between x and y will give the equa 
 tion of the curve. Now, as we have 
 already seen, the wall is subject to the 
 action of two forces, the weight of the 
 - dam P, which acts vertically downwards 
 through the centre of gravity and the 
 horizontal thrust T of the water. These 
 two forces produce a resultant R, which 
 cuts the base of the dam in this case at 
 the point H. This resultant, therefore, 
 may be regarded as applied directly to 
 
52 
 
 the point H and resolved into two com 
 ponents, HP and H 0, respectively, par 
 allel to O P and O T. We have also seen 
 by equation 5 that the horizontal thrust 
 of the water is equal to 
 
 F Ay=JA f ty ... 26. 
 
 Or replacing h by its value, and ly by its 
 value #, then T, or the horizontal thrust 
 of the water, equals 
 
 T- dx * 07 
 
 ~2~ 
 
 And in the same way 
 
 Returning now to equations 24 and 25, 
 we find that the quantity I is equal to y, 
 and that we have therefore to determine 
 the value of u in functions of x and of 
 y. Now u equals H C and H C = K C 
 - K H. The triangles O P R and O K H, 
 moreover, being equiangular triangles 
 are similar, and have their like sides 
 proportional, and 
 
53 
 KH : PR;; OK : OP 
 
 or 
 
 KH OK 
 
 PR~~OP 
 
 or to express the equality in terms of T, 
 x and P, 
 
 KH -* 29 
 
 ~T~~3P 
 
 Replacing in the 29th equation the 
 values of T and P, as obtained in the 
 27th and 28th equations, we have : 
 
 KH_ x -KH- ^_ 
 
 **,/* , " ~ 3<r 
 
 Ok 
 
 Or, for brevity, representing - , by D, 
 
 TTTT Da;3 
 
 KH = -jar ... so. 
 
 6 / 11 d x 
 </ 
 
54 
 
 This gives us the value of K H in the 
 expression 
 
 u=KC-KH. ... 31. 
 
 But KG is evidently equal to y B K, 
 in which B K is the distance from the 
 centre of gravity of the surface ABC 
 to the vertical axis of x or A B. This 
 distance is equal to the sum of the mo 
 ments of the areas such as a b c d, or 
 
 or again, 
 
 /y 
 if dx 
 
 BK= 
 
 y 
 Hence KC = y-BK=y- 
 
 y 
 
 2 / y dx 
 
 */ o 
 
 y 
 y*dx 
 
 /y 
 y dx 
 
 0* 
 
 2 yj ydwJ y*dx 
 
 32. 
 
 f ydx 
 
55 
 
 Substituting in equations 31, the values 
 of K H and K C obtained in equations 
 30 and 32, we have : 
 
 rv f* a D^ 3 
 
 2V ydx-J y dx -- 
 
 u = ~~ 6 / y dx 
 
 /y */ /, 
 
 T/cZcc 
 
 
 
 Or, reducing to a common denominator, 
 and subtracting, 
 
 /y py 
 
 ydx3J y*dxDx* 
 
 u _ _ t 33, 
 
 Thus, then, we have the value of u in 
 functions of x and y, and substituting 
 this value for u in equation 24, and re 
 membering that 1= 7/, we have : 
 
 4yP 6 ^P 1 
 
 ~^?~~ 
 
 dx36 y $ 
 
56 
 
 -18 tf /V dx-Q Da? S fy 
 /y 
 
 J y $ x 
 
 24 $ yf y if dx3Q y d f\fdx 
 
 o 
 
 i 
 A 
 
 + 6 Dec 3 c^ 
 
 
 
 Dividing both members of the last equa 
 tion through by 6 &yf y dx, wes hall 
 
 
 
 have, after bringing all terms containing 
 y into the first member, 
 
 - 2 )/y j; tf a + 3 f y y dx + Dec 3 A/ = 
 
 o o 
 
 34. 
 
 By making the proper substitutions in 
 equation 25, 
 
57 
 
 /y 
 ydx 
 J 
 
 
 {yj ydx 3y y*dx T>x 3 
 
 
 > =7 
 
 /J 
 
 6 / ydx 
 
 ^ o J 
 
 
 ., fv 7 
 26 J ydx- 
 
 18 8\yf y ydK--W\f y y*fo-- 3 XT)x : 
 
 
 
 /2/ 
 y dx 
 
 18 S ! \ 
 
 
 
 Transposing, after dividing each member 
 by 3 tf A, we have : 
 
 6 l^yy dx + 3 A 
 
 + X Dx 3 . . . 35. 
 
 But here a new difficulty presents it 
 self, for no sooner do we attempt to in 
 tegrate equations 34 and 35, than we see 
 
58 
 
 it is quite impossible to perform the in 
 tegration by any exact method. We 
 may, however, obtain an approximately 
 correct solution by finding the value of 
 y in a series of functions x. Treating 
 equation 34 by this method we obtain, 
 says M. Delocre, for y the value 
 
 r i t t V v 
 
 y=ax + bx + cx + dx + ex+fx+<&c. 36. 
 While equation 35 gives : 
 
 37. 
 y a x + b x* + c x 3 + d x* + e x" +f x % + &c. 
 
 These equations, as it is quite apparent, 
 are of no earthly value for practical pur 
 poses, and we shall, therefore, drop all 
 further consideration of them. Indeed, 
 if it were possible to obtain the equations 
 of the curve AmC, by a short and sim 
 ple process of integration, a moment s 
 reflection will show that such a profile 
 as that illustrated in Fig. 7 would not be 
 suitable for practical use. For this pro 
 file has been calculated on the hypothesis 
 that the dam is always to support a head 
 of water equal to its height, and in this 
 
59 
 
 case the pressure on any horizontal sec 
 tion as m n will, it is quite true, not ex 
 ceed the limit R. But as it happens 
 that the dam is very likely to be at times 
 empty, the profile must be such that, full 
 or empty, the pressure on any section 
 as m n shall not be greater than R. We 
 know that this limit will not be exceeded 
 for the face of the wall bounded by 
 A m C, and it thus remains to consider 
 only the vertical face A B. On reference 
 to the calculations we have made rela 
 tive to the profile of walls having only 
 their own weight to support, it becomes 
 noticeable that the limit will soon be 
 passed if the wall is slightly raised. 
 Supposing this limit to be reached at the 
 point n^ we are forced for the sake of 
 stability to depart from the vertical be 
 low this point, to give the water face a 
 swelling or bulging surface, and -thus 
 adopt a profile similar to that illustrated 
 in Fig. 8. This profile is supposed to 
 fulfill the conditions that, at any section 
 as de, taken below mn^ the pressure at 
 the point e, the dam being full, will be 
 
60 
 
 less than or equal to the limit R, and the 
 dam being empty, the pressure at d re- 
 
 FIG, 8. 
 
 DC B 
 
 suiting from the weight of the structure, 
 will also be less than or equal to the 
 same limit of pressure, R. 
 
 This last modification, moreover, is one 
 of no small importance, as it enables us 
 to correct some of the chief errors in 
 which the theoretical consideration has 
 unavoidably led us, and thus to approach 
 nearer to the end in view ; the determi 
 nation of a profile of equal resistance suit- 
 
61 
 
 able to practical requirements. If the two 
 curves mel> and ndT) could be readily 
 obtained by the above formula^he profile 
 of Fig. 8 would answer almost all necess 
 ary conditionsas testability and economy; 
 but they cannot. It therefore remains to 
 do the next best thing, and to replace the 
 curved surfaces, by polygonal surfaces 
 of as small sides as possible in order 
 that they may approach reasonably near 
 to the curves and then determine the 
 equations of these sides of the polygons; 
 or to adopt a similar method to find the 
 equations of the two curves in question. 
 This we shall now endeavor to do. It 
 is, how r ever, to be remarked that there 
 are two notable instances of the use of 
 the form of profile, shown in Fig. 8 ; 
 that of the dam at Furens, and that con 
 structed on the Ban, a tributary of the 
 Gier, by M. Mongolfier. Each of these 
 we shall consider later. 
 
 As this form of profile, therefore, has 
 been illustrated, and its economy, dura 
 bility and strength fully tested in the 
 case of the dam at Furens, and in 
 
62 
 
 that over the Ban, we shall now under 
 take its investigation, and determine 
 a series of formulae for the calculation 
 of the logarithmic curves forming the 
 inner and outer face of the dam, and, 
 finally, the establishment of a profile 
 type suitable for dams of various heights. 
 Our investigation, moreover, is to be 
 based on the practical experience of 
 MM. Graeff and Mongolfier, in the con 
 struction of the dams of Furens and 
 over the Ban, and the brief but thorough 
 report of Professor Rankine on this form 
 of profile, to many parts of which we 
 are greatly indebted. 
 
 In the first place, as to the limit of 
 pressure, two questions naturally present 
 themselves: first, what shall be the great 
 est limit of pressure we may with safety 
 assume ? and secondly, is the same limit 
 to be adopted for the inner as for the 
 outer face of the structure ? As regards 
 the first question, it becomes evident at 
 a glance that the limit R , to which any 
 point in the dam may be subjected with 
 out thereby endangering stability, will de- 
 
63 
 
 pend, to no small extent, on the nature 
 of the stone, cement, or mortar used. 
 Yet here, as in other cases where mason 
 ry is used, it is possible to assign a gen 
 eral limit, based upon practical experi 
 ence, which should not in any case be 
 overstepped, and if possible rarely equal 
 ed. In the two dams to which we have 
 above alluded, the limit of the pressure 
 was taken at 6 kilogrammes per square 
 centimetre, or 60,000 kilogrammes to the 
 square metre, or taking the kilogramme 
 as equal to 2.20485 pounds, 132.291 Ibs. 
 per square metre, which in turn is equal 
 to 1.1954 square yards. In Spain, how 
 ever, and indeed, we believe in some in 
 stances in France, the limit of pressure 
 has been taken so high as 14 kilogrammes 
 per centimetre,and the dam found to stand 
 well, but in the majority of cases at from 
 6 k. to 8.50k., generally at 6 k., per square 
 centimetre. We may express this press 
 ure in another form much more familiar 
 to English engineers, and take as the 
 limit of pressure for each square foot or 
 square yard, a column of masonry hav- 
 
64 
 
 ing that area for a base and a height of 
 160 feet. This is also based on experi 
 ence, as it is well known that good rubble 
 masonry will, when laid in strong hy 
 draulic cement, bear with safety the 
 pressure arising from the weight of a 
 column 160 feet in height. Taking,, 
 again, the density of masonry as double 
 that of water, this pressure would be 
 equaled by a water column 320 feet 
 high, or a pressure per square foot of 
 20,000 pounds. 
 
 The next question as to whether the 
 limit of pressure should be the same, 
 both for the inner and outer face of the 
 dam, seems to be viewed very differently 
 by different engineers, and to admit in 
 practice of a variety of solutions. In 
 the dams constructed by M. Graeff and 
 M. Mongolfier, and in the theoretical 
 profiles offered by M. de Sazilly and M. 
 Delocre, the same limit of pressure was 
 adopted for each face, and the discussion 
 of the formulae thus much simplified. 
 Yet there seems to be much ground for 
 departing from this observance and for 
 
65 
 
 adopting two limits, one for the outer 
 and one for the inner face, provided that 
 the dam has such a logarithmic curve of 
 profile as that we are considering. It is 
 evident that the vertical pressure along 
 these two faces is, at different times, un 
 equal ; that when the water is of great 
 depth behind the dam the outer face is 
 more severely strained than the inner, 
 and that when the water is very low, and 
 the dam has little more than its own 
 weight to resist, directly the opposite re 
 sult takes place and the severest strain 
 is found along the inner face. It is like 
 wise evident that the pressure at any 
 point along these faces must, in all cases, 
 be of necessity in the direction of the 
 tangent to the surface at that place. If 
 the face is vertical, the quantity we de 
 rive by the usual equations is the true 
 vertical pressure, or rather the entire 
 pressure. But when the surface slopes 
 off from the vertical, as it does in this 
 case, the pressure is in the direction of 
 the tangent, is inclined to the vertical, 
 and the quantity which the formula gives 
 
66 
 
 us is not the entire pressure, but only its 
 vertical component. The whole or real 
 pressure of course, exceeds this vertical 
 component, by a ratio which grows 
 greater and greater as we pass down the 
 face of the dam to parts where the bat 
 ter, or slope of the face, departs more 
 and more largely from the vertical. But 
 the outer face has a very much greater 
 batter than the inner, and the water be 
 ing high, is subjected to a much greater 
 strain, so that, to equalize matters, and 
 not allow the outer face, when the dam 
 is full, to suffer a- greater strain than 
 the inner face when the dam is empty, it 
 becc mes most expedient to take a lower 
 limit for the vertical pressure at the out 
 er than we do for the intensity of the 
 vertical pressure at the inner face. 
 
 Adopting this view, it remains to fix 
 these two limits of vertical pressure. On 
 the inner face, it is clear, where the slope 
 deviates so very little from the vertical 
 that, for all intents and purposes, it may 
 be safely neglected, we may take that we 
 have already fixed upon, namely, the 
 
67 . 
 
 weight of a column of masonry 160 feet 
 high. For the outer face, we may take 
 a pressure whose vertical component is 
 represented by the weight of a masonry 
 column 120 feet high, a pressure which 
 has been deduced from the practical ex 
 amples of M. Graeff. 
 
 The next matter to be taken into 
 account is that of tension, which must, 
 so far as possible, be avoided in every 
 portion of the dam. And this brings us 
 to the consideration of the " lines of re 
 sistance," of which in structures subject 
 ed to such varying pressure, there are of 
 necessity two ; one for the condition 
 that the dam or reservoir is full of water, 
 and one for the condition that it is empty. 
 As in the case of earth retaining walls 
 and buttresses, these are lines passing 
 through the centre of gravity of each 
 course of masonry, and may, when the 
 faces of the dam are rectilinear, be found 
 by any of the formulas used for such 
 purposes. They bear, therefore, intimate 
 relations to the stability of the dam, the 
 latter decreasing as they depart from the 
 
centre of thickness and near the faces. 
 They also bear relation to the tension, 
 and in order that the latter may not be 
 come appreciable in any part of the 
 structure, they must not deviate at any 
 point from the line passing through the 
 centres of thickness, either outward 
 when the dam is full, or inward when 
 empty, by a distance greater than one- 
 sixth of the thickness at that point. 
 
 With these conditions in view, we now 
 pass to the consideration of the profile. 
 
 PROFILE TYPE FOR DAMS HAVING CURVES 
 FOR BOUNDING FACES. 
 
 Let Fig. 9 represent the profile of a 
 dam bound by logarithmic curves, the 
 various equations relative to which we 
 wish to find. Let the vertical line A S 
 represent the asymptote of the curves, 
 and taking the origin of co-ordinates 
 at the top of the dam, represent by x 
 all horizontal, and by y all vertical 
 measurements, by b the breadth or 
 thickness of the dam across the top, 
 
69 
 
 18 
 
 .... 1-410 
 
 ~fr< 
 
 k.i|/v_--\50 
 
 ilA. 
 
 iW-\ 
 
 _Lv.-&-\9 
 
 FIG. 9. 
 
70 
 
 and by b the breadth at any other place 
 lower down. Also let s represent the 
 sub-tangent common to the two curves, 
 and represented in the figure by that 
 part of the asymptote contained between 
 F and G. As to the lines of resistance 
 let their deviation from the middle of the 
 thickness when the dam is full and empty 
 be expressed by the letters r and r re 
 spectively, and by R and R/ denote the 
 limits of pressure ; the first for the out 
 er, the second for the inner face. 
 
 Now, adopting Professor Rankine s 
 method of procedure, it becomes evident 
 that if the thickness across the top be 
 expressed by #, then the thickness at any 
 other portion of the dam lower down, 
 and at a distance y below the top, will 
 be expressed by the equation 
 
 b r = b.e- 38. 
 
 s 
 
 in which e is the modulus of the common 
 system of logarithms, or 0.434294. To 
 apply this equation therefore to practice, 
 it is necessary to know the value of the 
 
sub-tangent, the thickness across the top 
 and the vertical distances of different 
 points on the face of the dam below the 
 axis of X. These latter points are, of 
 course, assumed at random, and have in 
 the present case been taken five feet 
 apart. As to the thickness at the top it 
 has been taken at eighteen feet. In the 
 dams already alluded to (those of MM. 
 Sazilly and Mongolfier) with the height 
 of 50 and 42 metres respectively, and a 
 limit of pressure of 60,000 kilogrammes 
 per square metre, the thickness across 
 the top is, in the former, five, and in 
 latter, five and seven-tenths metres, 
 which, expressed in feet, gives for the one 
 16.4 and for the other 18.6 feet. But in 
 this instance we have slightly enlarged 
 on the thicknesses used by thflse engi 
 neers, in order to produce a profile suit 
 ed for a dam required to resist not only 
 the thrust of water, but also that of ice 
 when carried down by spring freshets. 
 The determination of the sub-tangent s 
 is not so obvious, but may be found by 
 
72 
 
 nj 
 
 giving to the exponent - of e an approxi- 
 
 s 
 
 mate value of , which substituted in 
 the formula of Prof. Rankine, gives a 
 corrected value of V, and a sub-tangent 
 equal to 80 feet. 
 
 If, then, adopting this breadth of 18 
 feet on top, we desire to find that at a 
 point thirty feet below, we may write 
 equation : 
 
 O A 
 
 # = log. + 0.434294X . 39. 
 80 
 
 = 1.255273 + 0.162858 = 26.19 feet, 
 
 which is to be measured off in such wise 
 that thirteen-fourteenths of it shall lie 
 on the down stream or outer side of the 
 asymptote, and the remaining one-four 
 teenth on the up stream or inner side. 
 Taking other values for y and proceed 
 ing in precisely the same way, we thus 
 obtain any desired number of points 
 through which must pass the logarithmic 
 curves that form the faces of the dam. 
 This done and the curve drawn, the next 
 step is to determine the lines of resist- 
 
73 
 
 ance when the dam is full and when it is 
 empty. To begin with the latter case, 
 the dam being empty, the deviation of 
 the line of resistance from the middle of 
 the thickness will evidently be inward 
 or towards the up stream side of the 
 dam. This deviation we have expressed 
 by the letter r , and if we wish to find 
 its value for a horizontal section of the 
 dam taken 50 feet below the top, we pro 
 ceed as follows. Let z denote the dis 
 tance hg or the deviation of the centre 
 line of the thickness outward from the 
 axis A S, and by z f the deviation of the 
 same line from the same axis at the top 
 of the dam. Referring to Fig. 9, the 
 distance we wish to find is evidently 
 equal to g h minus the deviation of the 
 centre of thickness of the top of the 
 dam from A S, divided by 2, or 
 
 r =^ 40 
 
 _ . ^rV 
 
 Because the dam having only its own 
 weight to carry, the line of resistance 
 must cut the line ghiu a point vertically 
 
74 
 
 below the centre of gravity of that part 
 of the structure above g A. 
 
 The thickness of the dam where y is 
 fifty feet is found from equation 39 to 
 be 33.63 feet ; the centre of thickness 
 16.81, and the value of z or the devia 
 tion of this centre from the axis A S is 
 14.41 feet. That of z r or the deviation 
 at the summit of the dam is 7.72 feet, 
 from which it follows that (eq. 40) r = 
 3.35 feet. It is in this way that the 
 values of r , given below in Table A, 
 have been calculated. 
 
 It is next necessary to determine an 
 equation from which to find the values 
 of r, or the amount by which the line of 
 resistance deviates outward from the 
 centre of thickness when the dam is full. 
 It is evident this deviation will depend 
 upon three things, the moment of the 
 horizontal thrust of the water, above the 
 section at which we wish to find r, the 
 weight of the dam above this same sec 
 tion, and the amount by which the line 
 of resistance is moved inward when the 
 dam has only its own weight to carry, so 
 
75 
 
 that if we divide the moment of the 
 thrust by the weight, and subtract the 
 quantity r , we shall at once have the 
 value of r. The thrust of the water 
 above any horizontal section of the dam 
 is, as we have already seen by equation 
 
 V 2 
 2, X 62.5 Ibs., and the moment is, 
 
 2t 
 
 therefore, ^-X 62.5 X-= 62.5 Ibs., or, 
 2 36 
 
 what is the same thing, if we express by 
 w the ratio in which the masonry is 
 heavier than the water, and take, as is 
 usual, this ratio as 2, we shall have for 
 the moment (expressed by m) of the 
 horizontal thrust of the water, 
 
 V 3 V s 
 m-^- - 41 
 
 Qw 12 
 
 The weight of any lineal unit of the 
 dam above the section may be found 
 most simply by the calculus. Thus giv 
 ing to y and b the same signification as 
 before, and taking the weight of a cubic 
 unit of masonry as the unit of weight, 
 
7G 
 
 the weight of each unit of length of the 
 wall above the section is expressed by 
 nr s*y 
 
 J v dy ... 42. 
 
 
 
 Integrating this between the limits y 
 
 and o, and remembering that V=be^ 
 
 $ 
 we have : 
 
 = * (* -&) . . . 43. 
 
 For r, therefore, we have : 
 
 w 
 
 y* 
 
 
 This equation gives for the value of r 
 at the distance fifty feet below the top, 
 the quantity 5.18 feet, which, as it falls 
 below one-sixth of the thickness at this 
 point, we are justified in considering the 
 
77 
 
 deviation as not too great to be perfect 
 ly consistent with stability. 
 
 But, to make assurance doubly sure, 
 we may apply a final test as to stability, 
 by calculating the amount of vertical 
 pressure at various points along both the 
 inner and outer faces, and comparing 
 the results with the limit of pressure, 
 which, it will be remembered, has been 
 fixed for the inner face at weight of a 
 column of masonry 160 feet in height, 
 and for the outer face at that of a col 
 umn 120 feet high. This matter we have 
 already considered at length, and have 
 deduced two equations, 13 and 14, which 
 as they are perfectly suited to the present 
 case, we shall not delay to deduce others, 
 but alter them to suit the notation of Fig. 
 9. Thus altered they are, calling p and 
 p the pressures at the outer and inner 
 face respectively, and P and P the lim 
 it at these same faces 
 
 and ^ 45. 
 
78 
 
 While for p r we have two others precise 
 ly similar, with the exception that P in 
 equation 45 is changed to P . It may, 
 perhaps, be well to again remark that 
 the first or second value of p in equtaion 
 45 is to be used according as the value 
 of it is greater or less than one-third of 
 the thickness, and that in all such pro 
 files as that of Fig. 9, the quantity u de 
 notes the distance from the outer face 
 to the line of resistance when the dam 
 supports a charge of water, and from the 
 inner face to the line of resistance when 
 the dam or reservoir is empty. To illus 
 trate by one example, let it be required 
 to find the vertical pressure at the point 
 C, on the outer face of the dam (Fig. 9), 
 situated fifty feet below the top. By re 
 ferring to Table A, we see that b is equal 
 to 33.63 feet, that the outward deviation 
 of the line of resistance is 4.98 feet, and 
 that u must therefore be 11.83 feet. The 
 quantity W=s ( ) is 1250.4. Since 
 
 u is here greater than = 11.21, we use 
 
 o 
 
 the first of equation 45, and, making 
 the substitution of values, we have : 
 
79 
 
 =2 (2 
 \ 
 
 33.637 33.63 
 Thus showing that the pressure is but a 
 little more than half the limiting press 
 ure. Precisely the same operation re 
 peated, with u equal to 13.46 feet, will 
 give the amount of vertical pressure at 
 the inner face at a point fifty feet below 
 the top, the dam supporting only its own 
 weight. This pressure is thus found to 
 be equal to a column of masonry 59.4 
 feet in height. 
 
 The area of the entire profile or of 
 any portion of it, included between two 
 horizontal sections, may be found by tak 
 ing the difference between the thickness 
 of the dam at these two sections, and 
 multiplying the difference by the sub- 
 tangent. For it is evident from the 
 figure that, if b equals the thickness of 
 a point y feet from the top, then this 
 thickness multiplied by the differential 
 of the height and integrated between 
 the limits y and zero, is the area, and 
 
 this expression / b f dy when integrat- 
 
80 
 
 ed, remembering that b is equal to be- 
 
 y v ^ 
 
 gives s be - s b, or replacing be - by Z> , 
 s s 
 
 the expression for the area becomes 
 s (b f b). In the notation we have used 
 b means the thickness of the dam across 
 the top, but in calculating the area of 
 any portion of the profile not bounded 
 by the top thickness, the quantity b is to 
 be understood to mean the smaller of 
 the two thicknesses which bound the 
 area. That is to say, if we wish to find 
 the area of that portion of the profile 
 included between horizontal sections 
 taken at thirty and eighty feet below 
 the top, b represents the thickness at 
 the former section, and we have 80 (48.93 
 26.19) = 1819.2 square feet. Having 
 the area, the solid contents and weight 
 for any length of the dam are of course 
 readily found. The areas for sixteen dif 
 ferent sections of the profile, each hav 
 ing the top of the dam for one side, have 
 been calculated in this way, and will be 
 found entered in the last column of 
 Table A. The first column of this table 
 
81 
 
 gives the distances in feet of the sec 
 tions estimated from the top downwards, 
 the second the thickness of the dam at 
 these sections, the third the deviation of 
 the line of resistance outward when the 
 reservoir is full, the fourth the deviation 
 inward when empty, and the last the 
 areas. 
 
 TABLE A. 
 
 
 
 r 
 
 r 
 
 Area 
 sq. feet. 
 
 
 
 18.00 
 
 
 
 
 
 o 
 
 10 
 
 20.40 
 
 .51 
 
 .18 
 
 192.00 
 
 20 
 
 23.10 
 
 1.09 
 
 .54 
 
 408.00 
 
 30 
 
 26.19 
 
 1.75 
 
 1.68 
 
 655.20 
 
 40 
 
 29.68 
 
 2.52 
 
 3.18 
 
 934.40 
 
 50 
 
 33.68 
 
 3.35 
 
 5.18 
 
 1254.40 
 
 60 
 
 38.10 
 
 4.53 
 
 6.66 
 
 1608.00 
 
 70 
 
 43.17 
 
 5.39 
 
 8.79 
 
 2013.60 
 
 80 
 
 48.93 
 
 6.62 
 
 10.52 
 
 2474.40 
 
 90 
 
 54.18 
 
 7.75 
 
 12.95 
 
 2894.40 
 
 100 
 
 62.97 
 
 9.63 
 
 13.53 
 
 3597.60 
 
 110 
 
 71.18 
 
 11.39 
 
 15.02 
 
 4254.40 
 
 120 
 
 81.79 
 
 13.62 
 
 14.59 
 
 5103.20 
 
 130 
 
 91.39 
 
 15.72 
 
 15.46 
 
 5871.20 
 
 140 
 
 103.60 
 
 18.34 
 
 15.05 
 
 6848.80 
 
 150 
 
 115.00 
 
 20.78 
 
 15.46 
 
 7440.00 
 
 160 
 
 133. GO 
 
 24.64 
 
 12.46 
 
 9200.00 
 
82 
 
 It is perhaps unnecessary to call at 
 tention to the fact, that this form of 
 profile has been calculated with a view 
 to its serving as a profile type for dams 
 of any height, great or small, whose 
 faces are logarithmic curves. For a 
 dam, then, of which the height is thirty 
 feet, that portion of Fig 9, above the 
 line marked 30, is the proper* profile : 
 for one eighty feet in height, that por 
 tion above the line marked 80, and so 
 for each succeeding section. It presents 
 again many strong points not found in 
 dams of the usual rectilinear profile, 
 which are especially deserving of con 
 sideration when damming a river or 
 valley of great breath and depth. Of 
 these not the least is its economy of 
 material, which, as we shall hereafter 
 see, is very great as compared with that 
 of stepped or sloping profiles ; while the 
 curves of the two faces are so gradual 
 that no great mechanical difficulty can 
 arise in cutting the facings. Another 
 matter, which, in the dams of Furens and 
 the Ban was not taken into account, 
 
83 
 
 that of tension, has here been considered 
 and the profile so determined that when 
 the reservoir is full the tension on the 
 outer face shall not at any point be 
 greater than it is on the inner face when 
 empty. 
 
 The profile of the Furens dam is given 
 in Fig. 10, and that constructed on the 
 
 /1G./0, 
 
 Ban, a tributary of the Gier, in Fig. 11. 
 The former has a height of fifty metres 
 with a breadth on top of 5.70 metres, 
 
84 
 
 A 
 
 FIG. 11. 
 
 and a limit of pressure of six kilogrammes 
 per square centimetre. The latter has a 
 height of forty-two metres, a thickness 
 on top of five metres, with the same 
 limit of pressure as the Furens dam. By 
 a comparison however, of the profile of 
 the former with that part of the profile 
 of the Furens which lies above the limit 
 A B we see that the thickness has been 
 very considerably reduced, while if we 
 extend the profile to fifty metres and 
 
85 
 
 then compare it with the Furens, we 
 find that the pressure nowhere exceeds 
 8 kilogrammes to the square centimetre. 
 To return now to the modifications of 
 which this type of profile is susceptible. 
 
 MODIFICATIONS OF THE LOGARITHMIC 
 PROFILE. 
 
 On a moments inspection of Fig. 8, it 
 is readily seen that, as the inner curve 
 does not anywhere depart very far from 
 the asymptot AS, the first and simplest 
 modification of this curve is to replace it 
 by a right line and thus make the inner 
 face vertical from top to bottom. But 
 the outer curve if treated in like manner, 
 and replaced by a right line, would give 
 us a form of profile which, though it 
 possessed no more thickness at the bot 
 tom than was absolutely necessary to 
 withstand the vertical pressure, would 
 at every other point, possess a thickness 
 greatly in excess of the requisite amount, 
 and thus occasion a prodigious waste 
 of masonry. We must therefore, break 
 this continuous slope and substitute for 
 
86 
 
 one long line two or more shorter ones 
 each of which makes a different angle 
 with the vertical. Limiting our atten 
 tion for the present to the first case, and 
 replacing the two logarithmic curves in 
 Fig. 9 by lines, the inner curve by one 
 vertical, and the outer by two inclined 
 we have produced for us a profile of the 
 form illustrated in Figs. 12 and 13. The 
 question that first presents itself in the 
 discussion of such a profile, is evidently 
 how far down the outer face the point C 
 is to be taken. It comprises indeed, the 
 entire discussion. Of course, it is a great 
 advantage, so far as the saving of ma 
 terial is concerned, to throw this point 
 as low as possible, but this is limited by 
 the condition, so necessary to secure 
 stability, that when the reservoir is full 
 the vertical pressure at C shall not be 
 greater than the limiting quantity R. 
 Having determined the thickness across 
 the top, which preserving our previous 
 notation, we will call b, the quantities to 
 be determined are first, the vertical dis 
 tance of the point C below the top, and 
 
FIG. 12. 
 
 second the thickness of the dam at this 
 point, or what is perhaps more easily 
 obtained the excess of the thickness at C 
 over the thickness at the top, A B. The 
 
88 
 
 distance, AD (Fig. 12) we will call y ; 
 78. 
 
 FIG. 13. 
 
 the total thickness D C we will represent 
 by 5 , and express the excess of thickness 
 by v. By W, denote the weight of the 
 part A B C D (Fig. 14), and by F, the 
 horizontal thrust of the water above D. 
 These two forces act through the centre 
 of gravity O, the former vertically 
 downward and represented in Fig. 14 by 
 
the line O P ; the latter horizontally 
 and represented in direction and inten 
 sity by O F. These two produce a re 
 sultant which cuts the base at V, 
 and this point may therefore be 
 regarded as the point of applica 
 tion. From this relation, as we have 
 
 seen, result two equations 2 /2 \ 
 
 P 2 P 
 
 - or<R , and or<R , which are 
 
 I O U 77 
 
 to be used according as u is > - or <-. 
 
 3 3* 
 
 In these equations P = W, is, accord 
 ing to the notation of Fig. 1 4, expressed 
 
90 
 
 by ( - - 1 y # , in which d is the 
 density of the masonry ; l=V = ~b + v and 
 
 K C may be found by the equation ex 
 pressing the relation that the moment of 
 the weight of A B C D, with respect to 
 C, is equal to the sum of the moments 
 of the two parts ABVD and BVC 
 into which the area of A B C D may be 
 divided. The moment of the weight of 
 A B C D, with respect to C, is evidently 
 
 (9 _j_ -y V 
 1 y d f multiplied by 
 
 KC ; that of ABVD by i^^ 
 
 itfd 
 and that of B Y C by J - . Hence, the 
 
 3 
 relation when expressed, becomes : 
 
 3 
 rn-(fr + 2)y* . 
 
 
91 
 
 ~~6~b + 3v 
 
 To find KY, we have from the two 
 similar triangles O K Y and O P R the 
 proportion 
 
 KY : KO;;PR : PC 
 
 whence 
 
 w KOxPR 
 
 K V _ or . . 48. 
 
 since P R is equal to the horizontal 
 thrust, which, as we see in the early part 
 
 if d 
 of our investigations, is equal to - ; 
 
 m 
 
 and since P is equal to the vertical 
 
 /2 Z?-}-v\ 
 pressure and this is equal to I 1 yd 
 
 we have finally for the value of K Y : 
 
 49. 
 - or 
 
 which latter equation is found by substi- 
 
 Ot 
 
 tuting for -^7 the letter 9. These values 
 
 given in equations 49 and 47 when re 
 placed in the expression 
 
02 
 
 _ 
 
 
 With this value of w we return to equa 
 tions 24 and 25, and, substituting it, we 
 obtain : 
 
 f 6 
 2<{ 2- 
 
 X 
 
 b A 
 
 /2 
 V 
 
 and 
 
 These, when reduced and made equal 
 to zero, give us two equations containing 
 two unknown qualities : 
 
 l>y hF = Q . 51. 
 * Q b /L v 
 . . 52. 
 
93 
 The first of which is to be used when 
 
 u > --, and the second when u < -. Each 
 3 3 
 
 of these equations express the relation 
 that when the reservoir is full the verti 
 cal pressure at the point C (Fig. 14) 
 shall be equal to the limit K. But we 
 must also take into consideration the 
 inner face, and find an equation express 
 ing the relation that the reservoir being 
 empty, the pressure at D r shall not ex 
 ceed the limit R. In this case, the face 
 being vertical, the pressure of the water 
 does not exist, and the force P, or the 
 weight of this portion of the dam, acts 
 downwards through the centre of gravi 
 ty, and 
 
 w=DK=DC-OK 
 
 2 v (v + 3 t>) + 3 5 2 
 
 _v(v + 3 b) + 3b z 
 
 3 
 With this value of u, we again return 
 
94 
 
 to equations 24 and 25, substitute in 
 each, and reducing, have : 
 
 54. 
 
 >/ y tf A -f 3 b y v2 I A v + tfy A 2 = 
 
 55. 
 
 By combining 51 and 54, or 52 and 55, 
 we may readily obtain the value of y 
 and v, which are the two quantities we 
 wish to find. It is moreover to be re 
 marked that A in the above equations is 
 found by dividing the limit of vertical 
 pressure at and D by the ratio in 
 which the masonry is heavier than 
 water. Thus in calculating the profile 
 of Fig. 12, we have first reduced the 
 limit of vertical pressure per unit of 
 surface from pounds to kilogrammes, 
 and taking the density of water, as given 
 in the French tables, as 1000 kilo 
 grammes and the density of masonry as 
 double that of water or 2000 kilo- 
 
 R 60,000k 
 grammes, we have A = = -^^ 
 
 or A = 30. We thus obtain for A, a 
 
95 
 
 very simple number, whereas had we re 
 tained the pressure as expressed in 
 pounds, we would have had a much 
 larger one to handle. In Fig. 13 how 
 ever, in order to produce a profile of 
 what may be considered as a type of the 
 greatest boldness consistent with safety, 
 we have taken the limit of vertical 
 pressure at 14 kilogrammes per square 
 centimetre, which as we have already 
 stated has been used in several instances 
 in France and Spain. This increases the 
 value of A to 70. The thickness across 
 the top is in each case the same as in 
 that of the profile illustrated in Fig. 9 ; 
 namely, eighteen feet, but the height of 
 that in Fig. 12 has been reduced to 
 ninety feet. The height AD of the 
 upper part A B C D and the value of v 
 corresponding to it have been found by 
 combining equations 51 and 54. The 
 lower part, by the same equation, by 
 substituting for y the difference between 
 the height A D of the upper part and the 
 entire height of the dam. 
 
 The deviation of the line of resistance 
 
96 
 
 when the reservoir is full may also be 
 found as follows. Let A B C D in Fig. 
 15, represent either the upper or lower 
 
 FIG. 15. 
 
 part of the dam whose profile is given 
 in Fig. 13, and let it be desired to find 
 the amount of deviation at any section 
 as E F. By O represent the centre of. 
 gravity of A B F E, then will O B, repre 
 sent the resultant of the two forces act 
 ing on this portion of the dam, and the 
 distance we wish to find will be E S. 
 We will suppose also, in order to cover 
 
97 
 
 all cases that the water stands at X. 
 Also let A B = 5 ; A X = Z. AE = y ; 
 E S = a; ; E W = A ; and the inclination 
 of the sloping side B C, to the vertical 
 be denoted by cc ; by # the density of 
 the masonry and by 6 that of the water. 
 Then by the similar triangles O P R and 
 O W S, we have : 
 
 _ 
 OW-QP 
 
 Now W S = E S - E W = a - A and 
 OW = TE=JXE because the centre 
 of pressure (T) of a rectangular plane 
 surface sustaining the pressure of water, 
 is at a point two-thirds the depth of its 
 immersion. Hence T E = J (y I). PR 
 or the horizontal thrust of the water 
 on XE is, as we know, expressed by 
 
 iZ -- L . an( j the pressure O P by 
 
 Zi 
 
 (Z> + b )y & (2 b+y tan. oc ) y & 
 
 x h _^b r (y Q 2 _ 56. 
 tan. oc & 
 
98 
 
 r/ 
 
 Then replacing by 6 
 
 o 
 
 7 _ 
 
 /I 
 
 an.c 
 + ty* tan - < 
 
 2 by + y* tan. oc 
 which, added to equation 56, gives : 
 
 58. 
 
 This value of & is, of course, to be 
 measured off from the vertical side. 
 When the water stands at the top of the 
 dam, the value of /, is zero, but when 
 the reservoir is empty, then I, is equal to 
 the entire height of the dam. The 
 simplest way, however, to find the devia 
 tion, is by means of Equation 50, ob 
 serving that the value of ^, when found 
 is to be laid off from the outer or slop 
 ing face of the dam ; and corresponds to 
 the distance FS in Fig. 15. 
 
 The second modification, then, of the 
 theoretical profile of equal resistance, 
 
consists in replacing the outer curved 
 face by a broken one composed of two 
 planes inclined at different angles to the 
 horizon. The principles, however, which 
 justify us in the use of such a modifica 
 tion, may be carried still further, and 
 the inner and vertical face replaced by 
 one almost a fac simile of the outer 
 broken one. Indeed the only essential dif 
 ference between them lies in the degree of 
 slope which we give to their two plane 
 surfaces. On the one side both are 
 sloping ; on the other that portion of 
 the face from the summit of the dam to 
 a point below, (where the pressure on 
 each unit of surface equals the assumed 
 limit of pressure,) the wall is vertical, 
 and from here to the base slope out 
 ward. This latter point moreover, must 
 be directly opposite that point on the 
 outer face at which the two sloping lines 
 of the profile intersect. Of a profile thus 
 constructed, some idea may be had from 
 the sixteenth figure. It does not present 
 any merit either as to beauty, strength, 
 stability or economy of material not 
 
100 
 
 possessed by that illustrated in Figs. 12 
 and 13. As to economy indeed, the 
 amount of material consumed is if any 
 thing greater in former than in the two 
 latter forms of dams, and it may be 
 justly doubted whether the additional 
 stability thus obtained, is a fair recom 
 pense for the additional outlay for 
 material and for cutting facing stones 
 for a third sloping face. 
 
 As to the mathematical calculations of 
 such a profile they are rather lengthy than 
 difficult. For the upper portion A B C D, 
 Fig. 16, we have already discussed the 
 principles at length, and obtained in 
 equations 51 to 55 the necessary formulae. 
 The value of A B or b is of course 
 known, as also that of AD or a which 
 is assumed, and is not to be greater 
 than A or the greatest height we can 
 with safety give to a wall with vertical 
 faces. That of the lower portion C D E F, 
 may also be conducted on the principles 
 previously laid down, and as it necessi 
 tates several eliminations of somewhat 
 startling length we shall consider it 
 
101 
 
 A B 
 
 merely in outline. Knowing the total 
 height of the dam, and the distance A D, 
 we of course know D G, or the height of 
 that portion of the dam C D E F, whose 
 breadth of base E F, we wish to find. 
 We also know from equations 51 and 54, 
 the breadth D C , and projecting this on 
 the base we at once obtain that portion 
 
102 
 
 of it between GandH. What there re 
 mains to be found is G E, and H F. The 
 former of these unknown quantities we 
 will denote by y, and the latter by z ; 
 the breadth E F, of the base by b, the 
 part G H, which is also equal to C D, by 
 # ; the height D G, of the lower section 
 of the dam by a, and that of the upper 
 section, or A D, by a f . Returning now 
 to the equations 15 and 16, which are 
 the general equations of stability for a 
 dam supporting the pressure of a head 
 of water, we find that the three unknown 
 quantities for which we wish to find 
 values in term of the known quantities we 
 possess are it, I, and p. The value of , 
 or the thickness E F, of the base is, 
 when expressed in terms of the above 
 notation. 
 
 l=y+b+z 
 
 While P is of course the area of the ir 
 regular polygon A B C F E D multiplied 
 by the weight per unit of volume, plus 
 the vertical component of the weight of 
 the water resting on the sloping face 
 
103 
 
 DE. The area of ABCD is ( 
 
 3 a . That of C D E F is y -?- + Z ^ 
 
 4- V & a. The vertical thrust of the 
 
 (2 a - 
 2" 
 
 o 
 The value of P, therefore, is V d f a + 
 
 y #, which reduces to the form 
 
 P= 59. 
 
 (2 
 
 2 
 
 Again, to find the value of u^ the first 
 step is to construct the diagram of 
 forces, as illustrated in the figure, O P 
 representing in direction and intensity 
 the vertical component P, or the weight 
 of the dam and the water, and O F the 
 horizontal component or the outward 
 thrust of the water behind the dam. 
 Then will F T represent u which is clearly 
 equal to 
 
 u=z + TLI IT . . 60. 
 
104 
 
 But by the two similar triangles we 
 
 O "F 1 
 
 have, as before, IT=OI X ~-p or since 
 
 O 1= - - and O F (equation 2) equals 
 
 3 
 
 \ 2 
 
 ( ) < 
 
 
 yd] 
 
 HI is to be obtained in precisely the 
 same manner as K C was obtained from 
 Fig. 14, by expressing the relation that 
 the moment of weight P (which includes, 
 it is to be remembered, that of the dam and 
 that of the water pressing on the inclin 
 ed face D E), with respect to the point 
 F is equal to the sum of the moments of 
 the components of this force. Obtaining 
 these moments in the same manner as 
 we obtained those for the equations de 
 duced from Fig. 14, and putting them 
 equal to the expression P X IF, or P x 
 (IH-f^), we have after reduction, the 
 equation 
 
105 
 
 H= 
 
 12 cc 
 
 (1 2 cc + Vd) + 6 a z 4- 6ay + 12 a ?/0 + Qayd 
 
 In which oc is a short expression for the 
 area of A BCD, and the distance 
 from C to the point where the perpen 
 dicular of the centre of gravity of 
 A B C D cuts C D, and this replaced in 
 equation 60, gives for the value of u 
 
 12z(oc + b 
 
 + 12oc/? + 6 2 a + 2a;?/(2/ + 36 ) + 3 (2 a 
 + a) (y + Zb^ye-Zatf-ie (a 
 
 12(oc +b f 
 
 Eq. 61. 
 
 The quantities P, u and I, being thus 
 obtained in terms of & , y, z, a and a , a 
 substitution in equations 15 and 16, will 
 furnish us with two equations of great 
 length, from which, by the process of 
 elimination, the values of x and y are 
 readily found. 
 
 To take but one example of this form 
 
106 
 
 of profile, let it be required to calculate 
 the dimensions of such a profile for a 
 masonry dam one hundred and seventy 
 feet in height and eighteen feet broad on 
 top, the limit of pressure being taken at 
 132,000 pounds. For this purpose we 
 have to determine beforehand the height 
 a! of the part A B C D. This, in the 
 present case, is taken at 80 feet, and may 
 in all cases be assumed arbitrarily. Now, 
 since the dam has one vertical face, we 
 have to determine but one quantity v, or 
 the difference between the thickness of 
 the dam at AB and that at CD, and 
 this value of v is readily obtained from 
 equation 51, which, modified to suit the 
 present notation, becomes 
 
 Solving this with reference to v, we 
 have 
 
 b*a + da" 72 
 
 v +2 o v= - -- 1} 
 A 
 
 And replacing the quantities by their 
 
107 
 
 values,remembering that A equals 98.4 ft., 
 and 9 (or the ratio in which the mason 
 ry is heavier than water) equals , the 
 result finally obtained is, 
 
 v = 53. 52 18 or 
 
 53.52 feet. 
 
 With this value of b we return to the 
 equations expressing the values of x and 
 y as deduced from equations 15 and 16, 
 after the substitution of the value of u 
 given in equation 61, and find that the 
 value of b ff =x + b + y is 178.42 feet. 
 
 Once more, we may carry this princi 
 ple one step further and produce a pro 
 file which is little more than a modifica 
 tion of that given in Fig. 16. If, for 
 instance, while preserving the same 
 height of structure, we divide each of 
 the three sloping faces into two parts, 
 and give to each part thus produced a 
 face inclined to the horizon, we shall 
 then have a profile of such shape as that 
 illustrated in the seventeenth figure. 
 
 A glance at this is sufficient to show that 
 it is in reality but a compound of the 
 
108 
 
 A B 
 
 II I 
 
 two preceding profiles, and that there 
 fore the principles to be observed in the 
 calculation of its parts are those already 
 discussed. The entire profile may thus 
 be considered as divided into three 
 pieces ; that from A to D, in which the 
 inner face is vertical throughout, and the 
 outer made up of two inclined faces, 
 constituting a profile exactly similar in 
 design to that of Fig. 12 : that from D 
 
109 
 
 to F, and that from F to H, in each of 
 which both the outer and inner faces are 
 sloping. The first part is, therefore, to 
 be calculated in the same manner as we 
 would calculate the thickness of a dam 
 having the profile of Fig. 1 2, and each 
 of the two remaining portions by the 
 equations deduced from Fig. 16. To 
 illustrate this by a case in point, let it be 
 required to find the thickness at various 
 points of a masonry dam, having such a 
 profile as that we are discussing, its 
 thickness across the top being 18 feet, 
 and the total height 170 feet. The 
 first thing that claims attention is the 
 determination of the vertical distances 
 between the points B and C ; C and E ; 
 E and G ; and finally G and I. These 
 may, of course, be chosen at pleasure, 
 just as we may select the number of 
 parts that each face is to be composed 
 of, and as in the present case the dam is 
 170 feet high, and the outer face divided 
 into four parts, we will for convenience 
 divide the dam first into two equal parts, 
 then divide the lower of these again into 
 
110 
 
 two equal parts, and the upper also into 
 two, but two unequal parts. The verti 
 cal distances between the sections will 
 then be, beginning at the bottom and 
 going up G I = 42,5 feet; E G = 42.5 
 feet ; C E = 45 ; and B C = 40 feet. Had 
 the dam, however, been one hundred and 
 fifty, or one hundred and eighty feel 
 high, or indeed any other number, then 
 the best arrangement would again have 
 been, to make the second vertical dis 
 tance that from C to D longer than 
 the remaining three, so that, if the dam 
 was one hundred and fifty feet high, the 
 best arrangement would be BC 30 ; 
 CE = 60; and E G and GI each thirty 
 feet ; if the height had been one hund 
 red and eighty feet, then B C = 40; CE 
 = 50 ; and the others each forty-five 
 feet. Although this arrangement may 
 seem to be somewhat arbitrary, it is in 
 reality based upon fixed principles,which 
 clearly show that where such a number 
 of divisions and such a profile as that 
 used in the present instance are employ 
 ed, the second part should be decidedly 
 
Ill 
 
 longer than either of the other three. 
 Those portions, moreover, which are 
 bounded on both sides by sloping faces 
 are in almost all cases made of equal 
 depth, nor does there seem to be any 
 reason whatever for not adhering to this 
 method. 
 
 With these distances thus determined, 
 we return to equations 51 and 54, and 
 from the first of these find the value of 
 v, as was done for equation 63, and sub 
 stituting for a r the value 40, and for b 
 the quantity 18 feet, we have 
 
 98.4 
 
 And, consequently, # = + ^ = 21.37 feet. 
 To find the value of b , however, it is 
 necessary to use equations 51 and 54, 
 from which by the common method of 
 elimination we may find an expression 
 
 6y* v*hy 3 byv = Q 
 
 from which by the substitution of the 
 proper values we obtained for a final value 
 of //, or the thickness of the base of this 
 
112 
 
 section, "=54.64 feet, or ^ = 33.27 feet. 
 The next step is to find the values of x 
 and ?/for the third section. As this, and 
 also the last section have both faces slop 
 ing, by substituting the value of u given 
 in equation 61, in equations 15 and 16, and 
 reducing and then eliminating, we obtain 
 two expressions for x and y, from which 
 we derive the thickness GF = 100.36, 
 and by a similar process find that for 
 I H to be 152.22 feet. 
 
 It is thus apparent, that as there is al 
 most no limit to the number of sections 
 into which a dam may, on this principle, be 
 divided, there are a great number of 
 different forms of profile, each of which, 
 satisfy the conditions of stability, but 
 vary somewhat as to economy. Theo 
 retically the dam whose outer face con 
 sists of the greatest number of these 
 sloping faces is the most economical, 
 because in that case its face approaches 
 nearest to the logarithmic curve which 
 bounds the theoretical profile of equal 
 resistance, and it therefore contains very 
 little more masonry than is absolutely 
 
113 
 
 necessary to insure safety. In practice, 
 however, such a dam would, in all proba 
 bility prove much more costly than one 
 consisting of a less number of section, 
 though containing more masonry, be 
 cause the angle of inclination of the 
 different sections of the outer face 
 changing so frequently would greatly 
 increase the cost of cutting the facing 
 stone. To avoid the mechanical difficul 
 ties also likely to arise in such cases, it is 
 sometimes well to depart altogether from 
 this style of profile, and instead of slop 
 ing the outer and inner faces, cut them 
 into notches or steps. 
 
 THE STEPPED PEOFILE. 
 
 The stepped profile has been reserved 
 to the last for consideration, because, 
 while it is a natural outgrowth of the 
 preceding modifications, it possesses 
 many merits whose importance cannot 
 be fully appreciated till a comparison is 
 instituted between it and the forms 
 just treated of. In point of simplicity 
 of construction for instance, it would be 
 
114 
 
 difficult to find any design of profile 
 that can surpass it. Wherever the faces 
 of the dam are curved as in Fig. 9, or 
 made up of a series of sloping surfaces 
 of various inclination as in Figs. 12, 16 
 and 17, the dimensions of every facing 
 stone that is set have to be most care 
 fully determined beforehand by the 
 rules of stereography, and this, when the 
 dam is an high one and the number of 
 stones consequently large, is of itself a 
 work of no small difficulty. In the 
 stepped dam however, all this is done 
 away with, as every facing stone, (unless 
 the dam is curved) possesses only a ver 
 tical or, if it happens to form the edge 
 of the step, a vertical and horizontal 
 face, and thus requires no pattern for the 
 stone cutter. A further advantage to 
 be derived from it, is, that it enables us 
 to approach much nearer the curved 
 form of profile than we can in any other 
 profile type. Indeed, when well designed 
 it is in reality nothing but the logarith 
 mic curved profile cut into steps or 
 notches, so that should we draw a con- 
 
115 
 
 tinuous line through the upper edges of 
 all the steps, or through the lower edges 
 of their vertical faces, this line would 
 form a logarithmic curve. 
 
 Here, as in the calculation of the 
 previous profiles, it is quite allowable to 
 assume arbitrarily either the breadth or 
 height of the step and from this one de 
 termine the other. Yet it is by far the 
 best plan to assume the vertical height 
 of the step and calculate the breadth. 
 For, it must be apparent, that by this 
 method of procedure, the quantity we 
 calculate is really the abscissa of the 
 curve, which we lay off at regular inter 
 vals perpendicularly to the vertical 
 axis of the dam, and in this way we are 
 enabled to preserve very closely the 
 logarithmic profile. The general appear 
 ance of the dam is, moreover, much more 
 pleasing when this arrangement is ob 
 served than when we assume a constant 
 breadth and calculate the depth, because 
 the breadth of the steps near the summit 
 of the dam is then very narrow and in 
 creases gradually as they approach the 
 
116 
 
 bottom, and the departure from the curve 
 is thus scarcely perceptible ; but when the 
 breadth is everywhere the same and the 
 depth varies, the whole face of the dam 
 has an extremely broken appearance, 
 which is anything but agreeable. 
 
 In this profile, as in all the others, the 
 inner face is made vertical for as great a 
 distance as the limit of pressure will al 
 low, and from that point down it is 
 stepped. The outer face is likewise 
 made vertical for a distance which de 
 pends in all cases on the thickness across 
 the top, being as a general thing very 
 nearly twice that dimension. In the de 
 termination of the following formulae, 
 the depth of the step has been assumed 
 as the same throughout the entire dam, 
 and the breadth has been taken as the 
 unknown quantity. Fig. 18 then rep 
 resents a portion of the profile of a dam 
 bound by a curved or sloping face, which 
 we wish to change into a stepped profile. 
 ABDC represents this section, and if 
 H F be taken as the vertical height of 
 the step, then will C H F represent the 
 
element with which we are especially 
 concerned, and its base CH the quantity 
 we are in search of, the breadth of the 
 step. The height B D of the section we 
 will denote by h\ and the density of the 
 masonry by d r ; and the greatest thick 
 ness FT or HD of the known element 
 ABTDHF by t\ from which three 
 quantities we may obtain an expression 
 for the weight P, of this element, which 
 must of course be accurately known, in- 
 
118 
 
 as much as the object of making the step 
 at this point being to lessen the amount 
 of vertical pressure on each superficial 
 unit, the breadth of the step will depend 
 very largely on the weight of that por 
 tion of the dam which is above it. The 
 weight which is plainly equal to 
 
 expressed by P, while that of the ele 
 ment C II F is equal to - , in which 
 
 a is the height of the step F H, and b 
 the breadth C H. The point of applica 
 tion of the thrust of the water is T 
 situated at two-thirds the depth of im 
 mersion. T and T 2 are the horizontal and 
 vertical components respectively. Then 
 will P represent the direction of the re 
 sultants of P and T 2 ; V V the resultant 
 
 of P, T 2 and the weight - of the ele 
 
 ment C H F, while the general resultant 
 of all the forces is R. Now, in this case, 
 as in the previous ones, the whole solu 
 tion of the problem depends on finding 
 
119 
 
 the value of C R, or the distance from 
 the outer edge C to the point where the 
 resultant cuts the base, and this we will 
 express as heretofore by the letter u. 
 Then from the figure 
 
 *4=CH + HY RV ... 64. 
 
 in which we know the value of C H=&, 
 and require that of H Y and R Y. But 
 
 R Y 
 
 ^r^f is equal to the tangent of the angle 
 
 which the general resultant R makes 
 with the vertical, or calling this angle 
 cc then 
 
 RY T 
 
 tan. cc = 
 
 YY 
 
 in which e is to be understood to express 
 
 the value of YY=^-FfT The distance 
 H Y may be found from the theorem of 
 
120 
 
 moments, by expressing the relation that 
 
 d tfa 
 
 M- 
 
 6 
 d Va 
 
 > + * 
 
 M denoting the moment of P with re 
 spect of H. As to C H, its value is &, 
 the quantity we are in search of. Re 
 placing these quantities in the equation 
 expressive of the value of u, we have 
 
 M - d W a 
 
 which, reduced to a common denomina 
 tor, becomes 
 
 65. 
 
 u = 
 
 6P + 3 6 ba 
 
 Having thus obtained an equation for 
 the value of u, the next step is to find by 
 means of it an expression for b the 
 
121 
 
 breadth of the step. For this purpose 
 draw from R, the point at which the 
 general resultant of all the acting forces 
 outs the base, a perpendicular R N" to the 
 resultant, and from N a perpendicular to 
 the base C D, thus forming a triangle 
 R N O. Then, since the two triangles 
 R Y Y and R 1ST O have their bases on 
 the same right line C D, and the side YR 
 of the one perpendicular to the side NR 
 of the other, and the sides YY and NO 
 parallel, the angles at Y and N are equal 
 and the triangles are similar. But by 
 the relation existing between the sides of 
 such similar triangles, we have the pro 
 portion 
 
 N O : R Y ; : R O : Y Y. 
 
 which gives for N O the equation 
 
 RVXRO T / 
 ~V~V^ ~ d ab 66. 
 
 ~T 
 
 in which /is the distance R O. But we 
 have another pair of similar triangles 
 which gives yet another value for N O, 
 which must be deduced and made equal 
 
122 
 
 to that just found. These triangles are 
 CO 1ST and C H F, and the proportion 
 derived from the relation of their sides 
 is, 
 
 NO:CO;:FH:HC or 
 
 _ 
 H C b 
 
 Equating equations 66 and 67, 
 
 COx=/X ^, 
 
 b p 6 ab 
 
 ~~ 
 
 T 
 
 ~ 
 
 And again, since if four quantities be 
 proportional they will be in proportion 
 by composition and division 
 
 > 
 
 p + - 2 - b 
 
 and reducing, 
 
.123 
 
 68. 
 
 But the condition of stability is (equa 
 tion 16) expressed by the relation 
 
 / = 
 
 3 d A 
 
 And equating these values given in equa 
 tions 68 and 69, 
 
 ua 
 
 Substituting for u its equivalent value as 
 given in equation 65, and dividing both 
 members of the resulting equation by 
 the common factor 2 P + d b a, there re 
 sults 
 
 <T A (6 b P f + 3 tfd a + 6 M - 6 T e - d Vd) 
 
 Solving this with respect to x b*, and ex 
 tracting the root, 
 
124 
 
 3 A- 
 
 9 T" 
 
 4- 
 
 (2 T r 
 6 a 
 
 
 2 a A - a 2 - 
 
 2 T 
 
 But this is capable of being yet further 
 reduced by dividing through by 
 
 9 T 
 
 3h-^p--2a 
 o a 
 
 2T 
 
 2 A- 
 
 c> 
 
 to the form 
 5= 
 
 P 
 
 70. 
 
 which is the expression for the breadth 
 
125 
 
 of the step. As to the meaning of the 
 letters it may once more be stated, that 
 P is the weight in pounds of A B D H F, 
 and d the density of the masonry. 
 The vertical height (F H), which we de 
 termine to give the step, is expressed 
 by a, that of the entire dam from the 
 top to the base of the step by A, and the 
 moment of the weight P, with respect to 
 the vertical F H forming the rise of the 
 step by M ; while by A, we mean, as in 
 all previous formulae, the greatest height 
 to which we can raise a vertical wall 
 without the pressure per unit of surface 
 on the base, becoming larger than the 
 limit R of pressure ; and by $, the ex 
 pression or the ratio in which the 
 
 density of the masonry exceeds that of 
 water. This value of 0, is safely taken 
 at . As to the height to be given to 
 the step, this is of course to be assumed 
 at pleasure, but the most pleasing effect 
 is produced when it is taken at six or 
 seven feet, for then, even in dams of one 
 hundred and sixty feet in height, con- 
 
126 
 
 structed of the heaviest stone, the breadth 
 of the step will rarely at any point be 
 materially greater than the rise. The 
 point on the outer face at which the 
 first step should begin, or in other words 
 the distance A B, in Fig. 19, is deter- 
 A 
 
127 
 
 mined, as in the other instances, by the 
 relation which the breadth on top bears 
 to the height. If the thickness t, across 
 the summit be assumed then 
 
 4 Z 4 2 
 
 8 ^6**.* 6 A 
 
 but if the height a be assumed the proper 
 thickness is to be had from the equation, 
 
 = a 4/ 6 A ^ 
 - 
 
 3 A - 4 
 
 When that point on the inner face is 
 reached, at which it becomes necessary 
 to begin stepping, the breadths l> and & , 
 of the outer and inner steps respectively, 
 may be had by substituting the value of 
 u, in equations 15 and 16, and from the 
 two resulting equations, finding by 
 elimination two expressions for b and V . 
 This calculation may, however, be avoid 
 ed, and considerable expense for cutting 
 facing stone saved, by making the inner 
 face vertical from top to bottom. Indeed 
 the matter of expense for dressing stone 
 is, perhaps, the most serious objection to 
 
128 
 
 the stepped profile, as it is necessary to 
 dress both faces of the step. 
 
 As regards the use of the formulae for 
 this form of profile, it is to be borne in 
 mind, that P includes the weight of the 
 water as well as the weight of the 
 masonry, so that in determining the 
 breadth of the fourth step, the weight 
 of the three columns of water resting, 
 one on the first, one on the second and 
 one on the third step, is to be added to 
 the pressure of the masonry. The press 
 ure of the water is readily obtained from 
 equation 1. 
 
 The principles that have now been es 
 tablished in connection w r ith the four 
 types of profiles treated of, are all that 
 are required to calculate the parts of any 
 profile that is ever likely to arise in 
 practice. They have, moreover, been 
 determined without regard to the length 
 of the dam, so that the structure will be 
 one of equal resistance, and withstand 
 the thrust of the water solely by its 
 own weight. There is, therefore, no 
 valid reason why a dam constructed with 
 
129 
 
 a profile of equal resistance should be 
 curved into the form of an arch, and 
 this holds good, whether it be high or 
 low, whether it obstructs a broad valley 
 or a narrow one. The only thing that 
 can be accomplished by curving a dam, 
 is to relieve it from severe strains, by 
 transmitting as large a part of the thrust 
 to the sides of the valley, but where the 
 profile is such that the dam is every 
 where equally strong, and equally capa 
 ble of resisting by its own weight the 
 severest strain it is ever subjected to, 
 there is surely nothing to be gained by 
 increasing its length in order to transmit 
 this thrust laterally to the sides of the 
 valley. It is true that in deep and nar 
 row valleys, some saving of material may 
 be affected by curving the dam, which 
 being thus relieved from a goodly por 
 tion of the thrust, may be diminished in 
 thickness. But in long dams, it is an 
 open question whether the saving thus 
 affected is not more than balanced by 
 the increased length. 
 
 One other matter which deserves the 
 
130 
 
 most careful attention, and which in 
 deed unless it is carefully attended to 
 will render the very best profile of no 
 account, it is the binding of the stones, 
 and the character of the inner filling. 
 As to the bond, it is undoubtedly the 
 wisest plan if the dam is to resist a great 
 pressure, to avoid laying the stones in 
 horizontal courses wherever such a thing 
 is practicable, and to place binders in 
 every possible direction. For assuredly, 
 if it is necessary for the stability of all 
 walls bearing a vertical load, that there 
 should be no continuous joints in the 
 direction of the pressure, it is just as 
 important that a dam should have no 
 continuous horizontal joints, because in 
 the case of such structures almost every 
 ounce of thrust they have to resist is 
 horizontal, and thus exactly coincides 
 with the joints. If the dam is curved, 
 then this matter of broken horizontal 
 joints is not of such vital importance, 
 because no layer can then slide until 
 some one of the stones has been crushed, 
 yet even here it cannot be too rigidly 
 
131 
 
 adhered to. By a strange inconsistency 
 on the part of engineers, we often see 
 this matter both regarded and disre 
 garded in the same dam. Many struc 
 tures of this class could be named, in 
 which the rock foundation is stepped 
 with the utmost care to preclude any 
 possibility of sliding where sliding is of 
 all places the least likely to occur, while 
 the courses from the foundation to the 
 top are laid with the most perfect kind 
 of horizontal joints. 
 
 The filling again must not be of too 
 different a character from the facing. 
 Where masonry consists of dressed stone 
 and rubble work, the amount of settling 
 is so different in each case that nothing 
 like a bond can be preserved. The 
 affect of such settling, we constantly see 
 illustrated in the most striking way in 
 canal locks. As is well known these are 
 generally cut stone facings with rubble 
 backing, but the latter settling more 
 than the former become detached from 
 the facings, when the water penetrating 
 between the two kinds of masonry, the 
 
132 
 
 cut stone facings fall with the first frost. 
 A good filling is that made of large 
 rough blocks of stone, set at regular in 
 tervals apart, (the distance increasing as 
 the top is approached) and the spaces 
 between and over them filled in with 
 beton of the first quality, a method, we 
 believe, lately adopted in the construction 
 of one of the Croton dams in this state. 
 But perhaps a yet better one is to replace 
 the beton by the French mixture known 
 as beton coigmt. Both of these fillings, 
 however, are good, as when well rammed, 
 they form a close connection with the 
 facing stones, and do away entirely with 
 joints of any kind. 
 
V Any booTc in this Catalogue sent free by matt on 
 receipt of price. 
 
 VALUABLE 
 
 SCIENTIFIC BOOKS, 
 
 PUBLISHED BY 
 
 D. VAN NOSTRAND, 
 
 23 MURRAY STREET AND 27 WARREN STREET, 
 NEW YORK. 
 
 FRANCIS. Lowell Hydraulic Experiments, being a 
 selection from Experiments on Hydraulic Motors, on 
 the Flow of Water over Weirs, in Open Canals of 
 Uniform Rectangular Section, and through submerg 
 ed Orifices and diverging Tubes. Made at Lowell, 
 Massachusetts. By James B. Francis, C. E. 2(1 
 edition, revised and enlarged, with many new experi 
 ments, and illustrated with twenty-three copperplate 
 engravings, i vol. 4to, cloth $15 oo 
 
 ROEBLING (J. A.) Long and Short Span Railway 
 Bridges. By John A. Roebling, C. E. Illustrated 
 with large copperplate engravings of plans and views. 
 Imperial folio, cloth 25 oo 
 
 CLARKE (T. C.) Description of the Iron Railway 
 Bridge over the Mississippi River, at Quincy, Illi 
 nois. Thomas Curtis Clarke, Chief Engineer. 
 Illustrated with 21 lithographed plans, i vol. 410, 
 cloth 7 50 
 
 TUNNER (P.) A Treatise on Roll-Turning for the 
 Manufacture of Iron. By Peter Tunner. Trans 
 lated and adapted by John B. Pearse, of the Penn- 
 
 I 
 
D. VAN NOSTKAND S PUBLICATIONS 
 
 ylvania Steel Works, with numerous engravings 
 
 wood cuts and folio atlas of plates. $ic oo 
 
 ISHERWOOD (B. F.) Engineering Precedents for 
 Steam Machinery. Arranged in the most practical 
 and useful manner for Engineers. By B. F. Isher- 
 \yood, Civil Engineer, U. S. Navy. With Illustra 
 tions. Two volumes in one. 8vo, cloth $2 50 
 
 GILLMORE (Gen. Q. A,) Practical Treatise on the 
 Construction or Roads, Streets, and Pavements. By 
 Q. A. Gillmore, Lt.-Col. U. ^ Corps of Engineers, 
 Brevet Major-Gen. U. S- Army, with 70 illustra 
 tions. i2mo, cloth 2 oo 
 
 1 Report on Strength of the Building Stones in the 
 
 United States, etc. 8vo, illustrated, cloth i 50 
 
 CAM PIN on the Construction of Iron Roofs. Ey 
 
 Francis Campin. 8vo, with plates, cloth 2 oo 
 
 COLLINS. The Private Book of Useful Alloys and 
 Memoranda for Goldsmiths, Jewellers, &c- By 
 James E. Collins. i8mo, cloth 75 
 
 CIPHER AND SECRET LETTER AND TELE 
 GRAPHIC CODE, with Hogg s Improvements. 
 The most perfect secret code ever invented or dis 
 covered, impossible to read without the key. By 
 C. S. Larrabee. i8mo, cloth i oo 
 
 COLBURN. The Gas Works of London. By Zerah 
 
 Colburn, C. E. i vol 12010, boards 60 
 
 CRAIG (B. F.) Weights and Measures. An account 
 of the Decimal System, with Tables of Conversion 
 for Commercial and Scientific Uses. By B. F. Craig, 
 M.D. i vol. square 32mo, limp cloth 5 
 
 NUGENT. Treatise on Optics; or, Light and Sight, 
 theoretically and practically treated ; with the appli 
 cation to Fine Art and Industrial Pursuits. By E. 
 Nugent. With one hundred and three illustrations. 
 i2mo, cloth 2 oo 
 
 FREE HAND DRAWING- A Gnide to Ornament* 
 al Figure and Landscape Drawing. By an Art Stu 
 dent. i8mo, boards, 5 
 
 2 
 
D VAN NO-STRAND S PUBLICATIONS. 
 
 HOWARD. Earthwork Mensuration on the Basis of 
 the Prismoidal Formulae. Containing simple and la 
 bor-saving method of obtaining Prismoidal contents 
 directly from End Areas. Illustrated by Examples, 
 and accompanied by Plain Rules for Practical Uses. 
 By Conway R. Howard, C. E., Richmond, Va. Il 
 lustrated, 8 vo, cloth i 50 
 
 GRUNER. The Manufacture of Steel. By M. L. 
 Gruner. Translated from the French, by Lenox 
 Smith, with an appendix on the Bessamer process in 
 the United States, by the translator. Illustrated by 
 Lithographed drawings and wood cuts. 8vo, cloth.. 3 50 
 
 AUCHINCLOSS. Link and Valve Motions Simplified. 
 Illustrated with 37 wood-cuts, and 21 lithographic 
 plates, together with a Travel Scale, and numerous 
 useful Tables. By W. S. Auchincloss. 8vo, cloth. . 3 oo 
 
 VAN BUREN. Investigations of Formulas, for the 
 strength of the Iron parts of Steam Machinery. By 
 J. D. Van Buren, Jr., C. E. Illustrated, 8vo, cloth. 2 oo 
 
 JOYNSON. Designing and Construction of Machine 
 
 Gearing. Illustrated, 8vo, cloth 2 oo 
 
 GILLMOKE. Coignet Beton and other Artificial Stone. 
 By Q. A. Gillmore, Major U. S. Corps Engineers. 
 9 plates, views, &c. 8vo, cloth 250 
 
 SAELTZER. Treatise on Acoustics in connection with 
 Ventilation. By Alexander Saeltzer, Architect. 
 
 ramO} cloth 2 oo 
 
 BUT 1 , tiR ( W. F. ) Ventilation of Buildings. By W. F. 
 
 Butler. With illustrations. i8mo, boards 50 
 
 DICTIONARY of Manufactures, Mining, Machinery, 
 and the Industrial Arts. By George Dodd. i2ino, 
 cloth 2 oc 
 
 BOW. A Treatisa on Bracing, with its application to 
 Bridges and other Structures of Wood or Iron. By 
 Robert Henry Bow, C. E. 156 illustrations, 8vo, 
 doth j 50 
 
 BARBA (J.) The Use of Steel for Constructive Pur 
 poses; Method of Working, Applying, and Testing 
 Plates and Brass. With a Preface by A. L. Holley, 
 C. E. i-zmo, cloth i 50 
 
D. VAN NOSTBAND S PUBLICATIONS. 
 
 DIEDRICH. The Theory of Strains, a Compendium 
 for the calculation and construction of Bridges, Roofs, 
 and Cranes, with the application of Trigonometrical 
 Notes, containing the most comprehensive informa 
 tion in regard to the Resulting strains for a perman 
 ent Load, as also for a combined (Permanent and 
 Rolling) Load. In two sections, adadted to the re 
 quirements of the present time. By John Diedrich, 
 C. E. Illustrated by numerous plates and diagrams. 
 Svo, cloth t 5 QQ 
 
 WILLIAMSON (R. S.) On the use of the Barometer on 
 Surveys and Reconnoissances. Part I. Meteorology 
 in its Connection with Hypsometry. Part II. Baro 
 metric Hypsometry. By R. S. Wiliamson, Bvt 
 Lieut. -Col. U. S. A., Major Corps of Engineers. 
 With Illustrative Tables and Engravings. Paper 
 No. 15, Professional Papers, Corps of Engineers. 
 i vol. 4to, cloth... > 15 oo 
 
 POOK (S. M.) Method of Comparing the Lines and 
 Draughting Vessels Propelled by Sail or Steam. 
 Including a chapter on Laying off on the Mould- 
 Loft Floor. By Samuel M. Pook, Naval Construc 
 tor, i vol. Svo, with illustrations, cloth 5 oo 
 
 ALEXANDER (J. H.) Universal Dictionary of 
 Weights and Measures, Ancient and Modern, re 
 duced to the standards of the United States of Ame 
 rica. By J. H. Alexander. New edition, enlarged, 
 i vol. Svo, cloth 3 50 
 
 WANKLYN. A Practical Treatise on the Examination 
 of Milk, and its Derivatives, Cream, Butter and 
 Cheese. By J. Alfred Wanklyn, M. R. C. 8., izmo. 
 cloth i oo 
 
 RICHARDS INDICATOR. A Treatise on the Rich 
 ?rds Steam Engine Indicator, with an Appendix by 
 <?. W. Bacon, M. E. i8mo, flexible, cloth x oo 
 
 PORTER (C. T.) A Treatise on the Richards Steam 
 Engine Indicator, and the Development and Applica 
 tion of Force in the Steam Engine. By Charles T. 
 Porter. Third edition, revised and enlarged. Svo, 
 
 illustrated, cloth 3 5 
 
 & 
 
D. VAN NOSTKAND S PUBLICATIONS. 
 
 POPE. Modern Practice of the Electric Telegraph. A 
 Hand Book for Electricians and operators. By Frank 
 L. Pope. Ninth edition, revised and enlarged, and 
 fully illustrated. 8vo, cloth $2 oo 
 
 " There is no other work of this kind in tha English language that con 
 tains in so small a compass so much practical information in the appli 
 cation of galvanic electricity to telegraphy. It should be in the hands of 
 everyone interested in telegraphy, or the use of Batteries for other pur 
 poses. 
 
 EASSIE (P. B.) Wood and its Uses. A Hand -Book 
 for the use of Contractors, Builders, Architects, En 
 gineers, and Timber Merchants. By P. B. Eassie. 
 Upwards of 250 illustrations. 8vo, cloth i 50 
 
 SABINE. History and Progress of the Electric Tele 
 graph, with descriptions of some of the apparatus. 
 By Robert Sabme, C. E, Second edition, with ad 
 ditions, i2mo, cloth i 35 
 
 BLAKE. Ceramic Art. A Report on Pottery, Porce 
 lain, Tiles, Terra Cotta and Brick. By W. P. Blake, 
 U. S. Commissioner, Vienna Exhibition, 1873. 8vo, 
 cloth 2 oo 
 
 BE NET. Electro-Ballistic Machines, and the Schultz 
 Chronoscope. By Lieut. -Col. S. V. Benet, Captain 
 of Ordnance, U. S. Army- Illustrated, second edi 
 tion, 4to, cloth o 3 oo 
 
 MICHAELIS. The Le Boulenge Chronograph, with 
 three Lithograph folding plates of illustrations. By- 
 Brevet Captain O. E. Michaelis, First Lieutenant 
 Ordnance Corps, U. S. Army, 410, cloth . . . . 3 oo 
 
 ENGINEERING FACTS AND FIGURES An 
 Annual Register of Progress in Mechanical Engineer 
 ing and Construction, for the years 1863, 64, 65, 66 
 67, 68. Fully illustrated, 6 vols. i8mo, cloth, $2.50 
 per vol., each volume sold separately 
 
 HAMILTON. Useful Information for Railway Men. 
 Compiled by W. G. Hamilton, Engineer. Sixth edi 
 tion, revised and enlarged, 562 pages Pocket form. 
 Morocco, gilt 2 oo 
 
 STUART (B.) How to Become a Successful Engineer. 
 Being Hints to Youths intending to adopt the Pro 
 fession. Sixth edition. i2mo, boards 50 
 
D. VAN NOSTRAND S PUBLICATIONS. 
 
 ELIOT AND STORER. A compendious Manual of 
 Qualitative Chemical Analysis. By Charles W. 
 tiiot and Frank H. Storer. Revised with the Co 
 operation of the authors. By William R. Nichols, 
 Professor of Chemistry in the Massachusetts insti 
 tute of Technology Illustrated, i2mo, cloth $i 50 
 
 RAMMELSBERG. Guide to a course of Quantitative 
 Chemical Analysis, especially of Minerals and Fur 
 nace Products. Illustrated by Examples By C. F. 
 Ramn?elsberg. Translated by J. Tovvler, M. D. 
 8 vo, cloth 2 23 
 
 DOUGLASS and PRESCOTT. Qualitative Chemical 
 Analysis. A Guide in the Practical Study of Chem 
 istry, and in the Work of Analysis. By S. H. Doug 
 lass and A. B. J rescott, of the University of Michi 
 gan. New edition. 8vo. In pre&8. 
 
 JACOB. On the Designing and Construction of Storage 
 Reservoirs, with Tables and Wood Cuts representing 
 Sections, &c., iSmo, boards 50 
 
 WATT S Dictionary of Chemistry. New and Revised 
 edition complete in 6 vols 8vo cloth, $62.00 Sup 
 plementary volume sold separately. Price, cloth. . . 9 oo 
 
 RANDALL. Quartz Operators Hand-Book. By P. M. 
 Randall. New edition, revised and enlarged, fully 
 illustrated. i2mo, cloth 200 
 
 SILVERSMITH. A Practical Hand-Book for Miners, 
 Metallurgists, and Assayers, comprising the most re 
 cent improvements in the disintegration amalgama 
 tion, smelting, and parting of the > recious ores, with 
 a comprehensive Digest of the Mining Laws Greatly 
 augmented, revised and corrected. By Juiius Sliver- 
 smith Fourth edition. Profusely illustrated. i2mo, 
 cloth 3 oo 
 
 THE USEFUL METALS AND THEIR ALLOYS, 
 including Mining Ventilation, Mining Jurisprudence, 
 and Metallurgic Chemistry employed in the conver 
 sion of Iron, Copper, Tin, Zinc, Antimony and Lead 
 ores, with their applications to the Industrial Arts. 
 By Scoffren, Truan, Clay, Oxland, Fairbairn, and 
 
 others. Fifth edition, half calf , . 3 7.* 
 
 JO 
 
D. VAN NOSTRAND S PUBLICATIONS. 
 
 JOYNSON. The Metals used in construction, lion, 
 Steel, Bessemer Metal, etc., etc. By F. H. Joynson. 
 Illustrated, 1 2mo, cloth $o 75 
 
 VON COTTA. Treatise on Ore Deposits. By Bern- 
 hard Von Cotta, Professor of Geology in the Royal 
 School of Mines, Freidberg, Saxony. Translated 
 from the second German edition, by Frederick 
 Prime, Jr., Mining Engineer, and revised by the au 
 thor, with numerous illustrations. 8vo, cloth 4 oc 
 
 GREENE. Graphical Method for the Analysis of Bridge 
 Trusses, extended to continuous Girders and Dra 1 
 Spans. By C. F. Greene, A. M., Prof, of Civil Eng: 
 
 . 
 
 Trusses, extended to continuous Girders and Draw 
 
 pans. By C. F. Greene, A. M., Prof, of Civil Engi 
 
 neering, University of Michigan. Illustrated by 3 
 
 folding plates, 8vo, cloth 
 BELL. Chemical Phenomena of Iron Smelting. An 
 experimental and practical examination ot the cir 
 cumstances which determine the capacity of the Blast 
 Furnace, The Temperature of the air, aiid the 
 proper condition of the Materials to be operated 
 upon. By 1. Lowthian Bell. 8vo, cloth ........... 600 
 
 ROGERS. The Geology of Pennsylvania. A Govern 
 ment survey, with a general view of the Geology of 
 the United States, Essays on the Coal Formation and 
 its Fossils, ai;d a description of the Coal Fields of 
 North America and Great Britain. By Henr\ Dar 
 win Rogers, late State Geologist of Pennsylvania, 
 Splendidly illustrated with Plates and Engravings in 
 the text. 3 vols , 410, cloth with 1 ortfolio of Maps. 30 oo 
 
 BUKGH. Modern Marine Engineering, applied to 
 Paddl2 and Screw Propulsion Consisting of 36 
 colored plates, 259 Practical V\ cod tut Illustrations, 
 and 403 pa^es oi descriptive matter, the whole being 
 an exposition of the present practice of James 
 Watt & Co.. J & G Rennie, R. Napier & Sons, 
 and other celebrated firms, by N. P. Burgh, Engi 
 neer, thick 4to, vol., cloth, $25.00; halfmor ....... 30 oo 
 
 CHURCH. Notes of a Metallurgical Journey in Europe. 
 
 By J. A. Church, Engineer of Mines, 8vo ; cloth ..... 2 oo 
 
 11 
 
D. VAN NOSTKAND S PUBLICATIONS. 
 
 BOURNE. Treatise on the Steam Engine in its various 
 applications to Mines, Mills, Steam Navigation, 
 Railways, and Agriculture, with the theoretical in 
 vestigations respecting the Motive Power of Heat, 
 and the proper proportions of steam engines. Elabo 
 rate tables of the right dimensions of every part, and 
 Practical Instructions for the manufacture and man 
 agement of every species of Engine in actual use. 
 By John Bourne, being the ninth edition of " A 
 Treatise on the Steam Engine," by the " Artizan 
 Clubo" Illustrated by 38 plates and 546 wood cuts. 
 4to, cloth ,,..$15 oc 
 
 STUART. The Naval Dry Docks of the United 
 Spates. By Charles B. Stuart late Engineer-in-Chief 
 of the U. S. Navy. Illustrated with 24 engravings 
 
 on steel. Fourth edition, cloth 6 oo 
 
 ATKINSON. Practical Treatises on the Gases met 
 
 with in Coal Mines. i8mo, boards 50 
 
 FOSTER. Submarine Blasting in Boston Harbor, 
 Massachusetts. Removal of Tower and Corwin 
 Rocks. By J. G. Foster, Lieut -Col. of Engineers, 
 U. S. Army. Illustrated with seven plates, 4to, 
 cloth 3 50 
 
 BARNES Submarine Warfare, offensive and defensive, 
 including a discussion of the offensive Torpedo Sys 
 tem, its effects upon Iron Clad Ship Systems and in 
 fluence upon future naval wars. By Lieut. -Com 
 mander J. S. Barnes, U. S. N., with twenty litho 
 graphic plates and many wood cuts. 8vo, cloth.. . . 5 oc 
 
 HOLLEY. A Treatise on Ordnance and Armor, em 
 bracing descriptions, discussions, and professional 
 opinions concerning the materials, fabrication, re 
 quirements, capabilities, and endurance of European 
 and American Guns, for Kaval, Sea Coast, and Iron 
 Clad Warfare, and their Rifling, Projectiles, and 
 Br eech- Loading ; also, results of experiments against 
 armor, from official records, with an appendix refer 
 ring to Gun Cotton, Hooped Guns, etc., etc. By 
 Alexander L. Holley, B. P., 948 pages, 493 engrav 
 ings, and 147 Tables of Results, etc., 8vo, half roan. 10 oo 
 12 
 
THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW 
 
 AN INITIAL FINE OF 25 CENTS 
 
 WILL BE ASSESSED FOR FAILURE TO RETURN 
 THIS BOOK ON THE DATE DUE. THE PENALTY 
 WILL INCREASE TO SO CENTS ON THE FOURTH 
 DAY AND TO $1.OO ON THE SEVENTH DAY 
 OVERDUE. 
 
 MM F7 B35 
 *> 5 1947 
 
 
 nt\ r\or 59 M C 
 
 
 2\3 VJ"^ Ja 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J. A. School bred and R. E. Day- 
 
THE VAN NOSTRAND SCIENCE SERIES. 
 
 No 72. 
 
 No. 73. 
 
 No. 74. 
 No. 75. 
 
 No. 76. 
 No. 77. 
 No . 78. 
 No. 79. 
 No. 80. 
 No. 81. 
 
 No. 82. 
 No. 83. 
 No. 84. 
 
 No. 85. 
 No. 86. 
 
 No. 87.- 
 
 -TOPOGRAPICAL SURVEYING. By Geo. J. Specht, 
 Prof. A. S. Hardy, John B. McMaster and H. F. 
 Walling. 
 
 SYMBOLIC ALGEBRA ; or The Algebra of Algebraic 
 Numbers. By Prof . W. Cain. 
 
 TESTING MACHINES; their History, Construction 
 and Use. By Arthur V. Abbott. 
 
 RECENT PROGRESS IN DYNAMO- ELECTRIC MA 
 CHINES. Being a Supplement to Dynamo-Elec 
 tric Machinery. By Prof. Silvanus P. Thompson. 
 
 MODERN REPRODUCTIVE GRAPHIC PRO 
 CESSES. By Lt. Jas. S. Pettit, U.S.A. 
 
 STADIA SURVEYING. The Theory of Stadia 
 Measurements. By Arthur Winslow. 
 
 -THE STEAM ENGINE INDICATOR, and its Use. 
 By W. B. Le Van. 
 
 -THE FIGURE OF THE EARTH. By Frank C. 
 Roberts, C. E. 
 
 -HEALTHY FOUNDATIONS FOR HOUSES. 
 Glenn Brown. 
 
 -WATER METERS : Comparative Tests of Acci 
 
 Delivery, etc. Distinctive Features of the 
 
 ington, Kennedy, Siemens and Hesse Met 
 
 Ross E. Browne. 
 -THE PRESERVATION OF TIMBER, , 
 
 of Antiseptics. By Samuel Bagster Bom 
 -MECHANICAL INTEGRATORS. By Pi 
 
 S. H. Shaw, C. E. 
 -FLOW OF WATER IN OPEN CHAN] 
 
 CONDUITS, SEWERS, &c. ; wit 
 
 P. J. Flynn, C. E. 
 -THE LUMINIFEROUS 
 
 son Wood . 
 -HANDBOOK OF MINER, 
 
 and Description of Mil 
 
 States. - 
 -TREATISE 
 
 STRUC" 
 
 No. 88.- 
 No. 8 r 
 ^ 
 No. 91, 
 
 .-B r 
 
 cription of MinX /* i * 
 
 By -Prof. J. C/ t A C M 
 
 ^HET J V V / I 
 
 
 S 
 
 \ \ 
 
 \ 
 
 JE 
 A&S for 
 
 jture, Prop- 
 kisser, U.S.A. 
 
 the Gyroscope. 
 , ) G. Barnard. 
 
 LEVELING : Barometric, Trigonometric and Spirit. 
 By Prof. L 0. Baker. 
 
Tl UF7V. 
 
 Q 
 o.ilM 
 
 P: XV _ BASIS 01 
 
 I -of. T. H, HUXL^ ;, _L.-b. F.E.S. "Wirl 
 io ~ /a Profess m Y\Ie College. 1 
 P 3.: * "ers. J rice -:3nts. 
 
 - -1-L-i.E cOR T w . . _ ..TCN 0^ V* 
 T :> .x iSICAL FORO E; I By Prof. Gteo* OT 
 I.D., of Yale College. 86 p^. Paper Covci 
 
 TITAS BEGABP^ PEOTOPJ 
 to Prof. Huxley s Physical Basif 
 HUTCHISON ^TIRLIN^, F.7^. ^,S. pp. 72. I 
 
 TV. ON THE F~ XX HESIS OF F/ { 
 Physical and Af#U,j y$i\al. By Proi 
 COPE, 12mo., 72 pp. >apsr Covers, pr 
 
 V. SCIENTIFJC ADDKESSES: j. 
 thoda and Tendvncif.s of Physical /> 
 Baz* and D> . 8. C<r>, the tictenUf 
 nation. By Prof. JO^N TYNDALL, F.K. 
 pp. Paper Covers. Price 25 cents. Flex. ( 
 
 NO. VLNATURAL SELECTIOX A 
 TO MAN. By ALFRED RITSSELT, WAI_ 
 pamphlet treats (1; of the Development 
 Haces under the^aw of selection ; (2) thelii 
 ural Selection a? applied" to man. 54 pp. Pi 
 
 NO.VlI.-SPECTKTJM ANALYSIS, 
 ture i b T, Prof a. Roscoe, Muggins, and Loci 
 88 pp. Paper Covers. Pri 
 - T I -TFil SulT. A sketch of 
 ""-V "/ jT^i^fi as regards this hodl 
 
 : . - " " . " recent discoveries and 
 o ar% ,f. f; A, YoiTNG, Ph.I 
 
 in ">rv. v ^ ,... Pap; 
 
 N(. , ...\. ?. ^H/ i 
 
 A. l-l. * A- - 7r ., n- Bte>e^8 j 
 prof car. .11 e i n; J vtu.e on ti 
 
 netism. ii x > t , >vere. J ri 
 
 ihle Cloth, 50 - 
 
 N( 
 EAR. 
 
 
 J :-:c Jo ceif 
 
 78TJJEIES OF THE YOll 
 ^ror. O. N. .Roop, Columbia Col] 
 
 York, f One -. f tha "Tiost interesting Jectorea 
 f ver deliverer!.. Orv.-il discoveries, brilliaJ 
 ments. Beaut luliy illu. S8 pp. Paper Co vei