LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class s$a^ HYDRAULICS OF RIVERS, WEIRS AND SLUICES THE DERIVATION OF NEW AND MORE ACCURATE FORMULAE, FOR DISCHARGE THROUGH RIVERS AND CANALS OBSTRUCTED BY WEIRS, SLUICES, ETC., ACCORDING TO THE PRINCIPLES OF GUSTAV RITTER VON WEX BY DAVID A. MOLITOR, C.E. MEM. AM. Soc. OF C.E.; MEM. NATIONAL GEOGRAPHIC Soc.; MEM. Ass. FOR ADV. OF SCIENCE; MEM. Soc. PROMOTION ENG. EDUCATION; DESIGNING ENGINEER, ISTHMIAN CANAL COMMISSION FfRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS LONDON: CHAPMAN & HALL, LIMITED 1908 COPYRIGHT, 1908, BY DAVID A. MOLITOR 5tanbope ipress F. H. GILSON COMPANY BOSTON. U.S.A. To THE MEMORY OF GUSTAV RITTER VON WEX THIS WORK IS DEDICATED Hofrat Gustav Ritter von Wex was aulic counsellor and chief director of the Danube River regulation and improvement at Vienna knight of several imperial orders ; and member of many scientific societies. He was born 1811 and died Sept. 26, 1892, in Ischl, Austria. iii 1 96462 PREFACE IT seems strange that the earnest efforts of so high a technical authority as Hofrat von Wex should have failed to interest hydrau- lic engineers the world over. A careful search through the lead- ing hydraulic literature, with one exception, did not reveal a single comment regarding either the man or his work. Prof. I. P. Church says in his "Mechanics of Engineering," 1906 ed., foot of page 688, " Herr Ritter von Wex in his Hydrodynamik derives formulae for weirs, in the establishing of which some rather peculiar views in the mechanics of fluids are advanced." The unquestioned ability of Hofrat von Wex and his very exten- sive practical experience along the lines he has treated, place his work in the front rank of technical achievement in the specialty of river hydraulics. His views and theories on this subject, while radically different from those of his time, are most rational and sound, and merit the respect and approval of all practical hydraulic engineers. With a thorough conviction of the high value of the Wex theories, the author has ventured to place them before his profession in a form which he hopes will prove most practical and acceptable. The general status of our knowledge respecting hydraulics generally, and particularly the subject of weirs, is very unsatis- factory, to say the least. Hence any progress, if it be real progress, means a radical departure from former conceptions. It is the rule that new and progressive ideas are received with more or less suspicion, which Professor Church expresses in the word "peculiar." As such ideas become more generally known they receive a more charitable reception and are soon tolerated. When all opposition fails they are crowned by final acceptance. May the present work advance to this final state. vi PREFACE At this age of water power development, this little book should enjoy a hearty welcome. It was prepared with the utmost care, and while theoretical in its nature, it was written for the practical man. Simplicity and clearness were the first requirements, fol- lowed by a logical and practical arrangement in the presentation of the subject. It was not deemed advisable to enlarge the work by the addi- tion of mathematical tables as is usually done. On the contrary, such tables are of little value when dealing with general problems and in the author's opinion the most useful aid to the solution of the formulae here given is a copy of Barlow's tables of squares, cubes, square roots and cube roots, and Zimmermann's Rechen- tafel, being a multiplication table of all numbers from i to 1000 by all numbers from i to 100. These universal tools should occupy a prominent place on every engineer's book shelf, and nothing better can be proposed here as labor saving devices. Attention is called to the valuable information collected in Appendix A, which constitutes a most complete exposition of all known older formulae for overfalls. This in itself is the best argument which can be presented in defense of the new formulae to which this work is devoted. Appendix B contains the solution of a novel problem in Hy- draulics, which, so far as known, has never heretofore been solved in any satisfactory manner. The solution there given is theoretically correct, and probably more accurate than the knowable accuracy of the empiric coefficients would really justify. In Appendix C, all the new formulae are arranged in tabulations for ready reference, thus avoiding loss of time in picking out special cases from the text. It is believed that this offers a very attractive summary of the most useful contents of the book. The author here wishes to acknowledge his indebtedness to Herr Wilhelm Engelmann, of Leipzig, Germany, for many favors extended and advice given prior to undertaking the present work. Also to Professor Gardner S. Williams and Mr. Allen Hazen, members Am. Soc. C. E., for their kind permission to use the PREFACE vii tabulations relating to effect of weir crests as published in their "Hydraulic Tables," pp. 71-75. In conclusion he wishes to express his obligation and thanks to Mr. Alex. Ilich Wolkowyski, C. E., Ass't Eng'r, Isthmian Canal Commission, for valuable assistance rendered in the preparation of this work, also to Messrs. John Wiley and Sons for the most excellent manner in which they have accomplished the publication. DAVID A. MOLITOR. WASHINGTON, D.C., December 5, 1907, CONTENTS PAGE INTRODUCTION . i CHAPTER I. FUNDAMENTAL EQUATIONS . . . Flow through Lateral Orifices. CHAPTER II. COMPLETE OVERFALL WEIRS u A. Derivation of new formulae for the discharge over complete overfall weirs, built normally to the direction of the current. B. Derivation of new formulae for the discharge over complete overfall weirs, built obliquely to a channel or represented in plan by a curved or broken line. CHAPTER III. INCOMPLETE OVERFALL WEIRS 28 Derivation of new formulae for discharge over incomplete overfalls or submerged weirs, and through contracted river channels. CHAPTER IV. SLUICE WEIRS AND SLUICE GATES 36 Derivation of new formulae for discharge over sluice weirs and through sluice gates for regulating works. Three cases. CHAPTER V. BACKWATER CONDITIONS 46 Discussion and formulae for backwater height and distance; also computations of the dimensions of weirs for obstructed flow when the quantity of flow and backwater height are fixed. Illustrated by problems. x CONTENTS CHAPTER VI. PAGE FLOW IN RIVERS AND CANALS 52 Derivation of formulae for discharge from rivers or lakes into water power canals or flumes, i. General Discussion. 2. Proposed general solution. 3. When all the available water is to be diverted. 4. When only a portion of the avail- able water is to be diverted, and the remainder to be discharged over a weir built normally to the river. 5. The same as the previous case for a weir built diagonally to the river. * CHAPTER VII. EMPIRIC COEFFICIENTS 75 i. Introductory. 2. Complete overfalls ; a, weirs normal to the channel and no wing walls, and b, weirs normal to the channel but contracted on the ends by wing walls. 3. Incomplete overfalls. 4. Sluice weirs and gates. 5. End contractions. 6. Weir crests. APPENDIX A. A COLLECTION OF WEIR FORMULAE PROPOSED BY DIFFERENT AUTHORS. 1. COMPLETE OVERFALLS 102 2. INCOMPLETE OVERFALLS 109 APPENDIX B. ON THE FLOW OVER A FLIGHT OF PANAMA CANAL LOCKS. A NOVEL HYDRAULIC PROBLEM. APPENDIX C. A TABULATION OF THE NEW FORMULAE, arranged for ready reference. DEFINITIONS OF TERMS USED THROUGH- OUT THIS WORK A = discharge area in square feet. Q = discharge quantity in cubic feet per second. v = mean velocity of approach in feet per second. V = mean velocity of discharge in feet per second. S = the unit pressure at any point of the discharge area. iSj and S 2 are special values of S. g = acceleration due to gravity, in feet per second = 32.09 at the equator, and 32.26 at the pole, both at sea level. 7 = the weight of a cubic foot of water, usually taken as 62.5 pounds. n = Dubuat's coefficient = 0.67. // = empiric coefficient for discharge into free air. / t = empiric coefficient for submerged discharge. T = total depth of approach channel in feet. T l = total depth of discharge channel in feet. H = depth of flow of approach over crest of weir. H l = depth of flow of discharge over crest of weir. H 2 = H it^ = difference between the approach and dis- charge surfaces. k = depth of water on upstream side of weir and below weir crest. B = width of approach channel. b = width of the discharge section, or length of weir crest. b' = width of a diversion channel. d = depth at entrance to a diversion channel. xii TERMS USED THROUGHOUT THIS WORK -^ = angle of inclination of the upstream face of a weir dam with the horizontal. $ = angle of inclination of any wing dam with the side of the channel. = angle of inclination of a diversion channel with the main channel. s = surface slope = fall divided by length, both in feet. ^ r = mean hydraulic radius = . w w = wetted perimeter. D = original depth of any river, previous to placing any obstruc- tion in its course. Z = total backwater height, measured above the original or natural surface slope. L = total backwater distance, measured from the crest of the weir or dam obstruction. / and z = co-ordinates of backwater surface referred to the original slope as the /-axis and origin in the vertical through the crest of the weir. C = either Bazin's or Kutter's coefficient for the Chezy formula. OF THE UNIVERSITY OF HYDRAULICS OF RIVERS, WEIRS AND SLUICES INTRODUCTION IN treating a subject of this nature it is very important that the language used should be clear and specific. To accomplish this end a few of the most important terms must be accurately defined so that there may be no doubt as to the particular meaning implied by them. Much confusion is often created by a promiscuous use of technical terms, especially when these have received a variety of definitions by different authors. The following definitions will be adhered to throughout the present work. Hydrostatic pressure is the pressure exerted by the weight of a column of water and acts with equal intensity in all directions. It implies pressure due to water in a state of rest. Hydrodynamic pressure is that produced by a stream or jet of water impinging on a surface, and may be less than, equal to or greater than the hydrostatic pressure. It is a function of velocity only. It is always expressed as a static head v*/2g. Hydraulic pressure is the resultant water pressure on any sur- face caused by any possible combination of both hydrostatic and hydrodynamic pressures. Velocity in all of the relations here considered will always be regarded as the mean velocity, being equal for all points of the same section normal to the direction of flow. Hence the quantity of discharge passing a given section A per second of time will be Q = Av. Velocity of approach, v, is the mean velocity at such a section of an approach channel where the effect of an overfall is still too small to produce an acceleration. 2 INTRODUCTION Velocity of discharge, V, is the mean velocity at such a section of a discharge channel where the flow is again uniform or normal after having been accelerated by passing over an overfall. An overfall is a vertical drop in the bottom of a channel and may be complete or incomplete accordingly as it is or is not sub- merged below the lower water level or lower pool. A complete overfall weir is a weir the crest of which is above the level of the lower pool. An incomplete overfall weir, usually called a submerged weir, is such a weir the crest of which is below the level of the lower pool. A sluice weir is a weir the flow over which is partially obstructed by a gate, thus producing a condition of flow resembling that through a lateral orifice in the side of a vessel. The ends of a weir are usually the side walls of the channel. The end walls will be supposed to be vertical while the front and back of the weir or dam may have various dimensions. The crest of the dam may have a variety of forms which are given special consideration in Chapter VII. Contraction is a term used to designate the diminution in the flow area just beyond the discharge area when the discharge proceeds into free air and is not confined after leaving the discharge section. Complete contraction is contraction around the entire periphery of the discharge section. Partial contraction is contraction on only one, two or three sides of the discharge section. End contraction is contraction on the two ends of a weir. This occurs only when the side walls of the channel are suddenly ended at the vertical through the weir crest. All the following formulae are derived without reference to end contractions or shape of weir crest, and the manner of dealing with these special features is discussed in Chapter VII. In the several chapters, I to VI, the new formulae for a great variety of weirs and sluices were derived and these were finally tabulated in usable form in Appendix C. Chapter VII is devoted INTRODUCTION 3 to a determination of empiric coefficients for the new formulae and these results are likewise included in the tabulations of Appen- dix C. The inconsistencies and irrational constitution of the older for- mulae are discussed at some length in Appendix A, to which special attention is called here as furnishing the real justification for the new formulae. It was not deemed desirable to interrupt the continuity of the argument by discussing old formulae at every opportunity, and hence these were collected into an Appendix. In this way the reader will better understand the criticisms after having become familiar with the subject matter of the book. CHAPTER I. FUNDAMENTAL EQUATIONS. Flow through Lateral Orifices. THE fact that the fundamental equations for flow through lateral orifices have been applied by many hydraulicians, without proper modification, to determine the flow through canals and rivers, contracted by the introduction of weirs or sluices, makes it desir- able to review the derivation of these equations, and to show that such application is not justifiable. The fundamental equations for the flow of water through lateral orifices in the vertical sides of a large reservoir, in which the water is perfectly quiet, and is maintained at a constant level, will be derived in the following. These equations are generally accepted and will be used in the derivation of the new weir formulae. In Fig. i, A BCD represents the vertical wall of the reservoir with the orifice EFGH, through which the water flows freely into the air. According to the principle of hydrostatics, the pressures at any _ppints, J or G in this orifice, are equal to the columns of water EJ and EG, respectively, and it has been experimentally FUNDAMENTAL EQUATIONS 5 established that the velocity with which the water flows through small openings at / or G, is_equal to the velocity of a body falling in air through the heights EJ and EG, respectively. Hence, the velocity at the point / is JK = V 2 gEJ 3 and the velocity at the point G, is GL =V2 gEG, in which g = acceleration of gravity per second of time. Using notation indicated in Fig. i, and calling x the ordinate of a filament of water ad) of height dx, and flowing with a velocity y, then the differential equation representing the quantity of water in this filament is dQ = b .dx . V2 gx. To obtain the total quantity of water flowing through the orifice per second, this equation must be integrated between the limits x = o and x = H, giving f*B _ _ f*H Q = I bdx \/2 gx =b \/2 g I x* dx, J J or Q = f x*b V~g + C. For x = o, Q becomes zero, hence the constant C is zero. Therefore, for x = H, the quantity of flow through the orifice b . H, per second, becomes Q - W? 1 V~g - } bHV^H, in which b . H equals the area of orifice, and f \/2 gH = mean velocity. Since the values of y vary as the square root of the corresponding values of x, the curve EKL is a parabola. However, experiments have shown that the friction existing between the particles of water among themselves and at the edges of the orifice, causes a retardation in this theoretical velocity. Hence, the actual quantity of flow will always be less than the above, and the true equation must contain an empiric coefficient to correct for the combined effect of friction and contraction. The 6 HYDRAULICS equation of flow through a rectangular orifice in the vertical side of a reservoir in which the water is perfectly quiet and retained at a constant level may then be written thus: Q = $f*bHV 2 gH = %f*bV 2 gH* . . . (i) Should the orifice EFHG be partly closed by a sluice gate, EFRJ, then the quantity of flow through the remaining orifice, JRHGj will be the total quantity as from Eq. (i), less the quantity which is prevented from flowing out by the gate. This quantity is represented by the body JRHGKPNL, Fig. i, and calling the height EJ = H lt equation (i) when applied to this case becomes Q=it*bV^~g(H* -H*) .... (2) Should the sluice gate be lowered to a line ik, leaving only an orifice of breadth b and height a, which latter is small in comparison to the height H, the water may be assumed to flow through such an orifice under an average pressure ( H J and the quantity of flow per second will become Q=l^ab\j2g\H-^ (3) These three fundamental equations have been derived by most hydraulicians, and their correctness having been established by many accurate experiments, they may be generally accepted. However, it must not be forgotten that these equations apply only to cases in which the orifice is very small compared to the size of the reservoir, so that the water in the latter may retain its height and remain unagitated. Before passing on to the derivation of the weir formulae, it will be advisable to make an exposition of facts which have been gen- erally neglected by other authors in treating of submerged weirs. Let Fig. 2 represent the longitudinal section of a river in which isjplaced a submerged weir LM having its crest below the surface FG by an amount H l and damming the water to a height H FUNDAMENTAL EQUATIONS 7 above the crest of the weir. The quantity of water Q, passing over this weir per second, is regarded as being made up of two parts, that flowing into the air through the upper portion EK of the entire section EL, which may be found from Eq. (i); and that flowing Fig. 2 through the submerged area KL under the uniform pressure head H H r This is the basis of the argument generally applied in deriving weir formulae and leads to very erroneous results, as will presently be shown. Regarding the first of these increments of quantity, the following criticism is offered. As the channel leading up to a weir is always of limited dimensions, and the weir and other possible obstructions, such as wing dams, etc., must affect the conditions of flow through the upper part EK of the section EL, Fig. 2; and since the theo- retical Eq. (i), when applied to this flow, does not involve in any way the dimensions of the reservoir, or the channel dimensions in the case of weirs, it follows that the theoretical equation cannot be adapted to the weir condition by the mere introduction of an empiric coefficient. It is also assumed that the velocity of approach exerts an hydraulic pressure only on the area of flow, while it is positively known that this velocity likewise affects the surface of the weir, and all other surfaces of the channel approaching the weir. These pressures are deflected into the area of flow in a manner dependent on the shape of weir and other parts of the channel. In regard to the second increment of flow, it is assumed that the lower pool exerts a back pressure on the part section KL, the 8 HYDRAULICS same as if the water was not in motion. To prove that this is not in accordance with the existing conditions, the following experiments are cited. Referring to Fig. 3, in which water flows freely through the irregular vessel ABCD, and the flow is supplied from a large reservoir, so that the level MM t remains con- jWt stant, then the resultant hydraulic pressure at any point of the vessel is equal to the hydro- static pressure at that point, less the velocity height of the water flowing past this same point. (See Ruehlmann, Hydromechanik, pp. 211 and 214.) If in a contracted section EF, the water flows with a very high velocity v l} such that Fi S- 3 the velocity height becomes greater than the hydrostatic head h at a certain point F, then the resulting hydraulic pressure on the above hypothesis becomes negative and equal to (h l V indicating a suction. Now, if a glass tubeFJ be connected with the vessel at JP, then the water contained in a vessel KL will be drawn up into the tube by an amount ab = ( - 1 h 1 } . Also, since the pressure at any section EF must \2 g I be equal in all directions, it is apparent that the same suction would be produced if the opening of the tube were on the lower side at e. Hence, the statement may be made that for the conditions of flow just described, the water flowing past an orifice e in the lower side of a tube FJ, will produce a suction in this tube equal to g To determine the force of impactof water flowing with velocity v through a flume against a disc CD, Fig. 4 (which question has not yet been satisfactorily solved), Dubuat made numerous experi- ments and found that the hydraulic pressure on the back face of FUNDAMENTAL EQUATIONS 9 the disc CD is equal to the hydrostatic head h on the surface, less v 2 0.67 , proving that in this case there is also a suction on the Fig. 4 V 2 surface CD equal to the velocity height o. 67 (See Ruehlmann, 2 g Hydromechanik, p. 596.) The suction in this latter case is, however, less than for the closed vessel, which is supposed to be due to the difference between a closed vessel and an open channel, and also that the eddies pro- duced behind the disc exert a certain impact opposing the suction and thereby diminishing the latter. Darcy found,* by experimenting with a Pitot tube, that when the orifice was pointed perpendicular to the direction of flow, the hydrostatic column in the tube was lowered by an amount v 2 h 2 = 0.678 -below the surface of the water, this being the result o of suction produced by the water in flowing past the orifice. When the orifice was pointed with the current this suction amounted to v 2 only h 3 = 0.434 . Since the suction in the latter case should undoubtedly be greater than in the former, it seems reasonable to suppose that in the second case the filaments of water are deflected by the tube in such manner as to diminish the suction effect on the orifice when the tube is pointed with the current. The fact that the value of the amount of suction found by Darcy on the Pitot tube is less than was obtained for the flow through the vessel in Fig. 3, and less than was found by Dubuat for the disc, * See Ruehlmann, Hydromechanik, p. 383. 10 HYDRAULICS Fig. 4, is probably due to the friction, cohesion and capillarity to be expected by the flow along the small conical pressure tube used. Until better experiments on these lines shall have become avail- able, the results of Dubuat as o. 67 may be safely accepted. Since the above experiments make it apparent that there is a suction on the lower surface of all incomplete weirs, submerged weirs, and sluice gates, which suction diminishes, the hydrostatic counterpressure of the part section KL, Fig. 2, by an amount i? 0.67 , it follows that a larger quantity of flow is permitted through the submerged section, than is assumed under the supposi- tion that the water in the lower pool is perfectly quiet and exerts an hydrostatic pressure on the submerged section over its entire height KL. Hence, the generally accepted basis for weir formulae is erroneous, and in the following chapters new formulas are derived on the basis of more rational assumptions. Complete overfall weirs are treated first as a matter of convenience. CHAPTER II. COMPLETE OVERFALL WEIRS. A. Derivation of New Formula for the Discharge over Com- plete Overfall Weirs built normally to the Direction of the Current. THESE formulae are derived for the following conditions, viz.: That the water reaches the weir section with a certain initial Fig. 5 velocity v\ that all the water in the channel must flow over the weir; that the weir is horizontal and has an inclined upstream face; 5 k --> \ i' , F ' ^' , Velocity v per sec. Ps Quantity Q per sec. -'''' \h ^> e *\ ^& m* Fig. 6 and that the direction of the weir be normal to the direction of flow. In the general case, a wing dam is assumed located on each side of the weir. (See Figs. 5 and 6.) ii 12 HYDRAULICS Let g= acceleration of gravity. ft= coefficient of flow. 7= weight of i cubic foot of water. Other notations as per diagrams. The breadth of the weir is &; that of the channel is E\ the wing walls extend above the water level. From the dimensions and depth of water Q = B (H + k) v = BTv ........ (4) v = = The projected length of each wing wall on the direction of the B-b weir is - . 2 The water flowing over the weir takes the surface curve COM, but the quantity of flow is the same as if the surface were GNM, the effective fall being the same in each case. The forces acting on the entire structure and those acting on the discharge area will now be determined. 1. The hydrostatic pressure on the section bH is rir>ce *,-7>f - ...... (6) 2. The hydrodynamic pressure, equally distributed over the area bH, and resulting from a prism of moving water of area bH and velocity v, is 3. The hydrodynamic pressure against a -fixed surface is found v 2 v to be = yF - = yq - ........ (8) g g COMPLETE OVERFALL WEIRS 13 by Weisbach's experimental law* viz.: "When water flows in a channel enclosed by three sides, the impact against a fixed surface in the channel will be equal to the impact of an isolated stream of water of same cross-section as the water in the channel. As found from the figure, the value of q in Eq. (8) is v I \H, and the cross-section of the water in the channel is ( }H = F. V 2 I This would indicate that the hydraulic pressure against a fixed surface would be twice that of a stream flowing against an opening of same cross-section. But since in the above case the water is not confined, and is thus easily deflected towards the area of flow, the quantity q must be assumed for the latter case. Hence the pressure against each wing v wall = P = y -- 2 g 2g IB - b\ - - ....... (9) \ 2 / which force may be assumed to act in the center of gravity of the obstructed prism. Regarding the divergence of this water, it should be considered that the filaments passing along the sides of the canal must be deflected through the angle <, while those passing along the line LF are not deflected appreciably; hence, the angle of divergence for the entire prism is taken as. 2 The force P s _==_eJ (see Fig. 6) may be resolved into the com- ponents // and nf, the former representing the pressure which is effective against the discharge area, and the latter that expended on the sides of the canal and wing walls. From Fig. 6, Tj = ejcos - = P 3 cos - =yq cos -. 2 2 2 g 2 * Weisbach Experimental Hydraulik, 423. Ed. 14 HYDRAULICS The quantity thus reaching the weir cannot continue its flow in this direction and is again deflected into the direction W, hence, (10) which represents the pressure effectively expended against the discharge area, while the force if produces a contraction of the flow in the flow area. Therefore, the total pressure against the discharge area from the quantity of water partially obstructed by the two wing walls is cos 2 . (ii) g 2 g \ 2 2 This pressure may be represented by a volume of area bH and length t, thus P 4 = ybHd from which P. v* IB - b\ 2 < t== ~T^r = T\~ - cos 2 ^. . . . (12) ybH bg\ 2 / 2 4. The pressure resulting from the volume of -water q^ = Bkv in the lower channel area Bk is now found. Since this water q^ flows in a three-sided channel its hydro- dynamic pressure on the sloped weir area may be taken as p *-ni--1-. Bk ..... (13) o o which pressure may be regarded as acting in the axis of the channel. The reasoning previously applied gives for this case an average angle of divergence for this water equal to - . Resolving P 5 into components ro 2 and uo 2 , Fig. 5, the latter being effective on the discharge area, it is found from the figure that no* = 0,0, cos - = P 5 cos- = 7 - Bk cos . . (14) 2 2 2 COMPLETE OVERFALL WEIRS 15 Since this force o 2 w tends to lift the overlying water over the weir, it must be deflected horizontally by the counteraction of the upper strata, thus giving the resultant P Q = WjW = u^w cos = P 5 cos 2 S or P Q = 7 - Bk cos 2 g (15) This pressure certainly has its maximum effect on the discharge area just at the surface of the weir, which effect gradually dimin- ishes until it becomes zero at the surface of the water. Hence, the total effective result of P Q on the area bk may be represented by a triangular prism of length b, height H and bottom breadth /?, thus: i/r - cos 2 * bgH 2 (16) The quantity of water, of depth k, striking the wing walls, can reach the flow area only by material deflection and is here neglected as being small and introducing too many complications. Fig. 7 The individual pressures P lt P v P 4 , and P e are now combined (See Fig. 7.) 1. The area OEE^ represents P 2 = bH by making OE = JE, = 1 6 HYDRAULICS 2. Again, by making the area OO^R^ will represent the pressure P 4 against the dis- charge area. 3. Also, by making O^R. = s^ = H, the area of the triangle _ Ifjz O^s will represent P x = jj. - 4. Lastly, by making sz = /? = cos 2 the area O.sz bgH 2 will represent P 6 = 7/9 2 In conclusion, the sum of the flow areas just designated will make the area EE^zO v and a prism of this area and the length b will represent the total resultant pressure on the discharge area E^K Now to compute the velocity from the pressure, continue the line zO l to U, and call y the hydrodynamic pressure at W, of a filament of water distant X, below the surface, and having a velocity F. Also call the surface pressure EO l = S, and the pressure at the crest of the weir = E^z = S r Then from similarity of triangles UEO^ and UE : EO, = UE l : E& SH or Z:5= (Z + H) iS.orZ = , - S' also from similarity of triangles UWV and UE^z UW : VW = UE l : E& or (Z + X) : y = (Z + H) : S.ory = S, Hence, V = /*v 2 gy COMPLETE OVERFALL WEIRS 17 The quantity dQ flowing through an element doc at the point W with velocity V and breadth b will be dQ = bVdx = pbdoc\J2gSi - . . . (17) To find Q it is necessary to integrate Eq. (17) between limits X o and X = H, and obtain when X = o, then Q = o, hence the constant becomes and by substituting this value for C in the above expression, and also substituting H for X, the discharge through the area bH, in cubic feet per second, is obtained as follows : or Q=vb\j i [(Z + H)-Zl (18) / 7T9 1 \ Now substituting for Z its value ( J and reducing, the fol- lowing form is obtained : 1 i -Sl]. . . . (19) Since 5 = EO 1} and 5 t = EjZ, these values become, in accord- ance with previous deductions, 18 HYDRAULICS S =* \- f ) cos 2 - from which, by substitution for 2g bg \ 2 / 2 Q and H / 2_Y \B (k + H)/ cos^ - . (20) I. For a straight weir perpendicular to the channel and with vertical face, Eqs. (19) and (20) become bgH \B (k + H)/ ' . . . (21) II. For same weir without wing walls, whence B = b, S ~-'-(k + H))' gH (22) III. B = b-,k = o;(f> = o; and ^ = o. (23) COMPLETE OVERFALL WEIRS 19 which is Weisbach's formula for complete overfall weirs in rivers. This coincidence is important as it proves conclusively that Weis- bach^s formula for complete overfall weirs is applicable only to the case shown in Fig. 8, in which the weir offers absolutely no obstruc- tion to the flow. Such weirs are not built. Fig. 8 IV. When a portion of the flow is diverted through a lateral channel. In the preceding formulae it is assumed that all of the approaching water must flow over the weir. Should a portion Q f of this water be wasted through a lateral outlet or over a weir, this quantity must first be found by the formulae given in the following, and be then subtracted from the entire approaching water Q, for which case Eq. (19) will take the form Q -C'-l In the equations for 5 X and S, the value of v = p /L is B (k + H ) true only when the quantity Q arrives immediately in front of the weir with a velocity equal to v. The hydrodynamic pressures against weir surfaces and wing walls previously found may, however, be modified for the waste water, through a lateral channel or submerged weir, when a definite disposition has been decided upon. That the above formulae may be generally applied to complete overfall weirs, of whatever kind, will now be shown. 20 HYDRAULICS B. Derivation of New Formula for the Discharge over Complete Over jail Weirs, built obliquely to a Channel or Represented in Plan by a Curved or Broken Line. When it becomes desirable to prevent excessive rise during high water stages, or when, during low water stages, the entire flow is to be utilized through a lateral flume, this may best be accomplished by a diagonal weir of sufficient length. The case to be treated is_shown in Fig. 9, in which A ADD represents the channel and EF the diagonal weir, which is also inclined vertically by an angle ^r with the horizontal. The nota- tion in the previous article will be retained. Fig. 9 The water over the weir crest, having a depth H, quantity BHv, and a velocity of approach v exerts an hydrodynamic pressure against the weir section equal to P = BH~. This pressure, being the same in all points of the section, may be considered as acting in the axis Oe and represented in magni- tude by the length ae = P. Since P acts on the weir under the_ angle <, the former may be resolved into components eg and ef, respectively perpendicular COMPLETE OVERFALL WEIRS 21 and parallel to the weir. The component eg then affects the direct flow over the weir and is found from the following equations; eg = ae sin < = yB - sin $ . 2 g If this resultant normal pressure on the weir section be repre- . _ T> sented by a rectangular prism of length EF = - , height H, and thickness /, then yBH sin c& = 7 - , from which 2g sin t = sin 2 (/> = S. 2 g This hydraulic pressure / is applied at the surface in front of the weir section and may, therefore, be taken equal to S, as was done in the derivation of Eq. (20). Taking the section of the weir as in Fig. 7, and considering this section normal to the direction of the weir instead of parallel to the axis of the_ channel as before, then the rectangle EOE^R (in which now EO = S) will represent the rectangular prism just mentioned. _ Since no wing walls were assumed in Fig. 9, the rectangle OO^RR^ in Fig. 7 becomes superfluous. The hydrostatic pressure of the advancing water on the weir EF is always normal to the weir and is This pressure may again be represented by a triangular prism, as was done in the derivation of Eq. (20) by the right angled equi- lateral triangle O^s in Fig. 7. According to the previous derivation of Eq. (20), the total hydrodynamic pressure against the weir between base and crest is P 5 = 7 Bk, which pressure may be assumed as acting along o _ the gravity axis Oe, Fig. 9, and is represented by the line a^. 22 HYDRAULICS This pressure is again resolved into components e^j^ and e^ and from the parallelogram of forces is found : e.g. = die* sin = 7 Bk sin cos 2 . 2 g 2 Since this pressure is a maximum at w (the crest of the weir), and diminishes toward the water's surface, it may be represented n by a triangular prism of the length - , height H, and bottom width /? found from the equation, _____ ______ '/> OM = u.w= 0,0, cos = 7 Bk sin 6 cos 2 g 2 w.w = 7 Bk sin cos 2 = 7 r-* 2 2 sin n as ? = =: sin > cos - . 2 If this width /? be applied in Fig. 7 as 52, then the triangle O^sz will represent a section of the prism of water in question. Accordingly the total hydrodynamic and hydrostatic pressures acting at the level of the weir crest, will be from Fig. 7, - 2 tfk & S=ER + Rs + sz = S + H + 2 - sin 2 cos 2 i. Having thus found the hydraulic pressures both at the water's surface and at the crest of the weir, the quantity of flow per second COMPLETE OVERFALL WEIRS over a diagonal weir is found by the process adopted in the deriva- tion of Eq. (20) as follows: (25) When it is desirable to construct a weir such that the maximum flow shall not exceed a certain assigned limit, that the weir may become safer and that the flow be more or less deflected away from the sides of the canal and towards its axis; or when it is necessary to supply lateral channels with equal quantities of flow during low and average stages of water, one of the following modifications may be adopted : 1. The weir consists oj two diagonal parts meeting in the axis of the channel and forming an angle with the apex pointing up- stream as in Fig. 10. Fig. 10 Fig. ii Fig. 12 Fig- 13 2. The weir is like the preceding but the two diagonal sections are separated by a central portion EF normal to the direction of flow, as in Fig. n. 24 HYDRAULICS 3. The weir is curved to the arc of a circle with convex surface upstream as shown in Fig. 12. The formulae for the computation of flow over such weirs as shown in Figs. 10, n, and 12, may be derived from equations (20) and (25). When the obliquity of the two halves of the weir, Fig. 10, is the same, Eq. (25) will apply without modification. For weirs in plan like Figs, n and 12, without wing_walls, the total flow is found by considering separately the parts AE, FD and EF, in which the curved portion, EF, Fig. 12, is considered straight and normal to the direction of flow. The flow_is_thus separated into three parallel filaments, the central one EF exerting a normal hydraulic pressure on the central area of flow, found from Eq. (22) as /i-S' 8 ] . . ( 2 6a) in which S/ and S' have the following values: The pressure of the flow approaching with a velocity v against v 2 the weir section is according to the previous derivation p 2 = ybH hence its linear magnitude at the surface per unit of width of weir s as represented in Fig. 7 by the line EO = E^R. The pressure OO lt resulting from the wing walls, disappears in the present case because no walls were assumed. Likewise the normal pressure of the approaching flow against the area of the weir below its crest was found to be 7/2 <\fr P 6 = 7 1L wfcop^JE, g 2 by substituting b for B, because in the present case the width b only is considered. COMPLETE OVERFALL WEIRS 25 The bottom width of the triangular prism representing this pressure was previously found as 2 v 2 Bk , -> P = 7, rr COS bgH 2 which, by substitution of b for B, becomes The magnitude of the total hydraulic pressure of the approach- ing water at the level of the weir crest is found from Fig. 7 by the equation 2 k ( Q or The quantity of flow per second over the middle section EF, Figs, ii and 12, may then be found from Eqs. (26), a, b and c. The flow over the oblique portions of the weir may now be found from Eq. (25) in the following manner: Since for an oblique weir the hydraulic pressure of the advancing flow at the surface of the water is equal to S and at the crest of the weir is equal to S from Eqs. (25), in which S and S t are found per unit of width of channel, it follows that these_equations are also applicable to the oblique weir parts AE and FD, Figs, n and 12. Also, the length of each oblique weir = ; - and for the two 2 sin n _ r T> = -: Then by substituting for - - in Eq. (25), the value sin 9 sin 9 T) 1 - , the quantity of flow over the oblique weir sections is sm 9 found to be 26 HYDRAULICS Hence, the total flow over a weir, polygonal or circular in plan, as in Figs, n or 12, is equal to Q = <2i + <2 2 ...... (26*) When it is required to construct a weir for a medium stage of water such that the high water mark shall not be materially raised, the disposition shown_in Fig. 13 has of late been applied, by making the oblique portion OP of sufficient length. The section NO = b is so chosen as to_provide for the partial flow of the mass AN Of) and the sections OP and PR are so dimensioned as to admit of a regular flow of the quantity O^OUR. By giving the crest of the weir section OP a slope equal to the surface slope above the weir, and placing the weir section PR level with the lower point P, the approaching flow will reach all parts of the weir with the same initial velocity and constant depth H. To comply with the conditions of uniform flow, it is necessary to first find the formulae representing the flow per jsecond over each section of the weir. It is assumed that NO = PR = b, and that the cross-section NU is regular. Since, for the case in hand, < = ^ = 90 for the sections NO and PR, the formula (22) will apply to these sections and the flow over each will be represented by the equation q - . pi, g -- [S, - S] . . . (27*) 3 VJj O/ Regarding the flow over the section OP, it must be considered that since the water flows almost parallel to this section with a velocity v, the hydrostatic pressure H on this portion of the normal 0.67 v 2 section would be diminished by an amount = - 2 _ _ But, since the water flowing between cross-sections OU and PR also flows over the weir OP diagonally and hence exerts an hydro- dynamic pressure against the discharge area which is somewhat less than for normal impact, it would seem probable that these COMPLETE OVERFALL WEIRS 27 two opposite effects of the advancing water would neutralize each other, and hence it is fair to assume that the advancing flow exerts only an hydrostatic pressure H, and the flow over the section OP may be found from the simple formula (2 7 b) The total flow over the weir NOPR will then be Q = 2 g + fc (27c) Such weirs of broken lines offer the advantage that the back water usually caused at times of high water may be reduced to one half that which would result from a weir built normally to a stream, also that the Opposite bank N U is not so susceptible to washouts as in the case of oblique weirs. CHAPTER III. INCOMPLETE OVERFALL WEIRS. Derivation of New Formula for Discharge over Incomplete Over- falls or Submerged Weirs, and through Contracted River Channels. THE same rational suppositions employed in the previous derivations will be followed in the present chapter. Accordingly, the hydrodynamic and hydrostatic pressures, acting on the dis- charge section, will be ascertained from considerations peculiar to the problem. These pressures will then be graphically combined to produce the resultant pressure areas on the discharge section, and from the latter, analytical formulae will then be developed. Wingwall p * i b f , V/L M D Wing,|f j r jfc I D Fig. 14* Fig. 140 represents the longitudinal section of a river or canal, ob- structed by a submerged weir and wing walls A'E and D'F, shown in plan Fig. 14^. All dimensions are indicated on the two figures. 28 INCOMPLETE OVERFALL WEIRS 29 The mean depth of the lower pool above the weir crest is H l = T l k, when the damming effect of the velocity on the lower level is neglected. Let Q be the quantity, per second, approaching the weir with a velocity v, which after being discharged over the weir proceeds with a mean velocity F, the amount of which velocity depending on the nature, form and slope of the bed. Also, let ^ be the coefficient of flow through the upper part section EE it Fig. 140, and /^ for the lower submerged part E^E 2 . The flow of approach exerts an hydrodynamic pressure against the discharge area which is uniform over the total depth H and hence may be represented by a rectangular prism EOE 2 R, of height H, length b, and width EO = Kfi = = 2g 2g\BT/ The hydrodynamic pressure, against the discharge area, result- ing from the flow of approach against the two wing walls and over the height H, is again represented by a rectangular prism OJK.^RO of length b and width Of> = R^R = r-f ) cos 2 ^ bg\ 2 / 2 The hydrostatic pressure exerted by the head H 2 against the v discharge area EE 2 is represented by the area O^S^, being the resultant of the active hydrostatic pressure O^T^R^ less the back pressure EJL^e from the quiescent lower pool. However, the lower water is moving away from the discharge area with a V 2 nV 2 velocity F, thus creating a suction = 0.67 = - on the -part 2 g 2 g section EJ V which will diminish the residual effect of the hydro- static counterpressure E^E 2 e by an amount represented by the rectangle E^E^a. The triangle /o^thus represents the remain- ing hydrostatic back pressure on j^E^ while the smaller triangle fE^a = s { a 2 s 3 represents the residual pressure effect on the discharge area EJ 1 due to the suction. Hence, by neglecting this small suction triangle / x a, the net area, E^aJ = s^^ss^ then represents the resulting total suction on the section E { E 2 . Thus the figure EO^s z sE 2 represents the total effective pressure on the 30 HYDRAULICS discharge area EE 2 , except that due to impact against the weir wallG. The hydrodynamic pressure against the discharge area resulting from the velocity of approach impinging on the weir wall E 2 G was found by Eq. (15) as g and is represented by the line w^w. Fig. 5. This pressure has a maximum effect on the strata along sE 2 and may be regarded as entirely dissipated at the level /^g, hence the triangle sTs 3 of _ / n yz \ height / 1 2 = (r i -^ -- ] and base ft may be taken to represent this pressure P Q . The value of ft is then found from and hence, ft = The area EO l s 3 TE 2 then represents the total effective pressure on the discharge area, and the unit pressures along certain filaments of the discharge may now be determined. The pressure along the surface element O^E = S becomes g The filament f^^ down to which there will be no counter- pressure from the lower pool, will represent a pressure 5 t evaluated - f 7I =5 + tf 2+ ^L (,8ft) INCOMPLETE OVERFALL WEIRS 31 Finally the pressure along the filament E 2 T, at the crest of the weir, is found as 2 i?Bk cos 2 * The flow through the part section Ej^ proceeds without counter- pressure as for a complete overfall into free air, and by observing nV 2 that H 2 H = 5j S, this discharge Q l becomes, according nV 2 to Eq. (19), when H 2 H is substituted for H, / 2g [5^_5 5 ] ( 2 &/) The discharge Q 2 through the lower portion of the section over the height f^E 2 = (T l k Jis found by integration as was done in the derivation of Eq. (19). The velocity along /^ 3 will now be \/2 gS lt and that along E 2 T will be \/2 gS 2 . Q 2 then becomes M-*-^ Q 2 = - pj> v x 2 g \ -2- I (5 2 S -S?) . ( 2 8) 3 X ^2~ ^1 ' If the immersed height j^E.^ is small, then the mean velocity / 7^ + 5 \ through this portion of the section may be taken asy 2 g I a J, corresponding to a pressure height l L , whence . ( 2 8/) From these values the total discharge is thus found to be HYDRAULICS Equations (28) may then be regarded as the fundamental equa- tions for incomplete overfalls and submerged weirs. By introduc- ing special values as was done in Eqs. (19) and (20), the following simpler forms are obtained : When v = o, then S = 2 g t + Q >. . (2 9 a) wherein H^ = T l k. For the discharge through a lateral orifice in the vertical 'wall oj a reservoir, into a reservoir oj lower level, so as to produce an incom- plete overfall, and assuming quiet water in each reservoir, then < = ty = 90 and v = V = o and calling the height in the lower pool above the bottom of the opening H v then Eqs. (28) become S = o] S l = S 2 = H 2 \ This formula coincides perfectly with Dubuat's formula, thus proving that the latter applies only for the special case here assumed. Figs. 15 and i6a represent a river, the flow through which is obstructed by two piers A^E and DJ? such that the natural surface PN is raised to the level EA. The pressures active on the discharge area for this case will now be determined. The hydrodynamic pressure, resulting from the velocity of INCOMPLETE OVERFALL WEIRS 33 approach v, is uniformly distributed over the entire depth, and is represented by a rectangle EORE 2 , in which EO = . B ^ 2 2 /) / Wlei, x^^^^ N 1 / IF [__,/ i \ \ s R:R Fig- I 5 The hydrodynamic pressure, due to velocity of approach and deflected into the discharge area by the piers, is represented by the v rectangle OOfi^R, the width of which is The suction caused by the velocity of discharge V is represented by the rectangle E^E^a^j which subtracted from the hydrostatic pressure due to T l and represented by the triangle Ef^e^ leaves A \ \ *jt p /A PI 4 D Fig. Fig. (after neglecting the small triangle aEJ), the net hydrostatic pressure area fa^e = s 3 Ts. The total hydrostatic pressure result- ing from the head T is given by the triangle OJR-^T from which the net counterpressure s 3 Ts is subtracted to leave the effective pressure area O^^R^. 34 HYDRAULICS The total pressure on the discharge area is thus represented by the figure EO l s 3 sE 2 from which unit pressures at any point of the discharge section may be obtained. The discharge, through the part section Ef lt takes place as for discharge into open air, while that through the lower portion f^E 2 flows with a velocity corresponding to the pressure head sE 2 . Hence, the following pressures are obtained: - 2 g & r 2l b and ..... (306) CnV 2 \ H 2 H -- 1 =5. S, 2 g / Eq. (19) again furnishes the value for the flow through Ej^ as nV 2 Also, the discharge through the part section / 1 2 = T 1 -- subjected to counterpressure and flowing under constant head, is Finally, the total flow through the contracted section, of height T = EE 2 , and width b = EF, Fig. i6a, becomes For the case of a river obstructed by piers, as in Fig. i6&, the above formulae may be employed by substituting the aggregate width of the piers for the width B b of the former wing walls. This is on the supposition that the piers are sufficiently near to INCOMPLETE OVERFALL WEIRS 35 each other so that their individual damming effects would result in a uniform elevation of the approaching surface. However, this distinction must be made. In the case of wings, only two contractions occur, while for pier obstructions these contractions are repeated in each opening, thus necessitating special values for the coefficients /* and /^ according to the observations of Francis. In closing this subject a few remarks are here added relative to the effect of obstructions in rivers, and the permanency of river beds. From many accurate measurements and gagings, it is seen that the velocity is maximum at about one third the depth and dimin- ishes towards the bottom in all cases of unobstructed uniform flow. The discharge curve is then of parabolic form as shown by E 1 SN V Fig. 146. On the other hand, for obstructed flow, especially as in Fig. 140, the velocity at the surface is always diminished. It increases gradually with the depth down to the point j v where the counter- pressure from the lower pool becomes active, and below this point it remains practically constant. This is illustrated by the discharge curve EfiE^ Fig. 146. This observation explains the cause for the extensive erosions which always follow the placing of a pier or other obstruction into a river channel. CHAPTER IV. SLUICE WEIRS AND SLUICE GATES. Derivation of New Formula for Discharge over Sluice Weirs and through Sluice Gate Openings. IN deriving formulae for discharge through sluice openings, it is usually supposed that the efflux takes place as for a vessel, Fig. 17, the inside surface of which is assumed to be frictionless and gradu- Fig. 17 Fig. i 8 ally contracted into a conical shape, so that the velocity V, in the enlarged section M 1 M, is gradually accelerated toward the opening ATjAT. The water in falling, through the height H, acquires a certain vis viva, none of which is supposed to be lost in the produc- tion of acceleration.* Now if the flow through such a vessel be compared with that in a river contracted by sluice weirs, Fig. 18, it is almost self-evident that this condition of flow in no wise resembles the above illustration in Fig. 17. * Ruehlmann, Hydromechanik, pp. 207, 208, 463 and 464. 36 SLUICE WEIRS AND SLUICE GATES 37 The water in the river impinging on the wing walls, projections, corners, etc., produces impact. Also, the flow does not arrive at the discharge area with a gradual motion, but rather suddenly, and is then deflected at various angles toward the openings. All this goes to prove that a considerable portion of the vis viva stored in the flow of approach, is necessarily lost by impact and friction. The older formulae were also based on the erroneous supposition that the discharge over a submerged weir was resisted by an hydro- static counterpressure on the discharge area just as for discharge into quiet water, thus disregarding the suction due to velocity of discharge. Formulae based on such discordant ideas cannot be expected to furnish correct values for discharge through sluices, etc. For this reason new and more rational formulae will now be developed with the aid of the principles formerly applied to the derivation of Eqs. (19), (20) and (28). 1. Formula for case illustrated in Fig. 19, being a sluice weir with submerged discharge. Fig. 19 Figures 18 and 19 represent the plan and longitudinal section, respectively ,_of_ a submerged weir AJ) V built in a river. The wing walls, A,E and Df, extend above water level and the space EF, Fig. 1 8, contains three gate openings. 38 HYDRAULICS When the gates are raised to a height E^E 2 = a, the water in the upper pool approaching with a velocity V, will be supposed to exert a pressure head H v causing a velocity of discharge F, in the lower pool. The quantity of discharge Q, passing through the three openings, as above illustrated, will now be found. The hydrodynamic and hydrostatic pressures, exerted on the discharge area, are separately determined for each of the three following prisms of flow : 1. The lower prism CGKE 2 , of height k. 2. The middle prism KEJE t J t of height a. 3. The upper prism JEJEA, of height Making the sum of the widths of the three gates equal to >, then the total width of the interposed obstruction, including piers and wing walls, will be B b. The hydrodynamic pressure against each of the orifices may now be represented by a rectangular prism of height a, length B b, v 2 and width , shown in Fig. 19, as pressure area I. 2 g The hydrodynamic pressure deflected by the wing walls and piers into the discharge area, as found in the derivation of Eqs. (19) and (20), may be represented by a rectangular prism II, of width / = However, the wing walls in the present case being perpendicular to the canal axis, < = 90 and hence, SLUICE WEIRS AND SLUICE GATES 39 The upper prism of flow, of depth H a, exerts an hydro- dynamic pressure against the three gates_and the upper part of the wing walls, active along the gravity axis de and over an area This pressure is then H 2 -(k + a)]. . . . (32) Since the lower strata / x of this prism undergo scarcely any deflection, while the upper strata along AE must be deflected 90, the average deflection for the prism may be taken as 45. This is represented in Fig. 19, by the direction EJ. _ If the pressure_j> be represented as a linear magnitude d^e lt decomposed into ej and e^g, then, ^/ =~d^ cos 45 = yB [r t + H 2 -(k + a)] ~- cos 45. But ej = E^e^ may be again resolved into components E^e and EjT , from which the required horizontal component is found as = E cos 45 = VB [7\ +H, - (k + a)] - . . (33) o This component exerts a uniform pressure on the discharge area and its effect is represented by a rectangular prism of length b, height a, and_width t v which latter may be found from Eq. (33) by making / = yabt v whence which determines the pressure area III, Fig. 19. The total hydrostatic pressure of the upper pool on the whole discharge area may be represented by the weight of a triangular prism o^T. The counterpressure from the lower pool is then given by the triangular prism UJ = i 2 sT, which, subtracted from 40 HYDRAULICS the former, gives the net effective hydrostatic pressure, represented by the area o^R^. The portion o^p 2 s 2 i 2y of this latter area, is taken up by the upper portion of the sluice gate, leaving only the effective pressure area o 2 R^ss 2 = vbH 2 a, shown in Fig. 19, as area IV of width R^s = H 2 . The discharge velocity F, in the lower pool, produces a suction - , represented by the pressure area V, of width The triangular prism VI, with base T^ 3 = /?, represents that part of the hydrostatic pressure on the weir E 2 G, which, after being twice deflected, finally becomes effective on the discharge area. In the derivation of Eqs. (19) and (20), the value of /? was found as . abg 2 The total hydraulic pressure of the upper pool on the discharge area, is then obtained from the summation of all the pressure areas above found. This gives the trapezoidal figure E lSl T,E 2 = I + II + III + IV + V + VI . . (35) from which the unit pressure on any point of the discharge section may be ascertained. Thus, the pressure in the upper filament s 1 E l S is made up of the combined widths of the several prisms I to V, and is evalu- ated as 2 g or =I++ 2 g\_ 2b 2 ab\ /J 2 g The unit pressure along the filament T 1 E 2 = S l will then be abg SLUICE WEIRS AND SLUICE GATES 41 By a process similar to that employed in deriving Eq. (19), the quantity of discharge is found to be When the sluice sill is level with the river bed, making k = o, and thus Sj = 5, then the flow through a proceeds with the uniform velocity \/ 2 gS, and the discharge may be found from the following simple equation : Q = ^ 2. Formula for the case illustrated in Fig. 20, for k = o and T l < a, will now be derived. The rectangles I, II, and III represent, as in Fig. 19, the cross sections of the pressure prisms effective on the discharge area and resulting from the hydrodynamic pressure due to velocity of Fig. 20 approach impinging on the discharge area and on the wing walls and upper portion of the gate. Hence, the expressions just found will apply to the present case when k = o is introduced. Considering the hydrostatic counterpressure of the lower pool PN, active up to a line sjn v then the effective pressure on the discharge area will be given by the triangle oJK.^T less the triangle o^oJK. less the triangle i 2 sT, leaving the net area o 2 R t si v which is composed of areas IV and V, Fig. 20. HYDRAULICS The area VI finally represents the suction on the discharge sec- tion produced by the discharge velocity F, and this suction being taken as uniformly distributed, its horizontal intensity becomes nV 2 S ^~ 2g ' Fig. 20 then furnishes the unit pressures along any horizontal filament. From this figure also, the following dimensions are found: Ef, = o^ = o 2 K = H 2 + 7\ - a 2g nV 2 E 2 m T _ ~~ (37) The discharge is evaluated in two partial quantities, Qifor the flow through the upper part section E^, which takes place as for discharge into free air, and Q 2 for the flow through the lower part section E 2 m, which is submerged. Accordingly the resultant hydraulic pressure along the filament li^K = 5, is found as or g The total pressure in the lower filament 5 t w, is equal to S 4- Ejn = S 19 or S,= -T l . . . (386) 2g SLUICE WEIRS AND SLUICE GATES 43 The discharge Q lt according to the fundamental Eq. (19), now becomes nV 2 Q l = a + -r, The resultant hydraulic pressure over the lower part section E 2 m is uniformly distributed over this depth and has the unit intensity S t , corresponding to a velocity \/2 gS v from which Q 2 becomes = ,6 ( r - The total discharge through the orifice of height a is then Q - Q, + C 2 ....... (38*) 3. Another case of discharge through sluices is illustrated in Fig. 21. Here the hydraulic pressure is supposed to be very great, Fig. 21 and the discharge area very small. The velocity at the orifice is greater than the discharge velocity and, in consequence of this retardation, an impact is produced which is expended in elevating the discharge surface into a wave just in front of the sluice opening. Regarding this phenomena and its effect on the discharge, little reliable information exists. The following solution is offered. In the previous case all the factors bearing on the discharge were 44 HYDRAULICS carefully considered and these apply equally in the present problem with the exception that the wave, when it exists, enters as a disturb- ing element in determining the particular value of T v which is likely to govern the discharge. The matter then resolves itself into finding such a suitable value for T r Obviously, the whole height of the wave cannot be taken as the level of the lower pool, because the wave is merely the result of impact of discharge, and because the wave-crest is immediately followed by a deep wave-hollow. Also, the velocity at different points of the wave-crest varies in magnitude and direction, and no particular value could be selected as the discharge velocity. When the normal flow is once established, the depth and velocity in the lower pool undoubtedly exert a marked influence on the production and shape of the wave, which effect is probably trans- mitted back to the discharge area. Slight changes in the conditions of flow, constantly occurring, may cause the wave to disappear entirely, thus reducing the surface of the lower pool to a level. It would thus seem most rational to disregard the wave and assume as the most probable lower pool level, the one resulting from normal flow. However, should the length of the flume be insufficient to develop a continuity of flow through the flume, then the mean height of the wave may be taken as the resisting depth of the lower pool, and the mean velocity resulting from this depth may then be used in the above formulae. The values of T l and F, in the formulae of the present chapter, are supposed to be known from gaugings or otherwise. However, when they are not known, as in the case of proposed sluices, then the methods suggested in Chapter VI must be employed. In the previous formulae, a certain velocity of approach v was included. When discharge takes place from a lake or other quiescent water, then v = o, and the various pressure areas I, II, and III disappear, thus greatly simplifying the formulae. SLUICE WEIRS AND SLUICE GATES 45 This special condition when introduced into the above equations furnishes the following formulae for discharge from quiescent water into a flume or canal. a. For a sluice gate at the inlet to a canal supplied from a lake, case Fig. 19, then v = o and Eqs. (36) give = 5, ^ (39) b. For a sluice gate at the inlet to a canal supplied from a lake, case Fig. 20, then v = o, and Eqs. (38) become S = H 2 +T 1 -a, . . . . ( 4 o) c. Should the water be discharged from a high reservoir into one of lower level, by means of a sluice gate, both reservoirs being quiescent and the lower level is above the bottom of the sluice nV 2 opening, then the function =o, and the hydrostatic counter- 2 g pressure from the lower reservoir then becomes effective. When the discharge of a river is known, then the dimensions of weirs, sluices, etc., as well as backwater height and distance, can be determined by a method of approximation to be discussed in Chapter VI, with the aid of the above formulae. By the application of - the principles here employed, in the derivation of new formulae in Chapters II, III, and IV, it will be possible to solve any similar problems by developing formulae appro- priate to such cases. The formulae of the present chapter are also applicable to small regulating gates. CHAPTER V. BACKWATER CONDITIONS. Discussion and Formula for Backwater Height and Distance. THE formulae in the previous chapters deal with problems of obstructed flow due to objects of assigned dimensions, and the backwater height enters as a given function. In the following, the reverse conditions will be treated under three cases. 1. When the dimensions of an existing or proposed weir or sluice are given together with the flow of approach, to find the backwater height. 2. The flow of approach and backwater height being given to find the dimensions of the weir or sluice such that the flow will proceed with this assigned height. 3. Given the backwater height to find the backwater distance and surface curve. When the flow of approach Q, is given, as for problems under i and 2, then one of the unknowns H 2 , k or b, can be found for assigned values of the others, by using one of the above for- mulae. When these formulae become too complicated for direct solution of an independent variable, then the method of approx- imation, by substitution of assumed values for this variable, must be resorted to. Problems coming under the third head may be solved by the somewhat complicated formulae given by Professor Ruehlmann, which in the absence of a better solution are here reproduced. Drift and sedimentation always enter as a disturbing element in river hydraulics, so that no formulae, however accurate, could be made to permanently satisfy all these changeable conditions. 46 BACKWATER CONDITIONS 47 Professor Ruehlmann gives tabulated values for the ready solu- tion of his formulae, and the results so obtained agree very well with those of Hagen, Weisbach and Heinemann. Let D = the original uniform depth of the river. s = the original natural slope of the river. Z = the total backwater height measured above the natural surface slope. L = the total backwater distance measured from the crest of the weir. z and / are coordinates of the backwater curve, referred to a point O on the natural slope line, vertically above the crest of the weir. (See Fig. 22.) ; ~ >j Fig. 22 f ) and / f Jare Ruehlmann's functions of and respec- tively, the values for which are given in Table I. Ruehlmann's formula is When D, s, Z and z are given, the problem is thus solved. The different values of (TV) and I J are found from one of the columns i, of Table I, and opposite these in column 2, are the corresponding functions /f-jand /nU These, when substituted in Eq. (41), give the abscissa / corresponding to the ordinate z, for any given D and s. 48 HYDRAULICS When Dj s, Z and / are given, the value of z is found from Eq. (41), by transformation, thus Z\ si (42) Then having found fl-J from Eq. (42), the corresponding value of (j\ is given by Table I finally z = j- . D. In the same manner Z may be found from the following Eq. (43) : ./Z\ si f /z\ f $rD+ J \f>) (43) For the total backwater distance L the three quantities z } ( j and / f - J are each zero. The Eq. (41) then becomes : *-?/ <> When Table I does not include exactly the values for any given ( n) OI m)' t * ien f ( n) or ^ (n)' corr esponding to tne exact given value, may be found from the following interpolation formula /( ) or / ( ~~" 1 == 'V : ^ 'V ~f" ( 'V "V ) I ^ I (A f \ \ 7~)/ V 7~)/ ^ I / y v I v"o/ in which ^c = the given value of f J or f J / z \ x, = the next smaller value of x or [- ) in Table I. x 2 = the next larger value of x or f J in Table I, y = the required function /rj>) fZ\ y 1 = the value /f- corresponding to x r fZ\ y 2 = the value /(-J corresponding to x v CONDITIONS 49 TABLE I. PROFESSOR RUEHLMANN'S BACKWATER FUNCTIONS FROM HYDROMECHANIK, PAGE 484. I z D z D '(1) ft) i Z D z D (I) 6) i Z D z D (I) ft) i Z D z D si ft) I z D z D '(i) fe) O.OIO 0.0067 0- 2 35 1.2148 0.460 i . 6032 0.685 1.9077 0.910 2.1800 0.015 0.1452 0.240 1.2254 0.465 i. 6106 0.690 1.9140 0.915 2.1858 O.020 0.2444 o. 245 2358 0.470 .6179 0.695 1.920;; 0.920 2.1916 O.025 0.3222 0.250 .2461 o-475 .6252 0.700 1.9266 0.925 2.1974 0.030 0.3863 o- 2 55 2 563 0.480 .6324 0.705 1.9329 0.930 2 . 2032 0.035 0.44II 0.260 .2664 0.485 .6396 0.710 1.9392 o-935 2 . 2090 O.O40 0.4889 0.265 .2763 0.490 .6468 0.715 1-9455 0.940 2.2148 0.045 0.5316 0.270 .2861 0-495 .6540 0.720 I-95 1 ? 0.945 2.22O6 0.050 0.5701 0.275 .2958 0.500 .6611 0.725 1-9579 0.950 2.2264 0.055 0.6053 0.280 3054 0.505 .6682 0.730 1.9641 0-955 2.2322 O.O60 0.6376 0.285 3149 0.510 -6753 o-735 I-9703 0.960 2 . 2380 0.065 0.6677 0.290 3243 0.515 .6823 0.740 I-9765 0.965 2-2438 O.070 0.6958 0.295 333 6 0.520 .6893 0-745 1.9827 0.970 2.2496 0.075 O.7222 0.300 .3428 0-525 .6963 0.750 1.9888 0-975 2.2554 0.080 0.7482 -35 35i9 0-53 .7032 o-755 1-9949 0.980 2.26ll 0.085 o. 7708 0.310 .3610 o-535 .7101 0.760 2.0010 0.985 2.2668 0.090 0.7933 o^S .3700 0.540 .7170 0.765 2.0071 0.990 2.2725 0.095 0.8148 0.320 -3789 o-545 7239 0.770 2.0132 0-995 2.2782 O. IOO 0.8353 0-325 .3877 0-55 .7308 0-775 2.0193 .000 2.2839 0.105 0.8550 0.33 -3964 o-555 -7376 0.780 2.0254 . IOO 2.3971 O.IIO 0.8739 o-335 .4050 0.560 .7444 0.785 2.0315 .200 2.5683 0.115 0.8922 0.340 .4136 0-565 7512 0.790 2.0375 .300 2.6l79 O. I2O 0.9098 0-345 .4221 0.570 .7589 0-795 2.0435 .400 2.7264 0.125 0.9269 o.35o .4306 0-575 .7647 0.800 2.0495 -5 2.8337 0.130 0.9434 0-355 -439 0.580 .7714 0.805 2-0555 .60 2 . 9401 0.135 o-9595 0.360 4473 0-585 .7781 0.810 2.0615 .70 3-045 8 o. 140 o-975i -3 6 5 455 6 0.590 .7848 0.815 2.0675 i. 80 3.1508 0.145 0.9903 0.370 .4638 o-595 .7914 0.820 2.0735 1.90 3.2553 0.150 i . 005 i o-375 .4720 0.600 .7980 0.825 2.0795 2.00 3-3594 0.155 1.0195 0.380 .4801 0.605 .8046 0.830 2.0855 2. 10 3-463I o. 160 1-0335 0.385 .4882 0.610 .8112 0-835 2.0915 2. 2O 3-5564 0.165 1.0473 0.390 .4962 0.615 .8178 0.840 2.0975 2.30 3.6694 0.170 i. 0608 0-395 .5041 o. 620 .8243 0.845 2.1035 2.4O 3.7720 o.i75 i . 0740 0.400 5H9 0.625 .8308 0.850 2.1095 2.50 3-8745 0.180 1.0869 0.405 5 J 97 0.630 .8373 0-855 2.II54 2.6o 3.9768 0.185 1.0995 0.410 5 2 75 o-635 .8438 0.860 2.I2I3 2.70 4.0789 0.190 1.1119 0.415 53S3 0.640 -8503 0.865 2.1272 2.80 4.1808 0.195 1.1241 0.420 543 0.645 -8567 0.870 2.I33 1 2.90 4.2826 0.200 1.1361 0.425 557 0.650 .8631 0.875 2.1390 3-o 4-3843 0.2O5 i. 1479 0.430 5583 0-655 .8695 0.880 2 . 1449 3-5 4.4891 O.2IO I-.J595 o-435 5 6 59 0.660 8759 0.885 2.1508 4.00 5-3958 0.215 1.1709 0.440 5734 0.665 .8823 0.890 2.1567 4-5 5.8993 0.220 1.1821 o.445 5809 0.670 .8887 0.895 2.1625 5.00 6.4I2O 0.225 1.1931 0.450 5884 0-675 .8951 0.900 2.1683 0.230 i . 2040 o-455 5958 0.680 1.9014 0.905 2.1742 The above table is applicable for any units of measurement, as feet or meters. 50 HYDRAULICS The application of Table I and formula (41) will now be illus- trated by citing two examples given by Professor Ruehlmann in his Hydromechanik. Example I. Given a river 80 feet wide and 4 feet deep with a slope s = 0.000623. A weir built in this river raises the water 3 feet at the weir site. At what distance back from the weir will the backwater height be 0.25 feet? Here (} = ^ = 0.75. From the table, find the value for / f ) = 1.9888. Also () = - = 0.0625 for which no \DJ \DJ 4X4 value can be found in column i, hence interpolation by Eq. (45) becomes necessary. Thus : Cf for ~- = x l = 0.060, column 2 gives /(^)= y l = 0.6376, for = x 2 = 0.065, column 2 gives /(# 2 )= y 2 = 0.6677. By substituting these values in Eq. (45) the value of / / J corre- sponding to = x = 0.0625 is found as ; =0-6526. Having the value of / r^rh the required backwater distance is found from Eq. (41) as / = 4_ [1.9888-0.6526] = 8579 feet. 0.000623 Also the total backwater distance as given by Eq. (44) is L = 4_ x I<9 888 = 12,770 feet. 0.000623 Example II. Given two points A and B, distant 2020 meters apart. What backwater height Z must be produced by a weir at A BACKWATER CONDITIONS such that the backwater height at B is 0.891 m., when the original depth of water is 1.59 m., the fall in the river bed between A Fig. 23 and B is 1.737 m -> an d the discharge is 158.52 m. 3 per second. See Fig. 23. Then si si 2020 X 2020 = 1.737 and the depth D= 1.59 m.; hence - = 1.0924 and -1 = D D 1.59 1.7444 and from Eq. o . 560. Then from Table I, = I -9 2 4+ 1-7444 2 . 8368 for which the table gives very nearly = i . 50, hence 2 = 1.50 X 1.59 = 2.385 m. CHAPTER VI. FLOW IN RIVERS AND CANALS. Derivation of Formula lor Discharge from Rivers or Lakes into Water power Canals and Flumes. 1. General Discussion. THE following formulae are intended to serve the engineer in designing flumes or canals for manufacturing or water-power purposes. It is proposed to discuss fully the various short- comings of older formulae, and to show the general applicability of the foregoing principles and ideas to problems relating to cross-sections, slopes and inlets necessary for the delivery of certain quantities of water through flumes and canals which are supplied either from rivers or lakes. Ruehlmann, Hydromechanik, p. 439-442, says, for the case of such canal inlets without regulating works, " from a scientific standpoint, the presentation of this problem still lacks all mathe- matical demonstration, and for practical purposes we must con- tent ourselves with a few observations made by Dubuat." Based on these observations Ruehlmann then offers the following doctrine : Fig. 24 " The mean velocity and area of flow in any canal or flume of constant width and uniform slope, are so related that velocity 52 FLOW IN RIVERS AND CANALS 53 height is equal to the difference in level between the supply reser- voir at the inlet and the water in the flume at a point where the velocity of discharge has just become uniform." Referring to Fig. 24 (which is Ruehlmann's Fig. 167) and calling V the mean velocity of discharge through the section AC, v the velocity of approach, m a coefficient of contraction and other dimensions as indicated on the figure, then, according to Ruehlmann, i. V 2 v 2 e e l = - ' (46) 2 gm 2 2 g When v is very small, then V 2 e- ^ = - 7 (47) 2 gm 2 Ruehlmann then introduces further quantities as follows : / = length of canal a^which uniform flow is reached ; h n = absolute : - slope of the surface KD and of the canal bottom AE\ rj = actual - fall at section DE, below the water level of the inlet]_a = sectional area, and w the wetted perimeter at the section DE. Then the required relative slope ratio for the flume, expressed as a function of the head (e e^), becomes, h n _ r)-(e-e,) T / (48) V* w jV 2 and '-T^ + a'-F (49) wherein k is the experimental coefficient in Chezy's well-known formula for flow in rivers and canals. This formula is Li a h v = k\ - . - Y w I (So) Ruehlmann thus advocates Eqs. (47), (48) and (49), for the solution of all practical problems of the kind here considered. 54 HYDRAULICS Regarding these formulae the criticism is made that they do not take account of section, slope and character of the feeding river, which factors determine the velocity of approach v, as seen from Eq. (46) when solved for V. Thus .... (51) by which V is expressed entirely as a function of (e ej and the velocity height . Formula (46) is, therefore, in error. 2g _ _ The water, in flowing from the section AC to the section DE, is subjected to a variable velocity. Then for the case of an accelera- tion, the surface might be represented by the dotted line CD. For a retardation this surface would be the dotted line CiD, while the^ special surface CiKD II AE could scarcely be expected. VVXAY The distance /, required for the velocity of discharge V to become constant and equal to the velocity in the flume, also the total fall y, cannot, in the light of our present knowledge of hydraulics, be computed with any degree of precision. These factors can be ascertained only for an existing flume by means of carefully con- ducted current meter observations and slope determinations. w V 2 The second term - / , of Eq. (49), presupposes that the sur- d KT face KD i, AE and that the mean velocity over the distance / is constantly equal to V. Both of these presumptions are certainly incorrect and, therefore, Eq. (49) is not strictly reliable. In presenting the case of a flume with sluice gate at the inlet from the river, Ruehlmann employs the experiments made by Lesbros on small troughs of o . 2 meter in width, placed in the pro- longed axis of the supply channel, which latter was 3 . 68 meters wide, with a supply orifice o . 2 meter wide. The troughs were 3 meters long and the experiments cover a range of slope from i : 20 to i : 2.9. Owing to the large dimensions of the supply channel as compared with those of the trough, the velocity of approach was FLOW IN RIVERS AND CANALS 55 necessarily very small. Lesbros employs the formula for dis- charge through a lateral orifice into air and finds for his experiments (52) wherein H = the fall between the upper level and the sill of the orifice and h = the same fall to the upper edge of the orifice or lower edge of the gate. From this formula Lesbros computes the values for p which satisfy all the conditions of his experiments. While it must be admitted that this work might serve a useful purpose within the scope of the observations, yet it is apparent that the above formula (52) is not applicable to the case nor can the discharge coefficients thus found be utilized for computations relating to large flumes or canals. Ruehlmann (p. 269, 104) presents another formula for the case illustrated in Fig. 25, where an obstruction is interposed at E, thus slightly damming the water. The area a v with water level rj above the center of the opening at MN, is supposed to be located Fig. 25 at such a point where the parallelism of the filaments of flow is again established. Taking areas, velocities and heights as indi- cated in the figure, then from the principle of the conservation of energy, the following equation may be written out : or 2g (53) 56 HYDRAULICS Herein M represents the mass of water passing any section per second. For the case where A is very large in comparison with a, Poncelet first developed the following formula from Eq. (53) : i + (A - \aa (54) In the preceding chapters, on the derivation of new weir formulae, it is conclusively shown that the energy stored in a moving mass of water is not expended entirely in the production of flow through a contracted orifice, but that a large portion of that energy is lost as impact against all objects causing obstruction or contrac- tion, and hence the principle of the conservation oj energy is not applicable to a mass of water approaching a sluice or orifice. By inspection of Eq. (53) it is readily seen that the difference in height (h ^) furnishes the pressure height producing the accel- eration necessary to increase v to the value of F x and also to over- come the resistance encountered in reducing V to the value F,. / F 2 v 2 \ Also, the expression [ - ) is incorrect because the velocities \2g 2gl V 'j and v are in no wise related, since v is first accelerated to the value V and is then retarded to the value F r Furthermore, the pressure height necessary for the acceleration of v becomes greater in proportion to the increased resistance to flow through the area a due to details of design. And lastly, the velocity F t is not depend- ent entirely on the pressure height (h ??), but largely on the slope, width, depth and frictional resistances in the discharging flume. Hence for two different cases even though v and V 1 remained the same, the controlling height (h - vj) might be very different, a condition which cannot be rectified in the equation as it stands. The last expression I- ^-M from Eq. (53), represents the pressure height corresponding to a retardation F -- V 1 for the FLOW IN RIVERS AND CANALS 57 case of a sudden contraction in a closed pipe, according to Carnot's principle. However, this principle is not applicable to open flumes, because the water is not confined on all sides and at least a portion of the head (h y) can be expended in raising the level of the discharge surface. The velocity V is not suddenly changed to V 1 but the change is gradual and hence little of the vis viva is lost. All this is contrary to the underlying principles of the formula, and hence both Eqs. (53) and (54), are wrong and irrational. 2. Proposed General Solution. It is thus seen that no reliable formulae exist for any of the cases of flow just discussed. However, such problems can be solved with reasonable accuracy by employing the formulae previously derived for submerged and sluice weirs in combination with such formulae as those of Ganguillet and Kutter for flow of water in rivers and canals. Thus the problem presented in the above Fig. 25, may be solved with the aid of Eqs. (36), by taking the highest point of the dis- charge surface in the flume as the governing level of the lower pool and assuming the velocity of flow for this point equal to V r In the derivation of the previous formulae it was seen that the water in the lower pool influences the discharge quantity by exerting a hydrostatic counterpressure, and also by producing a suction on the area of flow. Hence the new formulae, which make allow- ances for these various conditions, are in every way applicable to a flume supplied from a lake or river, provided the depth and velocity of the water discharged thror ^h the flume is previously known. By referring to Eqs. (36), and' uie reasoning there followed, it is understood that the total head H 2 , between the upper and lower pools adjacent to a sluice gate, is not the governing factor deter- mining the discharge through the flume, but that the quantity of water discharged is determined by the dimensions and slope of the flume or canal. Hence, the formulae for flow in r* -rs and canals must be employed for the purpose of deciding the nee >ssary dimen- $8 HYDRAULICS sions and slope to deliver the required quantity, and this result is then used in the new formulae for flow through regulating works. From many gagings which have been conducted in the past, formulae of more or less accuracy are now available for flow through rivers, creeks, canals and flumes. These express the velocity or quantity of flow in terms of cross-section, slope and friction. For steep slopes in canals or flumes, Chezy's formula is probably most applicable. This formula, as revised by Bazin, in 1897, is = AV = ACVrs, where C = 22 for feet, and C = (55) and V = velocity in feet per second ; r = mean hydraulic radius in ^ feet = = area of flow divided by the wetted perimeter; 5 = slope w = fall, in feet, divided by length, in feet, over which the fall occurs. The experience coefficient m, depending on the roughness of the wetted surface, is given by Bazin (1897) as follows: For smooth cement or planed wood . . . . m = 0.06 For rough planks and brick m = 0.16 For masonry m = o . 46 For regular earth beds and slopes m = 0.85 For canals in good order .- . . m = 1.30 For canals in very bad order m = i . 75 These values are applicable to dimensions in feet and meters alike. For tabulated values of C, corresponding to values of r from one to ten, see tables 46 and 47, p. 565, Merriman's Hydraulics, 1904. Kutter's formula for the value of C in Eqs. (55) is probably FLOW IN RIVERS AND CANALS 59 the best in the present state of our knowledge for small slopes. This value, for r and V in feet, is .... ( 5 6) n I , < i + (41.65 + - Vr\ s For r and V in meters the formula becomes (57) s I The values of n in both formulae (56) and (57) are alike, and according to Kutter they are as follows : For well planed timber n = 0.009 For neat cement n = o.oio For cement with one third sand n = o.on For unplaned timber n = 0.012 For ashlar and brick work n = 0.013 For unclean sewers and conduits n = 0.015 For rubble masonry n = 0.017 For canals in very fine gravel n = 0.020 For canals and rivers free from stones and weeds n = o. 025 For canals and rivers with some stones and weeds n = o . 030 For canals and rivers in bad order n 0.035 Tabulated values for C, by Kutter's formulae, are given in Tables 44 and 45, p. 564, Merriman's Hydraulics, 1904; also in Trantwine, but most completely in Bellasis, p. 183 et seq. With the use of either Bazin's or Kutter's formulae then, the values of Q and V can be determined for any particular flume or canal. Or, suppose it is desired to design a flume to deliver an assigned quantity of water Q, then the required slope, the area 6o HYDRAULICS and the velocity may be so chosen as to fulfil the conditions for discharge. After thus assigning dimensions and slope to the flume, then the discharge area and head, necessary in the regulating works at the flume inlet, can be determined from one of the previously given weir or sluice gate formulae. Should the head H 2 , between the supply canal and the flume, be given, then the discharge area alone will require dimensioning. If the problem be stated thus : Given a regulating sluice gate of certain dimensions at the inlet of a flume, the slope s and width B of the latter being also given, to find the depth d, and quantity of discharge Q, through the flume, then the following solution is pro- posed : First compute the discharge through the sluice gate, using one of the formulae 30, 36 or 38 appropriate to the case in hand, and assuming that the discharge takes place into the air without any water in the flume. This value is naturally excessive and hence it can be reduced somewhat to obtain the first approximate value Q v which, when inserted into the formulae (55), will give ^4 preliminary values A 1 and F t and thus obtain a depth d l = ' Now by inserting the values A t and V 1 into the original sluice gate or weir formula first employed, a second quantity Q 2 is found as a second approximation. By a repetition of this process, using Q 2 in the Chezy formula ^4 to obtain A 2 and V 2 and finding d 2 = ? , and then employing A 2 and V 2 in the sluice gate formula to find a value <2 3 , finally obtain such a value Q as will satisfy both equations, from which * Q A = is found with reasonable accuracy. Perhaps a better way to carry on this successive approximation is to plot the curves representing the general relation between Q and dj for each of the two formulae, and these two curves will FLOW IN RIVERS AND CANALS 6l intersect in a common point, the Q and d of which will satisfy both equations. The foregoing discussion has been confined to cases where the axis of the flume is in continuation of the supply channel axis and the water in the flume extends upstream to the regulating works. 3. New formula for the case when all the available water of a lake or river should be diverted into a lateral canal or flume by a dam built normally to the river. A factory flume is required to furnish an assigned quantity Q, to be supplied from a river by interposing a dam. The flume is supposed to have its inlet just above the weir. This will require assigning dimensions to the regulating gate at the flume inlet and also to the flume itself and then give the flume such a slope as is necessary to deliver the quantity Q. Formulae for the solution of such problems have not hitherto been proposed, hence the following discussion is offered. / X \ /' n -E N N '/ r F ^ G D M "x-x^s -Xxv..;.,:;' " " ,. N .. 1 ., h \ D ^ZZ2$i$iii& Fig. 26 In Fig. 26, let EF represent a spillway, with_ regulating gate at GM , both normal to the flow of the river. JK is a lateral diver- sion channel making the angle 6 with the axis of the river. 62 HYDRAULICS A longitudinal section along A A, showing the flume inlet jja^a, is given in Fig. 27. The first case to receive consideration is when the water level of the river is even with the crest of the dam at E, and the entire discharge Q passes out through the flume. Fig 27 The hydrostatic pressure against the flume area is taken the same as against the dam, and the hydrodynamic pressure as found from Eq. (13) = p 5 = 7 Bk which (per unit of area) o V -. g becomes p 6 Using the dimensions indicated in the twojfigures, then ai = ae sin 0, and the dynamic pressure on the area ai is all that is trans- mitted to the discharge area. Hence the pressure on this entire area ^ = p=y- b l d sin 6. g Let the line Jr represent^ the magnitude of p 7 , which may be resolved into components Jg and Ja. Then the component Jg, normal to ae, is found from: tf g Jg = Jr cos (90 6} which may be represented by a rectangular prism of height d, in front of the discharge area. This prism will have a breadth b' ', and length sin 2 6. g FLOW IN RIVERS AND CANALS 63 The hydrodynamic pressure p e of the approaching water, acting normally to the banks, and hence deflected into the direction of the flume, maybe regarded active over the area b'd and in the direction JK. The hydrodynamic pressure deflected by the weir against the flume inlet, by analogy to Eq. (n), becomes p = 0.25 because H is now d, = - becomes 0.5 &', and = 90. This pressure p 4 , being equally distributed over the area b'd, may again be represented by a rectangular prism in front of the discharge v* area. Assume its height d, breadth V and length 0.25 - o The water flowing in the lower section of the river, exerts a dynamic pressure which is transmitted normally and horizontally against the flow area, and may be represented by a triangular prism of height (k d), a width = V sin 0, and a bottom length /?, ' : * ~~~7 | ^~^^^_ 4 v p/ 1 oil o i "^ -f _^ v 1 1 s * & \ ' "^ 1 r-i r z r e 3 1 1 ^^/^li^^ff^^^^^^/fi^///^//^^/^ i MUWJ!^ 1 Fig. 2 S which may be determined from Eq. (16) as follows: see Figs. 27 and 28. From Eq. (16) p = _ represented by oa^ = T. -- , in Fig. 28, and - is equal to Tr cos 2 -^ , in which the area VH is o'gti 2 64 HYDRAULICS om. Also the area Bk now_becomes (k d) sin 0, and since the resultant in the direction JK is wanted, the whole expression, for a unit width, finally becomes '*- All of these prisms are now combined in Fig. 28, as follows: _ ^ _ ^2 _ n y2 _ make ae = -sin 2 0; ee. = o. 25 - ; 0w = -- , and 8 = rr_; then 2 J? o o o the area e 2 e s #r represents the hydrostatic pressure and the final hydraulic pressure, active on the area a^e^e, is represented by the area ae 2 pr 2 a lf The resultant unit pressures along any horizontal filament of the pressure area are thus easily found from Fig. 28, and the following values are now derived: The hydraulic pressure along the surface filament is y S = ae + ee 2 = (0.25 + sin 2 6) . . . (590) o The hydraulic pressure along op, where the hydrostatic counter- pressure of depth 7\ becomes active, is S, = o^+^~p = S + d-^ = S + d-(T 1 -^~ \ (596) The hydraulic pressure at the bottom of the flume is Employing these values of S, S v and 5 2 , the discharge into free air, through the upper portion ao of the section, may now be found from Eq. (19), in which H = d T. + - , 2 g and s i -S = d-T 1 + , 2 g hence Q l = f FLOW IN RIVERS AND CANALS 65 In like manner the discharge through the submerged portion oa = T. , is found from Eq. (19) by making H = oa v 2 g then The total flow through the section Ifd = a^ae^e is then e = QI + e, Since the Chezy formula also furnishes a value for the uniform flow through the flume, then for any given case, for which the dimensions are assigned or determined from local conditions, the three unknowns can be found as previously indicated. When the flume is supplied from a lake, or river of sluggish flow, then the velocity of approach can be disregarded and the above formulae (59) reduce to simpler forms. Thus by making v = o, then n 5 = o, S 2 = S l = d T l H -- , and approximating Q 2 as was done in Eq. (28/), then Formulae (59) will also apply to the case illustrated 'in Fig. 9, because when no discharge takes place over the spillway, the approaching flow exerts equal pressure in all directions. The application of Eqs. (59) will now be illustrated by solving a problem such as is likely to occur in practice. The dimensions are all taken in the metric system, merely as a matter of convenience. 66 HYDRAULICS Problem. Given a river 10 m. wide, discharging 6 m. 3 per second. It is desired to build a factory flume having its inlet just above a spillway which is to be built normally to the river and of 3 m. height with vertical upstream face. The flume is to make an angle of 45 degrees with the axis of the river and must deliver the full river discharge to a factory, 2000 m. distant, when the water stands level with the top of the spillway. The section of the flume may be chosen as trapezoidal with i : i slopes and of such area that when the water is flowing with velocity V = i m. per second, it shall deliver the required Q when the depth of flow 7\, in the flume, is i m. The question then resolves itself into finding the width &', and depth d of the flume inlet, also the bottom width &", and slope 5 of the flume, necessary to discharge 6 m. 3 per second. The terms in Eqs. (59) will then have the following values: B = 10 m., Q = 6 m. 3 , k = 3 m., v = -^ =0.2 m. per second, Bk V = i m. per second, 2\ = i m., and 6 = 45. Also ^r = 90 when the river bank at the flume inlet is made vertical by a retaining wall. The four unknown quantities are b' ', d, b" and s, but as only three equations are available it will be necessary to make b' = b" which is equivalent to making the bottom width of the flume equal to the uniform width of the flume inlet. From Q=T 1 (b'+ 7\) V the value b'= -2-- T,= 5 m. = 6." The slope s, of the flume, is found by Chezy's formula, Eq. (55), wherein r= 2 o n = 0.7665 m. and Vr = 0.875. Taking 7.828 6 x x = 0.0326 or s = 0.001063, and the total fall required in the flume over the distance of 2000 m. will be 2. 12 m. By building FLOW IN RIVERS AND CANALS 67 the flume rectangular in section, with cement walls and bottom, to reduce friction, and maintaining the area 6 m. 3 = i X 6, and velocity V = i m. per second, this total fall could be reduced to about 0.4 m. Lastly, to find d from the formulae (59) the coefficients JJL and /^ must be known. In want of better data, these values are assumed each equal to 0.6, a value which is based on the experiments of Francis. Now substituting all of the foregoing values into the Eqs. (59) it will be found that the value d = i . 136 m. satisfies the three equations, making Q 1 = 0.64 m. 3 , Q 2 = 5.36 m. 3 , and Q = Q l + <2 2 = 6 m. 3 Hence, if the flume inlet be made 5 m. wide X 1.136 m. deep, thus making the water in the flume d T l = 0.136 m. lower than in the river, then the flume will discharge the required quantity Q = 6 m 3 . This example illustrates the points previously brought out in the present chapter, viz., that the slope of the flume is not a function of d 7\, but depends on the section and roughness of the flume. 4. New formula jor the case when a portion of the flow in a river should be diverted and the remainder be discharged over a weir built normally to the river. This case is exactly like the preceding, only that a portion of the water is discharged over the weir along a profile A^E^ Fig. 27, such that the depth H over the crest of the weir, may be main- tained aside from furnishing a stipulated flow through the power canal. Q will here represent the quantity to be diverted for power purposes. The problem of finding the hydraulic pressure active on the flume area a l aJJ 2 is very difficult, owing to the interference of cross currents, the retarding effect of which would depend largely on the relative quantities of flow in the two directions. Hence nothing better than an approximation could be expected. 68 HYDRAULICS The hydrostatic and hydrodynamic pressures exerted against the area of flow into the flume will be found as in the previous solution but modified as follows: 1. The flow of approach will not expend its entire hydro- dynamic pressure on the area of flow because the direction of the river current is past this area. 2. For this same reason, no hydrodynamic pressure will be deflected from the river banks against the area of flow. 3. That portion of the hydrodynamic pressure resulting from the flow against the area a^JJ^ Fig. 27, will now be divided so that in the present case this pressure will be assumed only half as great as it was with the water dammed. Fig. 29 These several conditions are shown on the profile, Fig. 29, where the graphic areas represent the pressures, exerted by various causes, against the area of flow. Figs. 26 and 27 still apply. The depth of flow into the flume inlet is now d + H, Fig. 27; the suction on the flow area produced by the water in the flume is represented by om = - - , and the uniform depth of flow in the flume is 7\. The pressure height a 2 o is a n o FLOW IN RIVERS AND CANALS 69 from which the quantity Q lt flowing through the discharge area a 2 o into free air, is found as In the derivation of Eq. (15), the hydraulic pressure exerted by the water flowing below the discharge section against this section, tf j, was found to be p Q = 7 Bk cos 2 . For the present case k d g 2 must be substituted for k, and b sin for B. Also the hydro- v 2 dynamic pressure becomes and retaining ^ as the angle of slope o of the river bank below the flume inlet, then the value p e becomes p ' = 7 -^ (k - d) V sin 6 cos 2 * . 2 g 2 Since this pressure may be assumed as varying with the depth, being a maximum at the bottom of the flume a l e v and zero at pm, it may be represented by a triangular prism of base /?, height - f T l --- J and length V. The previous equation for pj gives Hence the trapezoid opa l r 1 represents the total effective hydraulic pressure against the area of discharge ~oa^ into the flume. The pressure S l along the filament op and the pressure S 2 along the bottom of the flume a l r l are evaluated as = ej l = 7 - Bk cos , o directed towards the bank LM. FLOW IN RIVERS AND CANALS ^\ That portion p of the pressure ej lt Fig. 9, which is active over the height d, Fig. 27, of the flume inlet, is found from Fig. 26 as , p = y - db f - g sm by observing that LM = and MN = - - from Fig. o. tan sin But the pressure exerted normally to b', Fig. 26, is p cos (0 0) = 7 db' - n cos (0 0). g sin Now since p is uniformly distributed in the vertical direction, it may be represented by a rectangular pressure area of length &', height d, and breadth /? = - ^-^ cos (0 - 0) (63) g sin acting against the flume inlet. The hydrodynamic pressure exerted against the weir surface below the flume inlet is expressed by v* 1 g and the portion of p^ expended over the width of the inlet 3 / 3 , Fig. 27, against the river bank is v 2 ,, , sin This pressure p 2 acting against an inclined bank, making the angle ^ with the horizontal, causes a pressure against the flume discharge area tending to accelerate the discharge velocity. The amount of this effect is given from Eq. (15) as /L j\ T./ sn 2 ir p = y - (k d)b' T^ cos' g sin d 2 Since /> 6 is not uniformly active but creates maximum effects at the bottom of the flume, it is again represented by a triangular HYDRAULICS pressure area against the discharge opening. This pressure prism will have a length &', height ( J' 1 j and base /? x found from b' / ' - (T l - 2 \ 2g 7y mn 7 (k d) b' - '- cos" > g sm 6 2 whence The pressure areas, effective against the flume inlet, are graphic ally indicated in Fig. 30. Ai Fig. 30 The suction produced by the discharge velocity V on the section _ _ n _ _ a^ 2 is represented by om = - . The area aiaj represents the hydrodynamic pressure active on the flume discharge area, where ai = aj = /? from Eq. (63). The trapezoid # 3 /r^ represents the nV 2 total hydrostatic pressure plus the suction - active on the dis- charge area. Also, making rr l = /? 15 the triangle prr^ becomes the pressure area resulting from the flow below the bottom of the flume. Finally, transferring the triangle a 3 ei, representing the hydrostatic pressure due to the head H, to the position a^e^a in front of the discharge area, the whole hydraulic pressure will be represented by the figure a^ FLOW IN RIVERS AND CANALS 73 Since the pressure on the part section a 2 o, Fig. 30, undergoes an abrupt change e^e, which could not exist in reality and would necessitate some further complications in the resulting formulae, therefore, the pressure area opee^ may be replaced by the triangle opa r In this triangle the height and base have the following values : _ r n y* I _ _ _ a 2 o = \d + H + - T l and op = S = oi l + ij) y 2 g J making -7-, . (6 5 a) The quantity of discharge Q l through the part section a 2 o will then be 2 - S. . (656) Also, the resulting pressure a/ t = S t = a/ + rr l = S + p lt active at the bottom of the flume, will be c -s+ ^ k-d - fef) C G 2 T fAr/'^ Pi g nV 2 S . . (0$C) 2g The quantity of flow Q 2 through the lower portion a^o of the flume inlet will then become, after inserting the proper value for H into Eq. (19), nV 2 ! T 1 i " ^-S 1 ) . . (6 5 d) 3 The total discharge into the flume is then Should the angles (f> and 6 be equal, or nearly so, and the angle ^ be made 90 degrees, also making v = o, then these formulae (65) can be very much simplified as was done for Eqs. (60). 74 HYDRAULICS By combining these formulae with Chezy's formula, three unknown quantities may again be found as illustrated in previous examples. If a sluice gate were placed at the inlet to the flume then the hydrostatic and hydrodynamic pressures resulting from the back water in the river against the gate, may be found in the manner previously described, finally computing the discharge through the gate from one of the formulae (36) or (38) according to the case in hand. This closes the theoretical portion of the present treatise. CHAPTER VII. EMPIRIC COEFFICIENTS, i. INTRODUCTORY. IN the foregoing theoretical chapters no consideration was given to the various empiric coefficients employed. This subject is one of vital importance to the usefulness of any formulae, which latter can never be made to include all the disturbing elements always present when dealing with river hydraulics. The best that can ever be hoped for is the adaptation of rational formulae to the observed facts and to correct the shortcomings by the introduction of numerical coefficients. The effort here made in the direction of furnishing the rational forms for many of the complex problems frequently encountered in practice, by taking into account nearly all of the variable factors, should merit the appreciation of every hydraulic engineer. The attempt will now be made to evaluate the coefficients for these new formulae by employing all available experiments bearing directly on the subject. However, suitable and reliable experi- ments of the kind in question are not numerous. The very valua- ble and painstaking hydraulic experiments made by Messrs. A. Fteley and Frederic P. Stearns and published in 1883, Trans. Am. Soc. C.E.,Vol. 12, and the classic "Lowell Hydraulic Experiments," 1855, by Mr. James B. Francis, together with some more recent, but limited experiments, constitute about all the available data. Thanks then to the existence of these experiments, some reliable values for p in the new formulae have been found, although in many of the cases theoretically discussed, no empiric data are as yet at hand, from which to deduce coefficients. The rational formulae, previously derived, will still give values 75 OF THE UNIVERSITY 76 HYDRAULICS in excess of the actual when the coefficients are neglected. Hence these coefficients are really reduction factors to reduce theoretical to real discharge. The causes producing this effect may be thus summarized : a. The loss in its vis viva of the flow of approach due to impact and deflections near and at the discharge area. b. The friction and cohesion around the wetted surface at the discharge section, producing a retardation in the discharge velocity. c. The contraction in, and immediately adjacent to, the dis- charge section. This contraction, which is probably the most significant of the three named effects, is variable and depends on the dimensions of the overfall, on the interrelation of these dimensions, on the canal and its configurations near the weir, and on the velocity and depth of the flow of approach. In the following, two forms of contraction are distinguished, complete and partial. Complete contraction takes place when the discharge section is suddenly reduced on all sides of the orifice, as for dis- charge from a lateral orifice. Partial contraction occurs when the approaching flow is confined only on three sides, and the discharge section is contracted on three sides. The total correction for all the foregoing retarding influences will then be made by introducing an empiric reduction factor. This factor for discharge into free air will be called //. and for discharge through an entirely submerged section it will be called /* r For any given set of experiments the values of either or both of these coefficients are then determined by inserting all the observed quantities into the appropriate formulae to find the theoretical quantity of discharge, which latter divided into the observed dis- charge gives the corresponding ^. This method will be applied in the following to all available observations covering the widest possible range in weir and canal dimensions. For the most part the observations from the experiments of Messrs. Francis, Fteley, and Stearns, have been used. Aside from these the Cornell University experiments on standard sharp crested weirs made in 1899, and a few old experiments made by EMPIRIC COEFFICIENTS 77 Lesbros, in 1829 to 1834, and some made by Boileau, in 1845, were employed merely to extend the range of the former observations. It should be noted that the values ft are alike for United States and metric units, so long as all dimensions are expressed in terms of one such unit only. That is, ft is merely a ratio between two volumes, which ratio remains the same independent of the unit chosen. Regarding the coefficient in the formula for discharge through lateral orifices it should be mentioned that the usually accepted value is really the product of the numerical coefficient f and ft, so that the theoretical part of the formula is Q = %bH V 2 gH. However, the old values have been so widely adopted that a change would greatly confuse matters ; therefore, in all the follow- ing tables ft and f ft have been tabulated together. 2. COMPLETE OVERFALLS. (a.) Coefficients fi s for Eqs. (19) to (22), for weirs normal to the channel and no wing walls, hence B = b . See observations, Tables II and III. In all these observations B = b, $ = 90 degrees, and ^ = 90 degrees except for observations 46 to 50, where ^ = 20 degrees and a wide crest was used. The values for f ft and ft are found from Eqs. (19) to (22) and inserted in these Tables II and III. Since it is necessary to provide values for ft for any particular problem in hand, the following empiric formula (66) is offered for its computation. In this formula, ft is made to depend on b, k and H, and a constant as expressed by Eq. (66), = - W (66) The factor p is the experience factor for any crest compared with the standard sharp crest, and the quantity inside the parenthesis will represent f fi 5 which is the special value of ft for sharp crested 78 HYDRAULICS weirs. Hence for standard sharp crested weirs p = i, while for all other crests, p becomes a multiplier, the value of which depends entirely on the nature of the crest. The constants of Eq. (66) were derived from the values f p found from the tabulated experiments i to 22, Table II. When inserted in Eq. (66) and calling p = i, because all these experi- ments were made on sharp crested weirs, then for dimensions in feet: + 0.0001326 (67) / H \ , 0.00106 7 , <0 . 0.40105-0.00453 f J + - + 0.00043 b (68) and for dimensions in meters: 2 *' The values of f /* were then computed from Eq. (67) for each of the tabulations i to 32. The resulting errors, expressed in percentage, were entered in the last column of Table II. The tabulations 41-45, Table III, were similarly treated and comparatively close agreements were found even though the experi- ments of Boileau were not conducted with the same degree of accuracy, nor under the same conditions as those in Table II. The tabulations 33 to 40, in Table III, do not fit Eq. (67), because the values of k and H, and in fact all of the dimensions, are too small to promise any results which are comparable with those of Table II. However, the values of f i*. s were found from the new formula (22),* and for these values Eq. (66) becomes for metric units: (TT \ - ) 0.8738 H + 0.00048 b (690) d+k/ when the observations 33 to 37 only are included. For the remain- ing three observations 38 to 40 the following equation was found : (TT \ ^j k ) -0.63 H + 0.00048 b . (696) * The flume in these experiments widened out suddenly just beyond the dis- charge section, hence (b -f 0.042 H) was used in place of b in Eq. (22) as was originally suggested by Francis. EMPIRIC COEFFICIENTS 79 TABLE II. DETERMINATION OF p FOR EQS. (19) TO (22) WHEN B = b. ANDp=i. No. Original Experi- ment No. Measured Values. Computed from Eqs. (22) Error by Eq. (67) Per cent. B b k H V Q i* f*s ft. ft. ft. ft. ft. cu. ft. FROM TABLE XHI, EXPERIMENTS BY J. B. FRANCIS, 1852. I 67-71 9.992 9-995 5.048 0.79518 0.4071 23.790510.4068 0.6102 O. 2 2 44-5 9.992 9-995 5.048 0.97900 0-5403 32.56160.4052 0.6078 O.O 3 51-55 9.992 9-995 5.048 1.00026 0.5538 33.494610.4051 0.6077 o.o FROM TABLE XXVIH, EXPERIMENTS BY FTELEY AND STEARNS, 1878. 4 i & 5 5> 5 . 0048 3-56 0.1509 0.054 1.007 0.4246 0.6369 o.o 5 6 ' 10 5-o 5 44 3-56 0.23035 0.098 1.8685 0.4164 0.6246 0.0 6 ii ' 17 5-o 5-oo45 3-56 0.33685 0.168 3.284 0.4121 0.6181 O.I 7 18 ' 21 5-o 5-oo43 3.56 0.42425 0.233 4-6365 0.4097 0.6145 O.I 8 22 ' 27 5-o 5 49 3-56 0.4305 0.237 4.736 0.4096 0.6143 O.I 9 28 ' 34 5-o 5.0047 3-56 0.5116 0.301 6.134 0.4083 0.6124 + 0.1 10 36 5-o 5 . 0046 3-56 o-5477 o.33i 6.796 0.4076 0.6114 o.o ii 37,41,44 5- 5 0040 3-56 0.60076 o-375 7.8093 o. 4070 0.6105 o.o 12 46 &47 5-o 5 . 0042 3-56 0.69245 o-455 9-677 0.4063 0.6095 0. I 13 53 5-o 5-oo3 8 3-56 0.8047 0-55 6 12.147 0.4052 0.6078 o.o FROM TABLE XV, EXPERIMENTS BY FTELEY AND STEARNS, 1879. 14 10 19.0 18.997 6-55 0.4685 o. 151 20.178 0.4081 0.6122 + 0.6 15 9 I9.O 18.997 6-55 0.6460 0.239 32 . 685 o . 4066 0.6099 + 0.4 16 8 I9.O 18.997 6-55 0.8191 o-334 46. 760 0.4058 0.6087 + 0.3 17 7 I9.O 18.997 6-55 0-9853 o-433 62.023 0.4065 0.6097 o.o 18 6 I9.O 18.997 6-55 0.9873 o-433 62 . 061 0-4055 0.6083 0.2 19 5 I9.O 18.997 6-55 i. 1456 o-53 2 77.783 0.4052 0.6079 + 0.2 20 3 19.0 18.997 6-55 1.2981 0.632 94.192 0.4055 0.6082 0.0 21 2 I9.O 18.997 6-55 1.4546 o-737 112.066 0.4054 0.6081 O.O 22 I 19.0 18.997 6-55 1.6038 0.840 130.117 0-4053 0.6080 O.O FROM TABLE XIV, EXPERIMENTS BY FTELEY AND STEARNS, 1877. 23 30 5-o 4.996 3-i7 0.0746 0.023 0.3652 0.4450 0.6675 + 1.0 24 2 9 5-o 4-996 3-i7 0.0991 0.034 0.5498 0.4370 0-6555 + 0.1 25 24 5-o 4.998 3-i7 o. 1225 0.046 0.7526 0-4345 0.6518 -0.9 26 20, 21 5-o 4-9965 3-i7 0.16385 0.069 I-I536 0.4287 0.6431 -i-3 27 I7 19 5- 5-o 3-i7 0.21826 0.1033 I-75073 0.4219 0.6328 i.o 28 14, 15 5-o 4-999 3-17 0.25325 o. 1265 2-15975 0.4168 o. 6252 0.4 29 12, 13 5-o 4.996 3-17 0.32605 o. 180 3-14475 0.4143 0.6215 -0.6 3 6, 8, 9 5-o 4.998 3-17 0.48443 0.3117 5-696 0.4112 0.6168 -0.7 3i 4,5 5-o 4.999 3-i7 0.6737 0.488 9-376 0.4092 0.6138 -0.7 32 3 5-o 5-o 3-i7 0.8118 0.627 12.466 0.4088 0.6132 -0.9 NOTE. All above weirs were normal to the flow, had vertical faces making \l/ = 90 and had sharp crests consisting of a vertical, i inch planed, steel plate with beveled edge down stream. The above quantities represent means of the several observations bearing the observer's numbers in column 2. 8o HYDRAULICS DETERMINATION OF TABLE III. FOR EQS. (19) TO (22) WHEN B = b. No. Original Experi- ment No. Measured Values. Computed From Eq.(22) Error byEq. (6Q) Per cent. B m. b k H H Q ^s M, k + H m. m. m. cu. m. FROM TABLE XXm, EXPERIMENTS BY LESBROS, 1829-34. $= 33 i9*f o. 202 O. 2O2 0.043 -955 0.68953 0.012905 0.4045 0.6067 o.o 34 i9ff O.202 0.202 0.048 o-955 0.66551 0.012561 0.3987 0.5981 + 0.6 35 19** 0.202 O. 202 O.OyO 0.088 0.55696 o. 010532 0.3924 0.5886 0.0 36 i9fl O.2O2 O.2O2 O. IOO 0.0805 0.44598 0.008638 0.3813 0.5720 +0.5 37 i9ff O. 2O2 O.202 0.130 0.0705 0.35162 0.006864 0.3788 0.5682 0.0 38 i9f* 0.202 O.220 O.02O 0.0432 0.68354 0.003486 0-3747 0.5062 0.0 39 i9H O.202 O.202 0.030 0.0378 0-55752 0.00244 0.3380 0.5069 o.o 40 *W O.2O2 0.2O2 0.050 0.0228 0.31319 o . 000843 0.2653 0.3979 + 1.6 FROM TABLE IX, EXPERIMENTS BY BOILEAU, 1845. ^=90. Eq. 68. 41 0.895 0.895 0.340 0.0577 0.14508 0.0226025 0.4031 o . 6046 -3-9 42 0.895 0.895 0.340 0.134 0.2827 0.0823787 o . 4008 0.6012 -1.9 43 0.895 0.895 0.340 0.219 0.3918 0.17702 0.4029 0.6042 -0.6 44 1.616 1.616 0.468 0.0937 0.1668 0.0861429 0.4070 0.6105 -!-3 45 ~ 1.616 1.616 0.468 O.IIO 0.1903 o. 108462 0.4013 0.6020 - 2 -3 FROM EXPERIMENTS BY J. B. FRANCIS ON AN OVERFALL WITH CREST 2.95 FEET WIDE AND FACE INCLINED. ^=20. DIMENSIONS IN FEET. B b k H V Q I* I* p 46 89 9-995 9-995 5.048 0.5872 0.238 13-385 0.3610 0.4078 0.8852 47 9 9-995 9-995 5.048 0.7904 0.358 20.892 0.3581 0.4062 0.8816 48 9i 9-995 9-995 5.048 0.9767 0.480 28.914 0.3584 0.4052 0.8845 4Q 92 9-995 9-955 5.048 i-3 2 5 2 0.725 46.183 0-3583 0.4041 0.8866 50 93 9-995 9-995 5.048 1-6338 0.963 64.346 0.3609 0.4034 o . 8946 NOTE. Experiments 41 to 45 were on sharp crested weirs and belong in all respects to the class of experiments in Table II, though made in metric measures. The experiments 33 to 40, while in all other respects like those in Table II, were made on very small flumes and for weirs of very small heights k, for which Eqs. (67) and (68) did not apply. Experiments 46 to 50 were made on a wide crested weir and with ^ = 20 and 0= 90 otherwise the weir dimensions are like those of experiments i to 3. Hence this last set was used to show the effect of wide crests and the values fis were found from Eq. (67) for sharp crested weirs, while the values f /i were computed from the experiments, using Eq. (20). The coefficients p were then obtained from fa and ^, and the values here found should be compared with those given at the end of this chapter. EMPIRIC COEFFICIENTS 8 1 To reduce Eq. (69) to feet units, the two last terms in each equa- tion must be divided by 3.281. A few experiments on a wide crested weir were made by Francis on the same flume used for the experiments i to 3, and this furnished a means of determining p for this particular case. The results are given in Table III, No. 46 to 50, see also note at foot of this table. The subject of weir crests and their effect on discharge is of such vital importance that it will be treated separately; suffice it to say here, that the discharge for different crests may vary 20 per cent either way from the values obtained for standard sharp crested weirs. The following Table IV contains results taken from the Cornell University experiments, made for the United States Board of Engineers on Deep Waterways, in June, 1899, by Mr. George W. Rafter. These experiments cover a wide range of head H, though B = b and k are constant throughout. The values ^ s , determined from the new formulae (22), are seen to be uniformly decreasing for increasing values of H, while Mr. Rafter's coefficients, obtained from Bazin's formula Q = mbH \/2 gH, show no regular law. The new coefficients thus found were used to determine the constants in Eq. (66) for p = i, and the following equation was thus obtained for dimensions in feet : |f*.= 0.3851 - 0.0258 H + k + 2jj* + 0.000132 b . (yea) and for dimensions in meters: 3 A careful examination of the results in Table IV will show a very close agreement with the coefficients obtained in the previous and it should be noted that the same law of increase in /* s for decrease in H is clearly demonstrated in all the experiments here given. 82 HYDRAULICS TABLE IV. CORNELL UNIVERSITY EXPERIMENTS NOS. 20 AND 21, JUNE 1899, G. W. RAFTER. STANDARD SHARP CRESTED WEIRS WITH FREE OVERFALL. Measured Values. Computed from Eqs. (22). Com- puted . _ t No. from Coef. m. B=b k H V Q 2 Eq. (70) ft. ft. ft. ft. cu. ft. Ps IPs I 6-53 5.26 o-7 0.327 12 -7335 o . 4066 0.5421 0.4171 0.4204 2 I.O 0.527 21-8755 o . 4060 0-5413 0.4058 0.4174 3 1.5 0.902 39-8333 0.3981 0.5308 0.3962 0.4136 4 2.0 1.283 60.8596 0.3912 0.5216 0.3909 0.4106 5 2-5 1.679 85.0859 0.3879 0.5172 0-3873 0.4094 6 3-o 2. 062 111.2059 0.3831 0.5108 0.3846 0.4094 7 3-5 2-459 140. 6562 0.3819 0.5092 0.3825 0.4099 8 4.0 2.851 172.3920 0.3806 0-5075 0.3808 0.4112 9 4-5 3.240 206.4786 0.3798 0.5064 0-3794 0.4125 10 5- 3-6I5 242.1977 0.3784 0.5045 0.3782 0.4133 ii 5-5 3-979 2 79-5493 0.3761 -5 OI 5 o-377i 0.4135 12 6.0 4-335 3 l8 -7 2 93 0.3756 0.5008 0.3762 0.4136 NOTE. The above experiments were made on a standard sharp crested weir with = $ = 90 and p = i. (b.) Coefficients p> 8 for Eqs. (19) to (22) for weirs, normal to the channel but contracted by wing walls, for which B>b. The experiments given in Table V were used in Eqs. (21) to find i* s for the cases when B>b, and the values thus found were entered in columns 8 and 9. The original equation (66) must now be modified to include variations between B and b, and may be expressed thus : However, it was found for the experiments available for this case, that this equation did not give more accurate results than when the factor of /? was taken as b/B, and therefore this latter value was finally adopted. EMPIRIC COEFFICIENTS TABLE V. FOR EQS. (19) TO (21) WHEN B>b. Measured Values. Computed from Eqs. (21). Errors Original by Eq. No. Experi- ment B b k H Q (72) Per No. **. ft cent. ft. ft. ft. ft. cu. ft. FROM TABLE XIII, EXPERIMENTS BY J. B. FRANCIS, 1852. 51 72-78 13.96 9-997 5.048 0.62355 16.2148 0.4047 0.6072 o. i 52 56-61 13.96 9-997 5.048 0.79899 23-4305 0.4014 0.6021 o.o 53 n-33 13.96 9-997 5.048 0.99732 32.5798 0.3992 0.5988 o.o 54 5-10 13.96 9-997 5.048 i- 2 4757 45-5 6 54 o-3979 0.5969 o.o 55 1-4 13.96 9-997 5.048 i 55079 62.6019 0.3929 0.5894 + 1.0 56 79-84 13.96 9-997 2.014 0.64928 17.4428 0.4023 0.6034 +0.3 57 62-66 13.96 9-997 2.014 0.82624 25.0410 0.4000 0.6000 + 0.2 58 3 6 -43 13.96 9-997 2.014 1-05033 36.0017 0.3989 0-5983 o.o FROM TABLE XXVUI, EXPERIMENTS BY FTELEY AND STEARNS, 1878. 59 4 5- 3.008 3-56 0-2155 1.007 0.4146 0.6220 + 1.0 60 9 5- 3.0081 3-56 0.3301 1.869 0.4062 0.6092 o.o 61 14 5- 3.008 3-56 0.4843 3.284 0.3990 0-5985 O.I 62 3 1 5- 3.007 3-56 0.7398 6.134 0.3930 0.5894 o.o 63 40 & 45 5-o 3.0101 3-56 0.8708 7.8075 0.3904 0.5856 + 0.2 64 16 5- 2.3132 3-56 0.5824 3.284 0.3941 0.5911 -0.5 65 24 5- 2.3125 3-56 0.7478 4.736 0.3899 0.5848 O.2 66 35 5-o 2.3126 3-56 0.9548 6.796 0.3867 o. 5801 O.O 67 !9 5- 4.0058 3-56 0.4978 4.636 0.4038 0.6057 O.O 68 38 &39 5-o 4.007 3-56 0.70595 7-8i5 0-3974 0.5916 + 0-4 69 20 5- 3-3io7 3.56 0.5678 4-637 0.4018 0.6027 -0.9 70 3 2 5- 3-3ioi 3-56 0.6860 6.134 0-3993 0.5989 -0.9 7i 48 &49 5-o 3-3095 3-56 0.9307 9.6485 0-3956 0.5934 -0.7 NOTE. All of the above experiments were made on sharp crested weirs placed normal to the flow and vertical, so that = i/' = 90. While B > b, this differ- ence was equally divided on both sides for all experiments 51 to 66, though for the five last lines, 67 to 71, the channel contraction was all on one side of the flume. 8 4 HYDRAULICS The constants were thus determined from the experimental values 51 to 66 and furnished the following for dimensions in feet: 0.007328 . x -~ ~ +0.00093 b - (7 2 ) z. / \ +0.00305 b . (73) 2 / b - K = 0.3655 + 0.02357 ^- and for dimensions in meters: 2 /b\ 0.002384 - /i. = 0.3655 + 0.02357^ + -- g^ - In the same table all the values for f /*, were recomputed by Eq. (72) and the resulting percentage errors entered in the last column. This gave very concordant results. However, for weirs in which k becomes an important factor, it may eventually be necessary to employ the more complicated form (71). For weirs built diagonally to the flow, the above coefficients must necessarily answer the purpose until new experiments on such weirs shall have been made. 3. INCOMPLETE OVERFALLS. Very few experiments have ever been made on incomplete over- falls. In these the discharge takes place partly through a sub- merged area and partly as into free air, and coefficients i^ l and /* have been assigned to the two discharges respectively. Strangely enough, the submerged section has usually received Fig. 31 the larger /* and as this seems contrary to all reason the following explanation is offered. Let Fig. 31 represent the discharge section in which A BCD is the actual opening and AecdjB is the contracted discharge section EMPIRIC COEFFICIENTS 85 as for discharge into free air, making ft = 0.72. Now if the lower portion of the section, EFCD, is submerged, then most of the con- traction belongs to this portion, while the upper area AEFB includes little of the contracted area. If these areas retain any- thing like this relation after the lower portion of the section is sub- merged, then certainly the ratios which the contracted areas bear to the respective flow areas must be less for the submerged portion. Hence it will be natural to assume ^ < ft, and as there is great liability of meeting with unforeseen influences on a partially sub- merged section, the former values ^ for complete overfalls cannot safely be employed here. The experiments of Francis, given in Table VI, offer a valuable contribution to the present subject and from them values of /^ and ft were determined for Eqs. (28). The final equation (28) when solved for /^ or ft gives Q-^b\H 1 -^\Vg(S l + S 2 ) . . (740) (74&) & \H. - either of which may be used when all the other terms are deter- mined from experimental data, but a single set of observations will not be sufficient to solve one equation with two unknowns. The following process was employed for the determination of ft x and ft. Values were substituted in Eq. (740) for two sets of observa- tions having approximately the same dimensions for H y Now from the behavior of f ft for complete overfalls, it is found that for similar values of H 2 , f ft is practically the same. Hence, each such pair of equations may be equated so as to involve only fi lt which would then represent the mean value for the two sets of observations. Then using this value of ft 1 in each of the pairs of equations, the mean value f ft was found. 86 HYDRAULICS CQ . o 00 NMOONON^^J-oOt^ CSCNMHMHMOO O OO t^-OO NO O OOOOOOOOOOO ddoooooo O OOOOOOMMOHH ON ^" N ^- loO t^ 1 * O^ O^ w d o d d o o w ON t^O OO CO vo O ^ t^. W H NO OO t^- O ^~ M O M N M Q Q OOOOWHOOO M ON ON ON ON N O) t^OO -* x js Hi bfl *3 ^"^ fi g W "frt >-( O^ o O pq ^5 . C o w nJ 0, etc., and careful attention should be given to measure such dimensions as will make possible the separate determination of /^ and //. It was impossible to fit any formula to the values /^ and /* of the first three experiments. For heights H 2 , between 0.643 1.119 f eet > tne following formulae were adopted: For dimensions in feet: . O.OI038 . , T /* = 0.4001 + 77^- + 0.000146 . . (760) jM t = 0.5274 + O.OOOI46 0, and for dimensions in meters: 0.00316 j = 0.5274 H- 0.00048 b. f 0.00048 b _ (766) EMPIRIC COEFFICIENTS 89 When the pressure height H 2 becomes larger, the following for- mulae are preferable: For dimensions in feet: f /JL = 0.4001 + a 799 + 0.000146 b LO (77*) H i= = 0.5346 + 0.000146 &, and for dimensions in meters: o , 0.00244 , 4 IJL = 0.4001 H -f 0.00048 o ^2 ^ i= = 0.5346 + 0.00048 ft. All the above formulae (76) and (77) are for sharp crested weirs for which p = i. For any other kind of crest the values for p must be determined from experiments still remaining to be made. 4. SLUICE WEIRS AND GATES. Among the available experiments which can be used for the determination of JJL in Eqs. (36), there are only a few which are entitled to any confidence. Those given in Table VII comprise about all for which sufficient data were given to permit of their use. The largest experiments of this class are those made on the Danube Canal, 1876 to 1883, by Freiherr v. Engerth. These three experiments are tabulated as Nos. i, 2 and 3 and the /^ were found by inserting the observed values in the new formulae (36), using n = 0.67. For further values /^ the experiments given in the table were used. These were all made prior to 1880, by the following: No. 4-11 by Lesbros; No. 12-22 by Boileau; No. 23-34 by Weisbach; and No. 35-41 by Bornemann. The computation of values /^ for these experiments in Eqs. (36) resulted in figures entered in the table, and the percentage differences between these and the values found from Eqs. (78) to (81) are given in the last column. 90 HYDRAULICS / """ s illl^ s -^ Q j> M M~M iw^ -^ S *i| O t^ co - 00 ONt^QvOOO ONI^ VO VO VOVO VO VO VO VO dodddddd fIS odd 01 if* ^ vo *^ S S odd co O co voO t^.00 co rf vo O M O O t^cd ON t-OO t^ ^- CO M O\ vOvOOO^OOO vo M O\vO ON CO W 00 vO OS ONOO O t ON ONOO \O vo vovO vo vo vo vo vo t^ 10 00 t^. M vo vO t oooooooo OOOOOOOO O O O Measured Values. All in Meters. O* ON co co 0* !>. CO \O t^ O\OO co M vo t^ M ** VOOO M O ^ M M COCOM VON t^vo t^HC4MO\OOcO vo t^ co t^vO O vo cooo t^. (S 00 O "* ON N vO t^GO Tt- t^ Tf t^O M t^vocowvo coO co O^S^ vO co ON 00 ON ON O O ** t-~ O O O M N ^j-O OO oooooooo OOOOOOOO 000 nf t^ <+ 2 ON HI Tj- O M M CO 1OO MO MWt^-O vo t^ O t^OO l^- co ON . vd t^ vo vo v/-. O M CO ON M M Tt 10 n- d d d a *> M CM O vo r^ oo" f^ 00 O vo O ON ON 10 00 CM Tt VOVO VO odd M O odd OOOOOOOO dodddddd o o o odd VOVOVOVOVOVOVOVO dodddddd d to w \r> M ro o CO W CO cocovovoO O O O OOOOttW^J--. COOO 00 ^xCO \oo M oo OwO co M 00 vO CO M CO M 10 Tj- O t^-VQ M M M * 1 1 ' N ** (N M VO a M H N CO * vovO t^-00 ON O M N co rl- vovO f00 ON S S Iq juauiuadxg | -i^jaSug sojqsai nTOipg 8 EMPIRIC COEFFICIENTS * * S% xo Tf to 04 O t"- ft- CO M M O C* * O O O O M M 1 + 1 1 1 + 1 1 1 1 I 1 co t^ W vo VO l* O + + 1 1 1 1 1 co *** O co ON O r>- too oo O t-~ 04 Tf 00 00 O O J>. t^O t t^oO XOVO XO XO 0* t>. ON t^vO t^ t^OO vO O 000000 O O O O O O O O O O O O vO Tf co ON xo IO t^- 04 XO O O4 t- t^oo oo t^> * 00 Tf to Tf M vo to ON M Tf Tf O t"^ t^ t^^OO t^OO w xo co * ON N co * O M ON M M tH oo t^oo *>. ON t^ r- O O O O O o o o o o o O O O O O O roO 04 O to 04 vO 00 xo co ON O rj- to O ** O T^- ON t^ "* xoO xo ON t>- ^ t^- O xo CO co CO TJ- 04 CO co O O O O O O O O O O O O O O O 666666 O O 666666 O O 6666666 eg\g s- - o 5 o 5 o 3 o 2 2 XOXO XO to xo xo f"*- ^* t^* O T}-OO t> M M O M O XO 04 ON PO M XO O4 M M O M M O O O O O O O O O O O O O O xo xo ON ON xo xo CO CO CO - 000000 O O O xo xo ON ON tO XO 666666 CO PO CO PO CO PO 666666 Tf Tf Tf Tf W 04 04 Tf Tf Tf Tf O O O to xo xo XOCO OO OO 6666666 M M * * POOO Tf O ON w M M M VO ON XO t~- ^ 04 MM M PO Tf XOVO t^OO ON O M M CO rj- toO f00 ON O M CO co co CO CO Tf Tf qow ISP.&. uumuaujog III g 1 1 -a ^^^^ 92 HYDRAULICS The coefficients /*, which the various observers found from the use of their own formulae, are given in the i2th column, and the following comments regarding these may not be out of place here. Excepting the first three cases, the general comparison shows the new values to be somewhat less than those given by the observers themselves, and this is undoubtedly so because the new formulae take into account several conditions not considered by the older formulae. According to the experiments of Engerth, Boileau and Weis- bach on sluices wherein the sill was level with the bottom of the flume or canal, or when k = o, H 2 varies with p v while accord- ing to Lesbros for k = 0.54 m. this relation is inverse. The only experiments which form exceptions to this rule are Nos. 2, 21, 31 and 33, for which reason it is fair to assume that some small errors were committed in those cases. Weisbach's experiments, 23-28, were made on a sluice without side and bottom contractions and a rounded edge on the gate, which explains the rather high values there obtained for // r It should also be observed that for these experiments the coeffi- cient jtj, for submerged discharge, is smaller than for free dis- charge, which is exactly the opposite of our previous findings in this regard. However, the present case is different inasmuch as the contraction, whether for free or submerged discharge, takes place all around the opening when dealing with a sluice gate. Also the suction produced by the water leaving the discharge section, tends to increase the quantity for submerged discharge as against free discharge where no suction occurs. Therefore, these phenomena would indicate just what the values // indicate to be the true condition. Hence these tabulations of /^ and /* show quite conclusively that the new formulae apply to a rather wide range of conditions and can, therefore, be expected to give very much greater accuracy than was possible to attain with the older formulae which did not include the many variable influences always attending hydraulic problems. EMPIRIC COEFFICIENTS 93 However, these experiments are of such nature that their prac- tical applicability is very limited and until more extensive experi- ments of this kind shall have been made these values must be accepted as the best information available at this time. The following formulae for /* and /* t are given with reserve because the data on which they are based is entirely inadequate for the deduction of such formulae. However, they are better than nothing and may occasionally serve a useful purpose in want of something better. Bornemann gave a formula for /^ which seems to give fairly accurate results. This formula with slightly modified coefficients and another member involving b was used. For the first three experiments by Engerth, there is not enough data to justify any formula; however, the following one is offered for submerged discharge through large sluice openings : For dimensions in feet: 0.2717 \/a ^ = 0.7069 + 0.00093 b (7&0 H-- and for dimensions in meters: 'a + 0.00305 b . .. . (78^) The other experiments on submerged discharge are too unreliable and, therefore, formulas could not be found except for the last experiments 35-41, by Bornemann, for which the following are given : For dimensions in feet : /*! = 0.4988 + ^^ ? + 0.00093 & - H-- and for dimensions in meters: 0.4988 + ' 149 5 2 + 0.00305 b . H-- 2 94 HYDRAULICS The experiments 4-11 by Lesbros, for free discharge with com- plete contraction, give for dimensions in feet : 0.0135=; \/a 0.0382 H = 0.5708 H -- OJJ + ^z: + 0.00131 b . (8oa) and for dimensions in meters : 0.01355 * a i 0.021 , , ,_ , x / =0.5708 + , H -- ;=-* + 0.00431 b . (806) For the experiments 12-19 by Boileau on free discharge through gates of same size as the flume, without bottom sills, thus allowing the discharge to proceed without side or bottom contraction, the formula becomes for dimensions in feet: 0.004Q2 , , , -i 3*- + 0.000146 b . a 2 and for dimensions in meters: 0.0l8o8 Aa 0.00144 , or /o I.\ -575 ! -- / + *" + 0.00048 6 . (810) For experiments 23-28 by Weisbach, for free discharge like experiments 12-19, without contraction but having the bottom edge of the gate rounded off, the following formula was found for dimensions in feet: 0.21036 \/a 0.00718 ,. , /o\ // = 0.8452 -- 1 H -- '- + 0.000146 b . (820) and for dimensions in meters: 0.21036 Va 0.00210 , , /0 , x H = 0.8452 -- ^ -i --- - + 0.00048 b . (826) EMPIRIC COEFFICIENTS 95 5. END CONTRACTION. In all the new formulae it is supposed that the discharge continues with a uniform width of section b which is true when the water, after leaving the weir crest, is confined by lateral walls. All of these cases are known as discharge without end contraction, the term end being applied to the ends of the weir adjacent to the sides of the canal. When the discharge is not thus laterally confined and is allowed to take place into free air, then end contraction takes place and the quantity of discharge is slightly reduced. The exact amount of reduction thus produced can only be estimated. Francis proposed a simple formula for this purpose in the form of a correction to the length b of the weir, by sub- tracting an amount o.i H for each such end contraction. Thus the new b = (b - o. i nH) . . . . . . . ... (83) where n = number of end contractions. Undoubtedly this covers the case in a very approximate manner but in the absence of other more extended experiments to determine the effect of end contractions on the discharge, the above advice must be followed. 6. WEIR CRESTS. Coefficient p for Different Styles of Crests, for Complete Overfalls. The following data have been taken from pp. 71-75, " Hydraulic Tables" by Professor Gardner S. Williams and Mr. Allen Hazen, published by Messrs. John Wiley and Sons, in 1905. For the kind permission granted by these gentlemen, to use this very val- uable collection of data, the author here repeats his expression of gratitude and thanks. The following tables were derived from experiments conducted under the personal supervision of Professor Gardner S. Williams at the hydraulic laboratory of Cornell University between 1899 and 1904. Some of these experiments were made for the United States Geological Survey, some for the United States Board of Engineers 96 HYDRAULICS on Deep Waterways, and those on the Croton Dam model were made at the request of Mr. John R. Freeman. The figures, as here given, represent the results deduced by Professor Williams, and are entitled to the utmost confidence. The triangular weirs with sloped face upstream, are omitted here because this effect is included in the new formulae. In general then, the coefficient p from Eq. (66) is found to be p = t where ft is the empiric coefficient for discharge over standard sharp crested weirs and coefficient /* is the for any other crest. Hence, having found ft from the foregoing, the /* for any particular kind of weir crest, other than the standard, may be found from /* = ftp, or f p = f ft p (84) This is equivalent to calling p a multiplier with which to multi- ply ft for standard sharp crested weirs, to find /* for any weir of exactly the same general dimensions and conditions of flow differing only in shape of the crest. Hence p may also be called the crest coefficient, which represents the change in the contrac- tion of the discharge section due to the change in the crest from the standard sharp crest. In order that p may be determined experimentally for different kinds of crests it is, therefore, necessary to make duplicate experi- ments first with the standard sharp crest and then with the experimental crest, all other conditions remaining exactly alike. This was not done by Mr. Rafter for the very elaborate experi- ments made by him at Cornell University in 1899. There all the experiments for standard sharp crested weirs were confined to varying the head H, all dimensions of weir and channel remaining constant. The experiments on other forms were widely varied in height k and other dimensions and hence the values expressing the relation intended to be expressed by p, did not contribute much of scientific value, because the cases thus compared were not strictly comparable. EMPIRIC COEFFICIENTS 9^ The following data from the above named "Hydraulic Tables" is believed to be the most accurate in existence at this time and aside from a few general remarks the tables are self explanatory. All the formulae previously presented do not include the element of weir crests and the above values for /, are for sharp crested weirs of standard type, so that the /*, once found for a particular standard weir, the multiplier p to be used to find // for any other than standard weir, is given below. " On all the models having vertical downstream faces, including model P, air was admitted to the space underneath the sheet. On models D and E, experiments were made with the space under- neath the sheet unaerated, so that a partial vacuum existed there, which is shown to increase the discharge about 5 per cent at the high heads. For the weirs with inclined downstream faces, models F to O inclusive, no air was admitted under the sheet. A com- parison of the results upon models G and H, shows the effect of rounding the upstream corner of the weir to be an increase in dis- charge of about 4 per cent at the high heads." 9 8 HYDRAULICS TABLE VIII. RECTANGULAR FLAT CRESTED WEIRS. VALUES OF p FOR SAME 6 AND k. H Feet. When w = 0.48 ft. 0.93 ft. 1.65 ft. 3.i7ft. 5.84 ft. 8.98 ft. 12.24 ft. 16.30 ft. o-5 0.902 0.830 0.819 0.797 0.785 0.783 0.783 0.783 I.O 0.972 0.904 0.879 0.812 0.800 0.798 o-795 0.792 i-5 I .000 0-957 0.910 0.821 0.807 0.803 0.802 0.797 2.O I. 000 0.989 0.925 0.821 0.805 0.800 0.798 o-795 2-5 I.OOO I.OOO 0.932 0.816 0.800 o-795 0.792 0.789 3- I. 000 I.OOO 0.938 0.813 0.796 0.791 0.787 0.784 3-5 I.OOO I.OOO 0.942 0.810 -793 0.787 0.783 0.780 4.0 I .000 I.OOO 0.947 0.808 0.790 0.783 0.780 0-777 H = depth of water flowing over the crest of the weir. TABLE IX. COMPOUND WEIRS. VALUES OF p FOR VARIOUS TYPES OF WEIRS F TO L. H Feet. Type F. Type G. TypeH. Type I. Type J. Type K. TypeL. o-5 0.964 0.932 0-934 0.968 0.971 0.971 0.971 I .0 1.026 0.982 I .000 1.008 1 .040 1 .040 0.983 !-5 1.064 1.015 1 .040 1.032 1.083 1.092 I. 012 2.0 1. 066 1.031 1.061 1.041 1 .105 1. 126 I.04O 2 -5 1.025 1.038 1.073 1.043 1.118 1. 146 I-57 3-o 0.992 1.044 1.082 1.044 1.128 1.163 I .072 3-5 0.966 1.049 i .090 1.045 1.136 I-I77 1.085 4-0 0.944 I -53 1.097 1.046 1.144 1.190 1.097 NOTE : See cuts on opposite page. EMPIRIC COEFFICIENTS 99 y//////M//////////^ ////////////M^^^ 100 HYDRAULICS TRAPEZOIDAL WEIRS. TABLE X. VALUES OF p FOR TYPES A TO E FOR SAME b AND H ft. Type A. Type B. Type C. Type D. D with Vacuum. Type E. E with Vacuum. o-5 0.968 i .060 1.043 1.069 1. 088 1.069 1.069 I.O 1.071 1.079 1.040 1.079 1.106 1.079 1.079 i-5 1.077 1.091 1-037 1.084 1.117 i. 088 1.092 2.0 1.081 1.096 1.027 I-OS7 1.092 1.063 1.083 2-5 1.077 1.093 1.015 1.041 1.079 1.049 1.081 3-o 1.074 1.090 1.005 1.028 1.068 1.039 i. 080 3-5 1 .071 1.087 0.996 1.018 1.059 i .029 1.079 4.0 1.069 1.085 0.989 1.009 1.051 I. 021 1.078 EMPIRIC COEFFICIENTS 101 COMPLEX WEIRS. -38.-S4- TABLE XI. VALUES p FOR TYPES M TO P. H ft. Type M. Type N. Type 0. Type P. -5 0.964 0.897 1.095 o .920 I .0 0.965 0.946 1.088 0.915 !-5 0.963 0.999 1.084 0.914 2 .0 0.949 1 .025 1.069 -935 2 -5 0-933 1.039 1 .051 0.950 3- 0.920 1.052 i-35 0.962 3-5 0.911 1.063 i .024 0.972 4.0 0.903 1 .072 1.014 0.982 APPENDIX A. A COLLECTION OF WEIR FORMULA PROPOSED BY DIFFERENT AUTHORS, WITH DISCUSSION. i. COMPLETE OVERFALLS. THE following, Fig. i, shows the lettered dimensions used in the formulae for complete overfalls. Fig. i Eytelwein in his "Handbuch der Mechanik fester Koerper und der Hydraulik," 1823; also, Weisbach in "Huelse's Maschinen-Encyclopaedie, " 1841, give the following formula: 2 g g Let A 2 L represent the level of a quiet reservoir such that JG l = 2 g ' 102 APPENDIX A 103 Then according to Eq. (i), Chap. I, the discharge through an opening of height H Q would be 3 and taking off the small quantity passing through LE and which is _2 bV /v*\* 3 W' the final discharge through E^E becomes Q = Q l q = the above formula (i). The Eytelwein-Weisbach formula is thus derived from the funda- mental Eq. (i), for flow through a lateral orifice, and, therefore, does not apply to discharge with initial velocity of approach. The four following proofs of the incorrectness of the Eytelwein- Weisbach formula are now presented. 1 . No consideration is given to width and depth of the approach channel; to the height k of the weir, nor to the shape of the weir. All of these circumstances are known to exert a very considerable influence on the discharge. 2. The flow of approach is supposed to exert an hydrodynamic pressure of making the total hydraulic pressure on the discharge section (H H ). It was shown, however, that this depends on V 2 gl H, b y B f and k, and may have any value between 4 ( ) and \2 gl lS (fg 3. These authors were undoubtedly cognizant of some of these defects in their formula and sought to correct the error by the coefficient /*. However, a rational formula must express the true influence of all the hydrostatic and hydrodynamic pressures and 104 HYDRAULICS weir dimensions on the discharge. The coefficient n should serve merely to rectify the theoretically correct discharge to cover the unknowable effects due to adhesion, cohesion, friction and contraction. 4. The values of ;*, being variable in character, are necessarily without value except within the scope of the experiments from which they were obtained. For rational formulae this should not be the case, at least not to any great extent. This becomes the more important when it is realized that all hydraulic experiments must be confined to reasonably small conditions, and unless a rational solution of hydraulic problems is made possible the ulti- mate solution applicable to extensive waterpower plants would remain impossible. The above Eytelwein-Weisbach formula was discussed at some length because it is in most general use and also because the objections cited will be found equally applicable to most other formulae. Navier proposed the following formula, wherein the head EO = 0.2753 & ( see Fig. i), a relation based on the doctrine of least work. The formula is Q = \lA VTg [i - (0.27S3) 1 ] H ! = 2.5261 ftbH* . . (2) This is so manifestly incorrect that scarcely any comment is necessary. Professor Ruehlmann remarks of this formula that it does not agree any better with experiments than do the Scheffler and Braschmann modifications of the Weisbach formula. Lesbros experimented with very small overfalls (b = & inches; k = 20 inches; B = 12 feet) and computed the values ft from the fundamental formula which applies only for discharge through a lateral orifice when there is no velocity of approach. He determined 2000 different values for /, all for the constant condition b/B = 0.054. These coefficients are entirely without value and it is difficult to under- stand how any person could expend the mentality required for APPENDIX A 105 these computations when the futility of the undertaking must have been apparent. Weisbach later continued his experiments, commenced in 1842, and established the phenomena of incomplete contraction. On these experiments, made with 8-inch wide openings through the thin wall of a 1 4-inch wide flume, he based the two following formulae: When b < B and when b = B Q. - + 0-3693 r ( 3 b) The coefficients f /* are those given in Poucelet-Lesbros' tables based on 8-inch wide overfalls. Here again the second factor represents the discharge through a lateral orifice and the first factor is a variable coefficient of irra- tional form. Besides the apparent incorrectness of the formula it is certain that coefficients derived from 8-inch openings are not applicable to Weisbach' s experiments. Boileau, in 1845, was induced to make other experiments on complete overfalls for cases where b = B = 0.94 to 5.31 feet. He gave the following formulae for discharge and /*: or . (4) where e = OE,, Fig. i. 106 HYDRAULICS It is clearly seen that Boileau did not consider the hydrodynamic pressure against the discharge area and the weir, nor the velocity of approach. Instead, his coefficient is made to cover the varia- tions due to these pressures, the weir dimensions and contractions, etc. As previously shown, such formulae cannot have any general applicability. Redtenbacher , in 1848, proposed an empiric formula which was based on Castel's experiments on complete overfalls 0.4 to 29 inches wide. He gave (5) Q = 0.381 + 0.062 bH V 2 gH and when b = B, Q = 0.443 bH \/2 gH. According to Redtenbacher these formulae are applicable only when BT = 5 bH, and b is at least equal to B/$ and k T 1 equals at least 2 H. Finally the weir must be sharp crested. No consideration is given to hydrodynamic pressures due to the flow of approach, nor to the dimensions of the weir. The first term, which takes the place of /*, is a constant for all values of b/B = constant. It is hardly necessary to point out the uselessness of this formula. J. B. Francis, in discussing his epoch-making " Lowell Hy- draulic Experiments," 1855, modified Weisbach's formula to obtain the following: Q = 3-33 0- o-io nH)H* (6) wherein n = number of end contractions; H = measured height of water above the weir crest; H = a pressure height corrected for velocity of approach and given by Francis as In these experiments B = 13.96 feet; k = 2.04 feet to 5.05 feet; and b = 9.995 to 9.997 feet. The weir crest was the sharp plate since adopted as the standard form for experimental weirs. OF THE UNIVERSITY | OF APPENDIX A lO/ The coefficient 3.33 in Eq. (6) is the average of values ranging between 3.3002 and 3.3617 and is a value for f /* \/2 g. Hence j = 0.6228 is considered constant for all weir dimensions and depths of water, which is certainly wrong. When there are no end contractions, b = B and n = o, and Eq. (6) becomes 2 = 3-33^0* . . .:. .,.;.. . (66) which cannot be regarded as a general law because when b = o. 10 nH in Eq. (6), then Q = o, which is an apparent contradiction. These objections, together with the assumption that the flow takes place over a height H while in reality the height is only H, render the formula quite valueless except in special cases resembling the Francis experiments. This in no wise vitiates the high value which the very accurate experiments of Mr. Francis possess, irrespective of any theoretic deductions which may now or have ever been drawn therefrom. Braschmann, in 1861, proposed a formula based on the principle of least work. It was a modified form of Navier's formula using Castel's and Lesbros' experiments for the determination of his coefficients. The general form is Q = fibH V 2 gH where ft = 0.3838 + 0.0368 - + t'^jj&> . ( 7 ) The objections to this formula are apparent from the preceding. Bornemann, in 1870, experimented with weirs for which b = TT B = 3.8 feet; H = 2.75 to 8.27 inches; and = 0.2 to 0.8. His formula is, for # < - T, Q= (0.5673 - 0.1239 y~) bH vTp for H > - r, Q = (0.6402 - 0.2862 y ^ b(H + z) (8) 108 HYDRAULICS Bornemann himself points out that his formulae are not appli- cable unless b = B, and expresses the hope that somebody may eventually succeed in deriving mathematically correct forms. The first formula does not include velocity of approach and the second formula does this by introducing the fictitious height H + z. G. Hagen in his book "Die Stroeme," 1871, adopts the Eytel- wein-Weisbach formula. M. Becker, in 1873, g ave tne following formula: For velocity of efflux: and for quantity .... . . (9) This formula is based on the incorrect assumption that the velocity of approach exerts an hydrodynamic pressure against the discharge area only and that the mean velocity of efflux over the total depth corresponds to a pressure head H + , which 9 2g cannot be generally admitted. K. Pestalozzi, gives the Eytelwein-Weisbach formula, which need not be repeated here. Ruehlmann expresses the opinion that the scientific value of all the formulas above given is very small. He believes, however, that they are applicable to cases closely resembling the experiments on which they were founded. After presenting numerous examples he shows that, even within range of the experiments, the discharges found by formulas 3, 4, 6, 7 and 8 differ by amounts varying from 12 to 19 per cent. Bazin. The objections previously cited with reference to the Weisbach formula apply equally to the following Bazin formula, 1898, Ann. d. Fonts et. Ch., p. 223, where (10) APPENDIX A 109 Cipolletti uses a modification of Weisbach's formula, also a simple form of the Francis formula, and, in a very unscientific manner, adapts these to some experiments of his own. The attempt to use these experiments for the determination of fj. in the foregoing chapter proved futile because the necessary weir measurements were not published in the report describing the experiments. Fteley and Stearns, as a result of their very extensive hydraulic experiments, made in 1877 to 1879, published the following modi- fication of the Francis formula: Q = 3.31 m* + 0.007 * (11) The addition of the last term alone distinguishes this from Eq. (66) and" hence the criticisms previously made to the Francis formula also apply here. 2. INCOMPLETE OVERFALLS. Dubuat, under suppositions discussed in Chapter I, derived the following formulae and gives /* = fi t = 0.633: Q = f f*bH t V^lT 2 + fiJfH, \/2~^T 2 . . (12) A CD Fig. 2 Redtenbacher says that the derivation of formulae for incom- plete overfalls is connected with unsurmountable difficulties and adopts Dubuat's formula, making f /i = 0.57 and /^ = 0.62 both constant. HO HYDRAULICS The first half of Eq. (12) is incorrect, because it is based on the assumption that discharge through the height H 2 takes place as for discharge into open air, which is not true. The second term does not include velocity of approach nor suction due to velocity of discharge. Ruehlmann advises against the use of these formulae. Lesbros based the following simple formula on his experiments made in 1829 to 1834. Q = -tibH ^H 2 ...... (13) where /J. is variable and made to depend on the ratio H 2 /H. In principle, this formula is entirely wrong, assuming as it does that the discharge takes place over the whole height H and with a uniform velocity \/2 gH 2 . Bornemann, as a result of his experiments, made from 1866 to 1872, on overfalls 22 to 45 inches wide, gives this formula: I'll (H in which /* = 0.702 0.2226 \ ~j-* + 0.1845 and ^ = ^() 2 . 2g 2g\bTl The first member of Eq. (14) would be true provided the upper discharge did take place through the height H 2 into free air. The second term* is incorrect because it assumes the submerged section as discharging into quiet water. G. Hagen gives no formula for incomplete overfalls but states that the upper layer of flow may be regarded as a complete over- fall and that the submerged portion is subjected to a uniform pressure, corresponding to the height H v thus leaving out of con- sideration the suction and assuming the counterpressure from the lower level active over the entire submerged area. M. Becker proposes the formula (15) APPENDIX A III The first term is based on the wrong assumption that the flow of approach exerts an hydrodynamic pressure on the discharge area only and that the mean discharge velocity corresponds to a pressure height ( H 9 + ). The second term neglects suction \9 2 gl and assumes that the counterpressure is active over the whole sub- merged section. K. Pestalozzi's formula, for which he gives /J. = 0.8 to 0.85 and /*! = 0.62, is patterned after the Weisbach's formula for complete overfalls. It is (16) The second term neglects suction effect and weir dimensions. Messrs. Fteley and Stearns contributed a new formula for incomplete overfalls which is purely empiric and assumes both the upper and lower pools in a quiescent state. This formula is H wherein c is a coefficient depending on the ratio H/H l and varies between 3.089 and 3.372. While it is not supposed that any empiric formula will apply outside of the limits of the experiments for which it was proposed, it is not necessary, in view of what has gone before, to say more than that this formula has served its purpose well. It was not until the experiments of Messrs. Francis, Fteley and Stearns had been made and given to the world that anyone had even the right to claim any great knowledge regarding the science of weir hydraulics. While there is still far more to do along the experimental line than the sum total work already accomplished, everyone must cherish a feeling of gratitude towards these gentle- men and express the fond hope that some day, some one else will continue this work. 112 HYDRAULICS Several large hydraulic laboratories have come into being during recent years and they should undoubtedly contribute something to our present knowledge which will furnish such data as are necessary for work of the kind here treated. In concluding this subject it may be stated, that without excep- tion, modern writers have adopted one or more of the formulae given in this Appendix, sometimes without mentioning the source, hence this review is considered sufficiently exhaustive to show, without question, the irrational constitution of all older formulae. The attempt here made to offer something in a progressive direction would, therefore, seem justified. However, it is frankly admitted that the subject treated is still a very imperfect branch of engineering science. APPENDIX B. ON THE FLOW OVER A FLIGHT OF^PANAMA CANAL LOCKS. Solution of a Novel Hydraulic Problem. IN the design of a movable dam for the head of the triple flight of locks at Gatun, one of the first and most perplexing problems encountered was to determine the conditions of flow which would obtain in the event of serious accident to any of the upper lock gates. Such a catastrophe, while highly improbable, is nevertheless not impossible, and in the remote case of its happening would cripple the operation of the entire Panama Canal until the result- ing torrent pouring down through the locks could be effectively checked. To meet this extraordinary contingency, it is proposed to erect a movable dam, probably of the swing bridge type. Omitting the essential details of the dam, it is sufficient to mention here that the strength of the structure and the power for its operation depend for their determination on the depth and velocity of flow through the section at which the dam is to be located. For some time this flow problem seemed to offer unsurmountable difficulties owing to the many unknown quantities which neces- sarily enter. A thorough search through the world's hydraulic literature did not add much encouragement, and it became manifest that a solution, if one were possible, would mean a radical depar- ture from any of the previously known methods for solving hydraulic problems. Also a solution, to be of any value in con- nection with the general question, must lay claim to considerable accuracy. The triple flight of locks at Gatun presents a succession of 1 14 HYDRAULICS weirs or overfalls which may be complete or incomplete, depend- ing on the profile of the locks and the total drop in the water levels between Gatun Lake and the Atlantic Ocean. For the discharge through the straight portions of the locks and canal the Chezy formula with Bazin's coefficients will be used. This is probably the most reliable for flow through open channels having steep slope. For small slopes, the well tested formula of Ganguillet and Kutter may be more accurate. Suppose now that acceptable formulae are at hand for finding discharge over any single drop, also for the straight portions of the canal and locks. This then furnishes one formula for each condition of flow throughout the entire stretch of canal under discussion. Hence there are as many possible equations as there are varieties of conditions of flow. Regarding the feasibility of solving any or all of these equations, it will be best to give the general forms and discuss them with relation to the known and unknown quantities. For the straight portions of the channel and locks the Chezy formula gives v = CVrs and Q = Av = AC V7s . . . (i) also sL = H, which, substituted into Eq. (i), gives Q = 4C-M . wherein C = - - .... (2) 0.552 + Q = quantity of discharge in cubic feet per second. A = discharge section in square feet. r = mean hydraulic radius in feet. L = length of straight channel in feet. H = fall in surface, in feet, over length L. C = the Bazin Coefficient, wherein m is an experience number the values of which are given by Bazin for all conditions of flow. The values of m vary from 0.06 to 1.75, see under 2, Chapter VI. APPENDIX B US For incomplete overfalls, the new formulae, Eqs. (28), give (3) ij 1 nV 2 wherein S = } S. = S + H 2 H -- 2 g 2 g Here b = uniform width of canal and overfall in feet. g acceleration of gravity = 32.16 feet. v = mean velocity of approach in feet per second. V = mean velocity of discharge in feet per second. n = coefficient of contraction = 0.67. k = height of weir above approach canal bottom. H l = depth of weir crest below lower pool. H 2 = depth of lower pool below upper pool. T depth of approach canal. 7\ = depth of discharge section. /JL and /^ are discharge coefficients for free and submerged discharge respectively. Then v= = . ; F = -;and J ff I = r-# 2 . . . (4) The known quantities are b = 100 feet; T = 50 feet; and k, being small compared with the depth T, is neglected. All other quantities are unknown and depend for their values on Q and H 2 . Hence, if Q and H 2 are regarded as the independent vari- ables, the other quantities may be expressed in terms of these two. See Plate I, left hand end of lower profile, for lettered dimensions. The upper profile of Plate I represents the condition prior to an accident, the lower profile gives the computed water surface down the flight of locks after a uniform condition of flow has been established. 1 1 6 HYDRAULICS While it is possible then" to substitute numerical values into equations (3), involving only Q and H 2 as the final unknowns, it is quite impossible to solve directly the complicated form which results from such substitutions. The approach velocity and quantity for the second drop is now represented by the discharge velocity and quantity from the first drop. Hence if the first case were solved the second could be solved in like manner, and so on for a third or fourth drop. Now the quantity of discharge for a continuous flow must be constant, hence there is only one finally unknown, Q, while there is an unknown H 2 for each drop. This then enables writing out one equation for each H 2 in terms of Q, in which Q is a function of H 2 and itself. Thus: Q = f(Q, H 2 ) for first drop; Q = f(Q, H 2 ') for second drop; . .V . (5) Q = T (Q, H 2 ") for third drop; wherein there is one more unknown quantity than the number of equations. Hence the problem is not solvable until one other equation, involving these unknowns, is given. In the same manner a series of equations may be written out for the horizontal stretches of the canal by using Eq. (2), thus: IY Tl f r r 13" I r" ~H'" =Ac \A-- = A'c' V^- = A'V V(L , etc., (6) wherein all the quantities are known except Q, H', H" and H'", and these equations also number one less than the number of unknowns. However, the ocean level and the level for Gatun Lake being fixed, relatively, the total difference in their levels being 87 feet, the final condition follows: 1H 2 + *H' = 87 ....... (7) Hence putting Eqs. (5), (6) and (7) together this will give as many equations as there are unknowns, so the problem is definitely APPENDIX B 117 solvable. But owing to the complexity of the equations there is no method known in algebra by which these equations can be solved for simultaneous values of the unknowns. The solution given in the following is believed to be new and is original, as nothing bearing on this point could be found in any literature extant. After much deliberation and study it was found that simulta- neous values for the unknowns were obtainable by a graphic representation of Eq. (7), inasmuch as it is a straight line equa- tion and depends for its fulfilment on a certain definite value of Q. Then if the values H 2 and H f are ascertained for all reasonable, assumed values of Q, and plotted as co-ordi- nates, there will result as many curves as there are equations less one. The missing equation is Eq. (7) and, by trial, such a vakie of Q can be found for which Eq. (7) will be satisfied, and this fur- nishes the final solution. The value of Q thus determined, all the particular values of H 2 and H' become known and the profile of the surface can be drawn. This also fixes the velocity for every section along the entire stretch of canal. To exemplify the above reasoning, the complete solution will now be illustrated by reference to Plates I and II. Since all of the equations (3) are so extremely involved and complicated that they are not directly solvable, it becomes neces- sary to assign values to Q and find values for H 2 by .successive trials, continuing this process until the equation is satisfied. While this is a laborious operation it is far easier than at first appears and with a little experience an average computer can soon learn to solve a point by two or three approximations. A rough idea as to the limits between which the unknown Q may be located, can be obtained by a preliminary inspection of the given conditions, and by observing that a maximum value for V = 0.67 \/2 gh. However, this may be twice as large as the real velocity. Il8 HYDRAULICS In the present problem it was considered safe to assume that the first velocity of approach would have a value somewhere between 20 and 25 feet per second, although the theoretic v would be about 37 feet. It was also reasonable to suppose that each overfall would be incomplete, as a careful inspection of the profile would lead one to foresee. Hence, the formula for incomplete overfalls was used. But this assumption might have been erroneous, a fact which would be clearly indicated by the values H 2 resulting from a few preliminary computations. In the latter event the formula for complete overfalls would have to be employed. Suppose then that we have chosen the appropriate Eq. (3) and that the required Q corresponds to some velocity between 20 and 25 feet. Also, assume values for v from 20 feet and up, one foot apart; this will give a sufficient number of points to plot a curve such as shown on Plate II, for the upper lock, drop i. From Eqs. (4), Q may be found for any assumed v when the section is known, and since the depth at the head of the canal must remain constant, the discharge section may be assumed constant at the entrance to the canal. Therefore, the assumed data, for which values of H 2 are sought, would be as follows : Case (i) v = 20 ft. T = 50 ft. b = 100 ft. Q = 100,000 cu. ft. Case (2) 21 50 loo 105,000 Case (3) 22 50 100 110,000 Case (4) 23 50 100 110,000 Case (5) 24 50 100 115,000 Case (6) 25 50 100 120,000 The complete computation for the first point will now be given, and this will serve as an illustration for all of the computations. The problem is to solve Eq. (3) for the case when Q = 100,000 cubic feet per second, as per case (i). (See Plate I.) The coefficient /^ and ft must first be found from Eq. (7 7 a) by substituting proper values for H 2 and b. APPENDIX B IIQ H 2 is not known but since the total of the three drops is 87 feet, it is sufficiently close to assume H 2 = 25 feet, as the term in Eq. (770) involving H 2 has very small influence on jj. and does not enter into /^ Then with b = 100 feet, which is the constant width of the locks, Eqs. (770) give . -, + 0.000146 b 0.4150 or fi = 0.5533, Pi = 0-5346 + 0.000146 b = 0.5492. From Table VIII it is seen that p would be less than unity, but as no experiments on submerged weirs were available and as the crest in our problem is really a somewhat narrow sill of the miter gates, it is on the safe side to assume p = i. Hence in the following computations these coefficients are used : n = 0.67, p = i and /* = /^ = 0.55. (1) Assume now that H 2 =15 feet when Q = 100,000 cubic feet. Then v = IOO?OO = 20 feet; -H. = T -H 2 = ^ feet; and 50 X 100 T = H. + 21 = 56 feet, from which V = ^- = 17.86; = bT, 2 g 3.32 and $\/*g = 441.375. Also S = - - = 6.20 and 5 t = nV 2 S + H 2 + - - = 6.20 + 15 + 3.32 = 24.52. When k = o, then 5j = S 2 and y l -- - 2 = \ // S l , hence Eq. (3) becomes -V- . (8) Substituting all the above values into Eq. (8) and solving, then Q = 441-375 [f (121.4 - 15-4) + (35 -3-3 2 ) ^24.52 ]= 100,420 . (9) (2) This indicates that the first assumption for H 2 was a little large and the operation is repeated for H 2 = 14.75 f eet - It should 120 HYDRAULICS be noted that this change does not affect S and that a second com- putation is much easier. The new value gives 5, = 5 + H 2 + = 24.24; H, = 35.25; 7\ = 56.25; and V = 17.78. Hence Q = 44i-375[f (H9-4 -i5-4) + (35- 2 5 -3-288) ^24.24] = 99989. (10) The exact value of H 2 may now be found by interpolation between the values (9) and (10) by correcting the last value for ii cubic feet, which gives H 2 = 14.75, because the correction would be in the fourth decimal. It should be mentioned here that interpolation is not permissible unless reasonably close values have been found on both sides of the true value. In this manner the values H 2 in Table I, for drop i, were found, and these when plotted gave the curve for discharge for upper lock at drop i, see Plate II. Now each one of the above assumptions for v and Q (which always fixes a definite value F, for velocity of discharge) fur- nishes the conditions for approach to the second drop, drop 2. This is very important and on this fact is based the simultaneous relation of Q with the several values H 2 subsequently found for each drop. However, the discharge in passing over the length of the upper lock must have some slope sufficient to continue the discharge from the first drop. This slope and fall H f over a distance L 1000 feet is now computed by Eq. (2) for each v above assumed, and these figures are given in Table I in the horizontal column Upper Lock (H f ) and plotted as curve AB, Plate II. The various depths T v being the depths of discharge from the first drop, are now reduced by amounts H f , giving new values T f = T l H' y from which the new velocity of approach for the second drop is found for each of the previously assumed values of Q. Hence, by using the values T' in place of the former value APPENDIX B 121 T, each case of Q may again be solved exactly as for the first drop. From Plate I the following values may be taken: _2_ v f = ^ ,; Hi = T - H 3 '; Tj = Hi + 31 and V = 100 T Whence the same computations are repeated for the second drop and results entered in Table I and plotted on Plate II, as the dis- charge curve for drop 2. By a repetition of this process to the third drop the new values become and after computing H" for each assumed Q, the discharge curve for drop 3 was plotted on Plate II. Finally, from Eqs. (2) or (6), the falls H f through the lower lock, the Approach Canal, and two miles of wide canal connect- ing with the ocean, may be found for each of the first assumed quantities of discharge, and a discharge curve may then be plotted for each channel. Having thus found the related discharge curves for each con- dition of flow, over the entire stretch of canal and locks, the final solution is easily accomplished. Since for any value of Q, the accompanying values H 2 and H' are simultaneous values, made so by the previous method of computation, then such a value can be found, by trial, which will make ^H 2 + 2H' = 87 feet and the problem is solved. Referring to Plate II, and the tabulation in Table I, the value Q = 115,570 cubic feet per second satisfies this final condition. The figures in the last column of Table I were read from the curves of Plate II, excepting a few of the small drops which could better be interpolated from the table. These several values of H 2 and H' were then used to plot the surface slope for the entire stretch of canal, as shown in the lower profile of Plate I. From this profile the depth and velocity of flow at any discharge section 122 HYDRAULICS may be found by dividing the area of the section into 115,570. The upper approach velocity is thus found to be 23.2 feet. Regarding the accuracy of the above method it is believed one- half per cent would cover all of the slight inaccuracies inherent in the solution of the equations, while the coefficients are to some extent speculative. It may be said then that the problem is solved with the greatest accuracy attainable with our present knowledge of the empiric coefficients, and these may be consider- ably in error when the differences in circumstances, for which they were determined, are considered. Hence, we may conclude that such problems as the above are susceptible to general solution within the knowable accuracy of the empiric coefficients. This then emphasizes the importance of conducting larger hydraulic experiments especially for cases of deep and submerged flow. APPENDIX B 123 *i H-K 00 M 8 rf VO t^. O N 00 O ON ON O ^- * | 1* 1 = M M CO ON O M oo o o in vO vo CO 8 M O ON ON M 00 ^ ^t * . vO o 01 i_i ON M M . - vo rf TJ- S 3 | in H 1-4 M M 00 tN O O M - i Q O o o M ON VO ON s t^ vo CO VO i~ in o 6 vO* M j O t^ 5i H w M JO vo' 8 M 00 CO t^ '.-.* f o" o 4 M o M O vo trf H w C * fc] ST % * N S S: > , a | I . . . 1 "c3 ' CO S .2 P .2 H .2 1 o 9 A i O cfl 8 .a g - 8 : 1 i 1 04 M a, 2 v i a. i Q *l? co ^ co co O 1 + + + t CO cn II II co O 11 (0 O **cV 5 > J w -^! M? t) ^T? ct 6 ^5 Co CO Co O O ^r "J t rv J fficients Weirs, w i II 6-8 + + 8 8 - -2 * 1^4 6 - 6 1 1 1 II erim or ii : There are no ch to determine rfalls. CO 1 + + 3 + o O co O . f 6 . Ifl X > 11 Bog 1 t E :.~~ -1 i 71*0 ' ^ + + + i > O> W i. ^ ,- ^ I I* -SO" ill! u rt rt o {111 O E ' 95 Examples for backwater problems 50 133 134 INDEX PAGE Experiments by Boileau 80, 90 Bornemann 90 Engerth 90 Francis 79, 80, 83, 86 Fteley and Stearns 79, 83 Lesbros 80, 90 Rafter 81 Weisbach 90 Williams 95 to 101 Cornell University 81, 82 Eytelwein's formula 102 Flow in rivers and canals 52 over flight of Panama Canal locks 113 to 125 through lateral orifices 4 Formulae for backwater height and distance 47 complete overfalls 18, 19, 128 curved complete overfalls 24, 129 incomplete overfalls 31, 32, 34, 130 empiric coefficients 77 to 101 oblique, complete overfalls 23 sluice weirs and gates 40, 41, 43, 45, 132 various forms of complete overfalls 23, 129 waterpower diversions 64, 65, 69, 70, 73, 129, 131 Francis' formula 106 Fteley and Steam's formula 109, 1 1 1 Fundamental equations 4, 6 Ganguillet and Kutter's formula 59 Hagen's formula 108, no Hydraulic pressure defined i Hydrodynamic pressure defined i Hydrostatic pressure defined i Incomplete overfall weirs v 2, 28, 84, 109 Lesbros' formula 104, no Modified forms of complete overfalls 23, 24, 26, 129 INDEX 135 PAGE Navier's formula 104 New formulae for diversions 52, 61, 67, 70 incomplete overfalls 28 normal, complete overfalls u oblique, complete overfalls 20 sluice weirs 36 Overfall defined 2 Oblique weirs 20 Panama Canal locks 113 to 125 Partial contraction 2, 76 Pestalozzi's formula 108, 1 1 1 Pier obstructions " 33, 34 Pressure, hydraulic i hydrodynamic i hydrostatic i Proposed solution for waterpower diversions 57 Redtenbacher's formula 106, 109 Ruehlmann's formula 108 Sluice weirs and gates 2, 36, 89 Solution of waterpower problems 57 Table for backwater functions 49 Terms defined ix Velocity i Velocity of approach i Velocity of discharge 2 Waterpower canals 52, 57 Weir crests 95 to ii Weir formulae, coefficients for 75 to 101 Weirs, curved 23 oblique 20 Weisbach's formula . . 102, 105 OF THE UNIVERSITY OF SHORT-TITLE CATALOGUE OF THE PUBLICATIONS OP JOHN WILEY & SONS, NEW YORK. LONDON: CHAPMAN & HALL, LIMITED. ARRANGED UNDER SUBJECTS. Descriptive circulars sent on application. Books marked with an asterisk (*) are sold at net prices only. All books are bound in cloth unless otherwise stated. AGRICULTURE. Armsby's Manual of Cattle-feeding I2mo, Si 75 Principles of Animal Nutrition 8vo, 4 oo Budd and Hansen's American Horticultural Manual: Part I. Propagation, Culture, and Improvement I2mo, i 50 Part II. Systematic Pomology i2mo, i 50 Elliott's Engineering for Land Drainage I2mo, I 50 Practical Farm Drainage I2mo, i oo Graves's Forest Mensuration .8vo, 4 oo Green's Principles of American Forestry 12010, i 50 Grotenfelt's Principles of Modern Dairy Practice. (Woll.) i2mo, 2 oo Hanausek's Microscopy of Technical Products. (Winton.) 8vo, 5 oo Herrick's Denatured or Industrial Alcohol 8vo, 4 oo Maynard's Landscape Gardening as Applied to Home Decoration i2mo, i 50 * McKay and Larsen's Principles and Practice of Butter-making 8vo, i 50 Sanderson's Insects Injurious to Staple Crops i2mo, i 50 * Schwarz's Longleaf Pine in Virgin Forest izmo, i 25 Stockbridge's Rocks and Soils 8vo, 2 50 Winton's Microscopy of Vegetable Foods 8vo, 7 50 Wo IPs Handbook for Farmers and Dairymen i6mo, i 50 ARCHITECTURE. Baldwin's Steam Heating for Buildings i2mo, 2 50 Bashore's Sanitation of a Country House i2mo, i oo Berg's Buildings and Structures of American Railroads 4to, 5 oo Birkmire's Planning and Construction of American Theatres 8vo, 3 oo Architectural Iron and Steel 8vo, 3 50 Compound Riveted Girders as Applied in Buildings 8vo, 2 oo Planning and Construction of High Office Buildings 8vo, 3 50 Skeleton Construction in Buildings. 8^0, 3 oo Brigg's Modern American School Buildings 8vo, 4 oo Carpenter's Heating and Ventilating of Buildings 8vo, 4 oo 1 Freitag's Architectural Engineering 8vo. 3 50 Fireproofing of Steel Buildings 8vo, 2 50 French and Ives's Stereotomy 8vo, 2 50 Gerhard's Guide to Sanitary House-inspection i6mo, i oo> Sanitation of Public Buildings , . i2mo, i 50 Theatre Fires and Panics I2mo, i 50 *Greene's Structural Mechanics ,. . . 8vo, 2 50 Holly's Carpenters' and Joiners' Handbook i8mo, 75 Johnson's Statics by Algebraic and Graphic Methods 8vo, 2 00= Kellaway's How to Lay Out Suburban Home Grounds 8vo, 2 oo Kidder's Architects' and Builders' Pocket-book. Rewritten Edition. i6mo,mor., 5 oo Merrill's Stones for Building and Decoration 8vo, 5 oo Non-metallic Minerals: Their Occurrence and Uses 8vo, 4 oo Monckton's Stair-building 4to, 4 oo Patton's Practical Treatise on Foundations 8vo, 5 oo Peabody's Naval Architecture 8vo, 7 50 Rice's Concrete-block Manufacture 8vo, 2 oo Richey's Handbook for Superintendents of Construction i6mo, mor., 4 oo- * Building Mechanics' Ready Reference Book : * Carpenters' and Woodworkers' Edition i6mo, morocco, i so> * Cementworkers and Plasterer's Edition. (In Press.) * Stone- and Brick-mason's Edition i2mo, mor., i 50- Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 oo Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, i 50 Snow's Principal Species of Wood 8vo, 3 50- Sondericker's Graphic Statics with Applications to Trusses, Beams, and Arches. 8vo, 2 oo Towne's Locks and Builders' Hardware i8mo, morocco, 3 oo> Turneaure and Maurer's Principles of Reinforced Concrete Construc- tion 8vo, 3 oo> Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo> Sheep, 6 50- Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo, 5 oo> Sheep, 5 50 Law of Contracts ; 8vo, 3 oo> Wilson's Air Conditioning, ( In Press, ) Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .8vo, 4 oo- Worcester and Atkinson's Small Hospitals, Establishment and Maintenance, Suggestions for Hospital Architecture, with Plans for a Small Hospital. lamo, i 25: The World's Columbian Exposition of 1893 Large 4to, i oo- ARMY AND NAVY. Bernadou's Smokeless Powder, Nitro-cellulose, and the Theory of the Cellulose Molecule 1 21110, 2 50- Chase's Screw Propellers and Marine Propulsion 8vo, 3 oo Cloke's Gunner's Examiner 8vo, i 50 Craig's Azimuth 4to, 3 50 Crehore and Squier's Polarizing Photo-chronograph 8vo, 3 oo * Davis's Elements of Law 8vo, 2 50 * Treatise on the Military Law of United States 8vo, 7 oo> Sheep, 7 50 De Brack's Cavalry Outposts Duties. (Carr.) 241110, morocco, 2 oo Dietz's Soldier's First Aid Handbook i6mo, morocco, i 25 * Dudley's Military Law and the Procedure of Courts-martial. . . Large I2mo, 2 50 Durand's Resistance and Propulsion of Ships 8vo, 5 oo> 2 * Dyer's Handbook of Light Artillery I2mo, 3 oo Eissler's Modern High Explosives 8vo, 4 oo * Fiebeger's Text-book on Field Fortification Small 8vo, 2 oo Hamilton's The Gunner's Catechism i8mo, i oo * HofF s Elementary Naval Tactics 8vo, i 50 Ingalls's Handbook of Problems- in Direct Fire 8vo, 4 oo * Lissak's Ordnance and Gunnery 8vo, 6 oo * Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II. .8vo, each, 6 oo * Mahan's Permanent Fortifications. (Mercur.) 8vo, half morocco, 7 50 Manual for Courts-martial i6mo, morocco, I 50 * Mercur's Attack of Fortified Places i2mo, 2 oo> * Elements of the Art of War 8vo, 4 oo- Metcalf's Cost of Manufactures And the Administration of Workshops. .8vo, 5 oo * Ordnance and Gunnery. 2 vols I2mo, 5 oo Murray's Infantry Drill Regulations i8mo, paper, 10 Nixon's Adjutants' Manual 24mo, i oo Peabody's Naval Architecture 8vo, 7 50 * Phelps's Practical Marine Surveying 8vo, 2 50- Powell's Army Officer's Examiner I2mo, 4 oo- Sharpe's Art of Subsisting Armies in War i8mo, morocco, i 50- * Tupes and Poole's Manual of Bayonet Exercises and Musketry Fencing. 24010, leather, 50' Weaver's Military Explosives 8vo, 3 oo- Wheeler's Siege Operations and Military Mining 8vo, 2 oo> Winthrop's Abridgment of Military Law 12010, 2 50 Woodhull's Notes on Military Hygiene i6mo, i 50 Young's Simple Elements of Navigation i6mo, morocco, 2 oo ASSAYING. Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe. i2mo, morocco, I 50 Furman's Manual of Practical Assaying 8vo, 3 oo Lodge's Notes on Assaying and Metallurgical Laboratory Experiments. . . .8vo, 3 oo Low's Technical Methods of Ore Analysis. ... 8vo, 3 oo Miller's Manual of Assaying i2mo, I oo Cyanide Process i2mo, i oo Minet's Production of Aluminum and its Industrial Use. (Waldo.) i2mo, 2 50 O'Driscoll's Notes on the Treatment of Gold Ores 8vo, 2 oo Ricketts and Miller's Notes on Assaying 8vo, 3 oo Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, 4 oo Ulke's Modern Electrolytic Copper Refining 8vo, 3 oo Wilson's Cyanide Processes i2mo, i 50 Chlorination Process i2mo, i 50 ASTRONOMY. Comstock's Field Astronomy for Engineers 8vo, 2 50 Craig's Azimuth 4 t o , 3 50 Crandall's Text-book on Geodesy and Least Squares 8vo, 3 oo Doolittle's Treatise on Practical Astronomy 8vo, 4 oo Gore's Elements of Geodesy .8vo, 2 50 Hayford's Text-book of Geodetic Astronomy 8vo, 3 oo Merrirran's Elements ot Precise Surveying and Geodesy 8vo, 2 50 * Michie and Harlow's Practical Astronomy 8vo, 3 oo * White's Elements of Theoretical and Descriptive Astronomy i2mo, 2 oo 3 BOTANY. Davenport's Statistical Methods, with Special Reference to Biological Variation. i6mo, morocco, i 25 Thome and Bennett's Structural and Physiological Botany i6mo, 2 25 Westermaier's Compendium of General Botany. (Schneider.) 8vo, 2 oo CHEMISTRY. * Abegg's Theory of Electrolytic Dissociation. (Von Ende.) i2mo, i 25 Adriance's Laboratory Calculations and Specific Gravity Tables i2mo, i 25 Alexeyeff's General Principles of Organic Synthesis. (Matthews.) 8vo, 3 oo Allen's Tables for Iron Analysis 8vo, 3 oo Arnold's Compendium of Chemistry. (Mandel.) Small 8vo, 3 50 Austen's Notes for Chemical Students i2mo, i 50 Beard's Mine Gases and Explosions. (In Press.) Bernadou's Smokeless Powder. Nitro-cellulose, and Theory of the Cellulose Molecule i2mo, 2 50 Bolduan's Immune Sera 12mo , i 50 * Browning's Introduction to the Rarer Elements. . 8vo, i 50 Brush and Penfield's Manual of Determinative Mineralogy 8vo, 4 oo * Claassen's Beet-sugar Manufacture. (Hall and Rolfe.) 8vo, 3 oo Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.). .8vo, 3 oo Cohn's Indicators and Test-papers i2mo, 2 oo Tests and Reagents 8vo, 3 oo Crafts's Short Course in Qualitative Chemical Analysis. (Schaeffer.). . .i2mo, i 50 * Danneel's Electrochemistry. (Merriam.) I2mo, i 25 Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von Ende.) I2mo, 2 50 Drechsel's Chemical Reactions. (Merrill.) i2mo, i 25 Duhem's Thermodynamics and Chemistry. (Burgess.) Svo, 4 oo Eissler's Modern High Explosives Svo, 4 oo Effront's Enzymes and their Applications. (Prescott.) . . Svo, 3 oo Erdmann's Introduction to Chemical Preparations. (Dunlap.) i2mo, i 25 * Fischer's Physiology of Alimentation Large I2mo, 2 oo Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe. I2mo, morocco, i 50 Fowler's Sewage Works Analyses i2mo, 2 oo Fresenius's Manual of Qualitative Chemical Analysis. (Wells.) Svo, 5 oo Manual of Qualitative Chemical Analysis. Part I. Descriptive. (Wells.) Svo, 3 oo Quantitative Chemical Analysis. (Cohn.) 2 vols Svo, 12 50 Fuertes's Water and Public Health i2mo, i 50 Furman's Manual of Practical Assaying Svo, 3 oo * Getman's Exercises in Physical Chemistry i2mo, 2 oo Gill's Gas and Fuel Analysis for Engineers. i2mo, i 25 * Gooch and Browning's Outlines of Qualitative Chemical Analysis. Small Svo, i 25 Grotenfelt's Principles of Modern Dairy Practice. (Woll.) i2mo, 2 oo Groth's Introduction to Chemical Crystallography (Marshall) i2mo, i 25 Hammarsten's Text-book of Physiological Chemistry. (Mandel.) Svo, 4 oo Hanausek's Microscopy of Technical Products. (Winton.) 8vo, 5 oo * Haskin's and MacLeod's Organic Chemistry 12mo, 2 oo Helm's Principles of Mathematical Chemistry. (Morgan.) i2mo, i 50 Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 Herrick's Denatured or Industrial Alcohol Svo, 4 oo Hind's Inorganic Chemistry ; 8vo, 3 oo * Laboratory Manual for Students I2mo, i oo Holleman's Text-book of Inorganic Chemistry. (Cooper.) Svo, 2 50 Text-book of Organic Chemistry. (Walker and Mott.) Svo, 2 50 * Laboratory Manual of Organic Chemistry. (Walker.) 12 mo, i oo 4 Holley and Ladd's Analysis of Mixed Paints, Color Pigments , and Varnishes. (In Press) Hopkins's Oil-chemists' Handbook 8vo, 3 oo Iddings's Rock Minerals 8vo, 5 oo Jackson's Directions for Laboratory Work in Physiological Chemistry. .8vo, I 25 Johannsen's Key for the Determination of Rock-forming Minerals in Thin Sec- tions. (In Press) Keep's Cast Iron 8vo, 2 50 Ladd's Manual of Quantitative Chemical Analysis 12010, i oo Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 oo * Langworthy and Austen. The Occurrence of Aluminium in Vegetable Products, Animal Products, and Natural Waters 8vo, 2 oo Lassar-Cohn's Application of Some General Reactions to Investigations in Organic Chemistry. (Tingle.) lamo, i po Leach's The Inspection and Analysis of Food with Special Reference to State Control 8vo, 7 50 Lob's Electrochemistry of Organic Compounds. (Lorenz.) 8vo, 3 oo Lodge's Notes on Assaying and Metallurgical Laboratory Experiments 8vo, 3 oo Low's Technical Method of Ore Analysis 8vo, 3 oo Lunge's Techno-chemical Analysis. (Cohn.) I2mo I oo * McKay and Larsen's Principles and Practice of Butter-making 8vo, i 50 Maire ' s Modern Pigments and their Vehicles . (In Press. ) Mandel's Handbook for Bio-chemical Laboratory i2mo, i 50 * Martin's Laboratory Guide to Qualitative Analysis with the Blowpipe . . i2mo, 60 Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.) 3d Edition, Rewritten 8vo, 4 oo Examination of Water. (Chemical and Bacteriological.) I2mo, i 25 Matthew's The Textile Fibres. 2d Edition, Rewritten 8vo, 400 Meyer's Determination of Radicles in Carbon Compounds. (Tingle.). .i2mo, oo Miller's Manual of Assaying i2mo, oo Cyanide Process i2mo, oo Minet's Production of Aluminum and its Industrial Use. (Waldo.) . . . . i2mo, 50 Mixter's Elementary Text-book of Chemistry I2mo, 50 Morgan's An Outline of the Theory of Solutions and its Results i2mo, oo Elements of Physical Chemistry I2mo, 3 oo * Physical Chemistry for Electrical Engineers i2mo, 5 oo Morse's Calculations used in Cane-sugar Factories i6mo, morocco, i 50 * Muir's History of Chemical Theories and Laws 8vo, 4 oo Mulliken's General Method for the Identification of Pure Organic Compounds. Vol. I Large 8vo, 5 oo O'Driscoll's Notes on the Treatment of Gold Ores 8vo, 2 oo Ostwald's Conversations on Chemistry. Part One. (Ramsey.) I2mo, i 50 " " " Part Two. (Turnbull.) i2mo, 200 * Palmer's Practical Test Book of Chemistry -. 12mo, 1 oo * Pauli's Physical Chemistry in the Service of Medicine. (Fischer. ) . . . . 12010, i 25 * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 8vo, paper, 50 Pictet's The Alkaloids and their Chemical Constitution. (Biddle.) 8vo, 5 oo Pinner's Introduction to Organic Chemistry. (Austen.) I2mo, I 50 Poole's Calorific Power of Fuels 8vo, 3 oo Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- ence to Sanitary Water Analysis I2mo, i 25 * Reisig's Guide to Piece-dyeing 8vo, 25 oo Richards and Woodman's Air, Water, and Food from a Sanitary Standpoint. . 8v o , 2 oo Ricketts and Miller's Notes on Assaying 8vo, 3 oo Rideal's Sewage and the Bacterial Purification of Sewage 8vo, 4 oo Disinfection and the Preservation of Food. 8vo, 4 oo Riggs's Elementary Manual for the Chemical Laboratory 8vo, i 25 Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, 4 oo 5 Ruddiman's Incompatibilities in Prescriptions 8vo, 2 oo> * Whys in Pharmacy lamo, i oo> Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo 3 00= Salkowski's Physiological and Pathological Chemistry. (Orndorff.) 8vo, 2 50- Schimpf's Text-book of Volumetric Analysis i2mo, 2 50 Essentials of Volumetric Analysis. . . , i2mo, i 25 * Qualitative Chemical Analysis 8vo, i 25 Smith's Lecture Notes on Chemistry for Dental Students ' . . . 8vo, 2 50- Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco 3 oo^ Handbook for Cane Sugar Manufacturers i6mo. morocco. 3 oo> Stockbridge's Rocks and Soils 8vo, 2 50 * Tillman's Elementary Lessons in Heat 8vo, i 50 * Descriptive General Chemistry 8vo 3 oo- Treadwell's Qualitative Analysis. (Hall.) 8vo, 3 oo Quantitative Analysis. (Hall.) 8vo, 4 oo Turneaure and Russell's Public Water-supplies 3vo, 5 oo Van Deventer's Physical Chemistry for Beginners. (Boltwood.) i2mo, i 50 * Walke's Lectures on Explosives 8vo ; 4 oo Ware's Beet-sugar Manufacture and Refining. Vol. I Small 8vo, 4 oo " " " " " Vol.11 bmallSvo, 500- Washington's Manual of the Chemical Analysis of Rocks 8vo, 2 oa Weaver's Military Explosives 8vo, 3 oo Wehrenfennig's Analysis and Softening of Boiler Feed-Water 8vo, 4. oo Wells's Laboratory Guide in-Qualitative Chemical Analysis 8vo, i 50 Short Course in Inorganic Qualitative Chemical Analysis for Engineering Students I2mo, i 50 Text-book of Chemical Arithmetic I2mo, i 25 Whipple's Microscopy of Drinking-water 8vo, 3 50 Wilson's Cyanide Processes I2mo, i 50 Chlorination Process I2mo, i 50 Winton's Microscopy of Vegetable Foods 8vo, 7 50 Wulling's Elementary Course in Inorganic, Pharmaceutical, and Medical Chemistry - i2mo, 2 oo CIVIL ENGINEERING. BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING RAILWAY ENGINEERING. Baker's Engineers' Surveying Instruments I2mo, 3 oa Bixby's Graphical Computing Table Paper io, 24^ inches. 25 Breed and Hosmer's Principles and Practice of Surveying 8vo, 3 oo * Burr's Ancient and Modern Engineering and the Isthmian Canal . . 8vo, 3 50 Comstock's Field Astronomy for Engineers 8vo, 2 50 * Corthell's Allowable Pressures on Deep Foundations I2mo, i 25 Crandall's Text-book on Geodesy and Least bquares : .0*0, 3 oo Davis's Elevation and Stadia Tables 8vo, i oo Elliott's Engineering for Land Drainage i2mo, i 50 Practical Farm Drainage I2mo, i oo *Fiebeger's Treatise on Civil Engineering 8vo, 5 oo Flemer's Phototopographic Methods and Instruments 8vo, 5 oo Folwell's Sewerage. (Designing and Maintenance.) 8vo, 3 oo Freitag's Architectural Engineering. 2d Edition, Rewritten 8vo, 3 50 French and Ives's Stereotomy 8vo 2 50 Goodhue's Municipal Improvements I2mo, i 50 Gore's Elements of Geodesy 8vo 2 5 * Hauch and Rice's Tables of Quantities for Preliminary Estimates I2mo, i 25 Hayford's Text-book of Geodetic Astronomy 8vo, 3 oo Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 Howe's Retaining Walls for Earth 12010, i 25 Hoyt and Graver's River Discharge 8vo, 2 oo * Ives's Adjustments of the Engineer's Transit and Level i6mo, Bds. 25 Ives and Hilts's Problems in Surveying i6mo, morocco, i 50 Johnson's (J. B.) Theory and Practice of Surveying Small 8vo, 4 oo Johnson's (L. J.) Statics by Algebraic and Graphic Methods 8vo, 2 oo Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) 12010, 2 oo Mahan's Treatise on Civil Engineering. (1873.) (Wood.) 8vo, 5 oo * Descriptive Geometry. 8vo, i 50 Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50 Merriman and Brooks's Handbook for Surveyors i6mo, morocco, 2 oo Nugent's Plane Surveying 8vo, 3 50 Ogden's Sewer Design i2mo, 2 oo Parsons's Disposal of Municipal Refuse. 8vo, 2 oo Patton's Treatise on Civil Engineering 8vo half leather, 7 50 Reed's Topographical Drawing and Sketching 4to, 5 oo Rideal's Sewage and the Bacterial Purification of Sewage 8vo, 4 oo Riemer's Shaft-sinking under Difficult Conditions. (Corning and Peele.). .8vo, 3 oo Siebert and Biggin's Modern Stone-cutting and Masonry 8vo, i 50 Smith's Manual of Topographical Drawing. (McMillan.) 8vo, 2 50 Sondericker's Graphic Statics, with Applications to Trusses, Beams, and Arches. 8vo, 2 oo "Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 5 oo Tracy's Plane Surveying I6mo, morocco, 3 oo * Trautwine's Civil Engineer's Pocket-book i6mo, morocco, 5 oo Venable's Garbage Crematories in America .8vo, 2 oo Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo Sheep, 6 50 Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo, 5 oo Sheep, 5 50 Law of Contracts 8vo, 3 oo Warren's Stereotomy Problems in Stone-cutting 8vo, 2 50 Webb's Problems in the Use and Adjustment of Engineering Instruments. i6mo, morocco, i 25 Wilson's Topographic Surveying 8vo, 3 50 BRIDGES AND ROOFS. Boiler's Practkal Treatise on the Construction of Iron Highway Bridges. .8vo, 2 oo Burr and Falk's Influence Lines for Bridge and Roof Computations 8vo, 3 oo Design and Construction of Metallic Bridges ". . . .8vo, 5 oo Du Bois's Mechanics of Engineering. Vol. II Small 4to, 10 oo Poster's Treatise on Wooden Trestle Bridges 4to, 5 oo Fowler's Ordinary Foundations 8vo, 3 50 Greene's Roof Trusses 8vo, i 25 Bridge Trusses 8vo, 2 50 Arches in Wood, Iron, and Stone 8vo, 2 50 Grimm's Secondary Stresses in Bridge Trusses. (In Press.) Howe's Treatise on Arches 8vo, 4 oo Design of Simple Roof -trusses in Wood and Steel 8vo, 2 oo Symmetrical Masonry Arches 8vo, 2 50 Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of Modern Framed Structures Small 4to, 10 oo Merriman and Jacoby's Text-book on Roofs and Bridges : Part I. Stresses in Simple Trusses , 8vo, 2 50 Part II. Graphic Statics >. ,* . . .8vo, 2 50 Part III. Bridge Design i-. '.'. 8vo, 2 50 Part IV. Higher Structures 8vo, 2 50 7 Morison's Memphis Bridge. , 4to, 10 o Waddell's De Pontibus, a Pocket-book for Bridge Engineers . . i6mo, morocco, 2 oo Specifications for Steel Bridges i2mo, 50 Wright's Designing of Draw-spans. Two parts in one volume 8vo, 3 50 HYDRAULICS. Barnes's Ice Formation 8vo, 3 oo> Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from an Orifice. (Trautwine.) 8vo, 2 oo- Bovey's Treatise on Hydraulics 8vo, 5 oo Church's Mechanics of Engineering 8vo, 6 oo- Diagrams of Mean Velocity of Water in Open Channels paper, i so- Hydraulic Motors. . 8vo, 2 oo Coffin's Graphical Solution of Hydraulic Problems i6mo, morocco, 2 50 Flather's Dynamometers, and the Measurement of Power 12010, 3 oo Folwell's Water-supply Engineering 8vo, 4 oo FrizelPs Water-power 8vo, 5 oo Fuertes's Water and Public Health i2mo, i 50 Water-filtration Works i2mo. 2 50 Ganguillet and Kutter's General Formula for the Uniform Flow of Water in Rivers and Other Channels. (Hering and Trautwine.) 8vo, 4 oo Hazen's Clean Water and How to Get It Large I2mo, l 5o Filtration of Public Water-supply .8vo, 3 oo Hazlehurst's Towers and Tanks for Water- works 8vo, 2 50- Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal Conduits 8vo, 2 oo * Hubbard and Kiersted's Water- works Management and Maintenance. 8vo, 4 co- Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.) 8vo, 4 oo- Merriman's Treatise on Hydraulics 8vo, 5 oo * Michie's Elements of Analytical Mechanics 8vo, 4 oo- Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- supply Large 8vo, 5 oo> * Thomas and Watt's Improvement of Rivers 4*0, 6 oo- Turneaure and Russell's Public Water-supplies 8vo, 5 oo Wegmann's Design and Construction of Dams. 5th Edition, enlarged. . .4to, 6 oo> Water-supply of the City of New York from 1658 to 1895 4to, 10 oo> Whipple's Value of Pure Water Large i2mo, i oo* Williams and Hazen's Hydraulic Tables 8vo, i 50 Wilson's Irrigation Engineering Small 8vo, 4 oo- Wolff's Windmill as a Prime Mover 8vo, 3 oo Wood's Turbines 8vo, 2 50* Elements of Analytical Mechanics 8vo, 3 oo- MATERIALS OF ENGINEERING. Baker's Treatise on Masonry Construction 8vo, 5 oo> Roads and Pavements 8vo, 5 oa Black's United States Public Works Oblong 4to, 5 oo- * Bovey's Strength of Materials and Theory of Structures 8vo, 7 50- Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 7 5* Byrne's Highway Construction 8vo, 5 oo Inspection of the Materials and Workmanship Employed in Construction. i6mo, 3 oo Church's Mechanics of Engineering 8vo, 6 oo> Du Bois's Mechanics of Engineering. Vol. I Small 410. 7 5<> *Eckel's Cements, Limes, and Plasters 8vo, 6 oo 8 Johnson's Materials of Construction Large 8vo, 6 oo Fowler's Ordinary Foundations 8vo, 3 50 Graves's Forest Mensuration 8vo, 4 oo * Greene's Structural Mechanics 8vo, 2 50 Keep's Cast Iron. 8vo, 2 50 Lanza's Applied Mechanics 8vo, 7 50 Martens's Handbook on Testing Materials. (Henning.) 2 vols 8vo, 7 50 Maurer's Technical Mechanics 8vo, 4 oo Merrill's Stones for Building and Decoration 8vo, 5 oo Merriman's Mechanics of Materials 8vo, 5 oo * Strength of Materials i2mo, i 0.0 Metcalf's Steel. A Manual for Steel-users i2mo, 2 oo Patton's Practical Treatise on Foundations 8vo 5 oo Richardson's Modern Asphalt Pavements 8vo, 3 oo Richey's Handbook for Superintendents of Construction i6mo, mor., 4 oo * Ries's Clays: Their Occurrence, Properties, and Uses 8vo, 5 oo Rockwell's Roads and Pavements in France i2mo, r 25 Sabin's Industrial and Artistic Technology of Paints and Varnish 8vc, 3 oo *Schwarz's Longleaf Pine in Virgin Forest ... i2mo, i 25 Smith's Materials of Machines i2mo, i oo Snow's Principal Species of Wood 8vo, 3 50 Spalding's Hydraulic Cement. i2mo, 2 oo Text-book on Roads and Pavements I2mo, 2 oo Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 5 oo Thurston's Materials of Engineering. 3 Parts.. 8vo, 8 oo Part I. Non-metallic Materials of Engineering and Metallurgy 8vo, 2 oo Part II. Iron and Steel 8vo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 Tillson's Street Pavements and Paving Materials 8vo, 4 oo Turneaure and Maurer's Principles of Reinforced Concrete Construction. .8vo, 3 oo Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.). . i6mo, mor., 2 oo * Specifications for Steel Bridges. . . , T. I2mo, 50 Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on the Preservation of Timber 8vo, 2 oo Wood's (De V. ) Elements of Analytical Mechanics 8vo, 3 oo Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. ... 8vo, 4 oo RAILWAY ENGINEERING. Andrew's Handbook for Street Railway Engineers 3x5 inches, morocco, i 25 Berg's Buildings and Structures of American Railroads 4to, 5 oo Brook's Handbook of Street Railroad Location i6mo, morocco, i 50 Butt's Civil Engineer's Field-book i6mo, morocco, 2 50 Crandall's Transition Curve i6mo, morocco, i 50 Railway and Other Earthwork Tables 8vo, i 50 Crockett's Methods for Earthwork Computations. (In Press) Dawson's "Engineering" and Electric Traction Pocket-book. . i6mo, morocco 5 oo Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 oo Fisher's Table of Cubic Yards Cardboard, 25 Godwin's Railroad Engineers' Field-book and Explorers' Guide. . .i6mo, mor., 2 50 Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- bankments 8vo, i oo Molitor and Beard's Manual for Resident Engineers i6mo, i oo Nagle's Field Manual for Railroad Engineers i6mo, morocco, 3 oo Philbrick's Field Manual for Engineers i6mo, morocco, 3 oo Raymond's Elements of Railroad Engineering. (In Press.) 9 Searles's Field Engineering i6mo, morocco, 3 oo Railroad Spiral. i6mo, morocco, z 50 Taylor's Prismoidal Formulae and Earthwork 8vo, x so * Trautwine's Method of Calculating the Cube Contents of Excavations and Embankments by the Aid of Diagrams 8vo, 2 oo The Field Practice of Laying Out Circular Curves for Railroads. i2mo, morocco, 2 50 Cross-section Sheet Paper, 25 Webb's Railroad Construction i6mo, morocco, 5 oo Economics of Raikoad Construction Large i2mo, 2 50 Wellington's Economic Theory of the Location of Railways Small 8vo, 5 oo DRAWING. Barr's Kinematics of Machinery 8vo, 2 50 * Bartlett's Mechanical Drawing 8vo, 3 oo * " " " Abridged Ed 8vo, i 50 Coolidge's Manual of Drawing 8vo, paper, i oo Coolidge and Freeman's Elements of General Drafting for Mechanical Engi- neers Oblong 4to, 2 50 Durley's Kinematics of Machines 8vo, 4 oo Emch's Introduction to Projective Geometry and its Applications 8vo, 2 50 Hill's Text-book on Shades and Shadows, and Perspective 8vo, 2 oo Jamison's Elements of Mechanical Drawing 8vo, 2 50 Advanced Mechanical Drawing. 8vo, 2 oo Jones's Machine Design : Part I. Kinematics of Machinery 8vo, i 50 Part II. Form, Strength, and Proportions of Parts 8vo, 3 oo MacCord's Elements of Descriptive Geometry 8vo, 3 oo Kinematics; or, Practical Mechanism 8vo, 5 oo Mechanical Drawing 4to, 4 oo Velocity Diagrams 8vo, i 50 MacLeod's Descriptive Geometry Small 8vo, i 50 * Mahan's Descriptive Geometry and Stone-cutting 8vo, i 50 Industrial Drawing. (Thompson.) 8vo, 3 50 Moyer's Descriptive Geometry 8vo, 2 oo Reed's Topographical Drawing and Sketching 4to, 5 oo Reid's Course in Mechanical Drawing 8vo, 2 oo Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oo Robinson's Principles of Mechanism 8vo, 3 oo Schwamb and Merrill's Elements of Mechanism 8vo, 3 oo Smith's (R. S.) Manual of Topographical Drawing. (McMillan.) 8vo, 2 50 Smith (A. W.) and Marx's Machine Design 8vo, 3 oo * Titsworth's Elements of Mechanical Drawing Oblong 8vo, Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. i2mo, Drafting Instruments and Operations i2mo, Manual of Elementary Projection Drawing i2mo, Manual of Elementary Problems in the Linear Perspective of Form and Shadow i2mo, Plane Problems in Elementary Geometry i2mo, oo 25 50 00 25 Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 3 50 General Problems of Shades and Shadows 8vo, 3 oo Elements of Machine Construction and Drawing 8vo, 7 50 Problems, Theorems, and Examples in Descriptive Geometry 8vo, 2 50 Weisbach's Kinematics and Power of Transmission. (Hermann and Klein.) 8vo, 5 o o Whelpley's Practical Instruction in the Art of Letter Engraving i2mo. 2 oo Wilson's (H. M.) Topographic Surveying 8vo, 3 50 10 Wilson's (V. T.) Free-hand Perspective 8vo, 2 50 Wilson's (V. T.) Free-hand Lettering 8vo, i oo Woolf's Elementary Course in Descriptive Geometry Large 8vo, 3 oo ELECTRICITY AND PHYSICS. * Abegg's Theory of Electrolytic Dissociation. (Von Ende.) i2mo, i 25 Anthony and Brackett's Text-book of Physics. (Magie.) Small 8vo, 3 oo Anthony's Lecture-notes on the Theory of Electrical Measurements. . . . i2mo, i oo Benjamin's History of Electricity 8vo, 3 oo Voltaic CelL 8vo, 3 oo Betts's Lead Refining and Electrolysis. (In Press.) Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.).Svo, 3 oo * Collins's Manual of Wireless Telegraphy 1 21110, i 50 Morocco, 2 oo Crehore and Squier's Polarizing Photo-chronograph 8vo, 3 oo * Danneel's Electrochemistry. (Merriam.) i2mo, i 25 Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 oo Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von Ende.) i2mo, 2 50 Duhem's Thermodynamics and Chemistry. (Burgess.). 8vo, 4 oo Flather's Dynamometers, and the Measurement of Power 121110, 3 co Gilbert's De Magnete. (Mottelay.) 8vo, 2 50 Hanchett's Alternating Currents Explained i2mo, i oo Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 Hobart and Ellis's High-speed Dynamo Electric Machinery. (In Press.) Holman's Precision of Measurements 8vo, 2 oo Telescopic Mirror-scale Method, Adjustments, and Tests. . . .Large 8vo, 75 Karapetoff's Experimental Electrical Engineering. (In Press.) Kinzbrunner's Testing of Continuous-current Machines. ... 8vo, 2 oo Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 oo Le Chatelier's High-temperature Measurements. (Boudouard Burgess.) i2mo, 3 oo Lob's Electrochemistry of Organic Compounds. (Lorenz.) 8vo, 3 oo * Lyons'? Treatise on Electromagnetic Phenomena. Vols. I. and II. 8vo, each, 6 oo * Michie's Elements of Wave Motion Relating to Sound and Light 8vo, 4 oo Niaudet's Elementary Treatise on Electric Batteries. (Fishback.) i2mo, 2 50 Norris's Introduction to the Study of Electrical Engineering. (In Press.) * Parshall and Hobart's Electric Machine Design 410, half morocco, 12 50 Reagan's Locomotives: Simple, Compound, and Electric. New Edition. Large i2mo, 3 50 * Rosenberg's Electrical Engineering. (Haldane Gee Kinzbrunner.). . .8vo, oo Ryan, Norris, and Hoxie's Electrical Machinery. Vol. 1 8vo, 50 Thurston's Stationary Steam-engines 8vo, 50 * Tollman's Elementary Lessons in Heat 8vo, 50 Tory and Pitcher's Manual of Laboratory Physics Small 8vo, oo Ulke's Modern Electrolytic Copper Refining 8vo, 3 oo LAW. * Davis's Elements of Law 8vo, 2 50 * Treatise on the Military Law of United States 8vo, 7 06 * Sheep, 7 SO * Dudley's Military Law and the Procedure of Courts-martial . . . Largre i2mo, 2 50 Manual for Courts-martial i6mo, morocco, i 50 Wait's Engineering and Architectural Jurisprudence 8vo, 6 oo Sheep, 6 50 Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo 5 oo Sheep, 5 50 Law of Contracts 8vo, 3 oo Winthrop's Abridgment of Military Law 121110, 2 50 11 MANUFACTURES. Bernadou's Smokeless Powder Nitro-cellulose and Theory of the Cellulose Molecule 1 2mo, 2 50 Holland's Iron Founder 1 2mo, 2 50 The Iron Founder," Supplement \j 2mo, 2 50 Encyclopedia of Founding and Dictionary of Foundry Terms Used in the Practice of Moulding 12010, 3 oo * Claassen's Beet-sugar Manufacture. (Hall and Rolfe.) 8vo, 3 oo * Eckel's Cements, Limes, and Plasters 8vo, 6 oo Eissler's Modern High Explosives 8vo, 4 oo Effront's Enzymes and their Applications. (Prescott.) 8vo, 3 oo Fitzgerald's Boston Machinist 12 mo, i oo Ford's Boiler Making for Boiler Makers i8mo. i oo Herrick's Denatured or Industrial Alcohol ,8vo, 4 oo Holley and Ladd's Analysis of Mixed Paints, Color Pigments, and Varnishes. (In Press.) Hopkins 's Oil-chemists' Handbook 8vo, 3 oo Keep's Cast Iron 8vo, 2 50 Leach's The Inspection and Analysis of Food with Special Reference to State Control Large 8vo, 7 50 * McKay and Larsen's Principles and Practice of Butter-making 8vo, i 50 Maire's Modern Pigments and their Vehicles. (In Press.) Matthews's The Textile Fibres. 2d Edition, Rewritten 8vo, 4 oo Metcalf's Steel. A Maunal for Steel-users i2mo, 2 oo Metcalfe's Cost of Manufactures And the Administration of Workshops ,8vo, 5 oo Meyer's Modern Locomotive Construction 4to, 10 oo Morse's Calculations used in Cane-sugar Factories i6mo, morocco, i 50 * Reisig's Guide to Piece-dyeing 8vo, 25 oo Rice's Concrete-block Manufacture 8vo, 2 oo Sabin's Industrial and Artistic Technology of Paints and Varnish ..... .8vo, 3 oo Smith's Press-working of Metals 8vo, 3 oo Spalding's Hydraulic Cement i2mo, 2 oo Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco, 3 oo Handbook for Cane Sugar Manufacturers i6mo, morocco, 3 oo Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 5 oo Thurston's Manual of Steam-boilers, their Designs, Construction and Opera- tion 8vo, 5 oo Ware's Beet-sugar Manufacture and Refining. Vol. I Small 8vo, 4 oo Vol.11 8vo, 5 oo Weaver's Military Explosives 8vo, 3 oo West's American Foundry Practice 1 2mo, 2 50 Moulder's Text-book i2mo, 2 50 Wolff's Windmill as a Prime Mover 8vo, 3 oo Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel 8vo, 4 oo MATHEMATICS. Baker's Elliptic Functions 8vo, Briggs's Elements of Plane Analytic Geometry i2mo, Buchanan's Plane and Spherical Trigonometry. (In Press.) Compton's Manual of Logarithmic Computations i2mo, Da vis's Introduction to the Logic of Algebra 8vo, * Dickson's College Algebra Large i2mo, * Introduction to the Theory of Algebraic Equations Large i2tno, Emch's Introduction to Projective Geometry and its Applications ..... 8vo, Halsted's Elements of Geometry 8vo, Elementary Synthetic Geometry 8vo, * Rational Geometry . i2mo, 12 5cr oo 50 50 50 25 50 75 50 50 * Johnson's (J. B.) Three-place Logarithmic Tables: Vest-pocket size. paper, 15 100 copies for 5 oo * Mounted on heavy cardboard, 8 X 10 inches, 25 10 copies for 2 oo Johnson's (W. W.) Elementary Treatise on Differential Calculus. .Small 8vo, 3 oo Elementary Treatise on the Integral Calculus Small 8vo, i 50 Johnson's (W. W.) Curve Tracing in Cartesian Co-ordinates 12010, i oo Johnson's (W. W.) Treatise on Ordinary and Partial Differential Equations. Small 8vo, 3 50 Johnson's Treatise on the Integral Calculus Small 8vo, 3 oo Johnson's (W. W.) Theory of Errors and the Method of Least Squares. i2mo, i 50 * Johnson's (W. W.) Theoretical Mechanics I2mo, 3 oo Laplace's Philosophical Essay on Probabilities. (Truscott and Emory. ).i2mo, 2 oo * Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other Tables 8vo, 3 oo Trigonometry and Tables published separately Each, 2 oo * Ludlow's Logarithmic and Trigonometric Tables 8vo, i oo Manning's IrrationalNumbers and their Representation bySequences and Series i2mo, i 25 Mathematical Monographs. Edited by Mansfield Merriman and Robert S. Woodward Octavo, each i oo No. i. History of Modern Mathematics, by David Eugene Smith. No. 2. Synthetic Projective Geometry, by George Bruce Halsted. Ko. 3. Determinants, by Laenas Gifford Weld. No. 4. Hyper- bolic Functions, by James McMahon. No. 5. Harmonic Func- tions, by William E. Byerly. No. 6. Grassmann's Space Analysis, by Edward W. Hyde. No. 7. Probability and Theory of Errors, by Robert S. Woodward. No. 8. Vector Analysis and Quaternions, by Alexander Macfarlane. No. o. Differential Equations, by William Woolsey Johnson. No. 10. The Solution of Equations, by Mansfield Merriman. No. n. Functions of a Complex Variable, by Thomas S. Fiske. Maurer's Technical Mechanics. 8vo, 4 oo Merriman's Method of Least Squares 8vo, 2 oo Rice and Johnson's Elementary Treatise on the Differential Calculus. . Sm. 8vo, 3 oo Differential and Integral Calculus. 2 vols. in one Small 8vo, 2 50 * Veblen and Lennes's Introduction to the Real Infinitesimal Analysis of One Variable 8vo, 2 oo Wood's Elements of Co-ordinate Geometry 8vo, 2 oo Trigonometry: Analytical, Plane, and Spherical 121110, i oo MECHANICAL ENGINEERING. MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS. Bacon's Forge Practice i2mo, i 50 Baldwin's Steam Heating for Buildings i2mo, 2 50 Barr's Kinematics of Machinery 8vo, 2 50 * Bartlett's Mechanical Drawing 8vo, 3 oo * " " " Abridged Ed 8vo, i 50 Benjamin's Wrinkles and Recipes i2mo, 2 oo Carpenter's Experimental Engineering 8vo, 6 oo Heating and Ventilating Buildings 8vo, 4 oo Clerk's Gas and Oil Engine Small 8vo, 4 oo Coolidge's Manual of Drawing 8vo, paper, i oo Coolidge and Freeman's Elements of General Drafting for Mechanical En- gineers Oblong 4to, 2 50 Cromwell's Treatise on Toothed Gearing i2mo, i 50 Treatise on Belts and Pulleys i2mo, i 50 13 Durley's Kinematics of Machines 8vo, 4 oo Flather's Dynamometers and the Measurement of Power i2mo, 3 oo Rope Driving i2mo, 2 oo Gill's Gas and Ful Analysis for Engineers i2mo, i 25 Hall's Car Lubrication i2mo, i oo Hering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 Button's The Gas Engine 8vo, 5 oo Jamison's Mechanical Drawing 8vo, 2 50 Jones's Machine Design: Part I. Kinematics of Machinery 8vo, i 50 Part II. Form, Strength, and Proportions of Parts 8vo, 3 oo Kent's Mechanical Engineers' Pocket-book. i6mo, morocco, 5 oo Kerr's Power and Power Transmission 8vo, 2 oo Leonard's Machine Shop, Tools, and Methods 8vo, 4 oo * Lorenz's Modern Refrigerating Machinery. (Pope, Haven, and Dean.) . . 8vo, 4 oo MacCord's Kinematics; or, Practical Mechanism 8vo, 5 oo Mechanical Drawing 4to, 4 oo Velocity Diagrams 8vo, i 50 MacFar land's Standard Reduction Factors for Gases 8vo, i 50 Mahan's Industrial Drawing. (Thompson.) 8vo, 3 50 Poole's Calorific Power of Fuels 8vo, 3 oo Reid's Course in Mechanical Drawing 8vo, 2 oo Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oo Richard's Compressed Air i2mo, i 50 Robinson's Principles of Mechanism 8vo, 3 oo Schwamb and Merrill's Elements of Mechanism 8vo, 3 oo Smith's (O.) Press- working of Metals 8vo, 3 oo Smith (A. W.) and Marx's Machine Design 8vo, 3 oo Thurston's Treatise on Friction and Lost Work in Machinery and Mill Work 8vo, 3 oo Animal as a Machine and Prime Motor, and the Laws of Energetics . i2mo, i oo Tillson's Complete Automobile Instructor i6mo, i 50 Morocco, 2 oo Warren's Elements of Machine Construction and Drawing 8vo, 7 50 Weisbach's Kinematics and the Power of Transmission. (Herrmann Klein.). 8vo, 5 oo Machinery of Transmission and Governors. (Herrmann Klein.). .8vo, 5 oo Wolff's Windmill as a Prime Mover 8vo, 3 oo Wood's Turbines 8vo, 2 50 MATERIALS OF ENGINEERING. * Bovey's Strength of Materials and Theory of Structures 8vo, 7 50 Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edition. Reset 8vo, 7 50 Church's Mechanics of Engineering 8vo, 6 oo * Greene's Structural Mechanics 8vo, 2 50 Johnson's Materials of Construction 8vo, 6 oo Keep's Cast Iron : 8vo, 2 50 Lanza's Applied Mechanics 8vo, 7 50 Martens's Handbook on Testing Materials. (Henning.) 8vo, 7 50 Maurer's Technical Mechanics 8vo, 4 oo Merriman's Mechanics of Materials 8vo, 5 oo * Strength of Materials i2mo, i oo Metcalf's Steel. A Manual for Steel-users i2mo, 2 oo Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 oo Smith's Materials of Machines i2mo, i oo Thurston's Materials of Engineering 3 vols., 8vo, 8 oo Part II. Iron and Steel 8vo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 14 Wood's (De V.) Treatise on the Resistance of Materials and an Appendix on the Preservation of Timber 8vo, a oo Elements of Analytical Mechanics 8vo, 3 oo Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel 8vo, 4 oo STEAM-ENGINES AND BOILERS. Berry's Temperature-entropy Diagram . izmo, i 25 Carnot's Reflections on the Motive Power of Heat. (Thurston.) 12 mo, i 50 Creighton's Steam-engine and other Heat-motors 8vo, 500 Dawson's "Engineering" and Electric Traction Pocket-book i6mo, mor., 5 oo Ford's Boiler Making for Boiler Makers i8mo, i oo Goss's Locomotive Sparks 8vo 2 oo Locomotive Performance 8vo, 5 oo Hemenway's Indicator Practice and Steam-engine Economy .i2mo, 2 oo Button's Mechanical Engineering of Power Plants 8vo, 5 oo Heat and Heat-engines 8vo 5 oo Kent's Steam boiler Economy 8vo, 4 oo Kneass's Practice and Theory of the Injector 8vo, i 50 MacCord's Slide-valves 8vo, 2 oo Meyer's Modern Locomotive Construction 4to, 10 oo Peabody's Manual of the Steam-engine Indicator i2mo i 50 Tables of the Properties of Saturated Steam and Other Vapors 8vo, i oo Thermodynamics of the Steam-engine and Other Heat-engines 8vo, 5 oo Valve-gears for Steam-engines Svo, 2 50 Peabody and Miller's Steam-boilers 8vo, 4 oo Pray's Twenty Years with the Indicator Large 8vo, 2 50 Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. (Osterberg.) .i2mo, i 25 Reagan's Locomotives: Simple, Compound, and Electric. New Edition. Large 12 mo, 3 50 Sinclair's Locomotive Engine Running and Management i2mo, 2 oo Smart's Handbook of Engineering Laboratory Practice i2mo, 2 50 Snow's Steam-boiler Practice 8vo, 3 oo Spangler's Valve-gears 8vo, 2 50 Notes on Thermodynamics i2mo, i oo Spangler, Greene, and Marshall's Elements of Steam-engineering . 8vo, 3 oo Thomas's Steam-turbines 8vo, 3 50 Thurston's Handy Tables 8vo, i 50 Manual of the Steam-engine 2 vols., 8vo, 10 oo Part I. History, Structure, and Theory. 8vo, 6 oo Part II. Design, Construction, and Operation 8vo, 6 oo Handbook of Engine and Boiler Trials, and the Use of the Indicator and the Prony Brake 8vo, 5 oo Stationary Steam-engines 8vo, 2 50 Steam-boiler Explosions in Theory and in Practice i2mo, i 50 Manual of Steam-boilers, their Designs, Construction, and Operation 8vo, 5 oo Wehrenfenning's Analysis and Softening of Boiler Feed-water (Patterson) 8vo, 4 oo Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) 8vo, 5 oo Whitham's Steam-engine Design 8vo, 5 oo Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .8vo, 4 oo MECHANICS AND MACHINERY. Barr's Kinematics of Machinery 8vo, 2 50 * Bovey s Strength of Materials and Theory of Structures 8vo, 7 50 Chase's The Art of Pattern-making I2mo, 2 50 15 Church's Mechanics of Engineering 8vo, 6 oo Notes and Examples in Mechanics 8vo, 2 oo Compton's First Lessons in Metal-working izmo, Compton and De Groodt's The Speed Lathe 'I2mo, Cromwell's Treatise on Toothed Gearing i2mo, Treatise on Belts and Pulleys i2mo, Dana's Text-book of Elementary Mechanics for Colleges and Schools. .12010, Dingey's Machinery Pattern Making i2mo, Dredge's Record of the Transportation Exhibits Building of the World's Columbian Exposition of 1893 4to half morocco, 5 oo Du Bois's Elementary Principles of Mechanics: Vol. I. Kinematics 8vo, 3 50 Vol. II. Statics 8vo, 4 oo Mechanics of Engineering. Vol. I Small 4to, 7 50 Vol. II Small 4to, 10 oo Durley's Kinematics of Machines 8vo, 4 oo Fitzgerald's Boston Machinist, i6mo, i oo Flather's Dynamometers, and the Measurement of Power i2mo, 3 oo Rope Driving i2mo, 2 oo Goss's Locomotive Sparks 8vo, 2 oo Locomotive Performance 8vo, 5 oo * Greene's Structural Mechanics 8vo, 2 50 Hall's Car Lubrication i2mo, i oo Hobart and Ellis 's High-speed Dynamo Electric Machinery. (In Press.) Holly's Art of Saw Filing i8mo, 75 James's Kinematics of a Point and the Rational Mechanics of a Particle. Small 8vo, 2 oo * Johnson's (W. W.) Theoretical Mechanics i2mo, 3 oo Johnson's (L. J.) Statics by Graphic and Algebraic Methods 8vo, 2 oo Jones's Machine Design: Part I. Kinematics of Machinery 8vo, i 50 Part II. Form, Strength, and Proportions of Parts .8vo, 3 oo Kerr's Power and Power Transmission 8vo, 2 oo Lanza's Applied Mechanics 8vo, 7 50 Leonard's Machine Shop, Tools, and Methods 8vo, 4 oo * Lorenz's Modern Refrigerating Machinery. (Pope, Haven, and Dean.)- 8vo, 4 oo MacCord's Kinematics; or, Practical Mechanism 8vo, 5 oo Velocity Diagrams 8vo, i 50 * Martin's Text Book on Mechanics, Vol. I, Statics i2mo, i 25 * Vol. 2, Kinematics and Kinetics . .I2mo, 1 50 Maurer's Technical Mechanics 8vo, 4 oo Merriman's Mechanics of Materials 8vo, 5 oo * Elements of Mechanics i2mo, i oo * Michie's Elements of Analytical Mechanics 8vo, 4 oo * Parshall and Hobart's Electric Machine Design 4to, half morocco, 12 50 Reagan's Locomotives : Simple, Compound, and Electric. New Edition. Large i2mo, 3 So Reid's Course in Mechanical Drawing 8vo, 2 oo Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 oo Richards's Compressed Air i2mo, i 50 Robinson's Principles of Mechanism 8vo, 3 oo Ryan, Norris, and Hoxie's Electrical Machinery. Vol. 1 8vo, 2 50 Sanborn's Mechanics : Problems Large i amo, i 50 Schwamb and Merrill's Elements of Mechanism 8vo, 3 oo Sinclair's Locomotive-engine Running and Management i2mo, 2 oo Smith's (O.) Press-working of Metals 8vo, 3 oo Smith's (A. W.) Materials of Machines I2mo, i oo Smith (A. W.) and Marx's Machine Design. 8vo, 3 oo Sorel' s Carbureting and Combustion of Alcohol Engines. (Woodward and Preston.) Large 8vo, 3 oo 16 Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo. 3 oo Thurston's Treatise on Friction and Lost Work in Machinery and Mill Work 8vo, 3 oo Animal as a Machine and Prime Motor, and the Laws of Energetics. I2mo, i oo Tillson's Complete Automobile Instructor , i6mo, i 50 Morocco, 2 oo Warren's Elements of Machine Construction and Drawing 8vo, 7 So Weisbach's Kinematics and Power of Transmission. (Herrmann Klein.). 8vo. 5 oo Machinery of Transmission and Governors. (Herrmann Klein.). 8vo. 5 oo Wood's Elements of Analytical Mechanics 8vo, 3 oo Principles of Elementary Mechanics 1 2mo, i 25 Turbines 8vo, 2 50 The World's Columbian Exposition of 1893 4to, I oo MEDICAL. * Bolduan's Immune Sera 12mo, 1 50 De Fursac's Manual of Psychiatry. (Rosanoff and Collins.). . . .Large i2mo, 2 50 Ehrlich's Collected Studies on Immunity. (Bolduan.) 8vo, 6 oo * Fischer's Physiology of Alimentation Large 12mo, cloth, 2 oo Hammarsten's Text-book on Physiological Chemistry. (Mandel.) 8vo, 4 oo Lassar-Cohn's Practical Urinary Analysis. (Lorenz.) i2mo, oo * Pauli's Physical Chemistry in the Service of Medicine. (Fischer.) .... 12010, 25 * Pozzi-Escot's The Toxins and Venoms and their Antibodies. (Conn.). i2mo, oo Rostoski's Sejum Diagnosis. (Bolduan.) i2mo, oo Salkowski's Physiological and Pathological Chemistry. (Orndorff.) 8vo, 50 * Satterlee's Outlines of Human Embryology i2mo, 25 Steel's Treatise on the Diseases of the Dog 8vo, 50 Von Behring's Suppression of Tuberculosis. (Bolduan.) i2mo, oo Woodhull's Notes on Military Hygiene i6mo, 50 * Personal Hygiene i2mo, oo Wulling's An Elementary Course in Inorganic Pharmaceutical and Medical Chemistry i2mo, 2 oo METALLURGY. Betts's Lead Refining by Electrolysis. (In Press.) Egleston's Metallurgy of Silver, Gold, and Mercury; Vol. I. Silver , 8vo, 7 50 Vol. II. Gold and Mercury 8vo, 7 50 Goesel's Minerals and Metals: A Reference Book. , . . . . i6mo, mor. 3 oo * Iles's Lead-smelting i2mo, 2 50 Keep's Cast Iron 8vo, 2 50 Kunhardt's Practice of Ore Dressing in Europe 8vo, i 50 Le Chatelier's High-temperature Measurements. (Boudouard Burgess. )i2mo, 3 oo Metcalf's Steel. A Manual for Steel-users 12010, 2 oo Miller's Cyanide Process I2mo, i oo Minet's Production of Aluminum and its Industrial Use. (Waldo.). , . . i2mo, 2 50 Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, 4 oo Smith's Materials of Machines I2mo, j oo Thurston's Materials of Engineering. In Three Parts 8vo, 8 00 Part II. Iron and Steel 8 VO) 3 5O Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 Ulke's Modern Electrolytic Copper Refining 8vo, 3 oo MINERALOGY. Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 2 50 Boyd's Resources of Southwest Virginia g vo , 3 oo 17 Boyd's Map of Southwest Virignia Pocket-book form. 2 oo * Browning's Introduction to the Rarer Elements 8vo, i s*> Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo, 4 oo Chester's Catalogue of Minerals 8vo, paper, i oo Cloth, i 25 Dictionary of the Names of Minerals Svo, 3 50 Dana's System of Mineralogy Large 8vo, half leather, 12 50 First Appendix to Dana's New " System of Mineralogy." Large bvo, i oo Text-book of Mineralogy 8vo, 4 oo Minerals and How to Study Them I2mo, I 50 Catalogue of American Localities of Minerals Large 8vo, i oo Manual of Mineralogy and Petrography ismo 2 oo Douglas's Untechnical Addresses on Technical Subjects I2mo, i oo Eakle's Mineral Tables 8vo, i 25 Egleston's Catalogue of Minerals and Synonyms 8vo. 2 50 Goesel's Minerals and Metals : A Reference Book ibmo, mor. 3 oo Groth's Introduction to Chemical Crystallography (Marshall) 12 mo, i 25 Iddings's Rock Minerals 8vo, 5 oo Johannsen's Key for the Determination of Rock-forming Minerals in Thin Sections. (In Press.) * Martin's Laboratory Guide to Qualitative Analysis with the Blowpipe. I2mo, 60 Merrill's Non-metallic Minerals. Their Occurrence and Uses 8vo, 4 oo Stones for Building and Decoration 8vo, 500 * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 8vo, paper, 50 Tables of Minerals 8vo, i 00 * Richards's Synopsis of Mineral Characters i2mo. morocco, i 25 * Ries's Clays. Their Occurrence. Properties, and Uses 8vo, 5 oo Rosenbusch's Microscopical Physiography of the Rock-making Minerals. (Iddings.) 8vo, 5 oo * Tillman's Text-book of Important Minerals and Rocks 8vo, 2 oo MINING. Beard's Mine Gases and Explosions. (In Press.) Boyd's Resources of Southwest Virginia 8vo, 3 oo Map of Southwest Virginia Pocket-book form. 2 oo Douglas's Untechnical Addresses on Technical Subjects i2mo, I oo Eissler's Modern High Explosives 8-73, 4 oo Goesel's Minerals and Metals ; A Reference Book i6mo, mor. 3 oo Goodyear's Coal-mines of the Western Coart of the United States i2mo, 2 50 Ihlseng's Manual of Mining. 8vo, 5 oo * Bes's Lead-smelting I2mo, 2 50 Kunhardt's Practice of Ore Dressing in Europe 8vo, i 50 Miller's Cyanide Process i2mo, i oo O'DriscolJ's Notes on the Treatment of Gold Ores Svo, 2 oo Robine and Lenglen's Cyanide Industry. (Le Clerc.) Svo, 4 oo Weaver's Military Explosives Svo, 3 oo Wilson's Cyanide Processes i2mo. i 50 Chlorination Process limo, i 50 Hydraulic and Placer Mining. 2d edition, rewritten i2mo, 2 50 Treatise on Practical and Theoretical Mine Ventilation I2mo, i 25 SANITARY SCIENCE. Bastiore's Sanitation of a Country House i2mo, i oo * Outlines ot Practical Sanitation I2mo, I 25 FolwelTs Sewerage. (Designing, Construction, and Maintenance.) Svo, 3 oo Water-supply Engineering Svo, 4 oo 18 Fowler's Sewage Works Analyses . . , 121113, 2 oo Fuertes's Water and Public Health i2mo, i 50 Water-filtration Works i2mo, 2 50 Gerhard's Guide to Sanitary House-inspection i6mo, i oo Sanitation of Public Buildings 12mo, 1 50 Hazen's Filtration of Public Water-supplies 8vo, 3 oo Leach's The Inspection and Analysis of Food with Special Reference to State Control 8vo, 7 50 Mason's Water-supply. (Considered principally from a Sanitary Standpoint) 8vo, 4 oo Examination of Water. (Chemical and Bacteriological.) 12010, i 25 * Merriman's Elements of Sanitary Engineering 8vo rf 2 oo Ogden's Sewer Design i2mo, 2 oo Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- ence to Sanitary Water Analysis i2mo, i 25 * Price's Handbook on Sanitation lamo, i 50 Richards's Cost of Food. A Study in Dietaries i2mo, i oo Cost of Living as Modified by Sanitary Science 1 2mo, i oo Cost of Shelter i2mo, i oo Richards and Woodman's Air. Water, and Food from a Sanitary Stand- point 8vo, 2 oo * Richards and Williams's The Dietary Computer 8vo, i 50 Rideal's Si wage and Bacterial Purification of Sewage 8vo, 4 oo Disinfection and the Preservation of Food 8vo, 400 Turneaure and Russell's Public Water-supplies 8vo, 5 oo Von Behring's Suppression of Tuberculosis. (Bolduan.) i2mo, i oo Whipple's Microscopy of Drinking-water 8vo, 3 50 Wilson's Air Conditioning. (In Press.) Winton's Microscopy of Vegetable Foods 8vo, 7 50 Woodhull's Notes on Military Hygiene iCmo, i 50 * Personal Hygiene I2mo, i oo MISCELLANEOUS. Association of State and National Food and Dairy Departments (Interstate Pure Food Commission) : Tenth Annual Convention Held at Hartford, July 17-20, 1906. ...8vo, 3 oo Eleventh Annual Convention, Held at Jamestown Tri-Centennial Exposition, July 16-19, 1907. (In Press.) Emmons's Geological Guide-book of the Rocky Mountain Excursion of the International Congress of Geologists Large Cvo, i 50 Ferrel's Popular Treatise on the Winds 8vo, 4 oo Gannett's Statistical Abstract of the World 24010, 75 Gerhard's The Modern Bath and Bath-houses. (In Press.) Haines's American Railway Management I2mo, 2 30 Ricketts's History of Rensselaer Polytechnic Institute, 1824-1894.. Small 8vo, 3 oo Rotherham's Emphasized New Testament Large 8vo, 2 OQ Standage's Decorative Treatment of Wood, Glass, Metal, etc. (In Press.) The World's Columbian Exposition of 1893 4to, i oo Winslow's Elements of Applied Microscopy I2mo, i 50 HEBREW AND CHALDEE TEXT-BOOKS. Green's Elementary Hebrew Grammar 1 2mo, i 25 Hebrew Chrestomathy 8vo, 2 oo Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. (Tregelles.) Small 4to, half morocco. 5 oo Letteris's Hebrew Bible 8vo, 2 25 19 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. IEf 20 1933 SEP 21 1933 ocr 2? 1 93S 6 1940 h SEP 15 I 40ct'55Ei LD 21-50m-l,'33