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REESE LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Received 
 
 Accessions No. ^/ 5<- Shelf No. 
 
 Jb 
 
HYDRAULIC 
 
 TABLES, COEFFICIENTS, AND lOBMULffi, 
 
 FOR 
 
 FINDING THE DISCHARGE OF WATER FKOM ORIFICES, 
 NOTCHES, WEIRS, PIPES, AND RIVERS. 
 
 BY 
 JOHN NEVILLE, CIVIL ENGINEEB, M.E.I.A., 
 
 COUNTY 6UBVEYOB OF LOUTH AND OF THE COUNTY OF THE TOWN OF DBOOHEDA. 
 
 WITH EXTENSIVE ADDITIONS, NEW FORMULA, TABLES, AND GENERAL 
 
 INFORMATION ON RAIN-FALL CATCHMENT-BASINS, DRAINAGE, SEWERAGE, 
 
 WATER SUPPLY FOR TOWNS AND MILL POWER, 
 
 " It ought to be more generally known, that theory is nothing more than the conclusions 
 of reason from numerous and accurately observed phenomena, and the deductions of the 
 laws which connect causes with effects ; that practice is the application of those general 
 truths and principles to the common affairs and purposes of life ; and that science is the 
 recorded experience and discoveries of mankind, or, as it has been well defined, ' the know- 
 ledge of many, orderly and methodically digested, and arranged, so as to become attainable 
 by one.'" AMERICAN QUABTEBLY 
 
 LONDON: 
 JOHN WEALE, 59, HIGH HOLBORN. 
 
 1860-1. 
 
TO 
 
 MAJOR-GENERAL SIR THOMAS AISKEW LARCOM, K.C.B., 
 
 LL.D., F.R.S., M.B.I.A., ETC., 
 
 OF THE 
 
 ROYAL ENGINEERS, 
 
 UNDER SECRETARY OF STATE FOR IRELAND, 
 THIS WORK IS INSCRIBED 
 
 BY THE 
 AUTHOR. 
 
CONTENTS. 
 
 Introduction to the Second Edition * *; &.* ;'-;..**< .-*> 
 Introduction to the First Edition -*. ' * ~ 4.-; - #T- *1 
 
 SECTION I. Application and Use of the Tables, Formulae, 
 &c. Extra Horse-power required in Pumping Engines 
 from Friction in the Pipes Table of Heads due to 
 Friction and of Discharges f . ~ w 
 
 SECTION II. ' Formulae for the Velocity and Discharge 
 from Orifices, Weirs, and Notches Coefficients of Ve- 
 locity, Contraction, and Discharge Practical Remarks 
 
 on the Use of the Formulae ; ^. . -.# i f*-j. 
 
 SECTION III. Experimental Results and Formulse Co- 
 efficients of Discharge for Orifices, Notches, and Weirs 
 
 SECTION IV. Variations in the Coefficients from the Posi- 
 tion of the Orifice General and Partial Contraction 
 Velocity of Approach Various Practical Formulae for 
 the Discharge over Weirs and Notches Central and 
 Mean Velocities . . . . ' . ". '" .' *~"l 
 
 SECTION V. Submerged Orifices and Weirs Contracted 
 River Channels . j 4 * . ,, . .. 
 
 SECTION VI. Short Tubes, Mouth-pieces, and Approaches 
 Alteration in the Coefficients from Friction by in- 
 creasing the Length Coefficients of Discharge for 
 Simple and Compound Short Tubes Shoots 
 
 SECTION VII. Lateral Contact of the Water and Tube 
 Atmospheric Pressure Head measured to the Dis- 
 charging Orifice Coefficient of Resistance Formula 
 for the Discharge from a Short Tube Diaphragms 
 Oblique Junctions Formula for the Time of the Sur- 
 face smking a given depth Lock Chambers *' : ' 
 
 SECTION VIII. Flow of Water in Uniform Channels 
 Mean Velocity Mean Radii and Hydraulic Mean 
 Depths Border Train Hydraulic 1 nclination Ef- 
 fects of Friction Formulse for calculating the Mean 
 Velocity Application of the Formulae and Tables to 
 the Solutions of three useful Problems 
 
 PAGES 
 v. to xii. 
 
 1 to!4 
 
 1643 
 
 4463 
 6395 
 
 96139 
 140150 
 
 150165 
 
 166181 
 
 162234 
 
11 
 
 CONTENTS. 
 
 SECTION IX. Best Forms of the Channel Regimen 
 
 Velocity Equally Discharging Channels . . . 235253 
 
 SECTION X. Effects of Enlargements and Contractions 
 
 Backwater Weir Case Long and Short Weirs . 254 271 
 
 SECTION XI. Bends and Curves Branch Pipes Dif- 
 ferent Losses of Head General Equation for finding 
 the Velocity Hydrostatic and Hydraulic Pressure 
 Piezometer Catchment Basins Rain-fall per annum 271 289 
 
 SECTION XII. Rain-fall Catchment Basins Discharge 
 into Channels Discharge from Sewers Loss from 
 Evaporation, &p. . . . ; , . . 289315 
 
 SECTION XIII. Water Supply for Towns -Strength of 
 Pipes Sewerage Estimates and Cost Thorough Drain- 
 ageArterial Drainage ^ > . . + . , p. 315364 
 
 SECTION XIV. Water Power Dynamometer, or Friction 
 Brake Overshot, Breast, and Undershot Vertical 
 Wheels Turbines and Horizontal Wheels Hydraulic 
 $am Water-pressure Engine Work of Flour Mills . 364399 
 
 TABLES NOT EMBODIED IN THE TEXT. 
 
 TABLE I. Coefficients of Discharge from Square and Dif- 
 ferently proportioned Rectangular Lateral Orifices in 
 Thin Vertical Plates . . ... . . 400401 
 
 TABLE II. For finding the Velocities from the Altitudes 
 
 and the Altitudes from the Velocities .... 402 411 
 
 TABLE III. Square Roots for finding the Effects of the 
 Velocity of Approach, when the Orifice is small in pro- 
 portion to the Head. Also, for finding the Increase in 
 the Discharge from an Increase of Head. (See pp. 101 
 and 107.) ......... 412413 
 
 TABLE IV. For finding the discharge through Rectan- 
 gular Orifices j in which n =-^. Also, for finding the 
 
 Effects of the Velocity of Approach to Weirs, and the 
 
 Depression on the Crest. (See pp.101 and 107.) . . 414 417 
 
 TABLE V. Coefficients of Discharge for different Ratios 
 
 of the Channel to the Orifice . . 418423 
 
CONTENTS. 
 
 Ill 
 
 TABLE VI. The Discharge over Weirs or Notches of One 
 Foot in Length, in Cubic Feet per Minute. (See pp. 
 IQQ and 111 to 127.) 424429 
 
 TABLE VII. For finding the Mean Velocity from the 
 MaximumVelocity'at the surface, in Mill-races, Streams, 
 and Rivers with Uniform Channels ; and the Maximum 
 Velocity from the Mean Velocity. (See p. 184.) y 430 
 
 TABLE VIII. For finding the Mean Velocities of Water 
 flowing in Pipes, Drains, Streams, and Rivers. (See 
 p. 195) ..."...,'' 431443 
 
 TABLE IX. For finding the Discharge in Cubic Feet, per 
 Minute, when the Diameter of a Pipe or Orifice, and 
 the Velocity of Discharge, are known, and vice versd . 444445 
 
 TABLE X. For finding the Depths on Weirs of different 
 Lengths, the Quantity discharged over each being sup- 
 posed constant. (See p. 271.) 446 
 
 TABLE XI. Relative Dimensions of Equally-Discharging 
 Trapezoidal Channels, with Slopes from to 1, up to 2 
 to 1. (See pp. 249 and 252.) ..... 447 
 
 TABLE XII. Discharges from the Primary Channel in 
 
 the first column of Table XI. (See p. 253.) ; . 448449 
 
 TABLE XIII. The Square Roots of the Fifth Powers of 
 Numbers for finding the Diameter of a Pipe, or Dimen- 
 sions of a Channel from the Discharge, or the reverse ; 
 showing the relative Discharging Powers of Pipes of 
 different Diameters, and of any similar Channels what- 
 ever, closed or open. (See pp, 31, 42, 228, &c.) . . |j 450 
 
 TABLE XIV. English and French Measures . . . 451454 
 
 TABLE XV. Weight, Specific Gravity, Ultimate Strength 
 
 and Elasticity of Various Materials . . . 655 
 
 COBRECTION. 
 
 2 2 
 
 For o = 3 c dN / 2# h, p. 120, line 18, read o =g-<W 2$r. 
 
((UNIVERSITY jj 
 
 INTRODUCTION 
 
 TO THE SECOND EDITION. 
 
 IN order to render this edition more valuable to 
 the hydraulic engineer it has been considerably ex- 
 tended by the insertion of several new tables, formulae, 
 experimental coefficients, examples, and general esti- 
 mates of cost. It is hoped that the great extent and 
 practical nature of the additions will render this 
 work far more useful than the first in the ever 
 varying requirements of the profession. The six 
 pages on catchment basins, rain-fall, and water power 
 in the first edition have been extended into three new 
 sections of one hundred and ten pages, embodying 
 the subjects of water supply, sewerage discharge, 
 drainage, and the method of determining the useful 
 effect of water employed in turning the various kinds 
 of water wheels. Considerable additions have also 
 been made in Sections I, III, IV, VIII, and IX; 
 the practical formula for gauging by weirs have been 
 extended through twenty-one extra pages of new 
 matter ; the portion treating of the conveying powers 
 of pipes and rivers has been also extended by the 
 insertion of new tables at pages 42, 152, 191, 220, 
 and 252, and of several new formulae, amongst which 
 we believe that at page 215 (119A), will be found, in 
 practice, the simplest and most accurate yet dis- 
 
VI INTRODUCTION TO THE SECOND EDITION. 
 
 covered for ranges of velocity varying between one 
 inch and twenty feet per second, and for all descrip- 
 tions of channels, pipes, and rivers with which the 
 engineer has to deal. 
 
 We have noticed at page 95 the erroneous notation 
 of Morin and other engineers in giving only two- 
 thirds of the co-efficient of discharge for weirs. 
 This notation assumes that the theoretical discharge 
 from a notch is the same as if all the particles of 
 water had the same velocity as those undermost, 
 which, being too large by one-third, the experimental 
 coefficient has to be reduced in the same proportion. 
 Mr. Blackwell and Mr. Hughes, in this country, lend 
 themselves to this nomenclature ; the latter gentle- 
 man says, page 328 of his useful treatise on Water 
 Works*, " Mr. Neville, in his tables of the discharge 
 over weirs, termed the theoretical discharge 321A^ 
 instead of 481/^ in cubic feet per minute." We 
 were correct in doing so, for a weir one foot long ; 
 and those who adopt the latter formula are in error : 
 this is a matter of demonstration, not of opinion. 
 We can see no reason for sanctioning a different 
 notation for notches, or orifices at the surface, and 
 sunk orifices. The coefficients when in thin plates, 
 with large cisterns, have nearly the same general 
 value, -615 to -628, and it tends to confusion to adopt 
 in one place a coefficient for a correct formula, and 
 in another a coefficient for an incorrect one ; although 
 the final result by an equality of contrary errors 
 may be the same in both. We may here observe 
 
 * Weale's Series. 
 
INTRODUCTION TO THE SECOND EDITION. Vll 
 
 how very general the coefficient of two-thirds, and 
 
 thereabouts, is for all orifices, notches, and also for 
 
 the useful effect derived from the application of 
 
 water power, as well as the relation of the velocity 
 
 due to the fall and the velocity of water wheels to 
 
 give a maximum result. The modifications of coeffi- 
 
 cients dependent on the position, thickness, form, 
 
 and approaches of an orifice are seldom understood. 
 
 The defects in the ordinary formula when the velo- 
 
 city of approach has to be considered are pointed 
 
 out at pages 100, 115, and 116, and it is to be re- 
 
 gretted that the authority of D'Aubuisson and others 
 
 has misled many as to the correct form. Before the 
 
 effective power of a water wheel, or water engine, 
 
 can be determined we must know how to gauge the 
 
 water supplied to it correctly. This can be done 
 
 only by the application of formulae varied to suit the 
 
 circumstances of the case under consideration. From 
 
 causes, which it is not necessary to enter into here, 
 
 this has seldom been done, and very little dependence 
 
 can be placed on results obtained by the formula in 
 
 common use when applied generally. It is pleasing 
 
 to follow Francis and Thomson through the steps by 
 
 which they get the effective power of their wheels, 
 
 and we have accordingly made considerable use of 
 
 their labors in Section XIV. 
 
 We have modified some of the old partial equations 
 for the velocity in pipes, beginning at (86), from the 
 
 (rs . 2 H a . , a 
 
 form v = - + - 2 - , mto v = c^rs ; 
 
 principally for the purpose of giving the numerical 
 values of c in the form v = c \/rs. Of course these 
 
Vlll INTEODUCTION TO THE SECOND EDITION. 
 
 modifications are only applicable when the value of 
 O-T is small compared with that of the velocity. 
 
 Some remarks will be found at page 224 respecting 
 M. Darcy's new formula for the velocity of water in 
 iron pipes, as given by Morin in his Hydraulique. 
 We have, only since those remarks were written, seen 
 the original Memoire, printed in Tome XY, Memoires 
 presentes par divers savants a I'Academie des Sciences 
 de rinstitut Imperial de France, Paris, 1858. This 
 Memoire extends through 263 quarto pages, of which 
 34 contain tables calculated from the formula. We 
 have not had time to do more than glance through it, 
 but the deduced formula appears to be entirely de- 
 rived from the author's experiments. This we con- 
 sider, to some extent, objectionable ; but, however 
 this may be, there can be no doubt that the value of 
 the coefficient c, in the formula v c \/ r s, increases 
 with the inclination, $, as well as with the diameter, 
 4 r, of the pipe ; and as M. Darcy's formula makes 
 the value of the multiplier, c, depend alone on the 
 value of r, or 4 r, there appears an omission, in making 
 the coefficient of friction entirely independent of the 
 hydraulic inclination, and dependent only on the size 
 of the channel. We shall give a few examples, taken 
 at discretion, to show how limited this formula must 
 be in its application. 
 
 1. Couplet's experiment, No. 43, p. 103, reduced 
 to feet, gives r = -3997 feet,* zz -0035, r s = -001339, 
 and the observed velocity v =2 3-478 feet =. 95 \/r~s 
 nearly. Darcy's formula would give v = 110-8 
 \/f r s, our formula 106 \/r~s nearly, and Weisbach's 
 
INTRODUCTION TO THE SECOND EDITION. IX 
 
 105 \/r^ nearly. The pipe was probably an old 
 one, and a deduction of about 10 per cent, might be 
 made for the state of the bore. We have here, how- 
 ever, no means of judging the effect of a change of 
 inclination on the multiplier, c. 
 
 2. From Du Buat's experiments with an inch 
 pipe, nearly, Nos. 50 and 51, p. 103, we get, after 
 reducing them to feet, in experiment 50, r zz -0222, 
 s zz -228 and v zz 6*33 feet zz 89-2 \/rs ; or, after 
 making the necessary deductions in the head for the 
 velocity and the orifice of entry with the coefficient 
 815, s zz -147 and v zz 6*33 feet = 111*4 \/r~s. In 
 experiment 51, we also get in feet r zz -0222, s zz 
 3074, and v zz 7-54 zz 92 \/r~s ; or, by making 
 allowance for the head due to the velocity and the 
 orifice of entry, as before, s zz *179, and v zz 7*54 
 feet zz 119*7 \/rs. Here we see how the velocity 
 or value of the inclination, s, affects the value of 
 the multiplier, the diameter remaining constant. 
 M. Darcy's formula, in each case, would only make 
 v zz 80-3 \/r~s. 
 
 3. In the excerpt proceedings of the Institution 
 of Civil Engineers, p. 4, 6th February, 1855, James 
 Simpson, president, in the chair, there is given for 
 the "Colinton pipe" 16 inches diameter, eight or 
 nine years in use, three observations. First, 29,580 
 feet long, a head of 420 feet and a discharge of 571 
 cubic feet per minute : these give v zz 6-816 feet zz 
 99*2 </7s nearly. Secondly, a length of 25,765 feet 
 a head of 184 feet, and a discharge of 440 cubic feet 
 per minute : these give v zz 5*252 feet zz 96*3 \/rs. 
 And thirdly, a length of 3,815 feet a head of 184 
 
X INTRODUCTION TO THE SECOND EDITION. 
 
 feet, and a discharge of 1,215 cubic feet per minute : 
 these give v zi 14-5 feet z= 115 \/r s nearly. In these 
 three examples, the diameter, castings, and age of 
 the pipes are the same. Yet we see, clearly, that the 
 inclination affects the multiplier of \/r s, which in* 
 creases with the inclination, s, although M. Darcy's 
 formula would make the multiplier the same in each 
 case, and for all inclinations, viz, v = 110 \/rs< 
 Making those allowances inseparable from the state 
 of the pipe, and all experimental observations, these 
 results, as well as those from Du Buat's experiments, 
 confirm the accuracy of our general formula (119 A) 
 page 215*, and those others we have given following 
 it, as well also as that of Weisbach. 
 
 Dr. Young's formula, page 207, bears a resem- 
 blance to that of M. Darcy, in making the multi- 
 plier of \/r s depend only on the diameter; but it 
 works in a contrary manner : for the high velocities 
 being derived from pipes, with small diameters in the 
 experiments at his command, the value of c in v 
 c \/r s, reduced from his formula, becomes larger in 
 general for small than for larger diameters. No 
 doubt an allowance should be made in small pipes 
 for a thin film of water adjoining the pipe with little 
 or no velocity ; but within the limits which the en- 
 gineer has to deal this may be neglected. Its effect, 
 
 * The form in which we first discovered this formula was 
 
 v = 1 140 7 rj [ X vTrs. For measures in metres it becomes 
 . ( (rs)) 
 
 v = 77-3 (rs)* 4-9 (rs)s; in which r is half the radius of a cy- 
 lindrical pipe, or the hydraulic mean depth of any channel. 
 
INTRODUCTION TO THE SECOND EDITION. XI 
 
 as well as that of all the other resistances, junctions, 
 contractions, deposits, &c., is greater in small than 
 in large pipes. We must refer to the body of the 
 work for further remarks on this subject, but from 
 lately appearing at such length in the Memoires of 
 the Imperial Academy of Sciences, M. Darcy's for- 
 mula called for especial notice here. 
 
 The Statistics of rain-fall and catchment-basins 
 have not yet received the attention which the subject 
 deserves. The distribution of rain gauges with re- 
 ference to elevation, contour, temperature, and iso- 
 thermal lines has not been sufficiently attended to. 
 The connexion of the rain-fall with the discharge 
 generally, for the whole catchment, for the tributary 
 catchments, and their sub-catchments, at the sea in 
 the middle districts and at the sources, noting the 
 geology, must be" observed for several years before 
 the questions of supply, discharge, absorption, and 
 evaporation in any climate can be answered. The 
 maximum and minimum discharges in each year and 
 series of years must be observed, as well as the 
 average mean discharges, and the maximums and 
 minimums of these also, before the physical con- 
 nexion of climate and catchment can be correctly 
 ascertained, and the engineer furnished with reliable 
 data. Heretofore observations, even when best, have 
 been partial or limited, and a wide field is here yet 
 open to competent physicists in connexion with our 
 drainage works. 
 
 The general items of cost given in SECTION XIII. 
 will be found of use ; they are intended, how- 
 ever, more as guides than as standards for other 
 
Xll INTRODUCTION TO THE SECOND EDITION. 
 
 works, the cost of which must depend on their own 
 circumstances. Those who have practical experience 
 of the differences between estimates, cost, and value, 
 and how they are affected by time, locality, quality, 
 and quantity, will estimate for each case in detail ; 
 but the discrepancies between estimates and cost, 
 even under the same circumstances, are too well 
 known to call for any remark here. 
 
 A few words about our publisher. Mr. Weale 
 having purchased our interest in this edition, at once 
 decided on adopting larger type and better paper, 
 at a heavy extra outlay to himself. We had reason 
 to be satisfied with the manner in which the first 
 edition was brought out, in this, however, he has 
 excelled, and we hope his enterprise will receive a fair 
 return. 
 
 Jocelyn Street^ Dundalk, 
 October, 1860. 
 
it f r inc r 
 
 f C NIVERSITY 
 
 X^UFORN!^ 
 
 INTKODUCTION 
 
 TO THE FIEST EDITION. 
 
 IN preparing the following work, we had three ob- 
 jects in view : first, a collection of useful hydraulic 
 formulae ; secondly, a collection of experimental re- 
 sults, and coefficients ; and, thirdly, a collection of 
 useful practical tables, some calculated entirely from 
 the formulae and experiments, and others for the 
 purpose of rendering the calculations more easy. 
 
 The TABLES at the end of the volume are all 
 original, with the exception of TABLE I., which con- 
 tains the well-known coefficients of PONCELET and 
 LESBROS ; but those are newly arranged, the heads 
 reduced to English inches, and the coefficients for 
 heads measured over and back from the orifice, 
 placed side by side, for more ready comparison. 
 The coefficients in the small Tables throughout the 
 work have been all calculated by us from the original 
 experiments ; the formulae have been carefully ex- 
 amined, and the continental ones reduced to English 
 measures some of them, as will be seen, for the 
 first time. No labour has been spared in preparing 
 the TABLES, and they are all purely hydraulic, though 
 some of them are capable of being otherwise applied. 
 We have filled no gap by the introduction of Tables 
 applicable to other subjects, and in e very-day use. 
 
2 INTRODUCTION TO THE FIRST EDITION. 
 
 The correction of some of the experimental for- 
 mulae, particularly the continental ones, as printed 
 in some English books, cost us some labour. Even 
 Du Buat's well-known formula is frequently mis- 
 printed; and in a late hydraulic work, \/d -1, one 
 of the factors, is printed \A/--1 in every page where 
 it is quoted. It is not always that such mistakes 
 can be avoided, but experimental formula are so 
 often copied from one work into another without 
 sufficient examination, that an error of this kind 
 frequently becomes fixed ; and when applied to prac- 
 tical purposes erroneous formula get the correct ones 
 into disrepute. See note to formula (91). 
 
 The TABLES of velocities and discharges over weirs 
 and notches have been calculated for a great number 
 of coefficients to meet different circumstances of ap- 
 proach and overfall, and for various heads from ^th 
 of an inch up to 6 feet. TABLE II. embodies the 
 velocities acquired by falling bodies under the head 
 of " theoretical velocity," and the velocities, suited 
 to various coefficients, for heads up to 40 feet. 
 
 The formulae for calculating the effects of the ve- 
 locity of approach to orifices and weirs, and the 
 necessary corrections for the ratio of the channel to 
 the orifice, as well as TABLE V., we believe to be 
 original. They will be found of much value in de- 
 termining the proper coefficients suited to various 
 ratios. The remarks throughout SECTION IV. are par- 
 ticularly applicable to the proper use of this TABLE. 
 
 TABLE VII. of surface and mean velocities will be 
 found to vary from those generally in use, and to be 
 much more correct, and better suited for practical 
 
INTRODUCTION TO THE FIRST EDITION. 3 
 
 purposes, particularly as applied to finding the mean 
 velocities in rivers. 
 
 We have extended TABLE VIII. so as to make it 
 directly available for hydraulic mean depths, from 
 jVth of an inch to 12 feet, and for various hydraulic 
 inclinations, even up to vertical, for pipes. The fall 
 in rivers seldom exceeds 2 or 3 feet per mile, or the 
 velocity 5 or 6 feet per second. The extension of the 
 Table for great inclinations, and consequently great 
 velocities, was made for purposes of calculation, and 
 to include pipes. It must be understood through- 
 out this TABLE that the velocities are those which 
 continue unchanged for any length of channel, viz., 
 when the resistance of friction is equal to the accele- 
 ration of gravity, the moving water and channel 
 being then in train. Several of Du Buat's experi- 
 ments were made with small vertical pipes. This 
 TABLE is equally applicable to pipes and rivers, and 
 gives directly either the hydraulic inclination, the 
 hydraulic mean depth, or the velocity when any two 
 of them are known. 
 
 Hydraulic formulae have been frequently rendered 
 unnecessarily complex, and unsuited for practical 
 application, by combining them with those of mere 
 mensuration in order to find the discharge. We have 
 therefore given formulae for finding the mean velocity 
 principally, unless in a few instances, as in orifices 
 near the surface, where the discharge itself is first 
 necessary to find the mean volocity ; this once deter- 
 mined, the calculation of the discharge becomes one 
 of simple mensuration. 
 
 We have preferred giving the mean velocity to the 
 
 B3 
 
4 INTRODUCTION TO THE FIRST EDITION. 
 
 discharge itself in TABLE VIII., because, while an 
 infinite number of channels having the same hydraulic 
 inclination (s) and the same hydraulic mean depth (r) 
 must have the same velocity (v), yet the sectional 
 areas, and consequently the discharges, may vary 
 upwards from 6 -2 83 2 r 2 , the area of a semicircular 
 channel, to any extent ; and the operation of multi- 
 plying the area by the mean velocity, to find the 
 discharge, is so very simple that any tabulation for 
 that purpose is unnecessary. Besides this, the banks 
 of rivers, unless artificially protected, remain very 
 seldom at a constant slope, and therefore any TABLES 
 of discharge for particular side slopes are only of use 
 so far as they apply to hypothetical cases. Indeed 
 we have seen, in new river cuts, the banks, cut first 
 to a given slope, alter very considerably in a few 
 months ; while the necessary regimen between the 
 velocity of the water and the channel was in the 
 course of being established. The velocity suited to 
 the permanency of any proposed river channel, 
 though too often entirely neglected, is the very first 
 element to be considered. 
 
 For circular pipes, however, TABLE IX. gives the 
 discharge in cubic feet per minute when the velocity 
 in inches per second is known, or found from TABLE 
 VIII., and is calculated for pipes from Jth of an inch 
 up to 12 inches in diameter. TABLE XII. gives also 
 the discharges in cubic feet per minute from the 
 different equivalent river channels in TABLE XI. 
 
 TABLE X., for finding the heads on weirs of different 
 lengths, TABLE XI., of equally discharging river 
 channels, and TABLE XII., of the actual discharges 
 
INTRODUCTION TO THE FIEST EDITION. 5 
 
 from the equivalents in TABLE XI., will be found of 
 great practical value when new weirs and water-cuts 
 have to be made. TABLES XI. and XII. are equally 
 applicable to channels having side slopes, the widths 
 being then the mean or central widths. 
 
 When the discharge and fall are known, and the 
 hydraulic mean depth and the dimensions of any 
 channel have to be determined, Problem III., section 
 8, as illustrated in EXAMPLE 17, section 1, gives a 
 new and perhaps the most practically useful solution 
 yet published. Tables XI., XII., and XIII. are par- 
 ticularly applicable to this problem. 
 
 A uniform notation is preserved throughout the 
 work, so that the different experimental formulae can 
 be compared without any further reduction. The 
 letter h is used in every instance for the head, c for 
 the coefficient, r for the mean radius or hydraulic 
 mean depth, and s for the sine of the hydraulic 
 inclination, unless it be otherwise stated. In order 
 to designate particular values, the primary letters 
 have deponent or initial letters below to explain them. 
 Thus h t is the head to the fop of an orifice, h^ the 
 head at the bottom, A w the head on a weir, h t the head 
 due to/riction, c d the coefficient of discharge, c v the 
 coefficient of velocity, c c the coefficient of contraction, 
 &c. When the whole head is made up of different 
 elements, such as the portions due to friction, velocity, 
 contractions, bends, &c., it is expressed by the capital 
 letter H. 
 
 Some writers and engineers appear to confound 
 the inclination of a pipe, simply so called, or the head 
 divided by the length, with the hydraulic inclination ; 
 
6 INTRODUCTION TO THE FIRST EDITION. 
 
 and consequently have fallen into error in applying 
 such of the known formulae as take into consideration 
 only the head due to the resistance of friction. When 
 pipes are of considerable length, and the water is 
 supplied from a reservoir at one end, the inclination, 
 found as above, and the hydraulic inclination, may be 
 taken equal to each other without sensible error ; but 
 for shorter pipes, of say up to 100 feet long, or even 
 longer, the greater number of formulae, as Du Buat's 
 and others, do not directly apply ; and it is necessary 
 to take into consideration the head due to the orifice 
 of entry, the velocity in the tube, and also to the 
 impulse of supply when there are junctions. These 
 separate elements, and their effects, will be considered 
 in the following pages ; but it will be of use to refer 
 here to some late experiments, and the imperfect 
 application of formulae to them, first premising that 
 a pipe may be horizontal, or even turn upwards, and 
 yet have a considerable hydraulic inclination. 
 
 Mr. Provisos valuable experiments* with li-inch 
 pipes, from 20 to 100 feet long, have been used in a 
 recent work for the purpose of testing the accuracy 
 of Du Buat's and some other formulae ; but the head 
 divided by the length is assumed to be the hydraulic 
 inclination throughout, and no allowance is made for 
 the head due to the orifice of entry and velocity in 
 the pipe. Of course the writer's conclusions are 
 erroneous. We have shown, SECTION I., page 30, 
 how very nearly the formulae and experiments agree. 
 
 The formulae appear to have been also misunder- 
 
 * Transactions of the Institution of Civil Engineers, Vol. II., 
 pp. 201210. 
 
INTRODUCTION TO THE FIRST EDITION. 
 
 stood by the surveyor who experimented for the 
 General Board of Health ; for the inclination of the 
 pipe in itself is assumed to be the hydraulic inclina- 
 tion, and no allowance is made for the head due to 
 the impulse of supply. We quote from the CIVIL 
 ENGINEER AND ARCHITECT'S JOURNAL, Vol. XV., 
 page 366, in which it is stated that "the chief results 
 as respect the house drains are thus described in the 
 examination of the surveyor appointed to make the 
 trials."* 
 
 " What quantity of water would be discharged through a 3-inch 
 pipe on an inclination of 1 in 120 ? Full at the head, it would 
 discharge 100 gallons in three minutes, the pipe being 50 feet in 
 length. This is with stone-ware pipe manufactured at Lambeth. 
 This applies to a pipe receiving water only at the inlet, the water 
 not being higher than the head of the pipe. 
 
 " What water was this? Sewage-water of the full consistency, 
 and it was discharged so completely that the pipe was perfectly 
 clean. 
 
 "At the same inclination what would a 4-inch pipe discharge 
 with the same distances ? Twice the amount (that I found from 
 experiment) ; or, in other words, 100 gallons would be discharged 
 in half the time. This likewise applies to a pipe receiving water 
 only at the inlet, and of not greater height than the head. In 
 these cases the section of the stream is diminished at the outlet 
 to about half the area of the pipe. 
 
 "Before these experiments were made, were there not various 
 hypothetical formulae f proposed for general use? Yes. 
 
 * Minutes of Information with reference to Works for the 
 removal of Soil, Water, or Drainage, &c., &c. Presented to both 
 Houses of Parliament, 1852. 
 
 f It is a mistake to call those formulae hypothetical, unless so 
 far as the hypothesis is founded on facts. Every formula with 
 which we are acquainted is founded on experiments, and has 
 been deduce^ from them, but those formulae are too often hypo- 
 thetically applied to short tubes without the necessary corrections. 
 It will be seen from SECTION VIII. that the experiments from 
 
8 INTRODUCTION TO THE FIEST EDITION. 
 
 " What would these formulae have given with a 3-inch pipe, and 
 at an inclination of 1 in 100 ? and what was the result of your 
 experiments with the 8-inch pipe ? The formulae would give 7 
 cuhic feet, the actual experiment gave 11 J cubic feet ; converting 
 it into time, the discharge, according to the formulae, compared 
 with the discharge found by actual practice, would be as 2 to 3. 
 
 " How would it be with a 4-inch pipe ? The formulae would 
 give about 14-7 cubic feet per minute, whereas practice gave 23 
 cubic feet per minute. 
 
 " Take the case of a 6-inch pipe of the same inclination? The 
 results, according to Mr. Hawkesley's formula, would be 40^ cubic 
 feet per minute ; from experiment it was found to be 63 cubic 
 feet per minute. 
 
 " Then with respect to mains and drainage over a flat surface, 
 the result of course becomes of much more value, as the difference 
 proved by actual practice increases with the diminution of the 
 inclination ? Certainly, to a very great extent. For example, the 
 tables give only 14-2 cubic feet per minute as the discharge from 
 a pipe 6 inches diameter, with a fall of 1 in 800 ; practice shows 
 that, under the same conditions, 47*2 cubic feet will be discharged. 
 
 " Will you give an example of the practical value of this when it 
 is required to carry out drainage works over a very flat surface ? 
 An inclination of 1 in 800 gives only 14 cubic feet per minute, 
 according to theory, while, according to actual experiment, and 
 with the same inclination, 47 cubic feet are given. 
 
 " Then this difference may be converted either into a saving of 
 water to effect the same object, or into power of water to remove 
 feculent matter from beneath the site of any houses or town? 
 It may be so. 
 
 " And also the power of small inclinations properly managed ? 
 Yes ; for example, if it was required to construct a water course 
 that should discharge, say 200 feet per minute, the formula would 
 require an inclination of 1 in 60=2 inches in 10 feet ; whereas, 
 experiment has shown that the same would be discharged at an 
 inclination of 1 in 200 =$ inch in 10 feet, thus effecting a con- 
 siderable saving in excavation, or a smaller drain would suffice at 
 the greater inclination." 
 
 which the formulae there given were derived, were in every way 
 greatly more extensive than those made by the directions of 
 the Board of Health. The formula named is, substantially, 
 Eytelwein's algebraically transformed. 
 
INTRODUCTION TO THE FIEST EDITION. 
 
 We have extracted and tabulated the results given 
 above, in the following Table, and also eight of the 
 experiments made for the Metropolitan Commissioners 
 of Sewers'* ; and assuming for the present, with the 
 surveyor, examined by the Commissioners, that the 
 inclinations of the pipes and hydraulic inclinations of 
 the formulae are the same, which is incorrect, we give 
 the calculated discharges, found by nieans of TABLES 
 VIII. and IX., in the last column of the Table. 
 
 Diameter of 
 pipe 
 in inches. 
 
 Inclination of 
 pipe. 
 
 Discharge in 
 cubic feet 
 per minute 
 by experiment. 
 
 Hypothetical 
 discharge by 
 Du Buat's 
 formula. 
 
 3 
 
 1 in 120 
 
 5-3 
 
 6-6 
 
 4 
 
 1 in 120 
 
 10-7 
 
 14 
 
 3 
 
 1 in 100 
 
 11-2 
 
 7-5 
 
 4 
 
 1 in 100 
 
 23 
 
 15-6 
 
 6 
 
 1 in 100 
 
 63-5 
 
 43-8 
 
 6 
 
 1 in 800 
 
 47-2 
 
 13-3 
 
 6 
 
 Iin60 
 
 75 
 
 59-3 
 
 6 
 
 1 in 100 
 
 63 
 
 43-.S 
 
 6 
 
 1 in 160 
 
 54 
 
 33-4 
 
 6 
 
 1 in 200 
 
 52 
 
 29-2 
 
 6 
 
 1 in 320 
 
 49 
 
 21-8 
 
 6 
 
 1 in 400 
 
 48-5 
 
 19-6 
 
 6 
 
 1 in 800 
 
 47-2 
 
 13-3 
 
 6 
 
 Level 
 
 46 
 
 00 
 
 Du Buat's formula, therefore, gives larger results 
 than the experiments in the two first cases, because 
 the water received at one end only barely filled it, 
 and the pipe was not full at the lower end ; but less 
 in the others. If in these the head due to the im- 
 
 * Adcock's Engineer's Pocket Book, 1852, pp. 261 and 262. 
 
10 INTRODUCTION TO THE FIRST EDITION. 
 
 pulse of entry, at the upper end, and at the side 
 junctions, were known, and the proper hydraulic 
 inclination determined by the experiments, the for- 
 mulae would be found to give larger approximate 
 results in every case, as might have been expected 
 from the sewage-water used. In the last eight ex- 
 periments it is stated*, that " the water was admitted 
 at the head of the pipe, and at five junctions or tri- 
 butary pipes on each side, so regulated as to keep the 
 main pipe full," and that " without the addition of 
 junctions the transverse sectional area of the stream 
 of water near the discharging end was reduced to 
 one-fifth of the corresponding area of the pipe, and 
 that it required a simple head of water of about 22 
 inches to give the same result as that accruing under 
 the circumstances of the junctions." It is also stated, 
 that " in the case of the 6-inch pipe, which discharged 
 75 cubic feet per minute, the lateral streams had a 
 velocity of a few feet per minute." 
 
 Now, the head of " about 22 inches" is wholly 
 neglected in the foregoing calculations, though in a 
 pipe 100 feet long it would be equal to an inclination 
 of 1 in 55 ! It however includes three elements at 
 least, viz. the portion due to the orifice of entry, 
 the portion due to the velocity in the pipe, and 
 the portion due to friction. Let us assume the 
 case of the horizontal pipe, which discharged 46 cubic 
 feet per minute^. This is equal to a mean velocity of 
 
 * Adcock's Engineer's Pocket Book, 1853, pp. 261 and 262. 
 | The horizontal pipe would discharge equally at both ends, 
 unless there was a head of water at either, or an equivalent in 
 
INTRODUCTION TO THE FIRST EDITION. 11 
 
 46-9 inches per second; with this velocity, we find 
 from TABLE VIII. the hydraulic inclination of a 6-inch 
 pipe to be 1 in 94, and, therefore, the head due to 
 friction in a pipe 100 feet long is 12-7 inches. As- 
 suming the coefficient for the orifice of entry and 
 velocity to be -815, we also find from TABLE II. a 
 head of 4^ inches due to these. We then have, 
 
 Head due to the velocity and orifice of 
 
 entry. - lJ . ' Mi . . . . 4-25 inches 
 
 Head due to the resistance of friction 12-70 
 
 Radius of pipe . . ^ ', 3-00 
 
 Total ^ "^ 19-95 
 
 which is about 2 inches less than the observed head : 
 this, however, is not stated definitely. It is therefore 
 evident, that the formula gives, if anything, larger 
 results than these experiments*, as might have been 
 expected, instead of less in the ratio of 2 to 3, as is 
 stated in the Report. 
 
 Wherever junctions are applied, as in the examples 
 above referred to, the formulae in general use require 
 correction; for the quantity of water then flowing 
 below each junction is increased. A certain amount 
 of error is, perhaps, inseparable from every calcula- 
 tion of this kind ; but before we condemn formulas 
 deduced from experiment by men every way quali- 
 
 the velocity of approach. Of course, a smaller pipe with a fall, 
 must be better than the larger one with none at all, in preventing 
 deposits. 
 
 * This is also true of the other formulae, for finding the dis- 
 charge from pipes, given in this work. 
 
12 INTRODUCTION TO THE FIRST EDITION. 
 
 fied for the task, it would be well that we should 
 learn to understand and properly apply them. 
 
 The diameter of a short pipe gives in itself the 
 means of increasing very considerably the surface 
 inclination of the fluid stream, by reducing the sec- 
 tion at the lower end. If we assume a horizontal 
 pipe 50 feet long and 6 inches in diameter, we per- 
 ceive, that if the receiving end be full, and the 
 discharging end one-third full, this inclination 
 
 6-2 1 
 will be 5Q x 12 = ^I5Q ; an( ^ ^ a ^ ^ e discharging en( i 
 
 cannot be kept full unless a head of several inches 
 be maintained at the receiving end, or an equivalent 
 from a lateral supply. When the pipe is about two 
 diameters long it becomes a short tube ; and when 
 the length vanishes, the transverse section becomes, 
 simply, a discharging orifice. 
 
 We have been led into the foregoing remarks, not 
 from any desire to find fault with a Report contain- 
 ing so much valuable information as the one referred 
 to, but for the purpose of defending from unmerited 
 reproach, in a Blue Book, the researches in this de- 
 partment, of 
 
 " Those dead but sceptred sovereigns who still rule 
 Our science from their urns " 
 
 Du Buat, Young, Eytelwein, Prony, and others. 
 
 We do not pretend to any particular accuracy in 
 the sketches scattered throughout the work ; they are 
 only intended to illustrate the text, and were sketched 
 while writing it, without further aim ; neither do we 
 pretend to have entered fully into the principles or 
 
INTEODUCTION TO THE FIEST EDITION. 13 
 
 practice of hydraulics, our object being to select, 
 construct, and arrange useful hydraulic formulae, ex- 
 periments, and Tables for the use of all classes of 
 engineers. We make, however, no apology for pre- 
 ferring formulae, in their simplicity, to any written 
 rules which may be deduced from them, as being 
 in every way more general, concise, and elegant. 
 In conclusion, it is hoped that any errors of con- 
 sequence in the work, will be found corrected in the 
 errata. 
 
 Roden Place, Dundalk, 
 January, 1853= 
 
ON THE 
 
 DISCHARGE OF WATER 
 
 FEOM 
 
 OEIFICES, WEIRS, PIPES, AND RIVERS. 
 
 SECTION I. 
 
 APPLICATION AND USE OF THE TABLES, FORMULA, &C. 
 
 To find the velocity of a falling body from the height 
 fallen, or the height fallen from the velocity. 
 
 RULE. MULTIPLY THE SQUARE ROOT OF THE 
 
 HEIGHT IN INCHES BY 27 '8, AND THE PRODUCT WILL 
 BE THE VELOCITY IN INCHES.* To FIND THE HEIGHT 
 FROM THE TELOCITY, SQUARE THE VELOCITY IN INCHES 
 AND DIVIDE THE SQUARE BY 772*84, THE QUOTIENT 
 
 WILL BE THE HEIGHT IN INCHES. See equation (1). 
 TABLE II., column 1, will give the velocity from the 
 height, found in the column of " altitudes," or the 
 height from the velocity, directly. 
 
 EXAMPLE 1. What is the velocity acquired by a 
 heavy body falling th of an inch ? In the Table 
 opposite to ith of an inch, found in the column 
 headed " altitudes /*," we find 9-829 in column 1, for 
 the required velocity, in inches per second. 
 
 EXAMPLE 2. What is the velocity acquired by a 
 
 * The square root of the height in feet multiplied by 8-OS5 
 gives the velocity per second in feet ; and the square of the 
 velocity in feet divided by 64-4 will give the height in feet. 
 
16 THE DISCHARGE OF WATER FROM 
 
 fall of II feet 3 inches? Opposite to 11 feet 3 
 inches, as before, we find 323-007 inches, for the 
 velocity required. 
 
 EXAMPLE 3. What height must a heavy body fall 
 through to acquire a velocity of 40J feet per second ? 
 Here 40J feet is equal 486 inches, opposite to the 
 nearest number to which, found in column 1, we find 
 25 feet 6 inches for the required fall. In this ex- 
 ample, the nearest number to 486 found in the Table 
 is 486*301. The difference *301 corresponds, very 
 nearly, to f ths of an inch in altitude, and, therefore, 
 the true head according to the rule would be 25' 6f"; 
 but for all practical purposes the difference is im- 
 material. 
 
 By means of TABLE II. we can find, directly, or 
 by simple interpolation, the velocity due to all 
 heights from TTO part of an inch up to 40 feet, and 
 the heights from the velocities. For a greater height 
 than 40 feet it may be divided by 4, 9, or some 
 square number s~, and the velocity found for the 
 quotient, from the Table, multiplied by 2, 3, or s, 
 the square root of the divisor, will give the velocity 
 required. 
 
 EXAMPLE 4. What is the velocity acquired by a 
 
 45 
 
 fall of 45 feet ? -j- = 11' 3", the velocity corre- 
 sponding to which, found from the Table, is 323-007. 
 Hence, 323-007 x</4 = 323-007 2 = 646"-014 = 
 53' 10"-014 is the velocity per second required. The 
 reverse of this example is equally simple. 
 
 Columns 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 in the 
 Table, give the values of \/Zgh multiplied by the 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 17 
 
 coefficients therein stated. These columns will be 
 found of great practical use in finding the mean 
 velocities in the vena-contracta, in the orifice, and in 
 short tubes ; and consequently also in finding the 
 mechanical force, as well as the discharge. An 
 examination of the coefficients in the small Tables 
 in SECTION III., and also of those in TABLES I. and 
 V., at the end of the work, will show how much they 
 vary ; but those most generally useful, and their 
 products by the theoretical velocity due to different 
 heads, up to 40 feet, are given in the columns 
 referred to. 
 
 EXAMPLE 5. What is the discharge from an 
 orifice 4 inches by 8 inches, the centre sunk 20 feet 
 below the surface of a reservoir ? From TABLE II., 
 we find 430-676 inches equal 35-89 feet for the 
 
 x A. 
 
 theoretical velocity of discharge : hence, 
 
 TT 
 
 2 
 
 35-89 = ^ x 35-89 = 7-976 cubic feet per second is 
 
 the theoretical discharge. If the discharge takes 
 place through a thin plate, or if the inner arrises 
 next the water in the reservoir be perfectly square, 
 and the water in flowing out does not fill the passage 
 so as to convert the orifice into a short tube, the 
 coefficient will be found from TABLE I. to be -603. 
 The true discharge then is 7-976 x -603 = 4-809 
 cubic feet per second. 
 
 For,, the determination of the coefficient suited to 
 any particular orifice, and the circumstances of its 
 position, we must refer generally to the following 
 pages. If in the example just given, the arrises 
 
 c 
 
18 THE DISCHARGE OF WATER FROM 
 
 next the reservoir were rounded into the form of the 
 contracted vein, see Fig. 4, the coefficient would 
 increase from -603 to -974 or -956, for a passage not 
 exceeding a couple of feet in length. With the 
 former the discharge would be 7-976 x -974 = 7-769 
 cubic feet, and with the latter 7-976 x -956 = 7-625 
 cubic feet. We may find from Table II. the latter 
 results otherwise. With a head of 20 feet and the 
 coefficient -974, the velocity is 419-48 inches = 34-957 
 
 '2 
 
 feet ; hence, the discharge is g x 34-957 = 7-768 
 
 cubic feet. With a coefficient of -956, the velocity 
 is 411-73 inches = 34-31 feet, and | x 34-31 = 7-624 
 
 cubic feet. These results are the same, practically, 
 as those previously found. 
 
 If the inner arrises be square, and the passage out 
 be from 18 inches to 2 feet long, the orifice will be 
 converted into a short tube, the coefficient for which 
 is -815. With this coefficient, and a head of 20 feet, 
 we find as before, from TABLE II., the mean velocity 
 of discharge equals 351 inches = 29-25 feet; hence, 
 
 2 
 the discharge now is ^ x 29-25 = 6-5 cubic feet per 
 
 second. 
 
 The velocities in inches per second, given in TABLES 
 II. and VIII., or elsewhere in the following pages, may 
 be converted into velocities in feet per minute, by mul- 
 
 7 60 
 tiplying by 5, equal j^ 
 
 EXAMPLE 6. The discharge from a small orifice 
 having its centre placed 1 feet below the surface of a 
 
OEIFICES, WEIRS, PIPES, AND RIVERS. 19 
 
 reservoir is 18 feet per minute, what will be the dis- 
 charge from the same orifice at a depth of 17 feet? 
 The discharges will be to each other as \/10 : \/W 9 
 or as 1 : N/F7; or, from TABLE III., as 1 : 1-3038, 
 whence we get the discharge sought equal 1-3038 x 
 18 = 23-4684 cubic feet. 
 
 EXAMPLE 7. What is the value of the expression 
 
 ( C z }* 
 
 c d \ 1 + 2 d g [ in equation (45), when c d = '617, and 
 m c a/ 
 
 m = 2 ? Here we have 
 
 * -2807 
 
 1052 ; 
 
 c 2 d -617 2 -3807 
 
 m z -c\ 4--617 2 3-6193 
 whence the first expression becomes equal to '617 
 (M052)* equal, from TABLE III, -617 x 1-0513 = 
 649, the value sought. TABLE V. contains the values 
 of this expression for various values of c d and m, 
 which latter, m, stands for the ratio of the channel 
 to an orifice ; and we can immediately find from it, 
 opposite 2 in the first column, and under the coeffi- 
 cient -617 in the sixth column, -649 the value sought. 
 When the head due to the pressure, and to the velocity 
 of approach, are both known, we can determine the 
 new coefficient of discharge by the above expression, 
 and thence the discharge itself. The coefficient suited 
 to the velocity of approach may however be found 
 directly in TABLE V. The usual methods for finding 
 the effects of the velocity of approach, given by 
 d'Aubuisson and others, are incorrect in principle, 
 see SECTION IV. 
 
 EXAMPLE 8. What is the discharge from an 
 orifice 17 inches long and 9 inches deep, having the 
 upper edge placed 4 inclies below the surface, and the 
 
 c3 
 
20 THE DISCHARGE OF WATER FROM 
 
 lower edge 13 inches? The expression for the dis- 
 charge is^x Av / 272xc d |(l +~)^-(-^) 2 [equa- 
 tion (43), in which we must take d = 9 inches; h t = 9 
 inches ;A = 17x9 = 153 square inches ; and\/2^^, 
 found from TABLE II. = 83-4 inches. We have, also, 
 
 = - = -444^ and hence the value of 
 
 (1-444)4 - (-444)2 = (from TABLE IY.) 1-44. 
 Assuming the coefficient of discharge to be -617, we 
 then have the discharge in cubic inches per second 
 equal to 
 
 | x 153 x 83-4 x -617 x 1-44 = 
 o 
 
 I x 12760-2 x -88848 = 7558. 
 Consequently, TI^ = 4-374 is the discharge in cubic 
 
 feet per second. Prom equation (6), we get the dis- 
 charge equal to 
 
 | x -617 x 27-8 x 17 x {13* 4*}. 
 
 But 13t - 4t = 46-872 - 8, from TABLE IY., equal 
 to 38-872, whence the discharge is 
 
 | x -617 x 27-8 x 17 x 38-872 = 114351 x 17 x 
 
 38-872 = 194-3967 x 38-872 = 7557 cubic inches 
 = 4*374 cubic feet, the same as before. 
 It is shown in equation (31), that by using the 
 
 mean depth for orifices near the surface, the discharge 
 
 /"'*' 
 will approximate very closely to the true discharge, 
 
 and that even for weirs the error will not exceed 
 6 per cent. The discharge is then expressed by 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 21 
 
 x 8i x 9 x 17 = (from TABLE II.) 50-01 x 
 153 = 7651-53 cubic inches = 4*427 cubic feet per 
 second. The head to the centre of the orifice is 
 here 8i inches, and the depth of the orifice 9 inches, 
 therefore, in equation (31), h = d very nearly ; and, 
 therefore, this result must be multiplied by *989, as 
 shown in that equation ; then -989 x 4-427 = 4-378 
 cubic feet, which gives a result differing from those 
 otherwise found, by a very small quantity, which, 
 practically, is of no value. By means of TABLE VI. 
 the discharge from rectangular orifices near the sur- 
 face can be found with very great facility. 
 
 We may always find the discharge from an orifice 
 near the surface with sufficient accuracy ', for practical 
 purposes, by taking the head to the centre, in the same 
 manner as if the orifice were sunk to a considerable 
 depth ; then by applying the corrections given in equa- 
 tion (31) ; or if the orifice be circular, those given in 
 equation (28) ; extreme accuracy ', according to the cor- 
 rect formula, is obtainable. 
 
 EXAMPLE 9. What is the discharge from a circular 
 orifice 4 inches in diameter ', having its centre placed 4 
 inches below the surface, when the coefficient of dis- 
 charge is *6L7 ? The area of the orifice is 4 x 4 x 
 7854 = 12-566 square inches. The velocity in the 
 orifice at the mean depth of 4 inches, with a co- 
 efficient of *617, is 34-31 inches, whence the discharge 
 is 12-566 x 34-31 = 431-139 cubic inches = -2496 
 cubic feet per second, or 14-97 cubic feet per minute. 
 By means of TABLE IX. the discharge in cubic feet 
 per minute can be found very readily when the velo- 
 city, 34-31 inches per second, is known. Thus, 
 
22 THE DISCHAKGE OF WATEK FEOM 
 
 Inches. Cubic feet. 
 
 For a velocity of 30-00 the discharge is 13-089 
 
 4-00 1-745 
 
 30 0-131 
 
 -01 0-004 
 
 ./ 34-31 ^ 14-969 
 
 By applying the coefficient found from equation 
 (28), which is -992, when the depth at the centre is 
 twice the radius, as it is in this example, we get 
 992 x 14-97 = 14*85 for the correct discharge in 
 cubic feet per minute. Here the difference in the 
 results is only 1 in 125. 
 
 The application of TABLE VI. will enable us to 
 find the discharge from rectangular orifices near the 
 surface very quickly. Resuming " EXAMPLE 8," the 
 discharge may be found from this Table for each foot 
 in length of the orifice, as follows. The discharge in 
 cubic feet per minute, when the coefficient is -617 for 
 a notch 1 foot long and 13 inches deep, is 223-323 ; 
 and for a notch of 4 inches deep, 38-116 ; therefore, 
 the discharge from an orifice 9 inches deep, with the 
 upper edge 4 inches below the surface, is 223-323 
 38-116 = 185-207 cubic feet per minute. But as the 
 length of the orifice is 17 inches, this must be multi- 
 
 17 
 
 plied by ^ an(i the product 262-377 is the dis- 
 charge in cubic feet per minute; this is equal to a 
 discharge of 4-373 cubic feet per second, and agrees 
 with that before found. This is the simplest way of 
 finding the discharge from rectangular orifices near 
 the surface. 
 
 EXAMPLE 10. What is the discharge in cubic feet 
 
OEIFICES, WEIES, PIPES, AND EIVEKS. 23 
 
 per minute, from an orifice 2 feet 6 inches long and 7 
 inches deep, the upper edge being 3 inches below the 
 surface, and the coefficient of discharge -628 ? From 
 TABLE VI. we find the discharge from a notch 1 foot 
 long and 10 inches deep to be 153-353, and for a 
 notch 3 inches deep, 25199. The difference, or 
 128-154, multiplied by 2i, will be the discharge re- 
 quired ; viz. 2 x 128-154 = 320-385 cubic feet per 
 minute. 
 
 EXAMPLE 11. The size of a channel is 2*75 times 
 the size of an orifice, what is the coefficient of dis- 
 charge when that for a very large channel in pro- 
 portion to the orifice is -628 ? We find from TABLE V. 
 the coefficient to be -645, when the approaching 
 water suffers full contraction. By attending to the 
 auxiliary Tables in the text, we find, for this case, 
 
 /> i 
 
 -r T = cTfTK = *36. We must, therefore, multiply 
 
 channel 2-75 
 
 2-75 by -857, which gives 2*36 for the ratio of the mean 
 velocities in the orifice and in the channel approaching 
 it. With this new value of the ratio of the channel to 
 the orifice, we find, as before, the value of the co- 
 efficient from TABLE V. to be -651. The remarks 
 throughout the work, with the auxiliary tables, will 
 be found of much use in determining the coefficients 
 for different ratios of the channel to the orifice, 
 notch, or weir, and the corrections suited to each. 
 If in this example we were considering, other 
 things being the same, the alteration in the coeffi- 
 cient for a notch, or weir, it would be found from 
 the Table, column 4, to be *672 instead of *645 
 found in column 3, for an orifice sunk some depth 
 
24 THE DISCHARGE OF WATER FROM 
 
 below the surface. For the corrections suited to mean 
 and central velocity, and to the nature of the approaches, 
 we must refer to the body of this work and to the 
 auxiliary tables therein at the end of SECTION IY. 
 
 EXAMPLE 12. What is the discharge aver a weir 
 50 feet long ; the circumstances of the overfall, crest, 
 and approaches, being such that the coefficient of 
 discharge is '617, when the head measured from the 
 water in the weir basin, 6 feet above the crest, is 
 17i inches? TABLE VI. will give the discharge in 
 cubic feet per minute, over each foot in length of 
 weir, for various depths up to 6 feet. It is divided 
 into two parts ; the first for " greater coefficients," 
 viz. -667 to -617; and the second for "lesser co- 
 efficients," viz. -606 to -518. The coefficient assumed 
 being -617, we find the discharge over 1 foot in 
 length, with a head of 17J inches, to be 348-799 
 cubic feet per minute ; hence the required discharge 
 is 50 x 348-799 = 17439-95 cubic feet. 
 
 The determination of the coefficient suited to the 
 circumstances of the overfall, crest, approaches, and 
 approaching section, will be found discussed else- 
 where through this work. The valuable Table de- 
 rived from Mr. BlackwelPs experiments will also be 
 of use ; but the heads being taken at a much greater 
 distance back from the crest than is generally usual, 
 the coefficients taken from it for heads greater than 
 5 or 6 inches, will be found under the true ones for 
 heads measured immediately at or about 6 feet, above 
 the crest. For heads measured on the crest, the 
 small Table of coefficients in SECTION III., applicable 
 to the purpose, will be of use. 
 
OEIFICES, WEIRS, PIPES, AND RIVERS. 25 
 
 EXAMPLE 13. What is the mean velocity in a large 
 channel, when the maximum velocity along the central 
 line of the surface is 31 inches per second? TABLE 
 VII. gives 25-89 inches for the required velocity, and 
 for smaller channels 24*86 inches. In order to find 
 the mean velocity at the surface from the maximum 
 central velocity, the latter must be multiplied by 
 914. 
 
 The velocity at the surface is best found by means 
 of a floating hollow ball, which just rises out of the 
 water, The velocity at a given depth is best found 
 by means of two hollow balls connected with a link, 
 the lower being made heavier than the upper, and 
 both so weighted by the admission of a certain quan- 
 tity of water that they shall float along the current, 
 the upper one being in advance but nearly vertical 
 over the other. The velocity of both will then be 
 the velocity at half the depth between them. The 
 velocity at the surface, found by means of a single 
 ball, being also found, the velocity lost at the half 
 depth is had by subtracting the common velocity due 
 to the linked balls from that of the single ball at the 
 surface. The velocity at any given depth is then 
 easily found by a simple proportion ; but the result 
 will be most accurate when the given depth is nearly 
 half the distance between the balls, which distance 
 can never exceed the depth of the channel. Pitot's 
 tube, Woltmann's tachometer, the hydrometric pendu- 
 lum, the rheometer, and several other hydrometers, 
 have been used for finding the velocity ; but these 
 instruments require certain corrections suited to each 
 separate instrument, as well as kind of instrument, 
 
26 THE DISCHAKGE OF WATEK FROM 
 
 and are not so correct or simple, for measuring the 
 velocity in open channels, as a ball and linked balls. 
 
 EXAMPLE 14. What is the discharge from a river 
 having a surface inclination of 18 inches per mile, or 
 1 in 3520, 40 feet wide, with nearly vertical banks, 
 and 3 feet deep ? The area is 40 x 3 = 120 feet, 
 and the border 40 + 2 x 3 = 46 feet ; therefore the 
 
 120 
 hydraulic mean depth is -TTT = 2-61 feet zz 2 feet 7-3 
 
 inches*. With this and the inclination we find from 
 TABLE VIII. 28-27 + 2-75 X ^ = 28-87 inches per 
 
 second =: 28-87 x 5 =z 144-35 feet per minute for 
 the mean velocity ; hence we get 144-35 x 120 = 
 17,322 cubic feet per minute for the required dis- 
 charge. For channels with sloping banks we have 
 only to divide the border, which is always known, 
 into the area for the hydraulic mean depth, with 
 which, and the surface inclination, we can always 
 find the velocity by TABLE VIII., and thence the 
 discharge. Unless the banks of rivers be protected 
 by stone pavement or otherwise, the slopes will not 
 continue permanent ; it is therefore almost useless to 
 give the discharges for channels of particular widths 
 and side slopes, When the mean velocity is once 
 known, the remaining calculations are those of mere 
 mensuration, and they should be made separately. 
 This example may also be solved, practically, by 
 means of TABLES XI. and XII. A channel 40 x 3 
 
 * For greater hydraulic depths than 144 inches, the extent of 
 the TABLE, divide hy 9, and find the corresponding velocity. 
 This multiplied by 3 will be the velocity sought. 
 
ORIFICES, WEIKS, PIPES, AND RITE 
 
 has the same conveying power as one 70 x 
 XL, which latter, TABLE XII. discharges with a fall 
 of 18 inches in the mile, 17,157 feet; or about one 
 per cent, less than that previously found. 
 
 EXAMPLE 15. The diameter of a very long pipe is 
 li inch, and the rate of inclination , or ivhole length 
 of the pipe divided by the whole fall, is 1 in 71^ ; 
 what is the discharge in cubic feet per minute ? The 
 
 1*5 
 
 hydraulic mean depth, or mean radius, is -j- = -375 
 
 3 
 inches = ^ inch. Consequently we find from TABLE 
 
 VIII. the velocity in inches per second equal to 
 25-09 - 1-92 x ^ = 25-09 - -29 = 24-80. The 
 
 discharge in cubic feet per minute for a IJ-inch pipe 
 is now found most readily by means of TABLE IX., 
 
 as follows : 
 
 Inches. Cubic feet. 
 
 For a velocity of 20-0 the discharge is 1-227 
 4-0 -245 
 
 8 -049 
 
 ,, 34-8 1-521 
 
 Whence the discharge in cubic feet per minute is 
 1-521. 
 
 For short pipes, of 100 or 200 feet in length, and 
 under, the height due to the velocity and orifice of 
 entry must be deducted from the whole height to find 
 the proper hydraulic inclination, and also the height 
 due to bends, curves, cocks, slides, and erogation. 
 The neglect of these corrections has led some writers 
 into mistakes in applying certain formulae, and in test- 
 
28 THE DISCHAEGE OF WATER FEOM 
 
 ing them by experimental results obtained with short 
 pipes. We shall now apply the TABLES to the deter- 
 mination of the discharge from short pipes, and 
 compare the results with experiment, referring gener- 
 ally to equation (153) and the remarks preceding it 
 for a correct and direct solution. 
 
 EXAMPLE 16. What is the discharge in cubic feet 
 per minute from a pipe WO feet long, with a fall or 
 head of 35 inches to the lower end, when the diameter 
 is 1 J inch ? Find also the discharge from pipes 80 
 feet t 60 feet, 40 feet, and 20 feet, of the same diameter 
 and having the same head. If the water be admitted 
 by a stop-cock at the upper end, the coefficient 
 due to the orifice of entry will probably be about -75 
 or less, -815 being that for a clear entry to a short 
 cylindrical tube. The approximate inclination is 
 
 100 x 12 
 or i= 1 in 34*3 ; but as a portion of the fall 
 
 must be absorbed by the velocity and orifice of entry, 
 we may assume for the present that the inclination 
 is 1 in 35. With this inclination and the mean radius 
 
 IL- 3 
 
 -jr zz g inches, we find the mean velocity from TABLE 
 
 VIII. to be 38-06 inches. Now when the coefficient 
 due to the orifice of entry and velocity is -75, we 
 find from TABLE II. the head due to this velocity to 
 be 3| inches nearly, whence 35 - 3f = 31f = 31-625 
 
 100 x 12 
 inches is the height due to friction, and ^^ 
 
 o J.*D^jO 
 
 equals 1 in 37-9, the inclination, very nearly. With 
 this new inclination we find, as before, from TABLE 
 VIII. the mean velocity of discharge to be now 36-35 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 29 
 
 inches ; and by repeating the operation we shall find 
 the velocity to any degree of accuracy in accord- 
 ance with the table, and the shorter the pipe is, the 
 oftener must it be repeated. The height due to 36-35 
 inches taken from TABLE II. as before, with a coeffi- 
 cient of -750, is 3^ = 3-125 inches. The corrected 
 fall due to the friction is now 35 - 3-125 == 31-875, 
 
 and 01.07- equal 1 in 37-6, the corrected inclina- 
 
 tion. With this inclination we find the corrected 
 velocity to be now 36-53 inches per second. It is 
 not necessary to repeat the operation again. The 
 discharge determined from TABLE IX. is as follows : 
 
 Inches. Cubic feet. 
 
 For a velocity of 30-00 the discharge is 1-841 
 
 6-00 -368 
 
 50 -031 
 
 03 -002 
 
 36-53 
 
 The experimental discharge found by Mr. Provis was 
 2-264 cubic feet per minute in one experiment, and 
 2-285 in another. The discharge from the shorter 
 pipes may be found in a similar manner, and we 
 place the results alongside the experimental ones 
 given in the work referred to below* in the following 
 short table : 
 
 * " Transactions of the Institution of Civil Engineers," vol. ii. 
 p. 203. " Experiments on the Flow of Water through small 
 Pipes." By W. A. Provis. The small Tables in SECTIONS VI. 
 and VIII. of this edition give at once the coefficient to be mul- 
 tiplied by^^n, or 8-v/lf, to find the velocity when the ratio of 
 the diameter to the length of the pipe is known. They will be 
 found of great advantage in calculating directly the velocity from 
 short pipes. For long pipes, see the TABLE pp. 42 and 43. 
 
30 THE DISCHARGE OF WATER FROM 
 
 EXPERIMENTAL AND CALCULATED DISCHARGES FROM SHORT PIPES. 
 
 1, 
 
 4 
 
 1 
 
 .a 
 
 
 
 !* 
 
 A 
 
 o 
 
 
 ^o 
 
 fa 
 
 1 
 
 .a 
 1 
 
 III 
 
 II 
 
 jji 
 
 II 
 
 11 
 w 
 
 I'l 
 
 if 
 
 If 
 
 3 
 
 w 
 
 
 
 w 
 
 
 
 
 
 100 
 
 35 
 
 2-275 
 
 37-082 
 
 3* 
 
 31| 
 
 37-6 
 
 36-53 
 
 2-242 
 
 80 
 
 35 
 
 2.500 
 
 40-750 
 
 3f 
 
 3H 
 
 30-8 
 
 41-18 
 
 2-521 
 
 60 
 
 35 
 
 2-874 
 
 46-846 
 
 5 
 
 30 
 
 24-0 
 
 48-02 
 
 2-946 
 
 40 
 
 35 
 
 3-504 
 
 57-115 
 
 n 
 
 27i 
 
 17-5 
 
 58-50 
 
 3-590 
 
 20 
 
 35 
 
 4.528 
 
 73-801 
 
 12* 
 
 221 
 
 10-7 
 
 78-61 
 
 4-824 
 
 The velocities in the fourth column have been cal- 
 culated by the writer from the observed quantities 
 discharged, from which the height due to the orifice 
 of entry and velocity in column 5 is determined, and 
 thence the quantities in the other columns as above 
 shown. The differences between the experimental 
 and calculated results are not large, and had we used 
 a lesser coefficient than 8 750 for calculating the re- 
 duction of head due to the velocity, stop-cock, and 
 orifice of entry, say -715, the calculated results, and 
 those in all of Mr. Provis's experiments in the work 
 referred to, would be nearly identical.* 
 
 EXAMPLE 17. It is proposed to supply a reservoir 
 near the town of Drogheda with water by a long pipe, 
 having an inclination of 1 in 480, the daily supply to 
 be 80,000 cubic feet ; what must the diameter of 
 
 * In a late work, " Researches in Hydraulics," the author is 
 led into a series of mistakes as to the accuracy of Dtf Buat's and 
 several other formulae, from neglecting to take into consideration 
 the head due to the velocity and orifice of entry when testing 
 them by the experiments above referred to. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 31 
 
 the pipe be ? The discharge per minute must be 
 * =56* cubic feet, nearly. Assume a pipe 
 
 whose "mean radius" is 1 inch, or diameter 4 
 inches, and the velocity per second found from TABLE 
 VIII. will be 14*41 inches. We then have from 
 TABLE IX., 
 
 Inches. Cubic feet. 
 
 For a velocity of 10-00 a discharge of 4-363 
 
 4-00 1-745 
 
 40 -175 
 
 01 -004 
 
 14-41 6-287 
 
 The discharge from a pipe 4 inches in diameter would 
 be therefore 6-287 cubic feet per minute. We then 
 have 
 4* : d$ : : 6'287 : 56, or 1 : rf* : : -196 : 56 : : 1 : 286; 
 
 therefore d^ =. 286, and d =. 9-61 inches, nearly, as 
 may be found from TABLE XIII., &c. This is nearly 
 the required diameter. It is to be observed that the 
 diameters thus found will not always agree exactly 
 with those found from Du Buat's or other formulas, 
 nor with each other, because the discharges are not 
 strictly as d^ ; but in practice the difference is imma- 
 terial, and the approximative value thus found can 
 be easily corrected. If we assumed a pipe whose 
 diameter is 1, the operation would have been more 
 simple ; for the velocity would then be, TABLE VIII., 
 at the given inclination, 6-4 inches ; and the discharge 
 175 cubic feet, TABLE IX. Hence we get d% 
 
 * Hydraulic Tables, Weale, 1854, give at once this discharge 
 for a pipe between 9 and 10 inches diameter, also the TABLE, p. 42. 
 
32 THE DISCHARGE OF WATER FROM 
 
 = 320, and, therefore, TABLE XIII., d 10 
 
 inches nearly, which differs about half-an-inch from 
 the former value, 9-6 inches, found by assuming a 
 pipe of 4 inches to calculate from. It is necessary 
 to understand that different results must be expected, 
 in working from practical formulae, for different 
 operations. When once an approximative value is 
 obtained, it can be easily corrected to any required 
 degree of accuracy. 
 
 Again the velocity in inches per second, from a cylin- 
 drical pipe 6 inches in diameter, is nearly equal to the 
 discharge in cubic feet per minute ; and as 6^ m 88*2, 
 we have 88*2^ : d^ : : the velocity in inches per 
 second from a 6-inch pipe : the discharge per minute 
 from a pipe whose diameter is d. Hence this pro- 
 portion would enable us to find, very nearly, the 
 discharge from the diameter and fall ; or the diameter 
 from the discharge and fall by finding the velocity 
 only, due to a 6-inch pipe. See TABLE pp. 42 and 43. 
 
 EXAMPLE 18. The area of a channel is 50 square 
 feet, and the border 20*6 feet ; the surface has an in- 
 clination of 4 inches in a mile ; what is the mean 
 
 50 
 velocity of discharge? ^g = 2*427 feet = 29-124 
 
 inches is the hydraulic mean depth ; and we get 
 
 from Table VIII. 12-03 - -- - = 12-03 - 
 
 19 = 11*84 inches per second for the required velo- 
 city. Though this velocity will be found under the 
 true value for straight clear channels, it will yet be 
 more correct for ordinary river courses, with bends 
 
OKIFICES, WEIES, PIPES, AND KIVERS. 33 
 
 and turns, of the dimensions given, than the velocity 
 found from equation (114). For a straight clear 
 channel of these dimensions, Watt found the mean 
 velocity to be 13-5 to 14 inches; that is to say, 
 17 at top, 10 at bottom, and 14 in the middle, Our 
 
 formula v = 140 (r sfi 11 (r sft gives v = 1*143 
 feet, or nearly a mean of these two. 
 
 EXAMPLE 19. A pipe 5 inches in diameter, 14,637 
 feet in length, has a fall of 44 feet ; what is the dis- 
 charge in cubic feet per minute ? The inclination is 
 
 1 i r* Q IT K 
 
 -jj = 332*7, and mean radius j = 1J. We then 
 
 find from TABLE VIII. the velocity equal to 19-81 + 
 
 L \*. 5 4 - = 19-81 + -16 = 19-97, or 20 inches per 
 
 second very nearly ; and by TABLE IX. the discharge 
 in cubic feet per minute is, as before found to be, 
 13-635. The TABLE, p. 42, gives, by inspection, 13-6 
 feet. 
 
 EXAMPLE 20. What is the velocity of discharge 
 from a pipe or culvert 4 feet in diameter, having a 
 
 fall of 1 foot to a mile ? Here s =: goftQ' and r = 
 
 1 foot. We then find the velocity of discharge from 
 TABLE VIII. to be 14-09 inches, equal to 1-174 feet per 
 second. By calculating from the different formulae 
 referred to below, we shall find the velocities, when 
 r s = -0001894, and \/Ts = -01376, as follows. 
 
 Velocity 
 in feet. 
 
 Eeduction of Du Buat's formula .... equation (81) 1-174 
 Girard's do. (Canals with aquatic 
 
 plants and very slow velocities) (86) -621 
 Prony's do. (Canals) .... (88) 1*201 
 
 D 
 
34 THE DISCHAKGE OF WATER FROM 
 
 Velocity 
 in feet. 
 
 Seduction of Prony's formula (Pipes) . . equation (90) 1-257 
 
 Prony's do. (Pipes and Canals) . (92) 1-229 
 
 Eytelwein's do. (Kivers) ... (94) 1-200 
 
 Eytelwein's do. (Kivers) ... (96) 1-285 
 
 Eytelwein's do. (Pipes) ... (98) 1-364 
 
 Eytelwein's do. (Pipes) ... (99) 1-350 
 
 Dr. Young's do. . . .>. . (104)1-120 
 
 *D'Aubuisson's do. (Pipes) . . (109)1-259 
 
 *D'Aubuisson's do. (Kivers) . . (111) 1-199 
 
 ,, The writer's do. (Clear straight 
 
 Channels with small velocities) (114) 1-268 
 
 Weishaeh's do. (Pipes) ... (119) 1-285 
 
 The author's, for Pipes and Kivers 1-295 
 
 We have calculated this example from the several 
 formulae above referred to, whether for pipes or rivers 
 in order that the results may be more readily com- 
 pared. The formula from which the velocities and 
 tables for the discharges of rivers are usually cal- 
 culated is, for measures in feet, v 94-17 </ r s. 
 This gives the mean velocity, for the foregoing ex- 
 ample, equal to 1-295 feet per second. This is the 
 same as is found from my general formula for all 
 velocities; but the particular expression, v = 99-17 
 \/ r s, is only suited for velocities of about 15 inches 
 per second; the results found from it for lesser 
 velocities are too much, and for higher velocities too 
 little, if bends and curves be allowed for separately- 
 For ordinary practical purposes the result of Du 
 Buat's general formula, equation (81), may be safely 
 adopted ; and we have, accordingly, preferred retain- 
 ing the results in TABLE VIII. calculated for our first 
 edition from it, notwithstanding the greater accuracy 
 and simplicity of our own general equation (119 A) 
 
 * These two formulae of D'Aubuisson's are, simply, adoptions of Eytel 
 vein's and Prony's. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 35 
 
 for the velocity in pipes and rivers, viz., v = 140 (rs)* 
 ll(r*)i. 
 
 Dr. Young's formula gives lesser results for rivers 
 and large pipes than Du Buat's, but they are too 
 small unless when the curves and bends are numerous 
 and sudden. Girard's formula (86) is only suited for 
 small velocities in canals containing aquatic plants, 
 and it is entirely inapplicable to rivers or regular 
 channels for conveyance of water. A knowledge of 
 various formulae, and their comparative results, ap- 
 plied to any particular case, will be found of great 
 value to the hydraulic engineer, and the differences 
 in the results show only an amount of error that may 
 be expected in all practical operations, and which be- 
 comes of less importance when we consider that by 
 increasing the dimensions of a channel every way, 
 by only one-third, we shall more than double its dis- 
 charging power. See TABLE XIII. 
 
 EXAMPLE 21. Water flowing down a river rises to 
 a height of 10 J inches on a weir 62 feet long ; to what 
 height will the same quantity of water rise^ on a weir 
 
 62 
 sim ilarly circumstanced, 120 feet long ? 
 
 nearly. In TABLE X. we find, by inspection, opposite 
 to -517, the ratio of the lengths, the coefficient *644, 
 rejecting the fourth place of decimals ; whence 10| 
 X *644 ~ 6-76 inches, the height required. When the 
 height is given in inches it is not necessary to take out 
 the coefficient to further than two places of decimals. 
 EXAMPLE 22. The head on a weir 220 feet long is 
 6 inches ; what will the head be an a weir 60 feet long, 
 similarly circumstanced ' f the same quantity of water 
 
 D 3 
 
36 THE DISCHAEGE OF WATER FROM 
 
 fiO 
 
 flowing over each ? o 273. As this lies between 
 
 27 and '28, we find from TABLE X. the coefficient 
 
 /* 
 4208; hence .TZZ 14*2 6 inches, the head required. 
 
 TABLE X. will be found equally applicable in 
 finding the head above the pass into weir basins, and 
 above contracted water channels. See SECTION X. 
 
 EXAMPLE 23. A river channel 40 feet wide and 
 4*5 feet deep is to be altered and widened to 70 feet ; 
 what must the depth of the new channel be so that the 
 surface inclination and discharge shall remain un- 
 altered? In "TABLE XL, OF EQUALLY DISCHARGING 
 
 RECTANGULAR CHANNELS," W6 find Opposite to 4*54, 
 
 in the column of 40 feet widths, 3 in the column of 
 70 feet widths, which is the depth required in feet. 
 
 EXAMPLE 24. It is necessary to unwater a river 
 channel 70 feet wide and 1 foot deep, by a rectangular 
 side cut Wfeet wide ; what must the depth of the side 
 cut be, the surface inclination remaining the same as 
 in the old channel? In TABLE XI. we find 4 -5 feet 
 for the required depth. When the width of a channel 
 remains constant, the discharge varies as \/f~s X d, 
 in which d is the depth ; and when the width is very 
 large compared with the depth, the hydraulic mean 
 depth r approximates very closely to the depth d, and 
 therefore d r; consequently the discharge then 
 varies as d^ x **, and when the discharge is given d% 
 must vary inversely as /; or more generally dr must 
 vary inversely, as /, when the width and discharge 
 remain constant. 
 
 In narrow cuts for unwatering, it is prudent to 
 make the depth of the water half the width of the 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 37 
 
 cut very nearly, when local circumstances will admit 
 of these proportions ; for then a maximum effect will 
 be obtained with the least possible quantity of exca- 
 vation ; but for rivers and permanent channels the 
 proper relation of the depth to the width must be 
 regulated by the principles referred to in SECTION IX. 
 
 TABLE XI. is equally applicable, whether the mea- 
 sures be taken in feet, yards, or any other standards 
 whatever. 
 
 EXAMPLE 25. A new river channel is to have a fall 
 of eighteen inches in a mile, and must discharge 18,700 
 cubic feet per minute, what shall the dimensions be ? 
 In TABLE XII., in the column of 18 inches per mile, 
 we shall find opposite to 18,766, that a primary chan- 
 nel 70 x 2*125 will be sufficient ; and opposite to 2*125 
 in TABLE XI. we shall find the equivalent rectangular 
 channels 60x2*37; 50x2*70; 40x3*19; 35x3*52; 
 30x3*96; 25x4*61; 20x5*58; 15x7*29; and 10x 
 11 *37, to select from. If the sides shall have any given 
 slopes, the discharge will not be practically affected 
 as long as the depth and area of the rectangular 
 channel and the one with sloping banks remain the 
 same. See SECTION IX. 
 
 EXAMPLE 26. A pipe WO feet long and 1 inch in 
 diameter has a head of 150 feet over the lower end, what 
 will be the discharging velocity? Here r rz: '020833 
 in feet, and s = 1*5, therefore r s = *03125. Hence 
 by formulse(119A) v = 140 x ('03125/ 11 X (-03125)* 
 = 140 x '1766 - 11 X '315 = 24*724 - 3*465 = 
 21*259 feet per second. If allowance is required 
 for the orifice of entry, the velocity is corrected as 
 follows. A square orifice of entry has a coefficient 
 of -815. The head due to this coefficient for a 
 
38 THE DISCHAKGE OF WATEK FKOM 
 
 velocity of about 20| feet, or 246 inches, is about 
 10 feet, TABLE II. The head due to friction is 
 
 therefore 150 - 10 zz 140 feet, and s = 
 
 r s now becomes 1-4 X '020833 zz -02917. Hence 
 v zz 140 \/~r~s 11 \/ rs now becomes 140 x *171 
 -11 X '308 nearly, equal to 23-940- 3-388 zz 20'552 
 feet, the velocity for a square junction. 
 
 EXAMPLE 27. A sewer 9 feet in diameter has a 
 fall of 2 feet per mile^ what will be the velocity and 
 discharge of water flowing through it when full ? 
 
 Hererzz 2-25 and s= 777^7^ therefore rs= -0008523, 
 
 Jo40 
 
 (r*)* = -02919 and (r*)* = '0948 ; and by formula 
 (119A) we have v zz 140 (r *)* - 11 (r *)* zz 140 x 
 02919 - 11 x '0948 zz 4-0866 - 1-0428 zz 3-0438 
 feet per second. Hence the discharge per minute is 
 9 2 x '7854 x 3-0438 x 60 zz 63-62 X 182-6 =11,617 
 cubic feet nearly. The velocity from a circular pipe 
 or sewer is however greatest when the circumference 
 is open for about 78 \ degrees at the top, but the 
 velocity of sewage matter would not be equal to that 
 of water. It would vary according to the dilution 
 in the sewer, and 50 per cent, should be allowed, 
 at least, in deduction, unless the dilution be very 
 considerable. 
 
 The TABLE for the values of r s and v, calculated 
 from the formula (119 A) SEC. VIII., will give the 
 velocity at once when r s is known, and r s when the 
 velocity is known, from the latter of which a definite 
 value of r or s can be fixed upon, when the other 
 may be found, by an operation of simple division. 
 
 EXAMPLE 28. Water is to be pumped through a 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 39 
 
 pipe 3000 feet long and 2 feet in diameter, with a 
 velocity not exceeding 4 feet per second, what head 
 must be allowed extra for friction in the pipe in 
 calculating horse power ? We shall find from our 
 TABLE of the values of the velocity and product of 
 the hydraulic mean depth and hydraulic inclination, 
 given near the conclusion of SECTION VIII. , that for 
 a velocity of 4 feet per second r szz -00142. The dia- 
 meter of the pipe is 2 feet, therefore r -5, whence 
 
 ,9 = -p zz -00284. and as the length of the pipe is 
 *o 
 
 3000 feet we get 3000 x '00284 zz 8*52 feet, the head 
 required. The TABLE, p. 43, would give 9-6 feet 
 nearly, which corresponds with Du Buat's formula. 
 If the velocity in the pipe was 10 feet instead of 
 4 feet per second, then, from our table, rs zz -007576, 
 
 arid zz s zz - = = -015152, and, therefore. 
 r -5 
 
 h = ls = 3000 x -015152 zz 45-456 feet, or about 
 six times as much as when the velocity was only 
 4 feet per second. The great loss of head arising 
 from pumping at high velocities, from friction alone, 
 is therefore apparent. Were the velocity double, or 
 8 feet per second, the head would be 30 feet nearly, 
 or from the TABLE, p. 43, 31-6 feet. 
 
 For velocities of about 2-1 feet per second, v is 
 equal to 100 \/ r s, and for velocities of about 5| feet 
 per second, v zz 110 \/~r~s. If / be the length of a pipe, 
 we would find in the former case the head h in feet due 
 
 Iv 
 
 to friction from the formula h zz -. Q zz Is ; and 
 
 Iv ' 
 
 in the latter h zz zz Is. 
 
 In questions of this kind, however, the diameter of 
 
40 THE DISCHARGE OF WATEE FKOM 
 
 a pipe, d should be used in preference to the 
 hydraulic mean depth, and as d = 4 r we shall find in 
 
 lv 
 
 the first case h = 2509.7 ^ s > an( ^ i n the second 
 
 lv* 
 
 If we wish to substitute the fall per mile for the 
 hydraulic inclination, the first of these will again 
 
 become h = -* = Is for the loss per mile ; and 
 
 in the second case, h = Y^. = i s f OT the loss per 
 mile in feet. 
 
 If the velocity were so low as about 1 foot per 
 second, then v = 90 \/ r s, and we should find 
 
 , lv j 
 
 -- 
 
 If for the inclination we substitute the fall per 
 
 mile, this will become h = -* = Is for the loss 
 
 per mile in feet. 
 
 The loss of head varies in the same pipe with the 
 velocity, and must be calculated differently, for small 
 and for high velocities, when using the common for- 
 mulae. The TABLE near the end of SECTION VIII. will 
 
 ?* ^ 
 
 always give the correct value of rs, and thence *=~' 
 
 In addition to the loss of head arising from fric 1 
 tion, losses also o'ccur from straight or curved bends, 
 from diaphragms, from junctions, and from the ori- 
 fices of entry and discharge ; these must be deter- 
 mined separately for each case, as is shown hereafter, 
 and added together and to the loss arising from fric- 
 tion, and the sum to the height the water is to be 
 raised, before the full or total head for determining 
 the power of an engine can be accurately known. 
 
OEIFICES, WEIES, PIPES, AND KIVEES. 41 
 
 The TABLE on the next two pages will be found of 
 great practical utility in solving all questions con- 
 nected with water-pipes and sewers discharging fully- 
 diluted sewage. In using it we can interpolate, by 
 inspection, for intermediate diameters or inclinations. 
 For greater diameters, divide those given by 4, and 
 multiply the corresponding velocity found in the 
 table by 2, and the corresponding discharge in the 
 table by 32. If the object be to find the size 
 of the channel, divide greater given velocities by 2, 
 and multiply the diameters or inclinations found from 
 the table by 4 ; also divide greater discharges by 32, 
 and multiply the diameters found from the table by 
 4. The small auxiliary table, p. 43, embodied in the 
 larger one, is of great use in making allowance for 
 the velocity and orifice of entry in short pipes, before 
 finding the head due to friction. The table also gives 
 the different diameters and inclinations which, taken 
 together, give the same velocity or discharge ; and it 
 enables us, from inspection, to select that relation of 
 diameter to declivity which is best suited for other 
 engineering aspects of the question. Taken in con- 
 nexion with TABLES VIII., XL, XII., and XIII., this 
 table completes the means of finding, by inspection, 
 the dimensions, inclinations, velocities, and discharges 
 of every class of water-channel or sewage-conduit 
 required in engineering practice. 
 
 TABLE XIV. gives the comparative values of Eng- 
 lish and French measures ; and TABLE XV. gives the 
 weight, specific gravity, and ultimate strength and 
 elasticity of various materials with which the engineer 
 has to operate. 
 
TABLE for finding, very nearly, the velocity and discharge from Cylindrical Water Pipes or Sewers, when the 
 diameter and fall are given. Any two of the four quantities, the velocity, discharge, diameter, and fall or 
 inclination, being given, the others can be found in THE TABLE from inspection. 
 
 sift 
 
 -w o 5 a 
 A * p,. 
 
 flj* 
 
 Mean by- 1 
 draulic incli-l 
 nation of 1 
 pipe or sewerj 
 
 The VELOCITY IN INCHES PER SECOND is given in the first horizontal line for each inclination 
 or fall ; and the DISCHARGE IN CUBIC FEET PER MINDTE in the next following one. 
 
 lin. 2 in. 1 3 in. 
 diameter diameter | diameter 
 
 4 in. 
 
 diameter 
 
 5 in. 
 diameter 
 
 6 in. 
 diameter 
 
 7 in. 
 diameter 
 
 8 in. 
 diameter 
 
 9 in. 
 diameter 
 
 10 in. 
 diameter 
 
 
 One in 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 5280 
 
 1-7 
 
 2-5 
 
 3-2 
 
 3-8 
 
 4-2 
 
 4.7 
 
 5-1 
 
 5-5 
 
 5-9 
 
 6-2 
 
 
 
 05 
 
 27 
 
 79 
 
 1-6 
 
 29 
 
 4-6 
 
 6-8 
 
 9-6 
 
 129 
 
 169 
 
 2 
 
 2640 
 
 2-5 
 
 3-8 
 
 4-7 
 
 5-6 
 
 6-3 
 
 6-9 
 
 7-5 
 
 8-1 
 
 8-6 
 
 9-1 
 
 
 
 07 
 
 41 
 
 1-2 
 
 2-4 
 
 4-3 
 
 68 
 
 101 
 
 14-2 
 
 191 
 
 24-9 
 
 3 
 
 1760 
 
 3-1 
 
 4-7 
 
 5-9 
 
 7-0 
 
 7-8 
 
 8-7 
 
 9'4 
 
 10-2 
 
 10-8 
 
 11-5 
 
 
 
 08 
 
 51 
 
 1-4 
 
 30 
 
 53 
 
 85 
 
 126 
 
 17-7 
 
 239 
 
 31-2 
 
 4 
 
 1320 
 
 3-6 
 
 5-5 
 
 6-9 
 
 8-2 
 
 9-2 
 
 10-2 
 
 ii-l 
 
 11-9 
 
 12-7 
 
 13-4 
 
 
 
 10 
 
 60 
 
 1-7 
 
 3-6 
 
 63 
 
 100 
 
 148 
 
 20-8 
 
 280 
 
 36-7 
 
 5 
 
 1056 
 
 4-1 
 
 6-2 
 
 7-9 
 
 9'3 
 
 10-4 
 
 11-6 
 
 12-5 
 
 135 
 
 14-4 
 
 15-2 
 
 
 
 11 
 
 68 
 
 19 
 
 4-0 
 
 7-1 
 
 11-3 
 
 16-8 
 
 236 
 
 31-7 
 
 41-5 
 
 6 
 
 880 
 
 4-6 
 
 6-9 
 
 8*7 
 
 10-3 
 
 11-5 
 
 12-8 
 
 13-9 
 
 15-0 
 
 15-9 
 
 16-9 
 
 
 
 12 
 
 76 
 
 2-2 
 
 45 
 
 79 
 
 126 
 
 186 
 
 261 
 
 351 
 
 460 
 
 7 
 
 754 
 
 5-0 
 
 7-5 
 
 9-5 
 
 11-2 
 
 1-2-6 
 
 14-0 
 
 15-1 
 
 16-3 
 
 17-4 
 
 18-4 
 
 
 
 14 
 
 82 
 
 2-3 
 
 49 
 
 86 
 
 13-7 
 
 20-2 
 
 285 
 
 383 
 
 502 
 
 8 
 
 660 
 
 5-4 
 
 8-1 
 
 10-2 
 
 12-0 
 
 13-5 
 
 15-0 
 
 16-3 
 
 17-6 
 
 18-7 
 
 20-0 
 
 
 
 15 
 
 89 
 
 2-5 
 
 53 
 
 92 
 
 14-8 
 
 21-8 
 
 306 
 
 41-3 
 
 541 
 
 9 
 
 587 
 
 5-7 
 
 8-7 
 
 11-0 
 
 12-9 
 
 14-5 
 
 16-1 
 
 17-4 
 
 18-8 
 
 20-0 
 
 21-2 
 
 
 
 16 
 
 95 
 
 27 
 
 56 
 
 99 
 
 15-8 
 
 233 
 
 328 
 
 441 
 
 57-7 
 
 10 
 
 528 
 
 6-1 
 
 9-2 
 
 116 
 
 13-7 
 
 15-4 
 
 17-1 
 
 18-5 
 
 19-9 
 
 21-2 
 
 22-5 
 
 
 
 17 
 
 1-00 
 
 29 
 
 60 
 
 92 
 
 167 
 
 24-7 
 
 348 
 
 46-8 
 
 61-3 
 
 11 
 
 480 
 
 6-4 
 
 9-7 
 
 12-3 
 
 14-4 
 
 16-2 
 
 18-0 
 
 19-5 
 
 21-0 
 
 22-4 
 
 23-7 
 
 
 
 17 
 
 11 
 
 3-0 
 
 6-3 
 
 111 
 
 17-7 
 
 261 
 
 36-7 
 
 49-4 
 
 64-7 
 
 12 
 
 440 
 
 6-7 
 
 10-2 
 
 12-9 
 
 15-2 
 
 17-1 
 
 18-9 
 
 20-5 
 
 22-1 
 
 23-5 
 
 24-9 
 
 
 
 18 
 
 11 
 
 32 
 
 6-6 
 
 11-6 
 
 186 
 
 27-4 
 
 386 
 
 519 
 
 679 
 
 13-2 
 
 400 
 
 7-1 
 
 10-8 
 
 13*6 
 
 16-0 
 
 18-0 
 
 20-0 
 
 21-7 
 
 23-3 
 
 24-8 
 
 26-3 
 
 
 
 19 
 
 1-2 
 
 33 
 
 69 
 
 123 
 
 196 
 
 28-9 
 
 40-7 
 
 548 
 
 71-7 
 
 15-1 
 
 350 
 
 7-7 
 
 11-6 
 
 14-7 
 
 17-2 
 
 19-4 
 
 21-6 
 
 23-4 
 
 25-2 
 
 26-8 
 
 28-4 
 
 
 
 21 
 
 1-3 
 
 36 
 
 75 
 
 33-2 
 
 21-2 
 
 31-2 
 
 439 
 
 591 
 
 77-4 
 
 17-6 
 
 300 
 
 8-4 
 
 12-7 
 
 16-0 
 
 18-8 
 
 21-2 
 
 23-5 
 
 25-5 
 
 27-5 
 
 29-2 
 
 31-0 
 
 
 
 23 
 
 1-4 
 
 3-9 
 
 82 
 
 14-4 
 
 231 
 
 34-1 
 
 40-8 
 
 646 
 
 845 
 
 21-1 
 
 250 
 
 9-4 
 
 14-1 
 
 17-8 
 
 20-9 
 
 23-5 
 
 26-1 
 
 28-3 
 
 30-5 
 
 32-5 
 
 34-4 
 
 
 
 26 
 
 15 
 
 4-4 
 
 92 
 
 160 
 
 25-7 
 
 378 
 
 533 
 
 71-7 
 
 93-8 
 
 26-4 
 
 200 
 
 10-6 
 
 16-0 
 
 20-2 
 
 23-8 
 
 26-8 
 
 29-7 
 
 32-2 
 
 34-7 
 
 36-9 
 
 39-1 
 
 
 
 29 
 
 1-7 
 
 5-0 
 
 104 
 
 18-2 
 
 29-2 
 
 431 
 
 60-6 
 
 816 
 
 1067 
 
 35-2 
 
 150 
 
 12-5 
 
 19-0 
 
 23-9 
 
 28-1 
 
 31-6 
 
 35-1 
 
 38-1 
 
 41-0 
 
 43-7 
 
 46-3 
 
 
 
 34 
 
 21 
 
 59 
 
 123 
 
 21-6 
 
 345 
 
 509 
 
 71-6 
 
 96-4 
 
 126-2 
 
 52-8 
 
 100 
 
 15-9 
 
 24-1 
 
 30-4 
 
 35-7 
 
 40-1 
 
 44-6 
 
 48-3 
 
 52-1 
 
 55-4 
 
 58-7 
 
 
 
 43 
 
 2-6 
 
 7-4 
 
 15-6 
 
 27-3 
 
 438 
 
 64-6 
 
 90-9 
 
 122-3 
 
 1601 
 
 587 
 
 90 
 
 16-9 
 
 25-6 
 
 32-3 
 
 38-0 
 
 42-7 
 
 47-4 
 
 51-4 
 
 55-4 
 
 58-9 
 
 62-5 
 
 
 
 46 
 
 28 
 
 79 
 
 166 
 
 29-1 
 
 46-6 
 
 687 
 
 96-7 
 
 130-2 
 
 170-3 
 
 C6- 
 
 80 
 
 18-1 
 
 27-5 
 
 34-6 
 
 40-7 
 
 45-8 
 
 509 
 
 55-2 
 
 59-4 
 
 63-2 
 
 67'0 
 
 
 
 49 
 
 30 
 
 8-5 
 
 17-8 
 
 312 
 
 499 
 
 73-7 
 
 103-7 
 
 1396 
 
 1827 
 
 754 
 
 70 
 
 19-6 
 
 30-0 
 
 37-5 
 
 44-1 
 
 49-6 
 
 55-1 
 
 59-7 
 
 64-4 
 
 68-4 
 
 72-5 
 
 
 
 53 
 
 32 
 
 9-2 
 
 192 
 
 338 
 
 54-1 
 
 79-8 
 
 112-3 
 
 1512 
 
 197-7 
 
 88- 
 
 60 
 
 21-5 
 
 32-6 
 
 41-1 
 
 48-3 
 
 54-4 
 
 60-4 
 
 65-5 
 
 70-6 
 
 75-1 
 
 79-5 
 
 
 
 59 
 
 36 
 
 101 
 
 211 
 
 371 
 
 59-3 
 
 87-5 
 
 123-2 
 
 165-8 
 
 2169 
 
 105-6 
 
 50 
 
 24-0 
 
 36-4 
 
 45-8 
 
 53-9 
 
 60-7 
 
 67-4 
 
 73-1 
 
 78-7 
 
 83-7 
 
 88-7 
 
 
 
 65 
 
 4-0 
 
 112 
 
 235 
 
 413 
 
 662 
 
 97-6 
 
 137-4 
 
 1849 
 
 242-0 
 
 132- 
 
 40 
 
 27-4 
 
 41-6 
 
 52-5 
 
 61-7 
 
 69-4 
 
 77-1 
 
 83-6 
 
 90-1 
 
 95-8 
 
 101-5 
 
 
 
 75 
 
 4-5 
 
 128 
 
 26-9 
 
 47-3 
 
 75-7 
 
 111-7 
 
 157-2 
 
 211-6 
 
 2768 
 
 176 
 
 30 
 
 32-6 
 
 49-5 
 
 62-5 
 
 73-4 
 
 82-6 
 
 91-7 
 
 99-5 
 
 107-2 
 
 1140 
 
 120-8 
 
 
 
 89 
 
 5-4 
 
 153 
 
 320 
 
 563 
 
 90-0 
 
 1329 
 
 1871 
 
 251-8 
 
 3294 
 
 212-2 
 
 25 
 
 36-4 
 
 55-3 
 
 69-8 
 
 8-2-0 
 
 92-2 
 
 102-4 
 
 lii-l 
 
 119-7 
 
 127-3 
 
 134-9 
 
 
 
 99 
 
 6-0 
 
 171 
 
 35-8 
 
 62-9 
 
 ICO 6 
 
 148-4 
 
 2089 
 
 2812 
 
 367-9 
 
 264-1 
 
 20 
 
 41-7 
 
 63-3 
 
 79-9 
 
 93-8 
 
 105-6 
 
 117-3 
 
 127-2 
 
 137-0 
 
 145-7 
 
 154-4 
 
 
 
 114 
 
 69 
 
 196 
 
 409 
 
 720 
 
 1151 
 
 1699 
 
 2392 
 
 3219 
 
 421-2 
 
 352- 15 
 
 49-6 
 
 75-3 
 
 95-0 
 
 111-7 
 
 125-6 
 
 1396 
 
 151-3 
 
 163-1 
 
 173-4 
 
 183-8 
 
 
 
 1-35 
 
 8-2 
 
 23-3 
 
 48-3 
 
 856 
 
 1370 
 
 202-2 
 
 284-6 
 
 383-1 1501-2 
 
 528 
 
 10 
 
 63-3 
 
 96-0 
 
 1212 
 
 142-4 
 
 160-2 
 
 178-0 
 
 192-9 
 
 207-9 
 
 221-1 1234-3 
 
 
 
 173 
 
 105 
 
 29-7 
 
 621 
 
 109-2 
 
 174-7 
 
 257-8 
 
 3629 
 
 4885 6390 
 
TABLE for finding, very nearly, the velocity and discharge from Cylindrical Water Pipes or Sewers, when the 
 diameter and fall are given. Any two of the four quantities, the velocity, discharge, diameter, or inclination 
 being given, the others can fee found in THE TABLE from inspection. 
 
 il 
 
 Mean hy- 1 
 draulic incli-l 
 nation of 1 
 pipe or sewer| 
 
 The VELOCITT IN INCHES PER SECOND is given in the first horizontal line for each inclination 
 or fall ; and the DISCHARGE IN COBIC FEET PER MINUTE in the next following one. 
 
 12 in. 1 14 in. 
 diameter) diameter 
 
 16 in. 
 diameter 
 
 18 in. 
 diameter 
 
 20 in. i 22 in. 
 diameter diameter 
 
 24 in. 26 in. 
 diameter| diameter 
 
 28 in. 
 diameter 
 
 30 in. 
 diameter 
 
 
 One in 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 5280 
 
 6-8 
 
 7-4 
 
 8-0 
 
 8-5 
 
 8-9 
 
 9-4 
 
 9-8 
 
 10-3 
 
 10-7 
 
 ll'l 
 
 
 
 26-9 
 
 40- 
 
 56- 
 
 75- 
 
 98- 
 
 124- 
 
 155- 
 
 189- 
 
 
 228- 
 
 271- 
 
 2 
 
 2640 
 
 10-1 
 
 10-9 
 
 11-7 
 
 12-5 
 
 13-2 
 
 13-9 
 
 14-5 
 
 15 
 
 1 
 
 15-7 
 
 16-3 
 
 
 
 40 
 
 58 
 
 82 
 
 110 
 
 144 
 
 183 
 
 228 
 
 278 
 
 
 336 
 
 400 
 
 3 
 
 1760 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 17 
 
 18 
 
 19 
 
 
 20 
 
 20 
 
 
 
 50 
 
 73 
 
 103 
 
 138 
 
 180 
 
 229 
 
 285 
 
 349 
 
 
 421 
 
 500 
 
 4 
 
 1320 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 
 23 
 
 24 
 
 
 
 58 
 
 86 
 
 120 
 
 163 
 
 212 
 
 269 
 
 335 
 
 410 
 
 
 494 
 
 588 
 
 5 
 
 1056 
 
 17 
 
 18 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 
 26 
 
 27 
 
 
 
 66 
 
 97 
 
 136 
 
 184 
 
 240 
 
 305 
 
 380 
 
 464 
 
 
 560 
 
 665 
 
 6 
 
 880 
 
 19 
 
 20 
 
 22 
 
 23 
 
 24 
 
 26 
 
 27 
 
 28 
 
 
 29 
 
 30 
 
 
 
 73 
 
 108 
 
 151 
 
 203 
 
 265 
 
 337 
 
 420 
 
 514 
 
 
 620 
 
 737 
 
 7 
 
 754 
 
 20 
 
 2-2 
 
 24 
 
 25 
 
 27 
 
 28 
 
 29 
 
 30 
 
 
 32 
 
 33 
 
 
 
 80 
 
 118 
 
 165 
 
 222 
 
 289 
 
 368 
 
 458 
 
 560 
 
 
 676 
 
 804 
 
 8 
 
 660 
 
 22 
 
 24 
 
 25 
 
 27 
 
 29 
 
 30 
 
 31 
 
 33 
 
 
 34 
 
 35 
 
 
 
 86 
 
 127 
 
 177 
 
 239 
 
 312 
 
 397 
 
 493 
 
 604 
 
 
 728 
 
 866 
 
 9 
 
 587 
 
 23 
 
 25 
 
 28 
 
 29 
 
 30 
 
 32 
 
 33 
 
 35 
 
 
 36 
 
 38 
 
 
 
 92 
 
 135 
 
 192 
 
 255 
 
 333 
 
 424 
 
 527 
 
 645 
 
 
 778 
 
 925 
 
 10 
 
 528 
 
 25 
 
 27 
 
 29 
 
 31 
 
 32 
 
 34 
 
 36 
 
 37 
 
 
 39 
 
 40 
 
 
 
 97 
 
 144 
 
 201 
 
 271 
 
 354 
 
 450 
 
 560 
 
 685 
 
 
 826 
 
 982 
 
 11 
 
 480 
 
 26 
 
 28 
 
 30 
 
 32 
 
 34 
 
 36 
 
 38 
 
 39 
 
 
 41 
 
 42 
 
 
 
 103 
 
 152 
 
 213 
 
 286 
 
 373 
 
 475 
 
 590 
 
 723 
 
 
 871 
 
 1036 
 
 12 
 
 440 
 
 27 
 
 30 
 
 32 
 
 34 
 
 36 
 
 38 
 
 39 
 
 41 
 
 
 43 
 
 44 
 
 
 
 108 
 
 159 
 
 223 
 
 300 
 
 392 
 
 498 
 
 620 
 
 759 
 
 
 916 
 
 1089 
 
 13-2 
 
 400 
 
 29 
 
 31 
 
 34 
 
 36 
 
 38 
 
 40 
 
 42 
 
 43 
 
 
 45 
 
 47 
 
 
 
 114 
 
 168 
 
 236 
 
 317 
 
 414 
 
 526 
 
 655 
 
 801 
 
 
 9i 
 
 36 
 
 1149 
 
 15-1 
 
 350 
 
 31 
 
 34 
 
 36 
 
 39 
 
 41 
 
 43 
 
 45 
 
 47 
 
 
 49 
 
 50 
 
 
 
 123 
 
 181 
 
 254 
 
 342 
 
 446 
 
 568 
 
 707 
 
 865 
 
 
 1043 
 
 1240 
 
 17-6 
 
 300 
 
 34 
 
 37 
 
 40 
 
 42 
 
 45 
 
 47 
 
 49 
 
 51 
 
 
 53 
 
 55 
 
 
 
 134 
 
 198 
 
 278 
 
 374 
 
 487 
 
 620 
 
 772 
 
 944 
 
 
 1139 
 
 1354 
 
 21-1 
 
 250 
 
 38 
 
 41 
 
 44 
 
 47 
 
 50 
 
 52 
 
 55 
 
 57 
 
 
 59 
 
 61 
 
 
 
 149 
 
 219 
 
 308 
 
 415 
 
 541 
 
 688 
 
 857 
 
 1048 
 
 
 1264 
 
 1504 
 
 26-4 
 
 200 
 
 43 
 
 47 
 
 50 
 
 53 
 
 56 
 
 59 
 
 62 
 
 65 
 
 
 67 
 
 70 
 
 
 
 169 
 
 249 
 
 350 
 
 472 
 
 616 
 
 783 
 
 975 
 
 1192 
 
 
 1438 
 
 1710 
 
 35-2 
 
 150 
 
 51 
 
 55 
 
 59 
 
 63 
 
 67 
 
 70 
 
 73 
 
 76 
 
 
 79 
 
 82 
 
 
 
 200 
 
 294 
 
 414 
 
 558 
 
 728 
 
 925 
 
 1152 
 
 1409 
 
 
 1700 2021 
 
 52-8 
 
 100 
 
 65 
 
 70 
 
 75 
 
 80 
 
 85 
 
 89 
 
 93 
 
 97 
 
 
 101 
 
 105 
 
 
 
 254 
 
 374 
 
 526 
 
 708 
 
 923 
 
 1174 
 
 1462 
 
 1788 
 
 2157 2565 
 
 58*7 
 
 90 
 
 69 
 
 74 
 
 80 
 
 85 
 
 90 
 
 95 
 
 99 
 
 Sop 
 
 Velocit 
 
 ies in inches, oer 
 
 
 
 270 
 
 398 
 
 559 
 
 753 
 
 982 
 
 1250 
 
 1556 
 
 c >t> 
 
 second, due to heads. 
 
 
 
 <D bo 
 
 66 
 
 80 
 
 74 
 
 80 
 
 86 
 
 91 
 
 96 
 
 101 
 
 106 
 
 
 -gg 
 
 
 
 '-*-* 
 
 
 
 
 290 
 
 427 
 
 600 
 
 808 
 
 1054 
 
 1340 
 
 1669 
 
 f~s 
 
 SEi 
 
 W 3 C 
 
 || 
 
 jllsj 
 
 75-4 
 
 70 
 
 80 
 
 86 
 
 93 
 
 99 
 
 105 
 
 110 
 
 115 
 
 rii^ 
 
 ? 
 
 
 5** 
 
 
 
 314 
 
 463 
 
 650 
 
 875 
 
 1141 
 
 1451 
 
 1807 
 
 i 
 
 27-8 
 
 22-7 
 
 20 
 
 88 
 
 60 
 
 88 
 
 95 
 
 102 
 
 109 
 
 115 
 
 120 
 
 126 
 
 2 
 
 39 
 
 32 
 
 28 
 
 
 
 344 
 
 506 
 
 712 
 
 959 
 
 1251 
 
 1591 
 
 1981 
 
 3 
 
 48 
 
 39 
 
 34 
 
 105-6 
 
 50 
 
 98 
 
 106 
 
 114 
 
 121 
 
 128 
 
 134 
 
 141 
 
 4 
 
 56 
 
 45 
 
 40 
 
 
 
 384 
 
 565 
 
 795 
 
 1070 
 
 1396 
 
 1775 
 
 2211 
 
 6 
 
 68 
 
 56 
 
 49 
 
 132 
 
 40 
 
 119 
 
 124 
 
 130 
 
 138 
 
 146 
 
 153 
 
 161 
 
 8 
 
 79 
 
 64 
 
 56 
 
 
 
 467 
 
 665 
 
 908 
 
 1224 
 
 1597 
 
 2031 
 
 2529 
 
 10 
 
 88 
 
 72 
 
 63 
 
 176 
 
 30 
 
 133 
 
 144 
 
 155 
 
 165 
 
 174 
 
 183 
 
 192 
 
 12 
 
 96 
 
 78 
 
 69 
 
 
 
 523 
 
 770 
 
 1082 
 
 1456 
 
 1900 
 
 2417 
 
 3010 
 
 24 
 
 136 
 
 111 
 
 97 
 
 212-2 
 
 25 
 
 149 
 
 161 
 
 173 
 
 184 
 
 195 
 
 204 
 
 214 
 
 36 
 
 167 
 
 136 
 
 119 
 
 
 
 584 
 
 860 
 
 1208 
 
 1627 
 
 2122 
 
 2699 
 
 3362 
 
 48 
 
 193 
 
 157 
 
 138 
 
 264-1 
 
 20 
 
 170 
 
 184 
 
 198 
 
 211 
 
 223 
 
 234 
 
 245 
 
 60 
 
 215 
 
 176 
 
 154 
 
 
 
 668 
 
 984 
 
 1383 
 
 1862 
 
 2430 
 
 3090 
 
 3848 
 
 72 
 
 236 
 
 192 
 
 169 
 
 352 
 
 15 
 
 202 
 
 219 
 
 236 
 
 251 
 
 265 
 
 278 
 
 292 
 
 84 
 
 255 
 
 208 
 
 182 
 
 
 
 795 
 
 1171 
 
 1646 
 
 2216 
 
 2891 
 
 3677 
 
 4579 
 
 96 
 
 272 
 
 222 
 
 194 
 
 528 
 
 10 
 
 258 
 
 279 
 
 301 
 
 320 
 
 338 
 
 355 
 
 372 
 
 108 
 
 289 
 
 235 
 
 207 
 
 
 
 1014 
 
 1496 2098 2826 
 
 3686 4688 
 
 5839 
 
 120 
 
 305 
 
 248 
 
 218 
 
44 THE DISCHARGE OF WATER FROM 
 
 SECTION II. 
 
 FORMULA FOR THE VELOCITY, AND DISCHARGE, PROM 
 ORIFICES, WEIRS, AND NOTCHES. COEFFICIENTS OF 
 VELOCITY, CONTRACTION, AND DISCHARGE. PRACTICAL 
 REMARKS ON THE USE OF THE FORMULAE. 
 
 The quantity of water discharged in a given time 
 through an aperture of a given area in the side or 
 bottom of a vessel, is modified by different circum- 
 stances, and varies more or less with the form, posi- 
 tion, and depth of the orifice ; but the discharge may 
 be easily found, when we have determined the velocity 
 and the contraction of the fluid vein. 
 
 VELOCITY. 
 
 If g be the velocity acquired by a heavy body 
 falling from a state of rest for one second, in vacuo, 
 then it has been shown by writers on mechanics, that 
 the velocity v per second acquired by falling from a 
 height k, will be 
 
 (1.) v \/ 2gh. 
 
 The numerical value of g varies with the latitude ; we 
 shall assume 2 g = 772-84 inches zz 64*403 feet. 
 These will give for measures in inches, 
 
 v = 27-8 v/V and h = jf ~ -001293t> 2 , 
 
 ' 
 
 and for measures in feet, 
 
 * The velocities for different heights are given in the column 
 number 1, TABLE II. 
 
ORIFICES, WEIES, PIPES, AND RIVERS. 
 
 45 
 
 v = 8-025 v/ h, and h = 
 
 64-403 
 If v be in feet, and h in inches, then 
 
 v = 2-317 
 
 h = 
 
 = -01553 v*. 
 
 = -1864 v\* 
 
 COEFFICIENT OF VELOCITY. 
 
 Let the vessel A B c D, Fig. 1, be filled with water 
 to the level E F : then it has been found, by experi- 
 
 ment, that the velocity of discharge through a small 
 orifice o, in a thin plate, at the distance of half the 
 diameter outside it, in the vena-contracta, will be very 
 nearly that due to a heavy body falling freely from 
 the height A, of the surface of the water E F, above 
 the centre of the orifice. The velocity of discharge 
 
 * The force of gravity increases with the latitude, and decreases 
 with the altitude above the level of the sea, but not to any con- 
 siderable extent. If A be the latitude, and h the altitude, in feet, 
 above the mean sea level, then we may, generally, take 
 
 g = 32-17 (1 -0029 cos 2A) x (I ), 
 
 in which E, the radius of the earth at the given latitude is equal 
 to 
 
 20887600 ( 1 + -0016 cos 2X). 
 
46 THE DISCHARGE OF WATER FROM 
 
 determined by the equation v v/2 g h, for falling 
 bodies, is, therefore, called the " theoretical velocity'' 
 If we now put v d for the actual mean velocity of dis- 
 charge in the vena-contracta, and c v for its ratio to 
 the theoretical velocity v, we shall get v d zz c v v ; and 
 by substituting for v, its value 
 (2.) v, 
 
 c v is termed "the coefficient of velocity ;" its numerical 
 value, at about half the diameter from the orifice, is 
 about *974 ; and, consequently, 
 
 t? d = -974 v/2 g h. 
 This for measures in inches becomes 
 
 v d = 27-077 
 and for measures in feet 
 
 v* = 7-816 
 
 The orifice o, is termed an horizontal orifice in Fig. 
 1, and in Pig 2 a vertical or lateral orifice. When 
 
 * The velocities for different heights calculated from this 
 formula, are given in the column numbered 2, TABLE II. It has 
 been latterly asserted in a Blue Book that theoretically v d = 
 f V 2 g h. It is not necessary here to combat this error, which 
 confounds the discharge with its velocity, and a single practical 
 fact, applicable only to a thin plate, with a theoretical principle. 
 The experimental^ discharge approximates to f */ % g h multiplied 
 by the area of the orifice ; but the theoretical velocity \/ 2 g h 
 always approximates to the experimental velocity, or *974 */ 2 gh, 
 obtained immediately outside the orifice in the vena-contracta. It 
 would be unnecessary to allude to this theory here if it were not 
 supported and put forward by three eminent engineers whose 
 authority may mislead others. Vide p. 4. Brief observations of 
 Messrs. Bidder, Hawksley, and Bazalgette on the answers of the 
 Government Referees on the METROPOLITAN MAIN DEAINAGE, or- 
 dered by the House of Commons to be printed 13th July, 1858. 
 
OEIFICES, WEIRS, PIPES, AND RIVERS. 47 
 
 small, each is found to have practically the same 
 velocity of discharge, when the centres of the con- 
 tracted sections are at the same depth, h, below the 
 surface ; but when lateral orifices are large, or rather 
 deep, the velocity at the centre is not, even prac- 
 tically, the mean velocity ; and in thick plates and 
 modified forms of adjutage, the mean velocities are 
 found to vary. 
 
 VENA-CONTRACTA AND CONTRACTION. 
 
 It has been found that the diameter of a column 
 issuing from a circular orifice in a thin plate, is con- 
 tracted to very nearly eight-tenths of the whole 
 diameter at the distance of the radius from it, and 
 that at this distance the contraction is greatest. The 
 ratio of the diameter of the orifice to that of the 
 contracted vein, vena-contracta, is not always found 
 constant by the same or different experimentalists. 
 
 Newton makes it 1 : -841, J md ' Before, that of the j. 707 
 
 I areas as 1 : ) 
 
 (7156 
 (622 
 1 : -6432 
 1 : -64 
 656 
 
 Poleni 
 
 1 : 
 
 f -846 
 \ -788 
 
 Borda 
 
 1 : 
 
 802 
 
 Michellotti 
 
 1 : 
 
 8 
 
 Bossut 
 
 1 : 
 
 f -81 
 
 {818 
 
 Du Buat 
 
 1 : 
 
 816 
 
 Venturi 
 
 1 : 
 
 798 
 
 Eytelwein 
 
 1 : 
 
 8 
 
 Bayer 
 
 1 : 
 
 7854 
 
 ,,1: '667 
 
 1: '637 
 
 1: -64 
 
 1 : '617 
 
 Bayer's value for the contraction has been deter- 
 mined on the hypothesis, that the velocities of the 
 particles of water as they approach the orifice from 
 all sides, are inversely as the squares of their 
 
48 
 
 THE DISCHARGE OF WATER FROM 
 
 distances from its centre ; and the calculations made 
 of the discharge from circular, square, and rectan- 
 gular orifices, on this hypothesis, coincide pretty 
 closely with experiments. 
 
 FORM OF THE CONTRACTED VEIN. 
 
 Let o R zz d, Fig. 3, be the diameter of an orifice ; 
 then at the distance R s zz -^ the contraction is found 
 
 to be greatest ; we shall assume the contracted di- 
 ameter o r zz '7854 d. If we suppose the fluid 
 column between o R and o r to be so reduced, that 
 the curve lines R r and o o shall become arcs of circles, 
 
 then it is easy to show 
 from the properties of 
 the circle, that the ra- 
 dius c r must be equal 
 to 1-22^. The mean 
 velocity in the orifice, 
 OR, is to that in the 
 vena-contracta, o r, as *617 : 1 ; and the mouth piece, 
 R r o o, Fig. 4, in which o p zz \ o R, and or zz -7854 x 
 o R, will give for the velocity of discharge at o r, the 
 vena-contracta, 
 
 v, zz -974 
 
 c-- 
 
 Fig. 4 
 
 zz 7-816 
 
 in feet very nearly. In 
 speaking of the velo- 
 city of discharge from 
 orifices in thin plates, 
 we always assume it 
 to be the velocity in 
 the vena-contracta, and not that in the orifice itself, 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 49 
 
 which varies with the coefficient of discharge, unless 
 in TABLE II., where the mean velocity in the latter, 
 as representing c d v/ 2 g h, is also given. 
 
 COEFFICIENTS OF CONTRACTION AND DISCHARGE. 
 
 If we put A for the area of the orifice o R, Pig. 3, 
 and c c x A for that of the contracted section at o r, 
 then c c is called the "coefficient of contraction." The 
 velocity of discharge v^ is equal to c v \/2 gh, equa- 
 tion (2). If we multiply this by the area of the 
 contracted section c c x A, we shall get for the dis- 
 charge 
 
 D =i c v x c c x A N/ 2 g h* 
 
 It is evident A \/ 2 g h would be the discharge if 
 there were no contraction and no change of velocity 
 due to the height h; c v x c c is therefore equal to the 
 coefficient of discharge. If we call the latter c d , we 
 shall have the equation 
 (3.) Ct = c v xc c , 
 
 and hence we perceive that the "coefficient of discharge" 
 is equal to the product of the coefficients of velocity 
 and contraction. In the foregoing expression for the 
 discharge D, h must be so taken, that the velocity at 
 that depth shall be the mean velocity in the orifice A. 
 In full prismatic tubes the coefficients of velocity and 
 discharge are equal to each other. 
 
 * The expression c v c c */2gh=:c d */2gh is the coefficient 
 of the area A, and, consequently, represents the mean velocity in 
 the orifice ; the coefficient of which is, therefore, equal to c d . 
 The values of the velocity c d \/2 g h t for different heights and 
 coefficients, are given in TABLE II. 
 
50 THE DISCHARGE OF WATER FROM 
 
 MEAN AND CENTRAL VELOCITY. 
 
 In order to find the mean velocity of discharge 
 from an orifice, it is, in the first instance, necessary 
 to determine the velocity due to each point in its 
 surface, and the discharge itself; after which, the 
 mean velocity is found by simply dividing the area 
 of the orifice into the discharge. The velocity due 
 to the height of water at the centre of a circular, 
 square, or rectangular orifice, is not strictly the 
 mean velocity, nor is the latter in these, or other 
 figures, that at the centre of gravity. When, how- 
 ever, an orifice is small in proportion to its depth in 
 the water, the velocity of efflux determined for the 
 centre approaches very closely to the mean velocity ; 
 and, indeed, at depths exceeding four times the depth 
 of the orifice, the error in assuming the mean velocity 
 to be that at the centre of the orifice is so small as to 
 be of little or no practical consequence, and for 
 lesser depths it never exceeds 6 per cent. It is, there- 
 fore, for greater simplicity, the practice to determine 
 the velocity from the depth h of the centre of the 
 orifice, unless in weirs or notches ; and the coeffi- 
 cients of discharge and velocity in the following 
 pages have been calculated from experiments on this 
 assumption, unless it shall be otherwise stated. 
 
 DISCHARGES THROUGH ORIFICES OF DIFFERENT FORMS IN 
 THIN PLATES. 
 
 The orifices which we have to deal with in practice 
 are square, rectangular, or circular; and sometimes, 
 perhaps, triangular or quadrangular in form. It will 
 
ORIFICES, WEIKS, PIPES, AND RIVERS. 
 
 51 
 
 be necessary to give here only the theoretical ex- 
 pressions for the discharge and velocity for each kind 
 of form, but as the demonstrations are unsuited to 
 our present purposes we shall omit them. 
 
 TRAPEZOIDAL ORIFICES WITH TWO HORIZONTAL SIDES. 
 
 Put d for the vertical depth of an orifice, h t for the 
 altitude of pressure at top, above the upper side, and 
 h b for the altitude at bottom, above the lower side, 
 we then get 
 
 ^b ^t = d. 
 
 Let us also represent the top or upper side of the 
 orifice A or c, Fig. 5, by / t , and the lower or bottom 
 
 lt + /b 
 
 side by / b , and put 
 
 = /. 
 
 Now, when l t = / b , the trapezoid becomes a paral- 
 lelogram whose length is I and depth d ; and putting 
 h for the depth to the centre of gravity, we get the 
 equation 
 
 *t + 2 = ^ 2 = * 
 
 The general expression for the discharge, D, through 
 a trapezoidal orifice, A, is then 
 
 B3 
 
52 THE DISCHARGE OF WATER FROM 
 
 (4.) D^v2 X 
 
 in which c d is the coefficient of discharge ; and when 
 
 the smaller side is uppermost as at c, 
 
 (5.) D= 
 
 PARALLELOGRAMIC AND RECTANGULAR ORIFICES. 
 
 When / t z= 4 = I, the orifice becomes a parallelo- 
 gram, or a rectangle, B, and we have for the discharge 
 
 (6.) D= 
 
 NOTCHES. 
 
 When the upper sides of the orifices A, B, and c, 
 rise to the surface as at A O , B O , and c , h t becomes 
 nothing, and we get, as h b = d, for the trapezoidal 
 notch A with the larger side up, 
 
 (7.) D = c d N/2^x 3"" 2 
 
 for the trapezoidal notch, c , with the smaller side up, 
 (8.) D = C 
 
 the same in form, but not in value, as the preceding 
 equation ; and for a parallelogramic or rectangular 
 
 notch B , 
 
ORIFICES, WEIRS, PIPES, AND RIVEES. 
 
 53 
 
 (9.) D = c, v/^7 X -Id = - cj N/ 2 ff . 
 
 o o 
 
 It is easy to perceive that the forms of equations (4) 
 and (5), and also of equations (7) and (8), are iden- 
 tical. The values for the discharge in equations (6) 
 and (9) are equally applicable, whether the form of 
 the orifice be a parallelogram or a rectangle, the only 
 difference being in the value of the coefficient of 
 discharge, c d , which becomes slightly modified for 
 each form of orifice. 
 
 TRIANGULAR ORIFICES WITH HORIZONTAL BASES, AND 
 RECTILINEAL ORIFICES IN GENERAL. 
 
 When the length of the lower side, 4 = 0, the 
 orifice becomes a triangle, D, Fig. 6, with the base 
 upwards. 
 
 Fig. 6 
 
 
 
 
 
 In this case, equation (4) becomes 
 (10.) .. 
 
 which gives the discharge through the triangular 
 orifice, D. 
 
 When / t = 0, in equation (5), the orifice becomes a 
 triangle, F, with the base downwards ; in this case, 
 we find for the value of the discharge, 
 
54 THE DISCHARGE OF WATER FROM 
 
 (11.) r, = c dN /2^xf />(*!-! X**-** 
 
 As any triangular orifice whatever can be divided 
 into two others by a line of division through one of 
 the angles parallel to the horizon ; and as the dis- 
 charge from the triangular orifice D or F is the same 
 as for any other on the same base and between the 
 same parallels, we can easily find, by such a divi- 
 sion, the discharge from any triangle not having one 
 side parallel to the horizon, and thence the discharge 
 from any rectilineal figure whatever by dividing it 
 into triangles. 
 
 If the triangle F be raised so that the base shall be 
 on the same level with the upper side of the trian- 
 gular orifice D ; if, also, the bases be equal, and also 
 the depths, we shall find, by adding equations (10) 
 and (11), and making the necessary changes indicated 
 by the diagram, 
 
 (12.) D = c d s2 X {+- 2 X 
 
 for the discharge from a parallelogram E with one 
 diagonal horizontal. Now this is the same as the 
 discharge from any quadrilateral figure whatever, 
 having the same horizontal diagonal, and also having 
 the upper and lower angles on the same parallels, or at 
 the same depths, as those of the parallelogram. If 
 the orifices D, F, and E rise to the surface of the 
 water, as at D O , E O , F O , we shall then have for the dis- 
 charge from the notch D O , 
 
 which for a right angled triangle becomes 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 55 
 
 : D = C d N X <. 
 
 For the discharge from the notch F O , 
 
 (14.) D = c dV /2^ x ^^: 
 
 and for the discharge through the notch E O , 
 
 (15.) D=c d v/2^x ] ^{4~2^} = c d v/2^x -9752/^1 
 
 When the parallelogram E O becomes a square / = 2 d, 
 and hence, 
 
 (16.) D=c dV / 2^X-9752/*x v/J = c d v/2^X'34478 l\ 
 The foregoing equations will enable us to find an 
 expression for the discharge from any rectilineal 
 orifice whatever, as it can be divided into triangles, 
 the discharge from each of which can be determined 
 as already shown in the remark following equation 
 (11.) The examples which we have given will be 
 found to comprehend every form of rectilineal orifice 
 which occurs in practice ; but for the greater number 
 of orifices, sunk to any depth below the surface, the 
 
 * In the Civil Engineer and Architect's Journal, 1858, p. 370, it 
 is stated that Professor Thompson, Belfast College, gave at the 
 British Association in Leeds for a right angled triangle, for 
 discharges of from 2 to 10 cubic feet per minute, the expression 
 Q = '317 H"ST, in which Q is the quantity in cuhic feet per minute, 
 and H the head in inches. Now the ahove equation for a coeffi- 
 
 cient of -617 becomes, for inch measures, D = 17-153 x T % d 2 = 
 
 _5 
 
 9-15 d2; or by multiplying by 60, and dividing by 1728, to 
 
 reduce the discharge to feet per minute, we get D = -317 d^, 
 identically the same as Professor Thompson derived from his 
 experiments. All sections of a triangular notch are similar tri- 
 angles, and hence the advantage of a triangular-notch-gauge, 
 where it can be used, as, probably, the coefficient remains constant. 
 Professor Thompson, I believe, first drew attention to this. 
 
56 
 
 THE DISCHAEGE OF WATEE FROM 
 
 discharge will be found with sufficient accuracy by 
 multiplying the area by the velocity due to the 
 centre. 
 
 CIRCULAR AND SEMICIRCULAR ORIFICES. 
 
 The discharge through circular and semicircular 
 orifices in thin plates can only be represented by 
 means of infinite series. Let us represent by s 1 the 
 sum of the series 
 
 Fig.7 
 
 -\ \x / v Y \ f 1 \ 
 
 = H 
 
 e^^-gjF^=^ 
 
 m 
 
 _f\ 1357 9\/l 1 3 5\/^ 
 ^2 4 6 o 10 12 2 4 6 o / 
 
 Let us also represent by s 2 the sum of the series 
 _M(i.IV . (I.I.* 
 
 3*1416( ^2 * 3^ A ^~ V2 ' 4 * 6 
 
 2*4*6*8 1(K ^3 5 
 
 then the discharge from the semicircle G, Fig. 7, with 
 
 the diameter upwards and horizontal, is 
 
 (17.) D = c d \/%g~h x 3-1416 r 8 (^ + s 2 ). 
 
 And the discharge from the semicircle i, with the 
 
 diameter downwards and horizontal, is 
 
 (18.) D = c d \/2gh x 3-1416 7-2 (s 1 .s 2 ). 
 
 If we put A for the area, we shall also have for the 
 
 discharge from a circle H, 
 
OKIFICES, WEIES, PIPES, AND EIVEKS. 57 
 
 (19.) v = c (i \2g/i X 
 
 In each of these three equations (17), (18), and (19), 
 h is the depth of the centre of the circumference 
 below the surface, and r the radius. 
 
 When the orifices rise to the surface, we have for 
 the discharge from a semicircular notch G O , with the 
 diameter horizontal and at the surface, 
 
 (20.) D = c d v2#^ X '9586 r 2 = c d v2#r X '6103 A ; 
 when the circumference of the semicircle is at the 
 surface, and the diameter horizontal, as at I , 
 
 when the horizontal diameter of the semicircle is 
 uppermost, and at the depth r below the surface, 
 (22.) D = c d v/27rx 1-8667 r^^v/^^x 1-1884 A; 
 and when the circumference of the entire circle is at 
 the surface, as at H O , 
 (23.) D = c d v/2<7>x 3-0171 r 2 = c d \/2^r x '9604A. 
 
 If we desire to reduce equations (20), (21), and 
 (22), to others in which the depth h of the centre 
 of gravity from the surface is contained, we have only 
 
 to substitute ^ for r in equation (20), and we 
 
 shall get, for the discharge from a semicircle with the 
 
 diameter at the surface, 
 
 (24.) D = c d \/2gh X '0367 A : 
 
 also, by substituting 7/yrinr for r in equation (21), we 
 
 get, for the discharge from a semicircle when the 
 
 circumference is at the surface and the diameter 
 
 horizontal, 
 
 (25.) D = c d V Zgh X -9653 A ; 
 
58 THE DISCHARGE OF WATER FROM 
 
 and when the horizontal diameter is uppermost, and 
 at the depth r below the surface r = i .4044 an ^ 
 
 (26.) D = c d v/2#A x -9957 A. 
 
 As A stands for the area of the particular orifice 
 in each of the preceding expressions for the dis- 
 charge, it must be taken of double the value, in 
 equation (23) for instance where it stands for the 
 area of a circle, that it is in equations (20), (21), or 
 (23), where it represents only the area of a semicircle. 
 
 MEAN VELOCITY. 
 
 The mean velocity is easily found by dividing the 
 area into the discharge per second given in the pre- 
 ceding equations. For instance, the mean velocity 
 in the example represented in equation (9), is equal 
 
 , which is had by dividing the area Id 
 
 into the discharge ; and in like manner the mean 
 velocity in equation (23) is -9604 c d \/ 2g r. 
 
 PRACTICAL REMARKS ON THE DISCHARGE FROM CIRCULAR 
 ORIFICES. 
 
 It has been shown, equation (19), that, for the 
 discharge from a circle, we have 
 
 x 2 A*! 
 
 in which h is the depth of the centre, A the area, 
 and s l the sum of the series 
 
 I 1 -/' 1 l \f l -r 1 l 3 5 \( l l 3 V- 
 II Vl'TAi'lJF UTi'lJUTOF 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 59 
 
 and it has also been shown, equation (23), that, when 
 the circumference touches the surface, this value 
 becomes 
 
 D = c d v/2^r x '9604 A. 
 
 Now when h is very large compared with r, it is easy 
 to perceive that 2 ^ = 1, and hence 
 
 (27.) D = c d \/%gh x A. 
 
 As this is the formula commonly used for finding the 
 discharge, it is clear, if the coefficient c d remain con- 
 stant, that the result obtained from it for D would be 
 too large. The differences, however, for depths 
 greater than three times the diameter, or 6 r, are 
 practically of no importance ; for, by calculating the 
 values of the discharge at different depths, we shall 
 find, when 
 
 h = r, that D = c d 
 
 (28.) 
 
 ;? 
 
 3 r 
 
 2' 
 
 7 r 
 
 X '960 A ; 
 X '978 A; 
 
 X -985 A; 
 
 X '989 A; 
 
 X '992 A ; 
 X '996 A; 
 X -998 A; 
 X '9987 A ; 
 X '9991 A. 
 
 These results show very clearly that, for circular 
 orifices, the common expression for the discharge 
 c d N/2^^ x A is abundantly correct for all depths 
 
 h 
 h 
 h 
 
 = 
 
 3r, 
 4r, 
 5r, 
 
 
 
 )> 
 
 
 
 D 
 D 
 D 
 
 = Cd 
 = C d 
 = C d 
 
60 THE DISCHARGE OF WATER FROM 
 
 exceeding three times the diameter, and that for 
 lesser depths the extreme error cannot exceed four 
 per cent, in reduction of the quantity found by this 
 formula. We shall show, hereafter, when discussing 
 the value of c d , that from the sinking of the surface, 
 and perhaps other causes, the discharge at lesser 
 depths is even larger than that exhibited by the 
 expression c d \/2 g h, x A, the value of the coefficient 
 of discharge, c d , being found to increase as the depths 
 h decrease. In fact, the sides of the orifice, the 
 rounding of the arrises, and the depth and position 
 with reference to the sides of the vessel, and surface 
 of the water, are of far greater practical importance 
 than extreme accuracy in the mathematical formula. 
 
 PRACTICAL REMARKS ON THE DISCHARGE FROM RECTANGULAR 
 
 ORIFICES. 
 
 It has been shown, equation (6), that the discharge 
 from rectangular orifices, with two sides parallel to 
 the horizon or surface of the water, is expressed by 
 the equation 
 
 D = c d x \ v/2~> x I (h\ - hj }, 
 
 o 
 
 in which / is the horizontal length of the orifice, A b 
 the depth of water on the lower, and h t the depth on 
 the upper, side. As it is desirable in practice to 
 change this form into a more simple one, in which 
 the height h of the centre and depth d of the orifice 
 
 only shall be included, we then have h b == h + and 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 61 
 
 h t = h --- By substituting these values of A b and 
 
 2 
 
 k t in the foregoing equations, and developing the 
 result into a series, the terms of which, after the 
 third, may be neglected, and putting A. for the area 
 I d, we shall find, 
 
 _ ( d* ^ 
 
 (29.) D = c d v/2 g h x A|l ^g-^ [very nearly. 
 
 We have therefore for the accurate theoretical dis- 
 charge 
 
 (30.) D = d -- 
 for the approximate discharge 
 
 D = c v2 X A 
 
 , 
 
 jl 
 
 and for the discharge by the common formula 
 
 D = c d v2 X A. 
 
 When the head (h) is large compared with (d) the 
 height of the orifice, each of the three last equations 
 gives the same value for the discharge ; but as the 
 common expression c d \/2 g h X A is the most simple ; 
 and as the greatest possible error in using it for lesser 
 depths does not exceed six per cent., viz. when the 
 orifice rises to the surface and becomes a notch, it is 
 evidently that formula best suited for practical pur- 
 poses. The following table and equations will show 
 more clearly the differences in the results as obtained 
 from the true, the approximate, and the common for- 
 mulce, applied to " lesser " heads ; and they will also 
 explain, to some extent, why " coefficients " deter- 
 mined from the common formula, and that used by 
 Poncelet and Lesbros, should decrease as the orifice 
 approaches the surface. 
 
62 
 
 THE DISCHARGE OF WATER FROM 
 
 (31.) 
 
 d 
 
 2> 
 
 5d 
 
 D=<V 
 
 x '9693 A. X '9733 A. 
 
 *= 
 
 36? 
 
 ?9 99 
 
 99 99 
 
 99 99 
 
 99 99 
 
 99 99 
 
 99 99 
 
 9? 99 
 
 X-9796A. 
 
 X-9854A. 
 
 X-9890A. 
 
 x '9953 A. 
 
 X-9974A. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x '9815 A. 
 
 x '9983 A. 
 X-9988A. 
 
 x -9864 A. 
 
 X-9896A. 
 
 x '9954 A. 
 
 x '9974 A. 
 
 x '9983 A- 
 
 X-9988A. 
 
 99 99 
 
 
 
 
 
 X'999lA. 
 X '9994 A. 
 
 X -9991 A. 
 
 ,, x *yyy'A. ,, ,, 
 
 x '9999 A. X '9999 A. 
 
 In the foregoing Table the first column contains 
 the head at the centre of the orifice expressed in 
 parts of its height d; the second contains the values 
 of the discharges according to equation (30) ; and 
 the third column contains 'the approximate values 
 determined from equation (29), the results in which 
 are something larger than those in column 2, derived 
 from the correct formula. The numerical coefficients 
 of A, at every depth, are less in both than one, the 
 constant coefficient according to the common formula. 
 The latter, therefore (as in circular orifices), gives re- 
 sults exceeding the true ones, but the excess is in- 
 
OKIFICES, WEIES, PIPES, AND EI\ T EKS. 63 
 
 appreciable at greater depths than h =z 3d, and for 
 lesser depths than this the error cannot exceed six 
 per cent. It may be useful to remark here, that 
 when the orifice rises to the surface and becomes 
 a notch, the "centre of mean velocity" is at four-ninths 
 of the depth, and the centre of gravity at two-thirds 
 of the depth from the surface. The former fraction 
 is the square of the latter. 
 
 SECTION III. 
 
 EXPERIMENTAL RESULTS AND FORMULAS. COEFFICIENTS 
 OF DISCHARGE. 
 
 We have heretofore dwelt but very partially on the 
 numerical values of the general coefficient of discharge 
 c d . In order to determine its value under different 
 circumstances more particularly, it will be now ne- 
 cessary to consider some of the experiments which 
 have been made from time to time. These do not 
 always give the same results, even when conducted 
 under the same circumstances and by the same par- 
 ties, and there appears to exist a certain amount of 
 error, more or less, inseparable from the subject. The 
 experiments with orifices in thin plates afford the 
 most consistent results ; but even here the differences 
 are sometimes greater than might be expected. In 
 many of the earlier experiments the value of the co- 
 efficient c d comes out too large, which arises, very 
 probably, from the orifices experimented with not 
 being in thin plates, and partaking, more or less, of 
 the nature of short tubes or mouth-pieces with rounded 
 
64 
 
 THE DISCHARGE OF WATER FROM 
 
 arrises, which, we shall see, give larger coefficients than 
 simple orifices. When an orifice is in the bottom of 
 a vessel, it would appear more correct to measure the 
 head from the surface to the vena-contracta than from 
 the surface to the orifice itself ; and as any error in 
 measuring the head in any experiment must affect the 
 value of the coefficient derived from such experiment, 
 so as to increase it when the error is to make the head 
 less, and vice versa, it appears that heads measured to 
 an orifice in the bottom of a vessel, and not to the 
 vena-contracta, must give larger coefficients from the 
 experimental results than, perhaps, the true ones. 
 The coefficients in the following pages have been al- 
 most all arranged and calculated, by the writer, from 
 the original experiments. 
 
 In 1739 Dr. Bryan Robinson made some experi- 
 ments on the discharge through small circular ori- 
 fices, from one-tenth to eight-tenths of an inch in 
 diameter, with heads of two and four feet*, which 
 give the following coefficients. 
 
 COEFFICIENTS FKOM DR. B. ROBINSON'S EXPERIMENTS. 
 
 Heads. 
 
 10 inch 
 diameter. 
 
 ^jinch 
 diameter. 
 
 30 inch 
 
 diameter. 
 
 To inch 
 diameter. 
 
 2 feet head 
 
 768 
 
 767 
 
 761 
 
 728 
 
 4 feet head 
 
 768 
 
 774 
 
 765 
 
 742 
 
 These results are pretty uniform, and the values 
 from which they are derived are said to be " means 
 taken from five or six experiments ;" as values of c d 
 they are, however, too high. The apparatus made 
 use of is not described ; but it is probable, from the 
 
 * Helsham's Lectures, p. 390. Dublin, 1739. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 65 
 
 results, that the plate containing the hole or orifice 
 was of some thickness, and that the inner arris was 
 slightly rounded. There is here, however, a very 
 perceptible increase in the coefficients for the smaller 
 orifices, but none for the smaller depth. 
 
 In a paper in the Transactions of the Royal Irish 
 Academy, vol. ii. p. 81, read March 1st, 1788, Dr. 
 Mathew Young determines the value of the coefficient 
 for an orifice ^ inch in diameter, with a mean head of 
 14 inches, to be *623. The manner in which this 
 value is determined is very elegant, viz. by comparing 
 the observed with the theoretical time of the water, 
 in the vessel^ sinking from 16 inches to 12 inches. 
 
 The following experiments by Michelotti, with 
 circular orifices from 1 to about 3 inches diameter, 
 and with from 6 to 23 feet heads, give for the mean 
 value c d 613 ; and for square orifices of from 1 to 
 9 square inches in area, at like depths, the mean 
 value of c d zz-628. The experiments are given in 
 French feet and inches, according to which standard 
 we have, in feet, D:z:7-77 A \/hx t, t being the time 
 in seconds/ As the time of discharge in these ex- 
 
 * The value of \/ % g h, equation (1), for measures in French 
 feet, is 7-77 ^/ h, and for measures in French inches, 36-9 \/ h, g 
 being equal to 30-2 feet, or 362-4 inches, French measure. One 
 French foot is equal to 1-06578 English feet, and the inches 
 preserve the same proportion. The resulting coefficients must be 
 the same, whatever standards we make our calculations from. 
 Many of the most valuable formulae and experiments in hydraulics 
 are given in French measures of the old style. As our object, 
 however, in the present section, is simply to determine from ex- 
 periment the relation of the experimental to the theoretical dis- 
 charge, it is not necessary to reduce the experiments to other 
 measures than those in the original ; but the value of the force of 
 
 P 
 
66 THE DISCHAKGE OF WATER FltOM 
 
 periments varies from ten minutes to an hour, and as 
 the depths are considerable, the results must be 
 looked upon as pretty accurate ; and it is worthy 
 of note that here the coefficients are larger for 
 square than for circular orifices. 
 
 We may remark here in passing how universal the 
 coefficients -613 to *6'28 are for all forms of orifices 
 in thin plates, or with the outside arrises chamfered. 
 Indeed, we may always use the coefficient '62 with 
 certainty, for practical purposes, for every orifice of 
 this kind, whether at the surface in the form of a 
 notch, or at the sides or bottom of a vessel, if the 
 section of the approaching water be large in propor- 
 tion to the area of the discharging orifice or notch. 
 By coefficient we of course here mean that decimal 
 which, multiplied by the theoretical value, gives the 
 practical result ; and this is substantially the same 
 for notches and orifices sunk below the surface, as 
 will appear farther on. There appears to us, how- 
 ever, an utter want of accuracy in using the coeffi- 
 cient '62 or thereabouts in gauging for all orifices, 
 weirs included, no matter what the thickness or form 
 of the orifice or crest of a weir may be, or area 
 of the approaching channel. These will cause the 
 coefficient to vary from '5 to 1 or more, and hence 
 the necessity for endeavouring to reduce this portion 
 of our subject to rule. 
 
 gravity, g, must, of course, be taken in those measures with which 
 the experiments were made. In the French decimal, or modern 
 style, the metre is equal to 3-2809 English feet, or 39-371 inches. 
 The tenth part of a metre is the decimetre, and the tenth part of 
 the decimetre the centimetre, as the names imply. TABLE 
 XIV. contains the weights and measures in general use in Great 
 Britain and France, with their general ratios to each other. 
 
OEIFICES, WEIKS, PIPES, AND RIVERS. 
 
 67 
 
 COEFFICIENTS FROM MICHELOTTl's EXPERIMENTS. 
 
 Description and 
 size of orifice, in 
 French inches. 
 
 Depth of the 
 centre of the 
 orifices in 
 French feet. 
 
 Quantity 
 discharged 
 in cubic feet. 
 
 Time of 
 discharge 
 in seconds. 
 
 Theoretical 
 time, 
 calculated 
 from 
 
 t= 5_ 
 
 7-77 A \A. 
 
 Resulting 
 coefficients 
 of discharge. 
 
 Square orifice, 
 3"X3" 
 
 6-613 
 6-852 
 11-676 
 11-818 
 21.691 
 21-715 
 
 463-604 
 566-458 
 516-785 
 612-118 
 415-437 
 499-222 
 
 600 
 720 
 510 
 600 
 300 
 360 
 
 371-3 
 445-6 
 311-4 
 366-6 
 183-7 
 220-6 
 
 619 
 619 
 610 
 611 
 612 
 613 
 
 Mean value of the coefficient ; square orifice 3" ) 
 
 <3" 
 
 614 
 
 
 Square orifice, 
 2"X2" 
 
 6-625 
 11-426 
 21-442 
 
 329-806 
 423-465 
 385-333 
 
 900 
 
 900 
 600 
 
 594- 
 580-4 
 385-7 
 
 660 
 646 
 643 
 
 Mean value of the coefficient ; square orifice 2" I 
 
 < 2" 
 
 649 
 
 
 Square orifice, 
 l" X l" 
 
 6-757 
 11-889 
 21-507 
 
 158-549 
 163-792 
 562-944 
 
 1800 
 1440 
 3600 
 
 1585- 
 880-6 
 2249-9 
 
 628 
 612 
 625 
 
 Mean value of the coefficient ; square orifice l" > 
 
 <l" 
 
 621 
 
 
 Circular orifice, 
 3" diameter 
 
 6-694 
 11-590 
 21-611 
 
 542-85 
 570-972 
 521-299 
 
 900 
 720 
 
 480 
 
 550-1 
 439-6 
 293-8 
 
 611 
 610 
 612 
 
 Mean value of the coefficient ; circular orifice 8" diameter . . -611 
 
 Circular orifice, 
 2" diameter 
 
 6-785 
 11-722 
 21-903 
 
 488-687 
 589-535 
 575-486 
 
 1800 
 1680 
 1200 
 
 1108-1 
 1016-4 
 725-9 
 
 616 
 605 
 605 
 
 Mean value of the coefficient ; circular orifice 2" diameter . . '609 
 
 Circular orifice, 
 1" diameter 
 
 6-875 
 11-743 
 22-014 
 
 247-354 
 324-11 
 444-535 
 
 3600 
 3600 
 3600 
 
 2227- 
 2233- 
 2237-2 
 
 619 
 620 
 621 
 
 Mean value of the coefficient ; circular orifice 1" diameter . . -620 
 
 F 3 
 
68 
 
 THE DISCHARGE OF WATER FROM 
 
 The experiments made by the Abbe Bossut, con- 
 tained in the following table, give the mean value of 
 c d , for both circular and square orifices, equal to -616 
 nearly ; and it may be perceived that, for the small 
 depth in the last experiment, the coefficient rises so 
 high as -649. These and other experiments led the 
 
 COEFFICIENTS FROM BOSSUT'S EXPERIMENTS. 
 
 Description, position, and size of 
 orifice, in French inches. 
 
 Depth of 
 the centre 
 of the 
 orifice in 
 French 
 inches. 
 
 Number 
 of French 
 cubical 
 inches 
 discharged 
 per minute. 
 
 Theoretical 
 discharge 
 per minute, 
 
 Eesulting 
 coefficients. 
 
 Horizontal and circular, ~" diameter 
 
 140-832 
 
 2311 
 
 3760-8 
 
 614 
 
 Horizontal and circular, 1" diameter 
 
 140-832 
 
 9281 
 
 15043-3 
 
 617 
 
 Horizontal and circular, 2" diameter 
 
 140-832 
 
 37203 
 
 60173-1 
 
 618 
 
 Horizontal and rectangular, 1" X k" 
 
 140-832 
 
 2933 
 
 4788-4 
 
 613 
 
 Horizontal and square, 1" X 1" 
 
 140-832 
 
 11817 
 
 19153-7 
 
 617 
 
 Horizontal and square, 2" X 2" 
 
 140-832 
 
 47361 
 
 76614-6 
 
 617 
 
 Lateral and circular, J" diameter . . 
 
 108- 
 
 2018 
 
 3293-3 
 
 613 
 
 Lateral and circular, 1" diameter . . 
 
 108- 
 
 8135 
 
 13173-3 
 
 617 
 
 Lateral and circular, " diameter . . 
 
 48- 
 
 1353 
 
 2195-5 
 
 616 
 
 Lateral and circular, 1" diameter . . 
 
 48- 
 
 6436 
 
 8782-2 
 
 616 
 
 Lateral and circular, 1" diameter . . 
 
 0-5833 
 
 628 
 
 968- 
 
 649 
 
 Abbe to construct a table of the discharges, at dif- 
 ferent depths, from a circular orifice 1 inch in dia- 
 meter, from which we have determined the following 
 table of coefficients. These increase, as the orifice 
 
 COEFFICIENTS DEDUCED FROM BOSSUT'S EXPERIMENTS. 
 
 Heads, 
 in feet. 
 
 Coefficients. 
 
 Heads, 
 in feet. 
 
 Coefficients. 
 
 Heads, 
 in feet. 
 
 Coefficients. 
 
 1 
 
 621 
 
 6 
 
 620 
 
 11 
 
 619 
 
 2 
 
 621 
 
 7 
 
 620 
 
 12 
 
 618 
 
 3 
 
 621 
 
 8 
 
 619 
 
 13 
 
 618 
 
 4 
 
 620 , 
 
 9 
 
 619 
 
 14 
 
 618 
 
 5 
 
 620 
 
 10 
 
 619 
 
 15 
 
 617 
 
OEIFICES, WEIRS, PIPES, AND EIVERS. 
 
 69 
 
 approaches the surface, from *617 to *621 ; and at 
 lesser depths than L foot other experiments show an 
 increase in the coefficient np to -650. The experi- 
 ments of Poncelet and Lesbros show, however, a 
 reduction in the coefficients for square orifices 8"x 8" 
 as they approach the surface from -601 to *572. 
 
 Brindley and Smeaton's experiments, with an orifice 
 1 inch square placed at different depths, give a mean 
 
 COEFFICIENTS CALCULATED FEOM BRINDLEY AND SMEATON'S EXPERIMENTS. 
 
 tean -637. 
 
 value for c d of -637. The last experiment, with an 
 orifice only \ inch by \ inch, gives so small a coeffi- 
 cient as *557 placed at a depth of 6 feet ! 
 
 For notches 6 inches wide and from 1 to 6J inches 
 deep, Brindley and Smeaton's experiments give the 
 mean value of c d 637. The coefficients of discharge 
 
 COEFFICIENTS FOR NOTCHES, CALCULATED FROM BRINDLEY AND SMEATON'S 
 EXPERIMENTS. 
 
 1 foot head : 
 2 feet head : 
 3 feet head : 
 4 feet head : 
 5 feet head : 
 6 feet head : 
 
 orifice 1" x 1" 
 orifice 1" x 1* 
 orifice 1" x 1" 
 orifice 1" x 1" 
 orifice 1" x 1" 
 orifice J" x J" 
 
 coefficient -639 
 coefficient *635 
 coefficient -648 
 coefficient -632 
 coefficient -632' 
 coefficient -557 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 92 to 1 
 
 6 X6 
 
 633 
 
 3-7 to 1 
 
 6X If 
 
 638 
 
 1-07 to 1 
 
 6 X 5| 
 
 571 
 
 4-4 to 1 
 
 6X1 
 
 654 
 
 1-2 to 1 
 
 6X5 
 
 609 
 
 4-8 to 1 
 
 exit 
 
 681 
 
 1-92 to 1 
 
 6 X 3* 
 
 602 
 
 6- tol 
 
 6X1 
 
 713 
 
 2-4 tol 
 
 6 X 31* 
 
 636 
 
 Mean value. '637 
 
 * The depth is misprinted 2^ inches in the Encyclopaedias, 
 the resulting coefficient for which would be *568 instead of '636 
 as above, for a depth of 2~ inches. 
 
70 
 
 THE DISCHAEGE OF WATEK FROM 
 
 for notches and orifices appear to differ as little from 
 each other as those for either do in themselves. The 
 results also show a general though not uniform in- 
 crease in the coefficients for smaller depths. 
 
 Du Buat's experiments with notches 18 '4 inches 
 long, give the mean value of c d =z-632, which differs 
 very little from the mean value determined from 
 Brindley and Smeaton's experiments. 
 
 COEFFICIENTS FOB NOTCHES, CALCULATED FKOM DU BUAT's EXPERIMENTS. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 2-72 to 1 
 3-94 to 1 
 
 18-4 X 6-753 
 18-4 X 4-665 
 
 630 
 627 
 
 5-75 to 1 
 10-3 to 1 
 
 18-4 X 3.199 
 18-4 X 1'778 
 
 624 
 
 648 
 
 Poncelet and Lesbros' experiments give the coeffi- 
 cients in the following table, for notches 8 inches 
 
 COEFFICIENTS FOR NOTCHES, BY PONCELET AND LESBROS. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 Ratio of the 
 length 
 to the depth. 
 
 Size of 
 notches 
 in inches. 
 
 Coefficients. 
 
 9tol 
 
 8X9 
 
 577 
 
 3-33 to 1 
 
 8 X2-4 
 
 601 
 
 1 tol 
 
 8X8 
 
 585 
 
 5 tol 
 
 8X1-6 
 
 611 
 
 1-3 to 1 
 
 8X6 
 
 590 
 
 6-7 tol 
 
 8X1-2 
 
 618 
 
 2 tol 
 
 8X4 
 
 592 
 
 10 tol 
 
 8 X 0-8 
 
 625 
 
 2-5 tol 
 
 8 X 3-2 
 
 595 
 
 20 tol 
 
 8X0-4 
 
 636 
 
 wide; the mean value of all the coefficients in these 
 experiments is *603. Here the coefficients increase 
 in every instance as the depths decrease, or as the 
 ratio of the length of the notch to its depth increases. 
 We shall have to refer to the valuable experiments 
 made at Metz, on the discharge from differently-pro- 
 portioned orifices immediately. 
 
 Rennie's experiments for circular orifices at depths 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 COEFFICIENTS FOR CIRCULAR ORIFICES, FROM RENNIE's EXPERIMEN 
 
 Heads at the 
 centre of each 
 orifice in feet. 
 
 J inch 
 diameter. 
 
 inch 
 diameter. 
 
 finch 
 diaiheter. 
 
 linch 
 diameter. 
 
 Mean 
 values. 
 
 1 
 
 671 
 
 634 
 
 644 
 
 633 
 
 645 
 
 2 
 
 653 
 
 621 
 
 652 
 
 619 
 
 636 
 
 3 
 
 660 
 
 636 
 
 632 
 
 628 
 
 639 
 
 4 
 
 662 
 
 626 
 
 614 
 
 584 
 
 621 
 
 Means 
 
 661 
 
 629 
 
 635 
 
 616 
 
 635 
 
 from 1 foot to 4 feet, and of diameters from i inch 
 to 1 inch, give the following coefficients. Here the 
 increase in the coefficients for lesser orifices and at 
 lesser depths exhibits itself very clearly, notwithstand- 
 ing a few instances to the contrary. The mean value 
 of the coefficient c d derived from the whole, is -635. 
 For small rectilineal orifices the coefficients were as 
 follows : 
 
 COEFFICIENTS FOR RECTANGULAR ORIFICES, FROM RENNIE's EXPERIMENTS. 
 
 Heads at the 
 centre of gravity, 
 in feet. 
 
 Square orifice, 
 1 inch X 1 inch. 
 
 Rectangular 
 orifice, longer side 
 horizontal, 
 2" X 4". 
 
 Rectangular 
 orifice, longer side 
 horizontal, 
 li" X |". 
 
 a 1! 
 iflj 
 
 
 
 Equilateral 
 triangle 
 of 1 square inch, 
 with base up. 
 
 1 
 
 617 
 
 617 
 
 663 
 
 }> 
 
 596 
 
 2 
 
 635 
 
 635 
 
 668 
 
 M 
 
 577 
 
 3 
 
 606 
 
 606 
 
 606 
 
 
 
 572 
 
 4 
 
 593 
 
 593 
 
 593 
 
 593 
 
 593 
 
 Means 
 
 613 
 
 613 
 
 632 
 
 593 
 
 585 
 
 The most valuable series of experiments of which 
 we are possessed are those made at Metz, by Poncelet 
 and Lesbros. They were made with orifices 8 
 inches wide, nearly, and of different vertical dimen- 
 sions placed at various depths down to 10 feet. The 
 discrepancies as to any general law in the relation of 
 the different values of the coefficient of discharge c d 
 
72 THE DISCHARGE OF WATER FROM 
 
 to the size and depth of the orifice in the preceding 
 experiments, have been remedied 'to a great extent by 
 these. They give an increase of the coefficients for 
 the smaller and very oblong orifices as they approach 
 the surface, and a decrease under the same circum- 
 stances in those for the larger square and oblong 
 orifices. There are a few depths where maximum 
 and minimum values are obtained : we use the terms 
 "maximum and minimum values" for those which 
 are greater in the one case and less in the other than 
 the coefficients immediately before and after them, 
 and not as being numerically the greatest or least 
 values in the column. We have marked with a *, in 
 the arrangement of the coefficients, TABLE I,, these 
 maximum and minimum values. The heads given in 
 this table were measured to the upper side of the 
 orifices, and by adding half the depth (d) to any 
 particular head, we shall obtain the head at the 
 centre. 
 
 As a perceptible sinking of the surface takes 
 place in heads less than from five to three times the 
 depth of the orifice, the coefficients are arranged in 
 pairs, the first column containing the coefficients for 
 heads measured from the still water surface some 
 distance back from the orifice, and the second ob- 
 tained when the lesser heads, measured directly at 
 the orifice, were used. A very considerable increase 
 in the value of the coefficients for very oblong and 
 shallow small orifices, may be perceived as they ap- 
 proach the surface, and the mean value for all 
 rectilinear orifices at considerable depths, seems to 
 approach to *605 or *606. 
 
OEIFICES, WEIKS, PIPES, AND RIVERS. 73 
 
 We have shown, equation (29), that the discharge is 
 
 approximately, in which expression d is the depth of 
 the orifice, and h the head at its centre. Now it is 
 to be observed, that it is not the value of c d simply, 
 which is given in TABLE I., but the value of c d x 
 
 2 g h, equation (29). 
 
 The coefficients in the table are, therefore, less than 
 the coefficients of discharge, strictly so called, by a 
 
 c d? 
 quantity equal to Q ^ , g . The value of this expression 
 
 is in general very small, and it is easy to perceive 
 from the first of the expressions in equation (31), p. 62, 
 that it can never exceed 4-2 per cent., or more correctly 
 0417 in unity. If we wish to know the discharge 
 from an orifice 4 inches square = 4" x 4", with its 
 centre 4 feet below the surface, which is equivalent 
 to a head of 3 feet 10 inches at the upper side, we 
 
 find from the table the value of c d jl ,A = 
 601 ; hence we shall get 
 
 D = -601 x A v/2# h = -601 x - x 8'025 x 2 = 
 601 x i x 16-05 = - x 9-646 = 1-072 
 
 9 9 
 
 cubic feet per second. In the absence of any expe- 
 riments with larger orifices, we must, when they 
 occur, use the coefficients given in this table ; and, 
 in order to do so with judgment, it is only necessary 
 to observe the relations of the sides and heads. For 
 example, if the size of an orifice be 16"x4", we must 
 
74 
 
 THE DISCHARGE OF WATER FROM 
 
 seek for the coefficient in that column where the ratio 
 of sides is as four to one, and if the head at the 
 upper side be five times the length of the orifice, we 
 shall find the coefficient *626, which in this case is 
 the same for depths measured behind, or at the 
 orifice. For lesser orifices, the results obtained 
 from the experiments of Michelotti and Bossut, 
 pages 67 and 68, are most applicable; and also 
 the coefficients of Rennie, page 71. It is almost 
 needless to observe, that all these coefficients are 
 only applicable to orifices in thin plates, or those 
 
 having the outside ar- 
 rises chamfered as 
 in Fig. 8. Very lit- 
 tle dependence can be 
 placed on calculations 
 of the quantities of 
 water discharged from 
 other orifices, unless where the coefficients have been 
 already obtained by experiment or correct inference 
 for them. If the inner arris next the water be 
 rounded, the coefficient will be increased. 
 
 NOTCHES AND WEIRS. 
 
 We have already given some coefficients, pages 69 
 and 70, derived from the experiments of Du Buat, 
 Brindley and Smeaton, and Poncelet and Lesbros, for 
 finding the discharge over notches in the sides of 
 large vessels ; and it does not appear that there is 
 any difference of importance between these and those 
 for orifices sunk some depth below the surface, when 
 the proper formula for finding the discharge for each 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 75 
 
 is used. If we compare Poncelet and Lesbros' co- 
 efficients for notches, page 70, with those for an 
 orifice at the surface, TABLE L, we perceive little 
 practical difference in the results, the head being 
 measured back from the orifice, unless in the very 
 shallow depths, and where the ratio of the length to 
 the depth exceeds five to one. The depths being in 
 these examples less than an inch, it is probable that 
 the larger coefficients found for the orifice at the 
 surface, arise from the upper edge attracting the 
 fluid to it and lessening the effects of vertical con- 
 traction, as well as from less lateral contraction. In- 
 deed, the results obtained from experiments with 
 very shallow weirs, or notches, have not been at 
 all uniform, and at small depths the discharge must 
 proportionably be more affected by movements of 
 the air and external circumstances than when the 
 depths are considerable. We shall see that in Mr. 
 Blackwell's experiments the coefficient obtained for 
 depths of 1 and 2 inches was -676 for a thin plate 3 
 feet long, while for a thin plate 10 feet long it in- 
 creased up to -805. 
 
 The experiments of Castel, with weirs up to about 
 30 inches long, and with variable heads of from 1 to 
 8 inches, lead to the coefficient -497 for notches ex- 
 tending over one-fourth of the side of a reservoir ; 
 and to the coefficient -664 when they extend for the 
 whole width. For lesser widths than one-fourth, the 
 coefficients decrease down to -584 ; and for those 
 extending between one-third of, and the whole width, 
 they increase from -600 to -665 and -680. Bidone 
 finds c d = -620, and Eytelwein c d = -635. It will be 
 
76 THE DISCHARGE OF WATER FROM 
 
 perceived from these and the foregoing results, that 
 the third place of decimals in the value of c d , and 
 even sometimes the second, is very uncertain ; that 
 the coefficient varies with the head and ratio of the 
 notch to the side in which it is placed ; and we shall 
 soon show that the form and size of the weir, weir- 
 basin, and approaches, still further modify its value. 
 
 When the sides and edge of a notch increase in 
 thickness, or are extended into a shoot, the coeffi- 
 cients are found to reduce very considerably ; and 
 for small heads, to an extent beyond what the in- 
 crease of resistance, from friction alone, indicates. 
 Poncelet and Lesbros found, for orifices, that the 
 addition of a horizontal shoot, 21 inches long, re- 
 duced the coefficient from *604 to '601, with a head 
 of 4 feet ; but for a head of only 4J inches, the 
 coefficient fell from -572 to -483, the orifice being 
 8" x 8". For notches 8 inches wide, with a hori- 
 zontal shoot 9 feet 10 inches long, the coefficient fell 
 from -582 to -479, for a head of 8 inches ; and from 
 622 to -340, for a head of only 1 inch. Castel found 
 also, for a notch 8 inches wide with a shoot 8 inches 
 long attached and inclined at an angle 4 18', that the 
 mean coefficient for heads from 2 to 4J inches was 
 only -527. Little dependence can be placed on ex- 
 perimental results obtained for shoots which partake 
 of the nature of short pipes, and should be treated 
 in like manner to find the discharge.* 
 
 We have obtained the following table of coefficients 
 from some experiments made by Mr. Ballard, on the 
 river Severn, near Worcester, " with a weir 2 feet 
 
 * Trait6 Hydraulique, par D'Aubuisson, pp. 46, 94 et 95. 
 
OKIFICES, WEIRS, PIPES, AND RIVERS. 
 
 77 
 
 COEFFICIENTS FOR SHORT WEIRS OVER BOARDS. 
 
 Heads measured on the crest. 
 
 Depths 
 in inches. 
 
 Coefficients. 
 
 Depths 
 in inches. 
 
 Coefficients. 
 
 Depths 
 in inches. 
 
 Coefficients. 
 
 1 
 
 762 
 
 3 
 
 801 
 
 5 
 
 733 
 
 i| 
 
 662 
 
 8J 
 
 765 
 
 H 
 
 713 
 
 ii 
 
 673 
 
 3* 
 
 748 
 
 i 
 
 735 
 
 u 
 
 692 
 
 31 
 
 740 
 
 5| 
 
 729 
 
 2 
 
 684 
 
 4 
 
 759 
 
 6 
 
 727 
 
 | 
 
 702 
 
 4* 
 
 731 
 
 7 
 
 716 
 
 ^ 
 
 756 
 
 4i 
 
 744 
 
 8 
 
 726 
 
 8* 
 
 786 
 
 4f 
 
 745 
 
 Mean 
 
 732 
 
 long, formed by a board standing perpendicularly 
 across a trough."* The heads or depths were here 
 measured on the weir, and hence the coefficients are 
 larger than those found from heads measured back 
 to the surface of still water. 
 
 Experiments made at Chew-Magna, in Somerset- 
 shire, by Messrs. Blackwell and Simpson, in 1850f, 
 give the following coefficients. 
 
 COEFFICIENTS DERIVED FROM THE EXPERIMENTS OF BLACKWELL AND SIMPSON. 
 
 Heads 
 in inches. 
 
 Coefficients. 
 
 Heads 
 in inches. 
 
 Coefficients. 
 
 Heads 
 in inches. 
 
 Coefficients. 
 
 Ito I 
 
 591 
 
 44 
 
 743 
 
 6 
 
 749 
 
 Ito 11 
 
 626 
 
 41 
 
 760 
 
 6 3 
 
 748 
 
 16 
 
 
 16 
 
 
 16 
 
 
 23 to 21 
 
 16 
 
 682 
 
 4 I 
 
 741 
 
 6fg to 6i 
 
 747 
 
 24 
 
 665 
 
 41 
 
 16 
 
 750 
 
 61 5 
 
 16 
 
 772 
 
 2 32 
 
 670 
 
 
 725 
 
 7 21 
 
 717 
 
 35 
 
 
 
 
 32 
 
 
 21 
 
 665 
 
 5 
 
 780 
 
 8 
 
 802 
 
 g29 
 
 653 
 
 51 
 
 781 
 
 8 to 8 13 
 
 737 
 
 32 
 
 
 16 
 
 
 16 
 
 
 215 
 
 654 
 
 5L 3 
 
 749 
 
 8 16 
 
 750 
 
 16 
 
 
 32 
 
 
 16 
 
 
 
 725 
 
 itoS 
 
 751 
 
 9 
 
 781 
 
 4 
 
 745 
 
 ^ 
 
 728 
 
 Mean 
 
 723 
 
 * Civil Engineer and Architect's Journal for 1851, p. 647. 
 f Civil Engineer and Architect's Journal for 1851, pp. 642 
 and 645. 
 
78 
 
 THE DISCHARGE OF WATER FROM 
 
 " The overfall bar was a cast-iron plate 2 inches 
 thick, with a square top." The length of the over- 
 fall was 10 feet. The heads were measured from 
 still water at the side of the reservoir, and at 
 some distance up in it. The area of the reservoir 
 was 21 statute perches, of an irregular figure, and 
 nearly 4 feet deep on an average. It was supplied 
 from an upper reservoir, by a pipe 2 feet in diameter 
 and of 19 feet fall ; the distance between the supply 
 and the weir was about 100 feet. The width of the 
 
 reservoir as it approached the overfall was about 50 
 feet, and the plan and section, Fig. 9, of the weir 
 and overfall in connection with it, will give a fair 
 idea of the circumstances attending the experiments. 
 For heads over 5 inches the velocity of approach to 
 the weir was "perceptible to the eye," though its 
 amount was not determined. We perceive that the 
 coefficient (derived from two experiments) for a depth 
 of 8 inches is *802, while the coefficient (derived 
 from three experiments) for a depth of 7J inches is 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 79 
 
 717, and for depths from 8 to 8H inches the mean 
 coefficient is -743 : as all the attendant circumstances 
 appear the same, these discrepancies and others must 
 arise from the circumstances of the case : perhaps 
 the supply, and, consequently, the velocity of ap- 
 proach, was increased while making one set of ex- 
 periments, without affecting the still water near the 
 side where the heads appear to have been taken. 
 By comparing the results with those obtained by one 
 of the same experimenters, Mr. Black well, on the 
 Kennet and Avon Canal, we shall immediately per- 
 ceive that the velocity of approach, and every 
 circumstance which tends to alter and modify it, 
 has a very important effect on the amount of the 
 discharge, and, consequently, on the coefficient. 
 
 The experiments made by Mr. Blackwell, on the 
 Kennet and Avon Canal, in 1850*, afford very valuable 
 instruction, as the form and width of the crest were 
 varied, and brought to agree more closely with actual 
 weirs in rivers than the thin plates or boards of 
 earlier experimenters. We have calculated and ar- 
 ranged the coefficients in the following table from 
 these experiments. The variations in the values for 
 different widths of crest, other circumstances being 
 the same, are very considerable ; and the differences 
 in the coefficients, at depths of 5 inches and under, 
 for thin plates and crests 2 inches wide, are greater 
 than mere friction can account for ; and greater also 
 than the differences at the same depths between the 
 coefficients for crests 2 inches thick, and 3 feet long. 
 
 * Civil Engineer and Architect's Journal, 1851, p. 642. 
 
80 
 
 THE DISCHARGE OF WATER FROM 
 
 When more than one experiment was made with the same head, and the results were pretty uniform, the resulting coefficients 
 are marked with a *. The effect of the converging wing-boards is very strongly marked. 
 NOTE. Francis' experiments give a coefficient of '565 for a level crest 3 feet wide, and a head slope of 3| to 1, see p. 121. 
 
 Crests 3 feet wide. 
 
 00 
 
 ||5 
 
 t- O . O . CO b- 
 CO OS .rH .'I* O 1 ', '. * 
 
 in 
 
 i-H OS 00 CO 00 CO O 
 00 fr- * rH rH \ CO OO O 
 
 co-^ oo ^i ^ ^p 
 
 { 
 
 !i 
 
 * * # * * 
 
 Gx t?* t^ t>- O O b* 
 
 CS OS m O OS * 00 CO CO 
 
 ^ Tj< O ^ ^ ^ -^ 
 
 
 ll 
 
 l^-COOSO ,rHt .00 
 CO CO CO O . CO CT 1 ^ * OS 
 TtlJOOT^ OO * ^ 
 
 
 fe 
 
 15 
 
 CO <M 
 
 OCOl>rHCO .COrHCJ 
 T^H -rH CO CO ^^ * rH OS OS 
 OOOrflO O^-rH 
 
 
 bo 
 
 Si 
 
 CO 
 
 C7<?C?rHOSCSrHOOOCO 
 OOOrJ<rHtOOOt~l.-~ * * ', 
 
 
 g 
 
 o 
 
 a 
 o 
 
 1 
 1 
 
 X 
 
 II 1 
 
 * O CO rH 
 
 
 It 
 
 o o 
 
 
 llliillllil 
 
 
 Planks 2 inches tl 
 
 I! 
 
 *-***** # 
 
 OSr-it^OrHOOOOOOSOSOOS 
 
 ocoost^ooocscocooi^ 
 
 rjiOOOCOCOCOOiOOOO 
 
 
 II 
 
 t-OSCOOSOOCO?.^ r 
 COOCO"+ODCSrHOO! 1 
 TjlOOOOOCOCOCOCO 
 
 
 Thin plates. 
 
 II 
 
 0,2 
 
 # * * 
 
 
 
 8 g* 
 
 t" O O l>* O5 CO ^ 
 
 
 -IIOAJ989I 9qj Ut 
 
 'saqout ut spB9H 
 
 rHdCO^OCOJ^OOCSOcMT* 
 rH rH rH 
 
OEIFICES, WEIRS, PIPES, AND RIVERS. 
 
 81 
 
 The plan and section, Fig. 10, will give a fair idea 
 of the approach to, and nature of the overfall made 
 
 The dotted lines on 
 Plan show the sub- 
 merged masonry ap- 
 pearing at C in Section. 
 
 use of in these experiments. The area of the 
 reservoir was 2 A. In. 3 OP., and the head was mea- 
 sured from the surface of the still water in it, which 
 remained unchanged between the beginning and end 
 of each experiment. The width of the approach A B 
 from the reservoir was about 32 feet ; the width at 
 a b about 13 feet, below which the waterway widened 
 suddenly, and again narrowed to the length of the 
 overfall. The depth in front of the dam appears to 
 have been about 3 feet ; the depth on the dam, next 
 the overfall, about 2 feet ; and the depth on the sunk 
 masonry in the channel of approach, about 18 inches. 
 Altogether, the circumstances were such as to in- 
 crease the amount of resistances between the reservoir, 
 from which the head was measured, and the overfall, 
 particularly for the larger heads, and we accordingly 
 
82 THE DISCHARGE OF WATER FROM 
 
 see that the coefficients become less for heads over 
 six inches, with a few exceptions. The measure- 
 ments of the quantities discharged appear to have 
 been made very accurately, yet the discharges per 
 second, with the same head and same length of over- 
 fall, sometimes vary ; for instance, with the plank 
 2 inches thick and 10 feet long, the discharge per 
 second for 4 inches head varies from 6-098 cubic 
 feet to 6-491 cubic feet, or by about one-sixteenth of 
 the whole quantity. Most of the results, however, 
 are means from several experiments. The quantities 
 discharged varied from one-tenth of a cubic foot to 
 22 cubic feet per second, and the duration of the 
 experiments from 24 to 420 seconds. If we compare 
 the coefficients for a plank 10 feet long and 2 inches 
 thick in the foregoing table with those for the same 
 overfall at Chew-Magna, we shall immediately per- 
 ceive how much the form of the approaches affects 
 the discharge. Indeed, were the area of the reser- 
 voir at Chew-Magna even larger than that for the 
 Kennet and Avon experiments, it would be found, 
 notwithstanding, that the coefficients in the former 
 would still continue the larger, though not fully 
 as large as those found under the particular cir- 
 cumstances.* 
 
 * There is a very important omission in all the preceding 
 experiments on weirs and notches. In Fig. 10, for instance, it 
 would have been necessary to obtain the heads at A B and a b in 
 each experiment, above the crest, and also the head on and a few 
 feet above the crest itself. These are, perhaps, best calculated by 
 means of the observed velocity of approach. They would indicate 
 the resistances at the different passages of approach, and enable us 
 to calculate the coefficients correctly, and thereby render them 
 more generally applicable to practical purposes. The coefficients 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 83 
 
 The following table gives the mean results of 88 
 experiments made by Francis, at the Lower Lock, 
 Lowell, Massachusetts, in 1852. The duration of 
 each of these experiments varied from 180 to 822 
 seconds. The coefficients in column 10 have been 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 1 
 
 it 
 
 il 
 
 >* 
 gi 
 
 fr* 
 
 11 
 
 iff 
 
 .s| 
 
 ^ 03 
 
 multiplier 
 ormula in 
 
 i 
 
 PH 
 
 . -4j 
 
 2 a 
 
 
 
 o 
 
 o -*^* ^ 
 
 r< 
 
 ** T 1 S 
 
 Q) tM 
 
 .2 p< 
 
 Si 
 
 \ 
 
 Observed n 
 over weir i 
 
 Observed di 
 cubical fe 
 cond. 
 
 Observed ve! 
 proach in 
 second. 
 
 5-^ A 
 
 Values of 
 formula it 
 
 | + o 
 
 Values of th 
 c in the 
 columnS. 
 
 Correspondi 
 the coefflc 
 charrge c< 
 
 1 
 
 9-997 
 
 1-55 
 
 62-6 
 
 78 
 
 1-56 
 
 1-56 
 
 62-6 
 
 3-32 
 
 621 
 
 2 
 
 9-997 
 
 1-24 
 
 45-6 
 
 59 
 
 1-25 
 
 1-25 
 
 45-4 
 
 3-33 
 
 623 
 
 3 
 
 9-997 
 
 1-00 
 
 33-4 
 
 44 
 
 1-00 
 
 1-00 
 
 32-5 
 
 8-32 
 
 621 
 
 4 
 
 7-997 
 
 1-01 
 
 26-8 
 
 36 
 
 1-02 
 
 1-02 
 
 26-3 
 
 3-36 
 
 628 
 
 5 
 
 9-997 
 
 1-05 
 
 36- 
 
 97 
 
 1-06 
 
 1-06 
 
 35-8 
 
 3-35 
 
 626 
 
 6 
 
 9-995 
 
 0-98 
 
 32-6 
 
 54 
 
 0-99 
 
 .98 
 
 32-4 
 
 3-34 
 
 624 
 
 7 
 
 9-995 
 
 1-00 
 
 33-5 
 
 55 
 
 1-01 
 
 1-00 
 
 33-3 
 
 3-33 
 
 623 
 
 8 
 
 9-997 
 
 0-80 
 
 23-5 
 
 33 
 
 80 
 
 80 
 
 23-4 
 
 3-32 
 
 621 
 
 9 
 
 9-997 
 
 0-82 
 
 25- 
 
 75 
 
 83 
 
 83 
 
 24-8 
 
 3-34 
 
 624 
 
 10 
 
 9-995 
 
 0-80 
 
 23-9 
 
 40 
 
 80 
 
 80 
 
 23-8 
 
 3-34 
 
 624 
 
 11 
 
 9-997 
 
 0-62 
 
 16-2 
 
 23 
 
 62 
 
 62 
 
 16-0 
 
 3-33 
 
 623 
 
 12 
 
 9-997 
 
 0-65 
 
 17-5 
 
 53 
 
 65 
 
 65 
 
 17-2 
 
 3-33 
 
 623 
 
 13 
 
 7-997 
 
 0-68 
 
 14-6 
 
 45 
 
 68 
 
 68 
 
 14-5 
 
 3-34 
 
 623 
 
 calculated by ourselves, and the other results con- 
 densed from the large table given in Francis' Book.* 
 
 in the two previous tables are not as valuable as they otherwise 
 would be from this omission. The level of still water near the 
 banks is below that of the moving water in the current, therefore, 
 heads measured from still water must give larger coefficients than 
 if taken from the centre of the current. This may account, to 
 some extent, for the larger coefficients in the first table, but apart 
 from this, the short contracted channel immediately above the 
 water-fall, Fig. 9, must increase the coefficients. 
 
 .* Lowell Hydraulic Experiments. New York, 1855. 
 
 G3 
 
84 THE DISCHAEGE OF WATEE FEOM 
 
 The heads given in the 6th column are those which 
 would give the observed discharge from the formula 
 
 o 
 
 As we have also equation (39) 
 
 we must, therefore, have 
 
 the values of which are given in column 6. The 
 values of h" in column 8 are those which would be 
 found by resolving the equation 
 
 D=O(/+ -Ink")*!* 
 
 n being the number of end contractions, and c a 
 multiplier varying from 3*32 to 3-36. 
 
 2 
 
 In this table the theoretical head ~ zz -0155 vjdue to 
 
 *& 
 the velocity of approach has been used and does not 
 
 exceed -02 of a foot. We are of opinion, however, 
 that the head is much greater, and should be taken 
 
 v 2 
 
 "2 ^o~~ = '04 vl or thereabouts. This would reduce 
 c d x &g 
 
 the values of the coefficient of discharge c d in the 10th 
 column. The differences between A, ti, and h" in 
 columns 3, 6, and 7 are here, practically, of little 
 moment, and the value of c d in column 10 would be 
 nearly the same derived from either. The crest of 
 the weir experimented upon was 1 inch thick. The 
 weir measuring 10 feet x 13 inches x 1 inch, the top 
 was rounded off at both arrises, leaving the central 
 horizontal portion one quarter of an inch wide. 
 The general result of these experiments verifies the 
 
OEIFICES, WEIKS, PIPES, AND KIVEES. 
 
 85 
 
 ordinary coefficient for notches in thin plates from 
 617 to -628 for the value of c d . 
 
 Professor Thomson's experiments with right- 
 angled triangular notches, in thin plates, give a mean 
 coefficient of -617. Vide Note p. 55. 
 
 HEAD, AND FROM WHENCE MEASURED. 
 
 By referring to TABLE I., we shall see that there 
 is a difference in the coefficients as obtained from 
 heads measured on or above the orifice. This dif- 
 ference is greater in notches, or weirs, than in orifices 
 sunk below the surface ; and when the crest of a 
 weir is of some width, the depths upon it vary. In 
 the Kennet and Avon experiments, the heads mea- 
 sured from the surface of the water in the reservoir, 
 and the depths at the "outer edge" (by which we 
 understand the lower edge) of the crest were as 
 follows : 
 
 DIFFEEENCE OF HEADS MEASUEED ON AND ABOVE WEIES. 
 
 I 
 
 Heads on crests 
 
 Heads on 
 
 II 
 
 2 inches thick. 
 
 crests 3 feet wide. 
 
 a -9 
 
 I 
 
 I 
 
 ao'3 
 
 l! 
 
 Jfc 
 
 !fe 
 
 |1 
 
 fi 
 
 bJDOJ 
 
 V 
 
 1 
 
 eo 
 
 1 
 D 
 
 11 
 
 w 
 
 11- 
 
 03 W 
 
 ll 
 
 || 
 
 "S-2.9 
 
 1 
 
 1 
 
 .. 
 
 7 
 16 
 
 .. 
 
 i 
 
 4 
 
 r 
 
 16 
 
 5 
 
 16 
 
 2 
 
 .. 
 
 
 i 
 
 
 
 
 
 
 
 1 
 
 16 
 
 3 
 
 . . 
 
 ll? 
 
 
 
 
 U to 1J 
 
 
 
 4 
 
 3to2g 
 
 3J 
 
 U 
 
 If 
 
 u 
 
 .. 
 
 .. 
 
 li 
 
 5 
 
 3* 
 
 3 r 
 
 2 i 
 
 14 
 
 .. 
 
 
 14 
 
 
 6 
 
 41 
 
 4 
 
 2| 
 
 .. 
 
 'i j'*:;; 
 
 24 
 
 2| 
 
 2i 
 
 7 
 
 .. 
 
 
 
 ft 
 
 2 
 
 
 
 .. 
 
 8 
 
 61 
 
 
 
 
 'J ;>". 
 
 ; : r 
 
 31 
 
 34 
 
 9 
 
 grV 
 
 .. 
 
 .. 
 
 ....... 
 
 .. 
 
 4i 
 
 N 
 
 .. 
 
 10 
 
 
 
 
 
 
 
 4 
 
 
 
 
 
 
 
 
 
I 
 86 THE DISCHARGE OF WATER FROM 
 
 No intermediate heads are given, but those registered 
 point out very clearly the great differences which 
 often exist between the heads measured on a weir, or 
 notch, and those measured from the still water above 
 it ; and how the form of the weir itself, as well as 
 the nature of the approaches, alters the depth pass- 
 ing over. On a crest 2 feet wide, with 14| inches 
 depth on the upper edge, we have found that the depth 
 on the lower edge is reduced to 11J inches, or as 1*26 to 
 1. The head taken from 3 to 20 feet above the crest, 
 where the plane of the approaching water surface 
 becomes curved, is that in general which is best 
 suited for finding the discharge by means of the 
 common coefficients, but a correct section of the 
 channel and water-line, showing the different depths 
 upon and for some distance above the crest, is neces- 
 sary in all experiments for determining accurately 
 by calculation the value of the coefficient of dis- 
 charge c d . 
 
 Du Buat, finding the theoretical expression for the 
 discharge through an orifice of half the depth h, 
 
 equation (6) 
 
 to agree pretty closely with his experiments, seems 
 to have assumed that the head h is reduced to ^ in 
 
 passing over. This is a reduction, however, which 
 never takes place unless with a wide crest and at its 
 lower edge, or where the head h is measured at a 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 87 
 
 considerable distance above the weir, and when a 
 loss of head due to the distance and obstructions in 
 channel takes place. When there is a clear weir 
 basin immediately above the weir, we have found 
 that, putting h for the head measured from the sur- 
 face in the weir basin, and h w for the depth on the 
 upper edge of the weir, that 
 (32.) h A w 
 
 for measures in feet, and 
 (33.) A A w = - 
 
 for measures in inches. The comparative values of 
 h and h w depend, however, a good deal on the par- 
 ticular circumstances of the case. Dr. Robinson 
 found* Am 1-111 A w , when h was about 5 inches. 
 The expressions we have given are founded on the 
 hypothesis, that h A w is as the velocity of discharge, 
 or as the \/A nearly. For small depths, there is a 
 practical difficulty in measuring with sufficient accu- 
 racy the relative values of h and A w . Unless for 
 very small heads the sinking will be found in general 
 
 to vary from to -, and in practice it will always 
 
 be useful to observe the depths on the weir as well 
 as the heads for some distances (and particularly 
 where the widths contract) above it. 
 
 In order to convey to our readers a more definite 
 idea of the differences between the coefficients for 
 heads measured at the weir, or notch, and at some 
 distance above it, we shall assume the difference of 
 
 A w A w .,1 A A w 
 
 the heads A A w zz ; then , , =r, and-zz r ? 
 
 * Proceedings of the Eoyal Irish Academy, vol. iv. p. 212. 
 
00 THE DISCHARGE OF WATER FROM 
 
 hence h = r *~ h w and A ff = ^_ k. 
 r r + J- 
 
 Now the discharge may be considered as that 
 which would take place through an orifice whose 
 depth is h w with a head over the upper edge equal to 
 
 A ^ w zz-^; hence from equation (6) the discharge 
 is equal to 
 
 and substituting for h% its value ( r ' A W V, we shall 
 
 \ r ' 
 
 find the value of 
 
 (34.) D = 
 
 As the value of the discharge would be expressed by 
 
 2 
 
 X 
 
 s 
 
 if the head h h^ were neglected, it is evident the 
 coefficient is increased, under the circumstances, 
 from c d to 
 
 or, more correctly, the common formula has to be 
 multiplied by (l + i)* (!)*, to find the true dis- 
 charge, and the value of this expression for different 
 values of - = n w m be found in TABLE IV. If we 
 
 suppose that 
 
 * z. ^w ., 1 1 
 h-hv = JQ, then - =j-g = n ; 
 
 and we find from the table(l + i) l ~( -)* = 1-1221. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 89 
 
 Now if we take the value of c d for the full head h to 
 be -628, we shall find 1-1221 x '628 =-705, rejecting 
 the latter figures, for the coefficient when the head is 
 
 1 2 
 
 measured at the orifice ; and if - = JQ n, we should 
 
 find in the same manner the new coefficient to be 
 1-2251 x '628 = -769 nearly. The increase of the 
 coefficients determined, page 77, from Mr. Ballard's 
 experiments is, therefore, evident from principle, as 
 the heads were taken at the notch ; and it is also 
 pretty clear that, in order to determine the true dis- 
 charge^ the heads both on, at, and above a weir should 
 be taken. Most of the discrepancies in the coefficients 
 determined from experiment have arisen from imper- 
 fect and limited observations of the facts. Amongst 
 these the velocity of approach should never be 
 neglected by observers, as its effect on the discharge 
 is often considerable in increasing the quantity. The 
 effect of the form of the weir and approaches is 
 scarcely ever sufficiently considered by professional 
 men. Most of the discussions which arose with 
 reference to the gaugings on the Metropolitan MAIN 
 DRAINAGE QUESTION would have been obviated if the 
 calculators, or engineers, had taken into account the 
 different circumstances attendant on it, instead of 
 applying generally a formula suited to a particular 
 case, namely, a thin crest, a small notch, and a large 
 body of water immediately above it ; and applied a 
 correct formula for finding the effects of the velocity 
 of approach. 
 
 The two following tables have been reduced to 
 English measures of feet, from Boileau's experiments ; 
 
90 
 
 THE DISCHARGE OF WATER FROM 
 
 they show the relation of the head to the depth on 
 the crest at the upper arris. The coefficient for the 
 head h being known, we may, from our equation (34), 
 calculate that due to h & on the weir. 
 
 TABLE showing the ratio of the head, h, to the depth., h^, on a Plank 
 Weir of the full width of the Channel, immediately at the upper 
 
 edge, or j, see equation (33), when the sheet of water is free 
 after passing over, with air under it. 
 
 Head h in 
 feet. 
 
 Values of the head h divided by the thickness of the sheet of water 
 passing over the weir immediately at the upper edge ; average r- = F 
 = 1-2 between heads of 3 and 14 inches. 
 
 Height of weir 
 in feet, 
 8&. 
 
 Height of weir 
 in feet, 
 1-07'. 
 
 Height, of weir 
 in feet, 
 1-33'. 
 
 Height of weir 
 in feet, 
 Ml*. 
 
 1 
 
 1-339 
 
 
 
 1-285 
 
 13 
 
 1-282 
 
 
 1-320 
 
 1-250 
 
 16 
 
 1-260 
 
 
 1-285 
 
 1-228 
 
 20 
 
 1-234 
 
 1-243 
 
 1-249 
 
 1-214 
 
 23 
 
 1-223 
 
 1-232 
 
 1-231 
 
 1-205 
 
 26 
 
 1-216 
 
 1-232 
 
 1-223 
 
 1-200 
 
 3 
 
 1-212 
 
 1-228 
 
 1-218 
 
 1-199 
 
 33 
 
 1-210 
 
 1-225 
 
 1-217 
 
 1-199 
 
 39 
 
 1-206 
 
 1-221 
 
 1-112 
 
 1-197 
 
 46 
 
 1-202 
 
 1-216 
 
 1-206 
 
 
 53 
 
 1-199 
 
 
 1-201 
 
 . . 
 
 59 
 
 1-196 
 
 
 1-195 
 
 .. 
 
 66 
 
 1-192 
 
 
 1-191 
 
 
 82 
 
 1-186 
 
 
 .. 
 
 
 99 
 
 1-184 
 
 . . 
 
 *''"*. '" 
 
 
 1-15 
 
 1-182 
 
 
 
 *W' C 
 
 
 
 If we were to use the head h w instead of h, to cal- 
 culate the discharge, when j- zi 1*2, then a coefficient 
 
 /i w 
 
 of -628 for the head h would become -769 for the 
 head A w in equation (34) : for - = '2, and, therefore, 
 TABLElV.,-628x(l'2)*-(-2)* = -628x1-2251 =-769. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 91 
 
 TABLE showing the ratio j , equation (33), when the sheet of water 
 
 ' l Vf 
 
 passing over is in contact with the crest and with the water im- 
 mediately below a Plank Weir. 
 
 
 h 
 Values of i for different heights of weirs and for different 
 "w 
 
 Head h 
 in feet. 
 
 heads : mean value for heads between 3 and 14 inches, equal 
 1=1-25. 
 
 Height of weir 
 in feet, 
 1-07'. 
 
 Height of weir 
 in feet, 
 1-1'. 
 
 Height of weir 
 in feet, 
 1-38'. 
 
 43 
 
 
 1-283 
 
 
 46 
 
 
 1-275 
 
 1-291 
 
 49 
 
 1-256 
 
 1-266 
 
 1-281 
 
 53 
 
 1-250 
 
 1-258 
 
 1-271 
 
 59 
 
 1-236 
 
 1-245 
 
 1-254 
 
 66 
 
 1-225 
 
 1-232 
 
 1-241 
 
 73 
 
 1-216 
 
 1-223 
 
 
 79 
 
 1-208 
 
 1-216 
 
 
 86 
 
 1-202 
 
 1-208 
 
 
 92 
 
 1-198 
 
 1-203 
 
 
 99 
 
 ,{!;* - 
 
 1-198 
 
 .:. 
 
 If we were to use the head h w instead of h to cal- 
 culate the discharge, when y- = 1-25, then a coeffi- 
 cient of -628 for the head h would become -799 for 
 the head h w in equation (34) : for - = -25 ; and, 
 
 therefore, the value of c d {(l + -)* - (-)*J,TABLE IV., 
 
 is -628 x (1-25)* - (-25)*-= -628 x 1'2725 = -799: 
 and so on we may calculate the value of the coeffi- 
 cient to be applied to the depth h w on the weir, for 
 any other ratios between h and h w by means of 
 equation (34). 
 
 Boileau made some valuable experiments at Metz, 
 
92 THE DISCHARGE OF WATER FROM 
 
 which were published in 1854. They give the fol- 
 lowing results for vertical plank weirs extending from 
 side to side of the channel, when the water passed 
 over without adhering to the crest : 
 
 Height of weir over bot- 
 torn of channel in feet. Head above Mean coefficient. 
 
 3- -2 to 1-6 -645 
 
 1-3 -16 to -5 -622 
 
 6 -15 to -25 -625 
 
 When the water passing over was joined to the crest, 
 and no air between the sheet passing over and the 
 water below the weir, the experiments gave 
 
 Height of weir over bot- -^ , , 
 
 torn of channel in feet. Mean coefficient. 
 
 2- 1- to 1-6 -694 
 
 1-3 -6 to 1-8 -690 
 
 6 -36 to 1-3 -675 
 
 When the plank weir leant up-stream 4 inches to a 
 foot, the mean value of c d was -620, the height of 
 weir being 1-5 foot, and with heads from -23 to *5 
 foot. When its crest was rounded to a semi-cylin- 
 der, the coefficient was, with a head of % 26 foot, '696, 
 and with a head of -52 foot, *843 ; the water adhering 
 to the crest. With a head of -6 foot the coefficient 
 was -867, and with a head of -85 foot, '840, when the 
 water passed over without air between it and the 
 water below the crest. The following tables give the 
 experimental and reduced coefficients for vertical 
 plank weirs of different heights, and with different 
 heads, when the water passes over in a full sheet, and 
 also when it is joined to the crest and lower water. Also 
 for plank weirs suitable for sluices, leaning up-stream 
 with a slope of one-third horizontal to one vertical. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 93 
 
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94 
 
 THE DISCHARGE OF WATER FROM 
 
 COEFFICIENTS of Vertical Plank Weirs at right angles to the Channel, 
 when the edge is chamfered at the lower arris, and when the head 
 passing over is in contact with the water at and below the Weir ; or 
 when the water immediately below the Weir rises to the crest. The 
 maximum coefficient -733 appears to obtain when the height of 
 the Weir is double the depth passing over the crest. 
 
 _g 
 
 Heights of weirs, in feet, over the bottom of the channel, and 
 
 a 
 
 +5 
 
 corresponding values of the coefficient of discharge c d in the 
 
 *;.J 
 
 13 
 
 formula v = c d X f A/2 g h. 
 
 1$ 
 
 w 
 
 66' 
 
 82' 
 
 99' 
 
 1-15' 
 
 1-32' 
 
 1-48' 
 
 1-65' 
 
 1-81' 
 
 1-98' 
 
 S 
 
 30 
 
 727 
 
 
 
 
 
 
 
 
 
 30 
 
 33 
 
 724 
 
 
 
 
 
 
 
 
 
 33 
 
 36 
 
 721 
 
 
 
 
 
 
 
 
 
 36 
 
 39 
 
 718 
 
 
 
 
 
 
 
 
 
 39 
 
 43 
 
 714 
 
 
 
 
 
 
 
 
 
 43 
 
 46 
 
 709 
 
 
 
 
 
 
 
 
 
 46 
 
 49 
 
 702 
 
 708 
 
 715 
 
 724 
 
 .. 
 
 
 
 
 
 49 
 
 53 
 
 694 
 
 699 
 
 708 
 
 718 
 
 . . 
 
 
 
 
 
 53 
 
 56 
 
 687 
 
 693 
 
 700 
 
 712 
 
 729 
 
 
 
 
 
 56 
 
 59 
 
 679 
 
 687 
 
 694 
 
 705 
 
 721 
 
 
 
 
 
 59 
 
 63 
 
 676 
 
 682 
 
 689 
 
 700 
 
 717 
 
 
 
 
 
 63 
 
 66 
 
 672 
 
 678 
 
 684 
 
 696 
 
 714 
 
 
 
 
 
 66 
 
 73 
 
 667 
 
 672 
 
 678 
 
 690 
 
 708 
 
 733 
 
 
 
 
 73 
 
 79 
 
 661 
 
 666 
 
 673 
 
 685 
 
 705 
 
 729 
 
 
 
 
 79 
 
 86 
 
 655 
 
 660 
 
 669 
 
 681 
 
 700 
 
 724 
 
 
 
 
 86 
 
 92 
 
 648 
 
 655 
 
 666 
 
 678 
 
 699 
 
 720 
 
 
 
 
 92 
 
 99 
 
 640 
 
 652 
 
 666 
 
 678 
 
 693 
 
 703 
 
 712 
 
 720 
 
 729 
 
 99 
 
 1-05 
 
 631 
 
 645 
 
 657 
 
 669 
 
 *681 
 
 691 
 
 702 
 
 "11 
 
 720 
 
 1-05 
 
 1-12 
 
 627 
 
 636 
 
 646 
 
 657 
 
 667 
 
 679 
 
 690 
 
 "00 
 
 711 
 
 1-12 
 
 1-19 
 
 625 
 
 636 
 
 646 
 
 657 
 
 666 
 
 675 
 
 685 
 
 694 
 
 703 
 
 1-19 
 
 1-25 
 
 625 
 
 636 
 
 646 
 
 657 
 
 666 
 
 675 
 
 682 
 
 690 
 
 696 
 
 1-25 
 
 1-32 
 
 625 
 
 636 
 
 646 
 
 657 
 
 666 
 
 673 
 
 679 
 
 685 
 
 691 
 
 1-32 
 
 1-39 
 
 
 
 
 
 666 
 
 672 
 
 678 
 
 682 
 
 684 
 
 1-39 
 
 1-45 
 
 
 
 t 
 
 i 
 
 664 
 
 670 
 
 675 
 
 679 
 
 684 
 
 1-45 
 
 1-52 
 
 ( 
 
 
 
 , 
 
 661 
 
 667 
 
 672 
 
 676 
 
 681 
 
 1-52 
 
 1-58 
 
 
 
 . 
 
 . 
 
 658 
 
 663 
 
 669 
 
 672 
 
 675 
 
 1-58 
 
 1-65 
 
 
 
 
 
 
 
 
 655 
 
 658 
 
 663 
 
 666 
 
 667 
 
 1-65 
 
 The following table gives the result of experiments 
 on chamfered plank weirs, for gauging, extending 
 across a channel at right angles to it, when the back- 
 water below was joined to the head- water at passing 
 over, and when there was no air between : 
 
 Height of weir over the bot- ") ' 
 torn of the channel below. . . J 
 
 Heads passing over the weir in } 
 each case, when absorbed 'at >'23 
 the crest into the back-water J 
 
 feet feet feet feet feet feet feet feet 
 66 -83 1-00 1-16 1-32 1-48 1'65 2-00 
 
 31 -38 -45 -51 -59 -66 
 
 92 
 
 which shows that the head was drowned (noyee) when 
 the depth of the lower channel below the crest of the 
 weir was less than 24 times the head passing over, 
 taking a general average. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 95 
 
 TABLE of Experimental Coefficients for Plank Weirs leaning up- 
 stream, when the crest has the down-stream arris rounded to a 
 quadrant; and when the crest is cylindrical and projecting up- 
 stream in the form of a knob. 
 
 Head h in 
 
 Plank weir leaning np-sti earns 
 one-third to oni- ; the lower arris 
 of crest rounded off to a quadrant 
 of a circle with a radius the full 
 thickness of the plank. 
 
 Plank weir leaning upwards one- 
 third to one, the crest rounded and 
 projecting in front beyond the 
 plank, so as to be thicker than it. 
 
 feet. 
 
 Water free from 
 curve of crest 
 13 foot thick. 
 
 Water in contact 
 with curve of 
 crest -17 foot 
 thick. 
 
 Water in contact 
 with curve of 
 crest -3 foot 
 thick. 
 
 Water in contact 
 with curve of 
 crest -33 foot 
 thick. 
 
 16 
 
 589 
 
 651 
 
 
 
 20 
 
 589 
 
 672 
 
 
 
 23 
 
 594 
 
 697 
 
 
 
 26 
 
 612 
 
 697 
 
 
 
 30 
 
 633 
 
 721 
 
 
 670 
 
 33 
 
 642 
 
 747 
 
 604 
 
 686 
 
 36 
 
 649 
 
 766 
 
 625 
 
 700 
 
 39 
 
 655 
 
 768 
 
 648 
 
 714 
 
 43 
 
 661 
 
 795 
 
 669 
 
 727 
 
 46 
 
 667 
 
 802 
 
 687 
 
 741 
 
 49 
 
 675 
 
 
 702 
 
 753 
 
 53 
 
 679 
 
 
 715 
 
 765 
 
 56 
 
 685 
 
 
 729 
 
 775 
 
 59 
 
 
 
 741 
 
 786 
 
 63 
 
 
 
 753 
 
 795 
 
 66 
 
 
 
 762 
 
 802 
 
 69 
 
 
 
 
 808 
 
 72 
 
 
 
 
 
 813 
 
 The effect of the form of the crest in increasing 
 the coefficients is distinctly observable in this Table, 
 although the weirs experimented on overhung the 
 water above, between the crest and the bottom of 
 the channel. 
 
 We must protest against the notation adopted by 
 Boileau and Morin, of giving only two-thirds of the 
 coefficient of discharge, c d , for weirs, instead of the 
 full and true value. The correct formula for the dis- 
 charge from a weir, is D zz - lh \/2gh. Now they 
 
 3 
 
 assume a coefficient due to an incorrect formula D zz 
 
 lh \/2gh, which reduces c d to - c d to give the same 
 
 3 
 
 final results. This leads also to an unnecessary dis- 
 tinction between the coefficients of orifices at the 
 surface, or notches, and orifices sunk to some depth, 
 which, practically, have the same, or nearly the same, 
 general value. 
 
96 THE DISCHARGE OF WATER FROM 
 
 SECTION IY. 
 
 VARIATIONS IN THE COEFFICIENTS FROM THE POSITION OF 
 THE ORIFICE. GENERAL AND PARTIAL CONTRACTION. 
 VELOCITY OF APPROACH. PRACTICAL FORMULAE FOR THE 
 DISCHARGE OVER WEIRS AND NOTCHES. CENTRAL AND 
 MEAN VELOCITIES. 
 
 A glance at TABLE I. will show us that the coeffi- 
 cients increase as the orifices approach the surface, 
 to a certain depth dependent on the ratio of the sides, 
 and that this increase increases with the ratio of the 
 length to the depth : some experimenters have found 
 the increase to continue uninterrupted for all orifices 
 up to the surface, but this seems to hold only for 
 depths taken at or near the orifice when it is square 
 or nearly so : it has also been found that the co- 
 efficient increases as the orifice approaches to the 
 sides or bottom of a vessel: as the contraction 
 becomes imperfect the coefficient increases. These 
 facts probably arise from the velocity of approach 
 being more direct and concentrated under the re- 
 spective circumstances. The lateral orifices A, B, c, 
 D, E, F, G, H, i, and K, Fig. 11, have coefficients dif- 
 fering more or less from each other. The coefficient 
 for A is found to be larger than either of those for 
 B, c, E, or D ; that for G or K larger than that 
 for H or i ; that for H larger than that for i ; and 
 that for F, where the contraction is general, least of 
 all. The contraction of the fluid on entering the 
 orifice F removed from the bottom and sides is com- 
 plete; it is termed, therefore, " general contraction;" 
 that at the orifices A, E, G, H, i, K, and D, is inter- 
 fered with by the sides ; it is therefore incomplete, 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 97 
 
 and termed "partial contraction'' The increase in 
 the coefficients for the same-sized orifices at the same 
 
 Fig-.ll 
 
 mean depths may be assumed as proportionate to 
 the length of the perimeter at which the contraction 
 is partial, or from which the lateral flow is shut off ; 
 for example, the increase for the orifice G is to that 
 for H as cd -{-de : de; and in the same manner the 
 increase for G is to that for E as cd + de : c d. If 
 we put n for the ratio of the contracted portion c d e 
 to the entire perimeter, and, as before, c d for the 
 coefficient of general contraction, we shall find the 
 coefficient of partial contraction to be equal to 
 (35.) c d + *09 n = c d -f- *1 n nearly, 
 
 for rectangular orifices. The value of the second 
 term *09 n is derived from various experiments. If 
 we assume *617 for the mean value of c d , we may 
 change the expression into the form (1 + *146 n) c d . 
 When n=%, this becomes 1-036 c d ; when n = |, it 
 becomes T073 c d ; and when n f , contraction is 
 prevented for three-fourths of the perimeter, and the 
 coefficient for partial contraction becomes l'L09c d . 
 The form which we have given equation (35) is, 
 however, the simplest ; but the value of n must not 
 exceed f . If in this case c d =. -617, the coefficient 
 
 H 
 
98 THE DISCHARGE OF WATER FROM 
 
 for partial contraction becomes '617 + -09 x f = 
 617 + -067 =z -684, Bidone's experiments give for 
 the coefficient of partial contraction (1 + -152n)c d ; 
 and Weisbach's (1 + -132 w) c d . 
 
 VARIATION IN THE COEFFICIENTS FROM THE EFFECTS OF 
 THE VELOCITY OF APPROACH. 
 
 Heretofore we have generally supposed the water in 
 the vessel to be almost still, its surface level un- 
 changed, and the vessel consequently large compared 
 with the area of the orifice. When the water flows to 
 the orifice with a perceptible velocity, the contracted 
 vein and the discharge are both found to be in- 
 creased, other circumstances being the same. If the 
 area of the vessel or channel in front exceed thirty 
 times that of the orifice, the discharge will not be 
 perceptibly increased by the induced velocity in the 
 conduit ; but for lesser areas of the approaching 
 channel corrections due to the velocity of approach 
 become necessary. It is clear that this velocity 
 may arise from either a surface inclination in the 
 channel, an increase of head, or a small channel of 
 approach. 
 
 We get equation (6) for the discharge from a 
 rectangular orifice A, Fig. 12, of the length /, with a 
 head measured from still water 
 
 in which A b and k t are measured to the surface at some 
 distance back from the orifice, as shown in the section. 
 The water here, however, must move along the channel 
 towards the orifice with considerable velocity. If A 
 be the area of the orifice, and c the area of the 
 
OEIFICES, WEIRS, PIPES, AND RIVERS. 99 
 
 channel, we may suppose with tolerable accuracy that 
 
 Fig. 12 
 
 this velocity is equal to -v , in which v represents 
 
 c 
 
 the mean velocity in the orifice. If we also represent 
 by v & the velocity of approach, we get the equation 
 
 (36.) '; : ,f;':'; : * = x... 
 
 and consequently the theoretical height (h a ) due to 
 it is 
 
 (37.) h & = ^ x - = '0155 
 o 2 c 
 
 in feet measures.* 
 
 The height A a may be considered as an increase of 
 head, converting A b into h b + h & , and h t into h t + h & . 
 The discharge therefore now becomes 
 
 (38.) D = 
 
 which, for notches or weirs, is reduced to 
 
 * When the approaching velocity passes through the orifice without 
 contraction, it is evident that the head h & required to produce that 
 
 A s v 2 
 
 velocity, in the orifice with contraction, must be h & = ^ x <> - 2 -^ 
 
 case equation (40) becomes 
 
 instead of h & = ^2 x cf~> 
 
 In like manner we must have h & = 
 
 ^ = c"a x 
 
 = '04 vl in feet measures when v & is the velocity of approach 
 
 and c d = -617. 
 
 H3 
 
100 THE DISCHARGE OF WATER FROM 
 
 (39.) D =1 c^^Tg { (Ab + Aj'r _ At} * 
 
 as h t then vanishes. As D is also equa to A x v , 
 equation (37) may be changed into 
 
 D 2 1 D 2 
 
 (40.) h a X - - -0155 -gin feet measures. 
 zg 
 
 If this value for h & be substituted in equations (38) 
 and (39), the resulting equations will be of a high 
 order and do not admit of a direct solution ; and in 
 (38) and (39), as they stand, h & involves implicitly 
 the value of D, which we are seeking for. By find- 
 ing at first an approximate value for the velocity of 
 approach, the height A a due to it can be easily found, 
 equation (37) ; this height, substituted in equation 
 (38) or (39), will give a closer value of D, from 
 which again a more correct value of h a can be de- 
 termined ; and by repeating the operation the values 
 of D and h & can be had to any degree of accuracy. 
 In general the values found at the second operation 
 will be sufficiently correct for all practical purposes. 
 It has been already observed that, for orifices, it is 
 advisable to find the discharge from a formula in 
 which only one head, that at the centre, is made use 
 of ; and though TABLE IV., as we shall show, enables 
 us to calculate the discharge with facility from either 
 formula, it will be of use to reduce equation (38) to 
 
 * The formula for the discharge over weirs, taking into account 
 the velocity of approach, D = 2-95 c d l*J h -f -115 vj, given by 
 D'Aubuisson, Traite Hydraulique, seconde Edition, pp. 78 et 95, 
 and adopted by some English writers and engineers, is incor- 
 rect in principle. In feet measures it becomes D = 5-35c d ZfeX 
 V h + -03494 t?J, which form, with alterations in the numerals 
 and measures, was used for calculating discharges of sewers 
 during the METROPOLITAN MAIN DRAINAGE discussion. 
 
ORIFICES, WEIES, PIPES, AND EIVERS. 101 
 
 a form in which only the head (h) at the centre 
 is used. The error in so doing can never exceed six 
 per cent., even at small depths, equation (31), and 
 this is more than balanced by the observed increase 
 in the coefficients for smaller heads. 
 
 The formula for the discharge from an orifice, h, 
 being the head at the centre, is 
 
 and when the additional head h a due to the velocity 
 of approach is considered, 
 
 which may be changed into 
 (41.) D= 
 
 Equation (39), for notches, may be also changed to 
 the form 
 
 (42.) D= 
 
 this is similar in every way to the equation 
 
 (43.) ^ 
 
 for the discharge from a rectangular orifice whose 
 depth is d, with the head h t , at the upper edge. 
 
 TABLE III. contains the values of j 1 + v- a | in equation 
 
 (41), and TABLE IV. the values of 
 in equation (42), or the similar expression in (43), 
 T- or -> being put equal to n; and we perceive that 
 the effect of the velocity of approach is such as to in- 
 
 crease the coefficient from c d to c d { 1 + j- & } for orifices 
 
 ft i 
 
102 THE DISCHARGE OF WATEE FROM 
 
 sunk some distance below the surface, and into 
 
 for weirs when h & is the height due to the velocity of 
 approach, h the depth of the centre of the orifice, 
 and A b the head on the weir. A few examples, show- 
 ing the application of the formulas (41), (42), and 
 (43), and the application of TABLES I., II., IIL, and 
 IV. to them, will be of use. We shall suppose, for 
 the present, the velocity of approach v & to be given, 
 and no extra head be required to maintain it through 
 
 v 2 
 the orifice : in other words when h= 
 
 = '017^ in feet measures nearly. 
 
 EXAMPLE I. A rectangular orifice, 12 inches wide 
 by 4 inches deep, has its centre placed 4 feet below the 
 surface, and the water approaches the head with a 
 velocity of 28 inches per second ; what is the dis- 
 charge ? For an orifice of the given proportions, 
 and sunk to a depth nearly four times its length, we 
 shall find from TABLE I. 
 
 *616 +'627 /jn-i i 
 
 c d -T 621 nearly. 
 
 As the coefficient of velocity, equation (2), for water 
 flowing in a channel is about -956, we shall find, 
 column No. 3, TABLE II. the height A a =l|izl-125 
 inch nearly, corresponding to the velocity 28 inches. 
 Equation (41), 
 
 now becomes 
 
 D=12 X4>/2^X -621 f 
 
OEIFICES, WEIES, PIPES, AND KIVEES. 103 
 
 We also find v/2# h = 192 -6 inches when h = 48 
 inches, TABLE II. ; therefore 
 
 D=12x4xl92-6X'62lfl+ }* 
 
 * 48 ' 
 
 =9244-8x'621{l + -0234p=9244-8x -621x1-0116, 
 (as {1-0234}>=1-0116 from TABLE III.) = 9244-8 x 
 628 nearly = 5 805 -1 cubic inches=3-36 cubic feet 
 per second. Or thus: The value of -621 x (1-0234) 
 being found equal -628, D=AX-628v/2#x48. Now 
 for the coefficient -628, and h = 48 inches, TABLE II. 
 gives us -628v/2^ x 48 =120-96 inches; hence we 
 get D=12 x 4x120-96=5806-08 cubic inches=3'36 
 cubic feet, the same as before, the difference -38 in 
 the cubic inches being of no practical value. If we 
 
 v 2 
 find h & from the formula h & = n 2 =2*6 inches. then 
 
 r 
 
 we shall get i>=3-41 cubic feet nearly. 
 
 If the centre of the orifice were within 1 foot of 
 the surface, the effect of the velocity of approach 
 would be much greater ; for then 
 
 Cd x x + * = (from TABLE L) . 623 
 
 = (from TABLE III.) -623 x 1-047 = -652 instead of 
 628. In this case the discharge is D = 12 x 4 x -652 
 
 X 12 = 12 x 4 x -652x96-3 (from TABLE II.) 
 = 12 X 4 x 62-8 = 3014-4 cubic inches = 1-744 
 cubic feet per second. Or we may find the value of 
 652 v/2# h directly from TABLE II. thus : 
 The value of -628 y/2^ x 12 = 60-48 -628 
 The value of -666 \/1g x 12 = 64-14 -652 
 
 38 ~T~ 3-66 ::~24: 2-31. 
 
 Hence -652 \/2^h = 60-48 + 2-31 = 62-79, and the 
 
104 THE DISCHAEGE OF WATER FROM 
 
 discharge zz 12 x 4 x 62-79 x 3013-92 cubic inches 
 zz 1-744 cubic feet per second, the same as before. 
 
 v 2 
 
 If we take A a zz f~ 2=: 2 -6 inches, we shall find DZZ 
 Agc<L 
 
 1-833 cubic feet nearly. 
 
 EXAMPLE II. A rectangular notch, 7 feet long, has 
 a head of 8 inches measured at about 4 feet above the 
 orifice, and the water approaches the head with a 
 velocity of 16 J inches per second ; what is the dis- 
 charge? For a still head we shall assume c d zz -628 
 in this case, and we have from equation (42) 
 
 As in the last example, we shall find from TABLE II. 
 (h & ) the height due to the velocity of approach (16J 
 
 inches) to be - zz 0-375 inch, assuming the coefficient 
 
 8 
 
 of velocity to be -956. We have, therefore, h & zz 
 375, h b = 8, c d zz -628, and A zz 7 x 12 x 8, or for 
 
 measures in feet - zz -047, h b zz -> and A zz 7 x - 
 n^ 3 
 
 hence 
 
 D zz - x 7 x ~^g X - X 628{(l-047) l -(-047) f }. 
 
 3 8 3 v / 
 
 The value of (1-047)* - (-047)* will be found from 
 
 TABLE IV. equal to 1-0612 ; the value of v 2# x ^ 
 
 will be found from TABLE II. equal to 6-552, viz. by 
 dividing the velocity 78-630, to be found opposite 8 
 inches, by 12 ; hence 
 
 Dzz-x7x-X 6-552 x '628 X 1'0612 
 
 3 3 
 
 zz x 7 X 4-368 x -628 x 1'0612 
 
OKIFICES, WEIKS, PIPES, AND RIVERS. 105 
 
 = - x 7 x 4-368 x -666 nearly 
 
 3 
 
 = - x 7 x 2-909 -1 X 1-939 
 
 3 
 
 = 13-573 cubic feet per second = 814-38 cubic feet 
 per minute. Or thus: From TABLE VI, we find, 
 when the coefficient is -628, the discharge from a 
 weir 1 foot long, with a head of 8 inches, to be 
 109-731 cubic feet per minute. The discharge for 
 
 a weir 7 feet long, when -r = *047 is therefore 
 
 109-731 x 7 x 1-0612 = 815-12 cubic feet per 
 minute. The difference between this value and that 
 before found, 814-38 cubic feet is immaterial, and has 
 
 arisen from not continuing all the products to a suffi- 
 
 v * 
 
 cientnumberof places of decimals. If A a :n 2 =. -87 
 
 &g c& 
 
 inch, then D zz 14-51 cubic feet per second nearly. 
 
 We have, in equations (36) and (37), pointed out 
 the relations between the channel, orifice, velocity of 
 approach, and velocity in the orifice, viz. 
 
 . = X t^andA. = X =, in which A. = 
 
 (neglecting, for the present, the coefficient of velocity 
 in passing through the orifice). As v is the actual 
 
 v 
 velocity in the orifice, must be the theoretical 
 
 velocity due to the head h + h^ and therefore 
 
 v* vl v\ 
 
 ;, and h =. -3 ; hence 
 
 ** 4A**=? 
 
 C C d A V & A 
 
106 THE DISCHARGE OF WATER FROM 
 
 We have hence 
 
 (44.) i a - C ' A " 
 
 \ / j - Q *2 Q ? 
 
 fl/ C "" Cft A 
 
 substituting this value in equations (41) and (42), 
 there results 
 
 (45.) 
 
 D = A v^AXCajl +^T 
 
 or 
 
 D = A < X 
 
 in which m = -, for the discharge from an orifice at 
 some depth, and for the discharge from a weir, 
 (46.) ^A 
 
 The two last equations give the discharge when the 
 
 o 
 
 ratio of the channel to the orifice - = m is known, 
 
 and also when the whole quantity of water passing 
 through the orifice, that due to the velocity of approach 
 as well as that due to the pressure, suffers a contrac- 
 
 tion whose coefficient is c d . When h & zz ^~ 
 
 "9 ^ C 
 
 is when the velocity of approach v & passes through 
 the orifice without contraction, we shall get 
 h. vl A 2 1 
 
 h ~ vl - vl -" c 2 - A 2 ^ m 2 - 1' 
 consequently, in this case, equation (45) becomes 
 
 (46o.) D rz A x/2^1 X c d x (l + i^if > ' T ^ 
 and equation (46) in like manner changes into 
 
OEIFICES, WEIES, PIPES, AND E1VEES. 107 
 
 The last members of these two equations are the 
 same as the like members in (45) and (46), when 
 c d? within the brackets = 1 ; consequently we shall 
 easily find their values for the coefficient 1 in the 
 last page of TABLE Y., for the respective values of 
 
 c 'h \ 
 
 m == - and also for those of ^ = 2 _ , When 
 
 c d zz 1, equation (45) may be changed into 
 
 Igh \\ 
 
 This is the equation of Daniel Bernoulli, and only a 
 particular case of the one we have given. 
 
 2 2 1 
 
 If we put n = gfc&^ the values of (l + gjtn^l)* 
 
 no o *| 
 
 and of {l + m ^ } - { m 2 C l_ ^2}^ respectively, can 
 
 be easily had from TABLES III. and IV. We have, 
 however, calculated TABLE V. for different ratios of 
 the channel to the orifice, and for different values 
 of the coefficient of discharge. This table gives at 
 once the values of 
 
 as new coefficients, and the corresponding value of 
 
 T~* or y- zz a o4f 
 h> h* nP cl 
 
 h i A S 
 
 * When T 5 = -5 T- sa ^ - -3 we shall have in EXAMPLE II. 
 
 /& Wl * X O^ - * A 
 
 -* = -11 and (l + ~* - ( 1-133, TABLE IV., (or 
 
108 THE DISCHARGE OF WATER FROM 
 
 It is equally applicable, therefore, to equations (41) 
 and (42) as to equations (45) and (46). For in- 
 stance, we find here at once the value of 628 
 {(1-047)1 --(-047)*} in EXAMPLE II., p. 104, equal 
 
 to -666, as T- a i= *047, and the next value to it for 
 
 the coefficient -628, in the table, is *046, opposite 
 to which we find -666, the new coefficient sought. 
 The sectional area of the channel in this case, as 
 appears from the first column, must be about three 
 times that of the weir or notch. 
 
 TABLE V. is calculated from coefficients, c d , in still 
 water, which vary from -550 to 1. Those from '606 
 to -650, and the mean value -628 are most suited for 
 application in practice. When the channel is equal 
 to the orifice, the supply must equal the discharge, 
 and for open channels, with the mean coefficient 
 628, we find, accordingly, from the table, the new 
 coefficient 1*002 for weirs ; or 1 very nearly as it 
 should be. We also find, in the same case, viz. when 
 A =z c, and c d zz -628, that for 
 short tubes, Fig. 13, the re- 
 sulting new coefficient be- 
 comes *807. This, as we 
 shall afterwards see, agrees 
 
 very closely with the experimental results. When 
 the coefficients in still water are less than -628, or 
 more correctly -62725, the orifice, according to our 
 
 TABLE V. for the coefficient 1.) Hence in this case *628 x 
 1-133 = -712 the new coefficient suited to the velocity of 
 
 v 2 
 approach. Here of course h & = ^^ (see Note p. 99). 
 
OEIFICES, WEIRS, PIPES, AND EIVEES. 109 
 
 formula, cannot equal the channel unless other re- 
 sistances take place as from friction in tubes longer 
 than one and a half or two diameters, or in wide 
 crested weirs ; and for greater coefficients the junc- 
 tion of the short tube with the vessel must be 
 rounded, Fig. 14, on one or 
 
 Fig. 14 
 
 more sides ; and in weirs or 
 
 notches the approaches must 
 
 slope from the crest and ends 
 
 to the bottom and sides, and 
 
 the overfall be sudden. The 
 
 converging form of the approaches must, however, 
 
 increase the velocity of approach ; and therefore 
 
 v a is greater than - x V Q when c is measured between 
 c 
 
 r o and R o, Fig. 14, to find the discharge, or new 
 coefficient of an orifice placed at r o. 
 
 As the coefficients in TABLE V. are suited for 
 orifices at the end of short cylindrical or prismatic 
 tubes at right angles to the sides or bottom of a 
 cistern, a correction is required when the junction is 
 rounded off as at nor o, Fig. 14. When the chan- 
 nel is equal to the orifice, the new coefficient in 
 equation (45) becomes 
 
 The velocity in the short tube Fig. 14 is to that in 
 
 ( 1 ) 
 the short tube Fig. 13 as 1 to c d y^ 2 nearly, or 
 
 vl C d ; 
 
 for the mean value c d zz -628, as 1 to -807. Now, as 
 - is assumed equal to - in the cylindrical or pris- 
 
J10 THE DISCHARGE OF WATER FROM 
 
 matic tube, Fig. 13, - - z=- in the tube Fig. 14 
 
 A V & 
 
 with the rounded junction, for v & becomes TOTI hence, 
 
 in order to find the discharge from orifices at the end of 
 
 the short tube, Fig. 14, we have only to multiply the 
 
 /t 
 numbers representing the ratio - in the first column, 
 
 ^L 
 
 TABLE V., by -807, or more generally by c d jj-^ 2} , 
 and find the coefficient opposite to the product. 
 Thus if c d =; *628, we find, when - zi l,c d j-j -- 2} 
 
 A. N JL c/d ) 
 
 zz -807 in the table. If, again, we suppose - z= 3, 
 
 A. 
 
 then 3 x '807 2-421, the value of - for the tube 
 
 Fig. 14, and opposite this value of -, taken in column 
 
 1, we shall find -651 for the new coefficient. For the 
 cylindrical or prismatic tube, Fig. 13, the new coeffi- 
 cient would be only -642. When the head h & is how- 
 
 v 2 
 ever equal to ^ * c z the results must be modified 
 
 accordingly (see Note p. 99).* 
 
 * Professor Rankine gives the value of the coefficient of dis- 
 charge, or contraction, for varying values of A and c at a diaphragm 
 in a pipe by the formula 
 
 618 
 
 When- = o, c d = 1; and when - = 1, c d = -618 ; as it should be 
 
 c c 
 
 very nearly for an orifice in a thin plate, to which only, and to A, in 
 the short tube, Fig. 14, the formula is suited (see SECTION X). 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. Ill 
 
 PRACTICAL FORMULA FOR THE DISCHARGE OVER WEIRS. 
 
 In order to reduce the preceding formulae for 
 weirs and notches to some of the forms in common 
 use, with definite combined numerical coefficients, 
 by substituting 8*025 for v/2^, equation (39) be* 
 comes for feet measures 
 
 (A.) D a =5-35 c d / {&+ A>-A a *}, 
 and for inch measures, as \f%g = 27*8, the dis- 
 charge, taken also in cubic feet, becomes 
 
 (B.) D a = -01072 c d I {(h b - A a )t - A>* } 
 When the length I is taken in feet and the depth in 
 inches, we shall have 
 
 (0.) D a =-1287c d /{(A,^AJ*-A>*}- 
 The three last equations being for seconds of time, 
 we shall get, when the time is taken in minutes for 
 feet measures, the discharge in cubic feet 
 
 ; (D.) D a -321 c d 
 for inch measures 
 
 (E.) D. = -6433c 
 and for lengths (I) in feet and depths in inches 
 
 , : (F.) * = W*Cil{(h, + h$-hJ}. (j If 
 The latter equation, when the coefficient of discharge, 
 c d , is taken at -614 becomes 
 
 ,D a = 4-74 / {(A b + A a )-A fc *}, and 
 (G.) JDn:4-74/^, when the velocity of ap- 
 
 ( proach vanishes. 
 For a coefficient of -617 
 
112 THE DISCHARGE OF WATER FROM 
 
 -^}' and 
 (H.) -p = 4*76// when the velocity of ap- 
 
 ( proach vanishes. 
 For a coefficient of -623 
 
 a 
 ' ' JD=:4'81/A with no perceptible approach. 
 
 For a coefficient of -628 
 
 - A.*}' and 
 
 a 
 
 ^ ' JDzr4-85 /A with no perceptible approach. 
 For a coefficient of -648 
 
 \ '* 1 D zz 5 I ti% with no perceptible approach. 
 For a coefficient of f or -667 
 
 D a iz5-14 / {(A b + A a )^ - Aj}' and 
 
 ' *' JDzz5*14 /At with no perceptible approach. 
 For a coefficient of -712 
 
 = 5-5 I ((Ah-1-A.Fr-A.IV' and 
 
 ' ' JDz=5.5 / 7^ with no perceptible approach. 
 And finally for a coefficient of -81 
 
 /D a = 6-3 /{(AH+AJ* A^Vand 
 (0.) 
 
 D 6-3 / A when the velocity of approach 
 
 vanishes. 
 
 The theoretical value of k & in each of the foregoing 
 equations is in terms of the velocity of approach v a 
 
 * * 
 
 h * - 27 
 
 in which 2 g must be taken equal to 64-403 for heads 
 in feet, and equal to 772-84 for heads in inches. But 
 it is evident that in order to produce the velocity 
 per second v & passing through the notch with a 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 113 
 
 nearly still-water basin above it, that A a must be 
 
 v l v l 
 increased from its theoretical value ^- to 2 -?in 
 
 * 9 C ^ L 9 
 which expression c d is the coefficient of discharge 
 
 due to the particular notch, or weir, and its attendant 
 circumstances ; whence we must take 
 
 /P v * Theoretical head 
 
 Now, unquestionably, the most general coefficient 
 both for notches and submerged orifices, in thin 
 plates, for gauging, whether triangular, rectangular, 
 or circular, is -617, when the orifice or notch is small 
 compared with the approaching channel ; whence for 
 measures in feet 
 
 A a zz -0408 v% and v a zz4'95 
 For measures in inches, 
 
 A a = -0034 t and v & = 17-2 
 And for measures in which v & is expressed in feet per 
 second, and 7 a in inches 
 
 h & zz '49 vl 9 and v & zz 1*43 v/A a . 
 
 By substituting these values of A a , found in terms of 
 the approaching velocity, according to the standards 
 used in the equations from (A) to (p) inclusive, and 
 also in equation (H), we shall be enabled to find the 
 proper discharge from a notch in a thin plate. The 
 values of A a , equation (p), can be found at once in 
 inches from the observed values of v & , to be also taken 
 in inches/for coefficients varying from -584 to -974, by 
 means of TABLE II. Thus, with a coefficient of -617, 
 we shall find, for an approaching velocity of 36 
 inches per second, that h & becomes 4f = 4-4 inches 
 nearly, while for a coefficient of -666, it is only 3|zz 
 
 i 
 
114 THE DISCHARGE OF WATER FROM 
 
 3-8 inches ; and for a coefficient of 1, the theoretical 
 head is but If zzl-7 inches nearly. 
 
 From the very nature of the case the approaching 
 velocity must continue nearly unimpaired through 
 the notch with but a very slight reduction arising 
 from the viscidity of the water when it enters the 
 aperture, and separates from the lateral fluid. But 
 in order to give this unimpaired velocity by means of 
 an extra head h & , it is evident that h & must be in- 
 creased above the theoretical value by the amount 
 due to the coefficient of discharge; or, as before 
 
 v 2 v* 
 
 stated. h a must be increased from ~ to -%- This 
 
 * 9 c &^9 
 
 value of h a is, perhaps, something too large, owing 
 to the reduction of v & at the moment it enters the notch 
 and is acted upon by the overfall, drawing it away, 
 as it were, from the lateral water above the crest. 
 
 The numerical results of the respective formula 
 from (A) to (o), inclusive, can be obtained by modify- 
 ing the form as in equation (42) into 
 
 (Q.) J 
 
 in which D is the discharge found, when there is no 
 velocity of approach, by the common form D = 5*35 x 
 c d / h*, for which separate values are given in equa- 
 tions from (H) to (o) inclusive ; and numerical values 
 
 in TABLE VI. ; and l+ "" a multi l )lier 
 suited to the velocity of approach, the values of 
 which can be found from TABLE IV. Suppose, for 
 
OEIFICES, WEIKS, PIPES, AND KIVEES. 115 
 
 example, D =: 158-1 cubic feet per minute, A b 10 
 inches, and h a = 4 inches, which is that due to an 
 approaching velocity of 3 feet per second with a 
 coefficient of -648 ; then the multiplier becomes 
 (l + -4)l_ -4! 1-4035, TABLE IV. Hence the dis- 
 charge due to an approaching velocity of 3 feet is 
 158-1 x 1*4035 = 221-9 cubic feet, or an increase of 
 about 40 per cent. Also, if the common formula 
 were used, it is plain that the coefficient -648 should 
 be increased to -648 x 1-4035, or to -909 nearly, 
 which approximates within 10 per cent, of the 
 theoretical value. Nothing can show more clearly 
 the necessity for varying the coefficients when the 
 ordinary formulae are used, even for a notch in a thin 
 plate : for other notches the coefficients, even for 
 still water above the crest, vary considerably. 
 
 The form of the equation used by D'Aubuisson 
 and several other writers is 
 (R.) D a = 
 
 in which c and c are numerical coefficients, and v a the 
 velocity of approach. This form is incorrect in 
 principle, although the values of c and c can be so 
 taken as to give resulting values for D a approximately 
 correct. For feet measures, and time in seconds, 
 Professor Downing makes, after D'Aubuisson, p. 37 
 of his translation, 
 
 D a = c d X 5-35 / \hl + -03494^. 
 Doctor Robinson* gives for like measures and time, 
 values varying from 
 
 D a = 3-55 / <hl + -1395^/ to 
 
 * Proceedings Royal Irish Academy, vol. iv. p. 212. -1395 vl is 
 nine times the theoretical head, and too much. 
 
 i3 
 
116 
 
 THE DISCHARGE OF WATER FROM 
 
 Mr, Taylor finds (for the Government Beferees, see 
 Report on the Main Drainage of the Metropolis, 
 13th July, 1858, p. 32) the discharge in cubic feet 
 per minute, when the depth is taken in inches, and 
 the length in feet to be, 
 
 and the Messrs. Hawksley, Bidder, and Bazalgette 
 assume, (p. 33 ibid,) for like measures, 
 D a = 5 
 
 which they consider is in " excess." The following 
 table, copied and extended from the report just 
 referred to, shows the results of the last two 
 formulae, and of our equations (L) and (N), in which 
 the depth, A b , must be taken equal to 10 inches, 
 and the length, /, equal to 1 foot, 
 
 Formulae. 
 
 Mean velocities approaching the notch in feet per second, 
 and discharges in cubic feet per minute. 
 
 
 
 5 
 
 1 
 
 1-5 
 
 2 
 
 2-5 
 
 . 3 
 
 
 158-1 
 
 158-5 
 
 159-5 
 
 161-4 
 
 164 
 
 167 
 
 171 
 
 D a = 5 <v // i 3 + -1875 v Jfe* 
 
 Equation(L)when the ' 
 
 
 
 
 
 
 
 
 head, 7& a , due to the ve- 
 
 
 
 
 
 
 
 
 locity of approach is 
 
 158-1 
 
 159-2 
 
 162-1 
 
 166-8 
 
 173 
 
 180 
 
 189 
 
 taken at only its theo- 
 
 
 
 
 
 
 
 
 retical value . 
 
 
 
 
 
 
 
 
 Equation (L) when the ' 
 
 
 
 
 
 
 
 
 head, h & , due to the ve- 
 
 
 
 
 
 
 
 
 locity of approach is in- 
 
 158-1 
 
 160 
 
 167 
 
 177 
 
 190 
 
 205 
 
 222 
 
 creased for the coeffi- 
 
 
 
 
 
 
 
 
 cient of velocity -648 
 
 
 
 
 
 
 
 
 Equation(N)when the ' 
 
 
 
 
 
 
 
 
 head, h a , due to the ve- 
 
 
 
 
 
 
 
 
 locity of approach is 
 
 173-9 
 
 175-1 
 
 178-3 
 
 183-5 
 
 190-1 
 
 198-3 
 
 207-5 
 
 taken at only its theo- 
 
 
 
 
 
 
 
 
 retical value . 
 
 
 
 
 
 
 
 
 Equation (N; when the ' 
 
 
 
 
 
 
 
 
 head due to the velocity 
 
 
 
 
 
 
 
 
 of approach is increased . 
 
 173-9 
 
 176 
 
 182 
 
 192 
 
 204 
 
 218 
 
 234 
 
 for the coefficient of ve- 
 
 
 
 
 
 
 
 
 locity -712 . 
 
 
 
 
 
 
 
 
 . = 6V*i + *..'*J 
 
 173-9 
 
 175-7 
 
 180-8 
 
 188-9 
 
 199-8 
 
 213 
 
 228 
 
OEIFICES, WEIKS, PIPES, AND EIVERS. 117 
 
 In equations (L) and (N) we can get, TABLE II., the 
 values of the head, A a , due to velocity of approach v & , 
 as follows : 
 
 ^a = '5, !, 1-5, -2, 2-5, 3-0 ; in feet per second. 
 
 7i a = '047, -186, -419, -745, 1-16, 1-68 ; theoretical head in inches. 
 
 Then 
 
 7^ = 'HI, -447, -997, 1-77, 2-76, 4- ; for a coefficient of -648. 
 
 and 
 
 7t a = -093, -366, -833, 1-47, 2-29, 3-31 ; for a coefficient of -712. 
 
 Whence as A b = 10 inches, we shall have in equa- 
 tion (Q), 
 
 ~ = -Oil, -045, -1, -18, -28, -4 ; for a coefficient of -648, 
 
 and 
 
 5l = -009, -037, -083, -15, -23, -33 ; for a coefficient of -712 ; 
 % 
 
 -m o 73 
 
 and hence, by means of TABLE IY. (l + ^f (jj^ 
 
 becomes of the following respective values suited to 
 the above velocities, 
 
 1-015, 1-059, 1-122, 1-205, 1-3, 1-403 ; for a coefficient of -648, 
 
 and 
 
 1-013, 1-049, 1-104, 1-175, 1-254, 1-344 ; for a coefficient of -712. 
 
 These latter values multiplied in order by the initial 
 values of the discharges, 158-1 and 173-9, in the 
 above table, give the discharges in the third and 
 fifth lines corresponding, due to the respective velo- 
 cities of approach. 
 
 The accordance between the results in the last 
 two lines of the table is remarkable. TABLE Y. 
 shows that if the coefficient be '617 when the water 
 above the crest is still, it will be increased to -712 
 when the approaching channel is about 1-83 times 
 the section of the water in the notch. If the arrises 
 of the two-inch thick waste board be rounded, 
 the coefficient must also be considerable, although 
 
!si 
 
 00 ^ 
 
 H? - -5 
 
 s * ^ 
 
 
 
 s ~s 
 
 i 
 
 <?* 9 9 
 
 O rH O 
 O *O rH 
 rH rH rH 
 
 00 
 
 i 
 
 9 
 
 H ^^ 
 
 >H*| 
 
 o 
 
 00 ? 
 
 ^ 
 
 999 
 o o oo 
 
 <N C< O 
 
 I 
 
 T* 
 
 
 g 5 
 
 
 
 
 J^ 
 
 
 'gS'a 
 
 "- 1 
 
 
 00 
 
 OJ 
 
 o 
 
 3 
 
 !i* 
 
 43 
 
 
 
 J^ 
 CO 
 
 M 
 
 rH 
 
 OS O <N 
 
 rH Cll OS 
 
 00 
 
 10 
 
 ifl 
 
 
 .9 m 
 
 S 
 
 >O O rH 
 
 9 
 
 
 
 5 it 
 
 *5 V 
 
 
 I- . 
 
 
 CO TH OS 
 
 o o oo 
 
 I I 
 
 10 
 
 9^-3 
 
 1 
 
 
 00 
 
 O O l> 
 
 
 
 CO 
 CO 
 
 *H 8 - 
 
 
 
 
 >O TH 00 
 
 CO 
 
 s 
 
 W ^ *. 
 
 s 
 
 +3 
 
 
 rH rH l> 
 
 
 
 H co 
 
 
 -S J 
 
 
 rH rH 
 
 1-1 
 
 
 *lf 1 
 H I "I 
 
 's 
 
 2 10 
 
 9 
 
 -rf rH OS 
 
 b 
 
 
 
 * J 5 
 
 S* % 
 
 
 
 
 
 rH (M O 
 00 00 t> 
 
 GO 
 
 lO 
 
 p|'S> 
 
 X 03 
 
 1 
 
 
 0< 
 
 00 CO OS 
 
 9 
 
 ^J 
 
 agg, 
 
 P> ^. g 
 
 H 
 
 1 1 
 
 I t 
 
 CO CO 1Q 
 
 
 
 
 
 g 
 
 J rH 
 
 CO 
 
 OS (N ^ 
 
 ,H 
 
 CO 
 
 w ^ ^ 
 
 a 
 
 ^ 
 
 
 H <yj CO 
 
 CO CO *) 
 
 CO 
 
 3 
 
 K S 
 
 
 
 
 
 
 
 s | i 
 
 S) 
 
 
 
 
 CO 00 CO 
 
 C<J 
 
 ^ 
 
 si8 
 || 
 
 ^^ a, 
 
 1 
 
 t/5 ^j 
 
 o c5 
 
 rH 
 
 5 s i 
 
 9 
 
 
 
 H >< ^* 
 
 -73 
 
 .a TO 
 
 00 
 
 rH <N TH 
 
 <2P 
 
 CO 
 
 *v? J 
 
 P3 ^ S 
 
 a 
 
 -* P 
 
 
 Tjl ^ 00 
 T^ T* CO 
 
 CO 
 
 
 
 > < r^ W 
 
 d 
 
 
 
 
 
 
 5 o . 
 s^ t 
 
 1 
 
 
 CO 
 
 10 l CO 
 
 OS 
 
 1 
 
 
 OQ 
 
 
 
 ^^ O *f) 
 
 (9b 
 
 *o 
 
 . H ^ 
 
 ? 
 
 o "S 
 
 
 CO CO Ol 
 
 d 
 
 
 !E^ 
 
 
 
 
 
 
 
 H^ 
 
 4 ec * < 
 c* ^ ** 
 
 fi ,^i ?S 
 
 "d 
 
 i 
 
 CO 
 
 CO 
 rH 
 
 o t* os 
 ct S S 
 
 cp 
 
 
 
 l^l 
 
 GO 
 
 '+3 
 
 
 P 
 
 CO O *H 
 
 CO 
 
 
 
 ^ **"* **^ 
 
 
 v 
 
 
 00 00 ^* 
 
 00 
 
 10 
 
 Pi 
 
 j* 
 
 Q O 
 
 S s 
 
 
 rH rH rH 
 
 rH 
 
 
 J rS.S 
 
 ** V 
 
 3 
 
 s 5 
 
 i 
 
 lO CO O 
 
 b b co 
 
 00 
 
 1 
 
 ^ 2 
 
 
 
 
 
 
 rH 
 
 
 
 d 
 
 
 
 
 
 
 ^c^l 
 
 
 
 - ^3 
 
 
 
 >O iO O 
 
 ^ 
 
 8 
 
 . ?^^ 
 
 S 
 
 d 
 
 
 O CO 3 
 
 CO 
 
 >o 
 
 cc ^ ?S 
 
 CO 
 
 S CO 
 
 CO 
 
 
 
 
 55 &2 ^ 
 
 w 
 
 
 
 >o o op 
 
 I> 
 
 l> 
 
 -.8 1 
 
 
 T 
 
 *? 
 
 ib >b -^ 
 
 
 
 
 
 ig the Metropolitan Sewage D 
 
 TROACH IN FEET PER SECOND. 
 
 nanner of obtaining the expe f 
 to -737 nearly. 
 
 scriptions and Formulae. 
 
 g 
 
 1 
 o 
 
 1 
 
 
 
 1 velocities of stream in \ 
 t per second . . . j 
 
 OJ O ^>_, O) 
 
 5 v 5 y fl 
 
 5 ^ 4 f 
 
 fJLii 
 
 : 4-8 \/A 3 H- -1875 Ji v* & ) 
 lal quantities measured . 
 
 2S of the coefficients, #, \ 
 the formula 
 
 = x \/h* + -8 A 2 v V 
 obtained from the actual 
 an titles discharged . ./ 
 
 , ' * 
 
 A 
 
 w 
 
 
 t ?|j 8 
 
 r2 .S 
 
 fi S c^ 
 
 
 
 H 
 
 c 
 
 < < o < 
 
 o ^ 
 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 119 
 
 uncertain ; but as the equation D a = 5'5\/hl +'81% hi 
 appears to have been framed by Mr. Taylor, to 
 express special experiments made for Mr. Simpson, 
 in which the quantities varied from 5 to 152 cubic 
 feet per minute, and for heads on a four-foot weir 
 varying from 1 inch to 8 inches,* we must conclude 
 the coefficient for heads measured from still water 
 above the crest in those experiments suited to the 
 form of the weir used, and its attendant circum- 
 stances, is -712. 
 
 The equations (39) and those from (A) to (o) may 
 be easily changed into forms in which only the 
 depth h b) the velocity of approach, and the coefficient 
 of velocity (in this case equal to that of discharge) c d 
 are introduced. It is, however, only necessary here to 
 reduce the general form (A) p. Ill, for feet measures, 
 which becomes, after, substituting for h & its value 
 
 c 2 x* 2 tf an( * ma ki n g some reductions, 
 
 and for time in minutes the discharge is 
 
 R91 
 (T.) D a = -- l{ (64-4 C\h, + tfl* - } 
 
 in which v & still continues the velocity in feet per 
 second, as determined from observation. These for- 
 
 * Vide p. 22, Letter dated 16th August, 1858, from the 
 Government Referees to the Eight Hon. Lord John Manners, 
 on the subject of the Metropolitan Main Drainage. 
 
120 THE DISCHARGE OF WATER FROM 
 
 mulge may be again reduced to many others. If we 
 take h b in inches (T) becomes 
 
 (U.) D a = 
 
 Mr. Pole, in a letter to Mr. Simpson and Captain 
 Gallon, already referred to, gives the special value, 
 
 D a = 1-06 / {(3 ^+^)t _<;*}, 
 
 which corresponds very closely with the experiments 
 made for Mr. Simpson. If we assume c d = -712, 
 which also closely corresponds with those experi- 
 ments, our equation (U) becomes for them 
 
 D a = 1-225 / {(2-72 h, + !)* J} ; 
 
 but the amount of the discharge must always depend 
 on the coefficient c d , equation (U) suited to the special 
 circumstances of the case under consideration. 
 
 The form of equation for the discharge proposed 
 by Mr. Boy den * includes the effects of the end con- 
 tractions : it is 
 
 D =z c {/ bnh^h^ 
 
 in which c zz f c d \/%gh, n the number of end con- 
 tractions, / the length of the weir, h^ the head 
 measured from the surface of the water above the 
 curvature of approach, and b a coefficient due to the 
 nature of the end contractions. The mean nume- 
 rical exponent of this formula, derived by Francis 
 from his experiments, is for feet measures, per second, 
 Dzz3-33 (I - 
 
 * Francis's Lowell Hydraulic Experiments, p. 74. 
 t Ibid, p. 119. 
 
OKIFICES, WEIRS, PIPES, AND RIVERS. 121 
 
 but the value of c varied from 3-303 to 3-3617. 
 These results give corresponding values of c d = -617 
 to -628, and when c = 3-33, c d = -623. The experi- 
 mental results compared with this formula have been 
 referred to at p. 83. 
 
 Francis's Lowell experiments on a wooden dam 
 10 feet long, level and 3 feet wide at the crest, with 
 a head slope of 3i to 1 in a channel 10 feet wide, 
 give, for heads between 6 and 20 inches, a mean 
 coefficient of -563 or -565. This for feet measures 
 would give for the discharge per second 
 
 Dm 3-02Ai 
 
 For greater depths, on this width of crest, the dis- 
 charge would probably rise as high as 3-lA^or 3-3^. 
 The section of the dam was the same as that erected 
 by the Essex Company across the Merrimack Kiver, 
 at Lawrence, Massachusetts. See, also, table of 
 coefficients, p. 80. 
 
 In equation (13), pp. 54 and 55, we have given a 
 general expression for the value of D through a tri- 
 angular notch. Professor Thomson, of the Queen's 
 College, Belfast, in a paper read at the British Asso- 
 ciation at Leeds in 1858, says : 
 
 " The ordinary rectangular notches, accurately ex- 
 perimented on as they have been, at great cost and 
 with high scientific skill, in various countries, with 
 the view of determining the necessary formulas and 
 coefficients for their application in practice, are for 
 many purposes suitable and convenient. They are, 
 however, but ill adapted for the measurement of very 
 variable quantities of water, such as commonly occur 
 to the engineer to be gauged in rivers and streams. 
 
122 THE DISCHARGE OF WATER FROM 
 
 If the rectangular notch is to be made wide enough 
 to allow the water to pass in flood times, it must be 
 so wide that for long periods, in moderately dry 
 weather, the water flows so shallow over its crest, 
 that its indications cannot be relied on. To remove, 
 in some degree, this objection, gauges for rivers or 
 streams are sometimes formed, in the best engineer- 
 ing practice, with a small rectangular notch cut down 
 below the general level of the crest of a large rectan* 
 gular notch. If now, instead of one depression being 
 made for dry weather, we use a crest wide enough for 
 use in floods, we conceive of a large number of de- 
 pressions extending so as to give the crest the 
 appearance of a set of steps of stairs, and if we 
 conceive the number of such steps to become in- 
 finitely great, we are led at once to the conception of 
 the triangular instead of the rectangular notch. The 
 principle of the triangular notch being thus arrived 
 at, it becomes evident there is no necessity for 
 having one side of the notch vertical, and the other 
 slanting; but that, as may in many cases prove 
 more convenient, both sides may be made slanting, 
 and their slopes may be alike. It is then to be 
 observed, that by the use of the triangular notch, 
 with proper formulas and coefficients derivable by 
 due union of theory and experiments, quantities of 
 running water from the smallest to the largest may 
 be accurately gauged by their flow through the same 
 notch. The reason of this is obvious, from consider- 
 ing that in the triangular notch, when the quantity 
 flowing is very small, the flow is confined to a small 
 space admitting of accurate measurement ; and that 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 123 
 
 the space for the flow of water increases as the 
 quantity to be measured increases, but still continues 
 such as to admit of accurate measurement. 
 
 " Further, the ordinary rectangular notch, when ap- 
 plied for the gauging of rivers, is subject to a serious 
 objection from the difficulty or impossibility of pro- 
 perly taking into account the influence of the bottom 
 of the river on the flow of the water to the notch* 
 If it were practicable to dam up the river so deep 
 that the water would flow through the notch as 
 if coming from a reservoir of still water, the diffi- 
 culty would not arise. This, however, can seldom be 
 done in practice, and although the bottom of the 
 river may be so far below the crest as to produce 
 but little effect on the flow of the water when the 
 quantity flowing is small, yet when the quantity 
 becomes great, the velocity of approach comes to 
 have a very material influence on the flow of the 
 water, but an influence which is usually difficult, if 
 not impracticable to ascertain with satisfactory ac- 
 curacy. In the notches now proposed of a trian- 
 gular form, the influence of the bottom may be 
 rendered definite, and such as to affect alike (or at 
 least by some law that may be readily determined 
 by experiment) the flow of the water when very 
 small, or when very great, in the same notch. 
 The method by which I propose that this may be 
 effected consists in carrying out a floor, starting 
 exactly from the vertex of the notch, and extending 
 both up-stream and latterally, so as to form a bottom 
 to the channel of approach, which will both be 
 smooth and will serve as the lower bounding surface 
 
124 THE DISCHARGE OF WATER FROM 
 
 of a passage of approach unchanging in form while 
 increasing in magnitude, at the places at least which 
 are adjacent to the vertex of the notch. The floor 
 may be either perfectly level, or may consist of 
 two planes, whose intersection would start from the 
 vertex of the notch> and would pass up-stream per- 
 pendicularly to the direction of the weir board ; the 
 two planes slanting upwards from their intersection 
 more gently than the sides of the notch. The level 
 floor, although theoretically not quite so perfect as 
 the floor of two planes, would probably for most 
 practical purposes prove the more convenient ar- 
 rangement. 
 
 " With reference to the use of the floor it may be 
 said, in short, that by a due arrangement of the 
 notch and the floor a discharge orifice and channel 
 of approach may be produced, of which (the upper 
 surface of the water being considered as the top 
 of the channel and orifice) the form will be un- 
 changed or but little changed, with variations of 
 the quantity flowing ; very much less certainly than 
 is the case with rectangular notches. 
 
 " Whatever may be the result in this respect, the 
 main object must be to obtain, for a moderate 
 number of triangular notches of different forms, 
 and both with and without floors at the passage 
 of approach, the necessary coefficients for the va- 
 rious forms of notches and approaches selected, 
 and for various depths in any one of them, so as 
 to allow of water being gauged for practical pur- 
 poses, when in future convenient, by means of 
 similarly formed notches and approaches. The util- 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 125 
 
 ity of the proposed system of gauging it is to be 
 particularly observed, will not depend upon a per- 
 fectly close agreement of the theory described with 
 the experiments, because a table of experimental 
 coefficients for various depths, or an empirical for- 
 mula slightly modified from the theoretical one, will 
 serve all purposes. 
 
 "To one evident simplification in the proposed 
 system of gauging, as compared with that by rect- 
 angular notches, I would here advert, namely, that 
 in the proposed system the quantity flowing comes 
 to be a function of only one variable namely, the 
 measured head of water while in the rectangular 
 notches it is a function of at least two variables, 
 namely, the head of water, and the horizontal width 
 of the notch ; and is commonly also a function of a 
 third variable very difficult to be taken into account, 
 namely, the depth from the crest of the notch down 
 to the bottom 'of the channel of approach, which 
 depth must vary in its influence with all the varying 
 ratios between it and the other two quantities of 
 which the flow is a function. 
 
 " The proposed system of gauging also gives facil- 
 ities for taking another element into account which 
 often arises in practice namely, the influence of 
 back water on the flow of the water in the gauge, 
 when, as frequently occurs in rivers, it is found 
 impracticable to dam the river up sufficiently to give 
 it a clear overfall free from the back or tail water. 
 For any given ratio of the height of the tail water 
 above the vertex of the notch to the height of head 
 water above the vertex of the notch, I would an- 
 
126 THE DISCHARGE OF WATER FROM 
 
 ticipate that the quantities flowing would still be 
 approximately at least, proportional to the \ power of 
 the head, as before ; and a set of coefficients would 
 have to be determined experimentally for different 
 ratios of the height of the head water to the height 
 of the tail water above the vertex of the notch. 
 
 " I have got some preliminary experiments made on 
 a right-angled notch in a vertical plane surface, the 
 sides of the notch making angles of 45 with the 
 horizon, and the flow being from a deep and wide 
 pool of quiet water, and the water thus approaching 
 the notch uninfluenced by any floor or bottom. The 
 principal set of experiments as yet made were on 
 quantities of water varying from about 2 to 10 cubic 
 feet per minute ; and the depths or heads of the water 
 varied from 2 inches to 4 inches in the right-angled 
 notch. From these experiments I derive the formula 
 
 Q = 0-317 H* 
 
 where Q is the quantity of water in cubic feet per 
 minute, and H the head as measured vertically in 
 inches from the still water level of the pool down to 
 the vertex of the notch. This formula is submitted 
 at present temporarily as being accurate enough for 
 use for ordinary practical purposes for the measure- 
 ment of water by notches similar to the one experi- 
 mented on, and for quantities of water limited to 
 nearly the same range as those in the experiments ; 
 but as being, of course, subject to amendment by 
 more perfect experiments extending through a wider 
 range of quantities of water." 
 
 In the first edition of this book we gave the gene- 
 ral form of the equation for the discharge through 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 127 
 
 triangular notches, and also showed the general appli- 
 cation of the coefficients -617 to -628 for all forms 
 of orifices and notches in thin plates. '617, as shown 
 in note p. 55, gives a result identical with the prac- 
 tical results of Professor Thomson's experiments. 
 The great advantage of the triangular notch for 
 gauging is, that the sections for all depths flowing 
 over are similar triangles, and therefore the coeffi- 
 cient probably remains constant, or nearly so, not 
 only for one but for all species of triangles, when the 
 depth at the open is not very little indeed in propor- 
 tion to the width flowing over at the surface. 
 
 The disadvantage of the proposed triangular form 
 of depression, if permanent in a dam, would be that 
 the angular point should be at a lower level than the 
 top of a horizontal crest to maintain the same level, 
 above, of the water during floods ; and therefore the 
 power of the water and head would be reduced at 
 the period when most required for mill-power or 
 navigation purposes ; that is, during dry weather. 
 For drainage purposes the winter level or that du- 
 ring floods, must evidently be kept down, unless 
 when the banks are steep, and along rapids ; but 
 these remarks do not apply to dams erected across 
 inillraces or streams where the banks are, generally, 
 considerably above floods. These remarks refer to 
 occasions for permanent gauging to find the relations 
 of evaporation, absorption, and discharge in given 
 catchment areas. In notch gauging to determine the 
 useful effect of water engines, rectangular forms in 
 thin plates have the coefficients already well deter- 
 mined, and the calculations are easy. 
 
128 THE DISCHAKGE OF WATEK FKOM 
 
 DIFFERENT EFFECTS OF CENTRAL AND MEAN VELOCITIES. 
 
 There is, however, another element to be taken 
 into consideration, and which we shall have to refer 
 to more particularly hereafter ; it is this, that the 
 central velocity, directly facing the orifice, is also 
 the maximum velocity in the tube, and not the mean 
 
 velocity. The ratio of these velocities is 1 : -835 
 
 /-i 
 nearly; hence, in the example, p. 110, where - 3, 
 
 A 
 
 we get 3 x *835 = 2-505 for the value of - in column 
 
 A 
 
 1, TABLE Y., opposite to which we shall find -649, 
 the coefficient for an orifice of one-third of the 
 section of the tube when cylindrical or prismatic, 
 Fig. 13; and 3 x '835 x '807 = 2-02 nearly, oppo- 
 site to which we shall get -661 for the coefficient 
 when the orifice is at the end of the short tube, 
 Fig. 14, with a rounded junction. We have, there- 
 
 c c* 
 
 fore, - x *835 equal to the new value of for 
 
 A A 
 
 finding the discharge from orifices at the end of 
 
 /i 
 cylindrical or prismatic tubes, and - x '835 x '807 
 
 A 
 C C 
 
 z= - x '67 nearly for the new value of J - when 
 
 A A 
 
 finding the discharge from orifices at the end of a 
 short tube with a rounded junction. 
 
 The ratio of a mean velocity in the tube to that 
 facing the orifice cannot be less than *835 to 1, and 
 varies up to 1 to 1 ; the first ratio obtaining when 
 the orifice is pretty small compared with the sec- 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 129 
 
 Fig 15 
 
 D 
 
 tion of the tube, and the other when they are 
 equal. If we suppose 
 the curve D c, whose 
 abscissae (A b) repre- 
 sent the ratio of the 
 orifice to the section 
 of the tube, and whose 
 ordinates (be) repre- 
 sent the ratio of the mean velocity in the tube to 
 that facing the orifice, to be a parabola, we shall 
 find the following values : 
 
 Katio of the orifice 
 
 to the channel, or 
 
 values of 
 
 AB 
 
 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 1-0 
 
 Values of 
 
 dc. 
 
 165 
 163 
 158 
 150 
 139 
 124 
 106 
 084 
 059 
 031 
 000 
 
 Ratio of the mean velocity 
 of approach in a tube or 
 
 channel to that 
 
 directly opposite the 
 
 orifice, or values of b c 
 
 835 
 837 
 842 
 850 
 861 
 876 
 894 
 916 
 941 
 969 
 1-000 
 
 These values of b c are to be multiplied by the cor- 
 
 /- 
 responding ratio - in order to find a new value, 
 
 opposite to which will be found, in the table, the 
 
 coefficient for orifices at the ends of short prismatic 
 
 K 
 
130 
 
 THE DISCHARGE OF WATER FROM 
 
 or cylindrical tubes ; and this new value again mul- 
 tiplied by -807, or more generally by c d (T 3}*, will 
 
 /~i 
 give another new value of -, opposite to which, in 
 
 A 
 
 the table, will be found the coefficient for orifices 
 at the ends of short tubes with rounded junctions. 
 
 EXAMPLE III. What 
 
 shall be the discharge 
 
 from an orifice A, Fig. 
 
 16, 2 feet long by 1 
 
 foot deep, when the 
 
 117 '945 
 12 
 
 value of ' is 3, and 
 
 A 
 
 the depth of the centre 
 of A I foot 6 inches 
 
 below the surface? We have D t zz 2 x 1 X 
 
 (TABLE II.) = 2 x 9-829 x 19-658 cubic feet per 
 second for the theoretical discharge. From the table 
 
 on last page the coefficient for the mean velocity, 
 
 /i 
 facing the ori&ce, is about -86 ; hence - x *86 = 3 x 
 
 A. 
 
 86 zz 2-58. If we take the coefficient from TABLE L, 
 we shall find it (opposite to 2, the ratio of the length 
 of the orifice to its depth) to be -617 ; and, for this 
 coefficient, opposite to 2-58, in TABLE V., or the next 
 number to it, we find the required coefficient -636 ; 
 hence the discharge is -636 x 19-658 = 12-502 cubic 
 feet per second. If we assume the coefficient in still 
 water to be -628, then we shall obtain the new co- 
 
 * See p. 106, with reference to the modifications of equations 
 
 * 
 (45) and (46) into (45 a) and (46a) suited to & a 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 131 
 
 efficient '647, and the discharge would be '647 x 
 19-658 = 12-719 cubic feet. If the junction of the 
 tube with the cistern be rounded, as shown by the 
 
 dotted lines, we have to multiply 2*58 by *807, which 
 
 / 
 gives 2-08 for the new value of -, opposite which we 
 
 A. 
 
 shall find, in TABLE V., when the first coefficient is 
 628, the new coefficient -659; and the discharge in 
 this case would be -659 x 19-658 zz 12-955 cubic 
 feet per second. 
 
 It is not necessary to 
 take out the coefficient 
 of mean velocity facing 
 the orifice to more than 
 two places of decimals. 
 For gauge notches in 
 thin plates placed in 
 streams and millraces, Fig. 17, the mean coefficient 
 628, for still water, may be assumed ; thence the new 
 
 /-! 
 
 coefficient suited to the ratio - may be found, as in 
 
 A 
 
 the first portion of EXAMPLE III. We shall leave 
 the working out of the results when h a is taken equal 
 
 to the student. 
 
 EXAMPLE IV. What shall be the discharge through 
 the aperture A, equal ^feet by I foot, when the channel 
 is to the orifice as 3-375 to 1, and the depth of the 
 centre is 1 *2 5 foot below the surface, taken at about 3 
 feet above the orifice? Here the coefficient of the 
 
 approaching velocity is -85 nearly, whence the new 
 /-( 
 
 value of - is 3-375 x -85 = 2-87 ; and as C A = -628, 
 
 K3 
 
132 THE DISCHARGE OF WATER FROM 
 
 we shall get from TABLE V. the new coefficient '644. 
 Hence 
 
 D = 2 x 1 x x -644 (TABLE !!.)= 2 x 8*972 x -644 
 
 12 V 
 
 = 17-944 x -644 = 11*556 cubic feet per second. 
 Weisbach finds the discharge, by an empirical 
 formula, to be 11*31 cubic feet. If the coefficient 
 be sought in TABLE I., we shall find it -617 nearly, 
 from which, in TABLE V., we shall find the new 
 coefficient to be *632 : hence 17-944 x *632 = 11-341 
 cubic feet per second. If the coefficient *6225 were 
 used, we should find the new coefficient equals *638, 
 and the discharge 11-468 cubic feet. Or thus: The 
 ratio of the head at the upper edge to the depth of 
 
 q 
 
 the orifice is = *75, and from TABLE IY. we find 
 12 
 
 (1*75)* (-75)* = 1-6655. Assuming the coefficient 
 to be *644, we find from TABLE VI. the discharge per 
 minute over a weir 12 inches deep and 1 foot long to 
 
 208-650 + 205'119 on/^oo^ i i j 
 
 be - - = 206*884 cubic feet nearly; and 
 
 2 
 
 as the length of the orifice is 2 feet, we have 
 
 2 X 206*884 X 1-6655 nn AC . n , . , ,. , 
 
 - = 11*482 cubic feet per second, which 
 
 60 
 
 is the correct theoretical discharge for the coefficient 
 644, and less than the approximate result, 11*556 
 cubic feet above found, by only a very small dif- 
 ference. The velocity of approach in this example 
 must be derived from the surface inclination of the 
 stream. The working out of this example and the 
 
 v 2 
 increase of the discharge when h & = a 2 will afford 
 
 practice to the student. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS.' 
 
 For notches or Poncelet weirs the approachin 
 velocity is a maximum at or near the surface. If the 
 central velocity at the surface facing the notch be 1, 
 the mean velocity from side to side will be -914. We 
 may therefore assume the variation of the central to 
 the mean velocity to be from 1 to '914; and hence 
 the ratio of the mean velocity at the surface of the 
 channel to that facing the notch or weir cannot be 
 less than -914 to 1, and varies up to 1 to 1 ; the first 
 ratio obtaining when the notch or weir occupies a 
 very small portion of the side or width of the channel, 
 and the other when the weir extends for the whole 
 width. Following the same mode of calculation as 
 at p. 129, Fig. 15, we shall find as follows : 
 
 Ratio of the Values of Values of 
 width of the notch J l. ~ 
 
 to the width of _. a 6 > _.*' C > 
 
 thechaunel. ]?lg. 15. Fig. 15. 
 
 -086 -914 
 
 1 -085 -915 
 
 2 -083 -917 
 
 3 -078 -922 
 
 4 -072 -928 
 
 5 -064 -936 
 
 6 -055 -945 
 
 7 -044 -956 
 
 8 -031 -969 
 
 9 -016 -984 
 
 1-0 -000 1-000 
 
 These values of b c are to be used as before in 
 
 /i 
 order to find the value of -, opposite to which in the 
 
 At 
 
134 THE DISCHARGE OF WATER FROM 
 
 tables, and under the heading for weirs, will be found 
 the new coefficient. 
 
 EXAMPLE V. The length of a weir is 10 feet; the 
 width of the approaching channel is 20 feet; the head, 
 measured about 6 feet above the weir, is 9 inches ; and 
 the depth of the channel 3 feet: what is the discharge ? 
 Assuming the circumstances of the overfall to be 
 such that the coefficient of discharge for heads, 
 measured from still water in a deep weir basin or 
 reservoir, will be -61 ?, we shall find from TABLE VI. 
 the discharge to be 128-642 x 10 = 128642 cubic 
 feet per minute; but from the smallness of the 
 channel the water approaches the weir with some 
 
 velocity, and - = = 8. We have also the width 
 
 A 10 X ^ 
 
 of the channel equal to twice the width of the weir, 
 
 and hence (small table, p. 133,) 8 x -936 = 7-488 for 
 
 P 
 the new value of -. From TABLE V. we now find the 
 
 A. 
 
 *fi24 
 
 new coefficient - = -623, and hence the dis- 
 
 2 
 
 charge is - - = 1298-93 cubic feet per minute. 
 
 Or thus : As the theoretical discharge, TABLE VI., is 
 2084-96 cubic feet, we get 2054-96 x -623 = 1298-93, 
 the same as before. In this example, however, the 
 mean velocity approaching the overfall bears to the 
 mean velocity in the channel a greater ratio than 
 1 : -936, as, though the head is pretty large in pro- 
 portion to the depth of the channel, the ratio of the 
 
 sections - = - is small. We shall therefore be more 
 
 C 8 
 
 correct by finding the multiplier from the small table, 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 135 
 
 /-i 
 
 p. 129. By doing so the new value of - is 8 x -838 
 
 A 
 
 = 6*704. From this and the coefficient *617 we shall 
 find, as before from TABLE V., the new coefficient to 
 be -627 ; hence we get 2084-96 x -627 = 1307-27 
 cubic feet per minute for the discharge. 
 
 The foregoing solution takes for granted that the 
 velocity of approach is subject to contraction before 
 arriving at the overfall or in passing through it ; 
 now, as this reduces the mean velocity of approach 
 from 1 to -784, TABLE V., when the coefficient for 
 heads in still water is '617, we have to multiply the 
 
 ri 
 
 value of- = 6-704, last found, by -784, and we get 
 
 A. 
 
 6-704 x -784 = 5-26 for the value - due to this cor- 
 
 A. 
 
 rection, from which we find the corresponding co- 
 efficient in TABLE V. to be -629, and hence the cor- 
 rected discharge is 2084-96 x -629 = 1311-44 cubic 
 
 /-i 
 feet. It is to be borne in mind that the value of - in 
 
 A. 
 
 TABLE Y. is simply an approximate value for the ratio 
 -of the velocity in the channel facing the orifice to the 
 velocity in the orifice itself ; and the corrections 
 applied in the foregoing examples were for the pur- 
 pose of finding this ratio of velocity more correctly 
 
 r\ 
 
 than the simple expression -gives it. The following 
 
 auxiliary table will enable us to find the correction, 
 and thence the new coefficient, with facility. Thus, 
 if the channel be five times the size of the orifice, 
 and a loss in the approaching velocity takes place 
 equal to that in a short cylindrical tube, we get 
 
136 
 
 THE DISCHAKGE OF WATEE FKOM 
 
 AUXILIARY TABLE, TO BE USED WITH TABLE V. FOR MORE NEARLY FINDING 
 THE COEFFICIENT OF DISCHARGE NEARLY SUITED TO EQUATIONS (45 a) 
 AND (46 a). 
 
 Eatio of the orifice 1 
 to the channel, or J 
 l 
 
 Multipliers due to I 
 the difference of I 
 the central and 1 
 mean velocity only.| 
 
 Multipliers for finding the new values of-^ in TABLE V., when 
 the water approaches and passes through the orifice, without 
 contraction or loss of velocity. 
 
 Coefflc*' 
 639 
 
 Coeffic*' 
 628 
 
 Coefflc 1 
 617 
 
 Coeffic*' 
 606 
 
 Coeffic* 1 
 595 
 
 Coeffic*' 
 
 584 
 
 Coeffic 1 
 573 
 
 
 
 835 
 
 69 
 
 67 
 
 65 
 
 64 
 
 62 
 
 60 
 
 58 
 
 1 
 
 837 
 
 70 
 
 68 
 
 66 
 
 64 
 
 62 
 
 60 
 
 59 
 
 2 
 
 842 
 
 70 
 
 68 
 
 66 
 
 64 
 
 62 
 
 61 
 
 59 
 
 3 
 
 850 
 
 71 
 
 69 
 
 67 
 
 65 
 
 63 
 
 61 
 
 59 
 
 4 
 
 861 
 
 72 
 
 69 
 
 68 
 
 66 
 
 64 
 
 62 
 
 60 
 
 5 
 
 876 
 
 73 
 
 71 
 
 69 
 
 67 
 
 65 
 
 63 
 
 61 
 
 6 
 
 894 
 
 74 
 
 72 
 
 70 
 
 68 
 
 66 
 
 64 
 
 62 
 
 7 
 
 916 
 
 76 
 
 74 
 
 72 
 
 70 
 
 68 
 
 66 
 
 64 
 
 8 
 
 941 
 
 78 
 
 76 
 
 74 
 
 72 
 
 70 
 
 68 
 
 66 
 
 9 
 
 969 
 
 81 
 
 78 
 
 76 
 
 74 
 
 72 
 
 70 
 
 68 
 
 1-0 
 
 1-000 
 
 831 
 
 807 
 
 784 
 
 762 
 
 740 
 
 719 
 
 699 
 
 5 x '842 zz 4-210 for the new value of -, opposite to 
 
 which, in TABLE V., will be found the coefficient 
 sought. If the coefficient for still water be '606, we 
 shall find it to be *6 12 for orifices and '623 for weirs. 
 But when the water approaches without loss of 
 velocity, we find from the auxiliary table '64 for the 
 multiplier instead of -842, and consequently the new 
 
 value of -becomes 
 
 A 
 
 5 x '64 zz 3-2, from which we 
 
 shall find *617 to be the new coefficient for orifices 
 and -636 for weirs. The auxiliary table is calculated 
 by multiplying the numbers in the second column 
 (see third column, table, p. 129) by the value of c d x 
 
 J 
 
 - 3-2! , which will be found from TABLE V., for 
 the different values of c d in the table, viz. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 137 
 
 639, -628, -617, -606, -595, -584., and -573, to be 
 831, -807, -784, -762, -740, -719, and -699 respec- 
 tively, as given in the top and bottom lines of figures. 
 
 c 2 
 When -3-^2- in equations (45) and (46) is equal to 
 
 , 2 _^j in equations (45a) and (46), then c d zz 1, 
 
 and c d (l + ^ - 2 } in equation (45) is equal to 
 ^ vn c d j 
 
 1 1 H -- 2 1 I i n equation (45a) ; and c d \(l + 
 
 in e( l uatio11 ( 46 ) is e( l ual to 
 
 l +^^=Tl""^=r ^ equation (460); and 
 therefore the coefficient found from TABLE V. for 
 c d = 1 will give the multiplier for d , outside the 
 brackets, in (45o) and (46a), to find the new coeffi- 
 cients. Thus in the last example m = 5, and hence 
 
 TABLE V. for c d = 1, we find { 1 +-^r l f = 1-021 
 
 and {(1 + ^f - fcj^r-i)*} - 1-OB5. Hence 
 
 1-021 x '606 = -619 nearly ; and 1-055 x '606 = -639 
 nearly, the new coefficients found from the other 
 method being *617 and -636, the difference by both 
 methods being of no great practical importance. 
 
 It is necessary to observe, that in equations (45), 
 (46), (45a), and (46o), the head due to the velocity of 
 supply or approach, A a , must be extra to the head k, 
 and no part of it, and that as is indicated by the 
 equations m can never be so small as unity. These 
 equations are not, therefore, strictly applicable to 
 orifices in the short tubes, Fig. 15 and Fig. 16, al- 
 
138 
 
 THE DISCHARGE OF WATER FROM 
 
 though they can be made practically so within definite 
 limits. The initial value of c d itself varies consider- 
 ably with the position and form of the orifice ; for a 
 mean value of *707 it changes according to the rela- 
 
 707 
 tion of c and A into -7271 an( ^ f r a value of 
 
 618 for an orifice, central in a thin plate, Professor 
 Bankine's formula,, p. 110, is applicable. 
 
 In weirs at right 
 angles to channels 
 with parallel sides, 
 the sectional area can 
 never equal that of 
 the channel unless it be measured at or above the 
 point A, where the sinking of the overfall commences ; 
 and unless also the bed c D and surface A B have 
 the same inclination. In all open channels, as mill- 
 races, streams, rivers, the supply is derived from 
 the surface inclination of A B, and this inclination 
 regulates itself to the discharging power of the over- 
 fall. When the overfall and channel have the same 
 width, and it is considerable, we have, as shall 
 appear hereafter, 91 \/ h s for the mean velocity in 
 the channel, where h is the depth in feet and s the 
 rate of inclination of the surface A B. We have also 
 
 - \/1gh for the theoretical velocity of discharge at 
 
 3 
 
 the overfall, of equal depth with the channel, and, 
 when both velocities are equal, 
 
 - \/!Tgh zz 5-35 \/~h = 91 \HTs ; 
 
 8 
 
 from which we find 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 139 
 
 ,= -=00348, 
 
 the inclination of B A when the supply is equal to 
 the theoretical discharge at the overfall. If the co- 
 efficient at the overfall were *628, or, which is nearly 
 the same thing, if a large and deep weir basin inter- 
 vene between the weir and channel, Fig. 19, A a 
 
 would be level, the velocity of approach would be 
 destroyed, and we should have 
 
 5-35 X '628 \/~h = 3-36 %/T= 91 \Ths ; 
 and thence the inclination of A B 
 
 s = =-00136 
 
 734 
 
 very nearly. When we come to discuss the surface 
 inclination of rivers, we shall see that the conditions 
 here assumed and the resulting surface inclinations 
 would involve a considerable loss of head. If the 
 quantity discharged under both circumstances be the 
 same, and h be the depth in the first case, Fig. 18, 
 we shall then have the head in the latter case, Fig. 
 
 19, equal ( ) T h = 1-36 h very nearly, from which 
 
 vJ'SG'' 
 
 and the surface inclination the extent of the back- 
 water may be found with sufficient accuracy. When, 
 
 in Fig. 19, the inclination of A B exceeds , the head 
 
 734 
 
 at a must exceed the depth of the river above A. 
 We must refer to pages further on, SECTION X, for 
 some remarks on the backwater curve. 
 
140 
 
 THE DISCHARGE OF WATER FROM 
 
 SECTION V. 
 
 SUBMERGED ORIFICES AND WEIRS. CONTRACTED 
 RIVER CHANNELS. 
 
 The available pres- 
 sure at any point in 
 the depth of the ori- 
 fice A, Fig. 20, is equal 
 to the difference of the 
 pressures on each side. 
 This difference is equal to the pressure due to the 
 height k, between the water surfaces on each side of 
 the orifice ; in this case, the velocity is 
 (47.) ^ 
 
 and the discharge 
 
 (48.) D = / c? c d \/2 g h ; 
 
 in which, as before, / is the length, and d the depth 
 of the rectangular orifice A. 
 
 When the orifice is 
 partly submerged, as 
 in Fig. 21, we may 
 put ^ b h = d<2 for 
 the submerged depth, 
 and k~h t = d l} the remaining portion of the depth ; 
 whence d l + d 2 = d is the entire depth. The dis- 
 charge through the submerged depth d 2 iscjd 2 ^x 
 \/ 2 g h, and the discharge through the upper portion 
 d l is 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 141 
 
 whence the whole discharge assuming the coeffi- 
 cient of discharge c d is the same for the upper and 
 lower depths is 
 
 (49.) D = c d /\/2^m 2 \/A-)- - (h? A t 2") L 
 We may, however, equation (31), assume that 
 
 very nearly, and hence 
 (50.) D = 
 
 Ash i + -=h -^ this equation may be changed 
 
 into 
 
 (51.) = c,ld. 2 
 
 In either of these forms the values of 
 
 c d v 2 ff A, 
 
 can be had from TABLE II., and the value of the dis- 
 charge D thence easily found. 
 
 When the water approaches the orifice with a 
 determinate velocity, the height h & due to that velo- 
 city can be found from TABLE II., and the discharge 
 is then found by substituting h + A a and h t + A a for 
 h and h t in the above equations. 
 
 In the submerged 
 weir, Fig. 22, A be- 
 comes equal to d^ and 
 h t =. ; the discharge, 
 equation (49), then be- 
 comes 
 
142 THE DISCHARGE OF WATER FROM 
 
 ( 52 
 
 When the water approaches with a velocity due 
 to the height h & , then h becomes h + A a , h t = h &) and 
 equation (49) becomes 
 
 (53.) D= 
 
 In the improvement of the navigation of rivers, it 
 is sometimes necessary to construct weirs so as to 
 raise the upper waters by a given depth, d^ The 
 discharge D is in such cases previously known, or 
 easily determined, and from the values of d^ and D 
 we' can easily determine, equation (52), the value of 
 
 (54.) 
 
 3 
 
 or, by taking the velocity of approach into account, 
 (55.) d 2 =- * 
 
 This value of d 2 must be the depth of the top of the 
 weir below the original surface of the water, in order 
 that this surface should be raised by a given depth, 
 d lt When h & is small compared with d 2 , we may take 
 
 = 2 _ x -in equation (55). 
 
 3 3 \/d l + A a 
 
 EXAMPLE VI. A river whose width at the surface 
 is 7 Q feet, whose hydraulic mean depth is k&feet, and 
 whose cross sectional area is 325 feet, has a surface 
 inclination of 1 foot per mile to what depth below, 
 or height above the surface must a weir at right angles 
 
\i 
 
 xv^V. 
 
 rv^v^ 
 
 
 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 143 
 
 to the channel be raised, so that the depth of water 
 immediately above it shall be increased by %\feet ? 
 
 When the hydraulic mean depth is 4-4 feet, and 
 the fall per mile 1 foot, we find from TABLE VIII. 
 that the mean velocity of the river is 29-98 or 30 
 inches very nearly per second. The discharge is, 
 therefore, 325x2izz812-5 cubic feet per second, 
 
 or 48750 cubic feet per minute. Hence, = 
 
 696*4 cubic feet, must pass over each foot in length 
 of the weir per minute. Assuming the coefficient 
 c d -628 in the first instance, we find from TABLE 
 VI. the head passing over a weir corresponding to 
 this discharge to be 27-4 inches ; but as the head is 
 to be increased by 3J feet, or 42 inches, it is clear 
 that the weir must be perfect ; that is, have a clear 
 overfall, and rise 42 27-4 = 14-6 inches over the 
 original water surface. In order that the weir may 
 be submerged, or imperfect, the head could not be 
 increased by more than 27-4 inches. Let us, there- 
 fore, assume in the example, that the increase shall 
 be only 18 instead of 42 inches; the weir then 
 becomes submerged, and we have, from equation (54), 
 
 d 2 = u " u * _ -i x 18" (as 7 = 1 foot). 
 
 628 \/18"x2# 3 
 
 The value of the first part of this expression is 
 found from TABLE VI. or TABLE II. equal to 
 
 696-4 696-4 
 
 = 07,1. ! = 1*88 feet = 22-56 m.; 
 
 - X-X 370-341 370 ' 341 
 
 18 2 
 
 net 
 
 hence 22-56 yzz 10-56 inches is the value of c? 2 ; 
 
144 THE DISCHARGE OF WATER FROM 
 
 that is, the submerged weir must be built within 
 10-56 inches of the surface to raise the head 18 
 inches above the former level. If, however, the 
 velocity of approach be taken into account, we shall 
 
 find this velocity equals ,o = 2 feet per second 
 
 very nearly ; and the height, or value of A a , due to 
 
 o 
 
 this velocity, taken from TABLE II., is - = -75 inches 
 nearly ; therefore, from equation (55), 
 
 d = 696-4 2 (18-75)* -(-76)* 
 
 628 v/2^ x 18-75" s X \Sl8-T5 
 
 The value of x 696 ' 4 = (from TABLE VI.) 
 
 628 x/2#x 18-75" 
 
 696-4 696-4 = 1 . 84fflet = 22H)8ilL5 
 
 393-75 378 ' 81 
 
 1875 
 
 2 (18-75)* -(-75)* 2 2 (-75)* 
 
 x ^- -i x 18 ' 75 -3 - 
 
 o fiK 
 
 = 12-5 --x ^ - 12-5- -1 =z 12-4. 
 3 4*33 
 
 Hence d 2 = 22-08 12-4 ~ 9*68 inches, or about 1 
 inch less than the value previously found from equa- 
 tion (54). The mean coefficient of discharge was 
 here assumed to be -628. Experiments on submerged 
 weirs show that the value of c d varies from -5 up 
 to *8, but as this coefficient would reduce the value 
 of d<i, or the depth of the top of the weir below the 
 surface, it is safer (where a given depth above a weir 
 
 * This is found from TABLE II. more readily. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 145 
 
 must be obtained) to use the lesser and ordinary 
 coefficients of perfect weirs, with a clear overfall, for 
 finding the crest levels of submerged weirs, when it is 
 necessary to construct them. If the coefficient -8 were 
 used in the previous calculation, we should have found 
 ^_ -628x^22-08 _ 124 _ 17 , 33 _ 12 . 4 = 4 . 93 in>j 
 
 or not much more than half the previous value ; 
 but this would only increase the whole height of the 
 weir by 9-68 - 4-93 = 4-75 inches. 
 
 As D = cJ\7Tg {(d, + h h] for a perfect 
 weir with a free overfall, it is clear that when D is 
 greater than -c^l \/'Yg {(^ + h^ hf}, the weir 
 
 3 
 
 is imperfect or submerged. For backwater curve 
 see SECTION X. 
 
 In the following table of coefficients from Lesbros* 
 d 2 is measured from that point below the weir 
 where its value is a minimum. On examining equa- 
 tion (52), it will be seen that the equation D zz c d / 
 (d l + d 2 ) v/2 g d l adopted by Lesbros is incorrect, 
 and can only be safely used within the limits of his 
 experiments. 
 
 * Vide p. 84, deuxieme Edition, Hydraulique, par Arthur 
 Morin. Paris, 1858. 
 
146 
 
 THE DISCHARGE OF WATER FROM 
 
 Values of 
 d l 
 
 pqy 
 
 Values of the 
 
 coefficient c , in the 
 d 
 formula 
 D = c 1 (d + d\ 
 d V i a/ 
 
 X-V/20^. 
 
 Values of 
 d i 
 
 Values of the 
 coefficient c, in the 
 d 
 Oin ula 
 D =- c I (d + d \ 
 d \ l */ 
 
 X*/20d~. 
 
 d i+ d * 
 
 001 
 
 227 
 
 060 
 
 519 
 
 002 
 
 295 
 
 080 
 
 517 
 
 003 
 
 363 
 
 100 
 
 516 
 
 .- 
 
 , 
 
 150 
 
 512 
 
 004 
 
 430 
 
 200 
 
 507 
 
 005 
 
 496 
 
 250 
 
 502 
 
 006 
 
 556 
 
 300 
 
 497 
 
 007 
 
 597 
 
 350 
 
 492 
 
 008 
 
 605 
 
 400 
 
 487 
 
 009 
 
 600 
 
 450 
 
 480 
 
 010 
 
 596 
 
 ^___ 
 
 _^__ 
 
 015 
 
 580 
 
 500 
 
 474 
 
 020 
 
 570 
 
 550 
 
 466 
 
 025 
 
 557 
 
 600 
 
 '459 
 
 030 
 
 546 
 
 700 
 
 444 
 
 035 
 
 537 
 
 800 
 
 427 
 
 040 
 
 531 
 
 900 
 
 409 
 
 045 
 
 526 
 
 1-000 
 
 390 
 
 050 
 
 522 
 
 
 
 > 
 
 The experimental values are those shown between the horizontal lines, the others above 
 the upper ones, and below the lower ones, were deduced from calculations bj Lesbros. 
 
 The true value of the discharge is expressed by 
 the equation D zz c d / j - di + d 2 r X v/2 g d 1} and 
 
 the values of c d in the above table are, therefore, 
 too small, applied to the correct formula. When 
 d^ = d 2 the table gives c d = -474. Now for weirs 
 in which the sheet passing over is " drowned," the 
 general value of the coefficient is about -67 ; this 
 would give the coefficient for the lower portion d 2 , 
 in the true formula, equal to -503, and a mean co- 
 efficient c d in the correct formula (52) equal to -569 
 nearly. When d 2 = 200 d l9 the apparent limits of 
 the experiments on the other side, then the mean 
 value of c d zz *496 nearly in equation (52). These 
 results would show that the coefficient due to the 
 submerged depth d 2 , in the first and last experiments, 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 147 
 
 is equal to about *5 nearly, (but varies to *6 nearly 
 
 in some of the middle experiments,) or thereabouts, 
 
 and, therefore, equation (52) for submerged weirs, 
 
 as the coefficient for the upper part d l is -67, would 
 
 become 
 
 (52A.) D = I X { '445 d, + -5 d 2 } x 
 
 which for feet measures would become again 
 (52B.) D = /X v/^ X {3-56 ^ + 44}* 
 for the discharge in cubic feet per second over a 
 submerged weir, Fig. 22. 
 
 CONTRACTED RIVER CHANNELS. 
 
 When the banks of a river, whose bed has a 
 uniform inclination, approach each other, and con- 
 tract the width of the channel in any way, as in 
 
 Fig. 23, the water will rise in the channel above the 
 contracted portion A, until the increased velocity of 
 discharge compensates for the reduced cross section. 
 If we put, as before, d for the increase of depth 
 immediately above the contracted width, and d 2 for 
 the previous depth of the channel, we shall find the 
 quantity of water passing through the lower depth, 
 d 2 , equal to cjd 2 \/2gd ly in which / is the width of 
 the contracted channel at A, and the quantity of 
 
 water overflowing through d l equal to - c d / d l \/2gd l ; 
 
 3 
 
 L3 
 
148 THE DISCHARGE OF WATER FROM 
 
 and hence the whole discharge through A is 
 (56.) D = c d / v/^M (d 2 + * d\ 
 
 \ O ' 
 
 When our object is to find the width I of the con- 
 tracted channel, so that the depth of water in the 
 upper stretch shall be increased by a given depth d l9 
 we shall find 
 
 (57.) /= 
 
 When the velocity of approach is considerable, or 
 when the height h & due to it becomes a large portion 
 of d 1} its effect must not be neglected. In this case, 
 as before, we find the discharge through the depth 
 
 d 2 equal to c^ld 2 v2#(<^i + A a ) ; and the discharge 
 through the depth d l equal to - c d / v/2^ {(d l + A a ) 7 
 Ay } ; and hence the whole discharge is 
 
 (58,) = cj^{d,( 
 from which we shall find 
 (59.) /= 
 
 If the projecting spur or jetty at A be itself sub- 
 merged, these formulae must be extended ; the man- 
 ner of doing so, however, presents no difficulty, as it 
 is only necessary to find the discharges of the 
 different sections according to the preceding formulae, 
 and then add them together ; but the resulting for- 
 mula so found is too complicated to be of much 
 practical value. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 149 
 
 HEADS ARISING FROM PIERS AND BACKWATER ABOVE BRIDGES. 
 
 Equations (56), (57), (58), and (59), are applicable 
 to cases of contraction of river channels caused by 
 the construction of bridge-piers and abutments, when 
 the width / is put for the sum of the openings between 
 them. The value of the coefficient c d will depend on 
 the peculiar circumstances of each case; we have 
 seen that it rises from -5 to -7 in some cases of 
 submerged weirs, and for cases of contracted chan- 
 nels it rises sometimes as high as -8, particularly 
 when they are analogous to those for the dis- 
 charge through mouth-pieces and short tubes. When 
 the heads of the piers are square to the chan- 
 nel, the coefficient may be taken at about *6 ; when 
 the angles of the cut-waters or sterlings are ob- 
 tuse, it may be taken at about '7 ; and when curved 
 and acute, at -8. With this coefficient, a head of 2| 
 inches will give a velocity of very nearly 36 inches, or 
 3 feet per second ; but as a certain amount of loss 
 takes place from the velocity of the tail-water being 
 in general less than that through the arch, also from 
 obstructions in the passage, and from square-headed 
 and very short piers, the coefficient may be so small 
 in some cases as -5, which would require a head of 
 6f inches to obtain the same velocity. This head is 
 to the former as 54 to 21. The selection of the proper 
 coefficient suited to any particular case is, therefore, 
 a matter of the first importance in determining the 
 effect of obstructions in river channels : we shall have 
 to recur to this subject again, but it is necessary to 
 observe here, that the form of the approaches, the 
 
150 
 
 THE DISCHARGE OF WATER FROM 
 
 length of the piers compared with the distance between 
 them, or span, and the length and form of the obstruc- 
 tion compared with the width of the channel, must be 
 duly considered before the coefficient suited to the 
 particular case can be fixed upon. Indeed, the coeffi- 
 cients will always approximate towards those, given 
 in the next section, for mouth-pieces, shoots, and short 
 tubes similarly circumstanced. For some further 
 remarks on contracted channels, see SECTION X. 
 
 SECTION VI. 
 
 SHORT TUBES, MOUTH-PIECES, AND APPROACHES. ALTERATION 
 IN THE COEFFICIENTS FROM FRICTION BY INCREASING THE 
 LENGTH. COEFFICIENTS OF DISCHARGE FOR SIMPLE AND 
 COMPOUND SHORT TUBES. SHOOTS. 
 
 The only orifices we have heretofore referred to 
 were those in thin plates or planks, with a few inci- 
 dental exceptions. It has been shown, page 48, Fig. 4, 
 that a rounding off, next the water, of the mouth- 
 piece increases the coefficient ; and when the curving 
 
 Fig. 24. 
 
 assumes the form of the vena-contracta, the coefficient 
 increases to -986, or nearly unity. The discharge 
 
OKIFICES, WEIES, PIPES, AND EIVEKS. 151 
 
 from a short cylindrical tube A, Fig. 24, whose length 
 is from one and a half to three times the diameter, is 
 found to be very 'nearly an arithmetical mean between 
 the theoretical discharge and the discharge through a 
 circular orifice in a thin plate of the same diameter 
 as the tube, or -814 nearly. If, however, the inner 
 arris be rounded, or chamfered off in any way, the 
 coefficient will increase until, in the tube B, Fig. 24, 
 with a properly-rounded junction, it becomes unity 
 very nearly. In the conical short tubes c and D the 
 coefficients are found to vary according to some func- 
 tion of the converging or diverging angles o, o, and 
 according as we take the lesser or greater diameter to 
 calculate from. When the length of the tube exceeds 
 twice the diameter, the friction of the water against 
 the sides may be taken into account. 
 
 The following table, calculated by us, for a coeffi- 
 cient of friction -00699, due to a discharging velocity 
 of about eighteen inches per second, see SECTION VIII., 
 shows the resistance arising from friction in pipes of 
 different lengths in relation to the diameter, and will 
 be found of considerable practical value. It will be 
 perceived that the calculations are made for three 
 different orifices of entry. First, when the arrises 
 are rounded, as in B, Fig. 24, with a coefficient of 
 986 ; secondly, when the arrises are square, as in A, 
 with a coefficient of -815 ; and, thirdly, when the pipe 
 projects into the vessel, when the coefficient of entry 
 becomes reduced to -715. The velocity is 
 
 v = c d v/2#A, 
 h being measured to the lower end of the tube. 
 
152 
 
 THE DISCHAEGE OF WATER FEOM 
 
 COEFFICIENTS FOE SHORT AND LONG TUBES. 
 
 Number of diame- 
 ters in the length 
 of the pipe. 
 
 Corresponding coefficients 
 of discharge, showing 
 the effects of friction. 
 
 Number of diame- 
 ters in the length 
 of the pipe. 
 
 Corresponding coefficients 
 of discharge, showing 
 the effects of friction. 
 
 2 diameters 
 
 986 
 
 814 
 
 715 
 
 650 diameters 
 
 228 
 
 225 
 
 223 
 
 5 
 
 936 
 
 779 
 
 690 
 
 700 
 
 220 
 
 217 
 
 215 
 
 10 
 
 884 
 
 747 
 
 668 
 
 750 
 
 213 
 
 211 
 
 209 
 
 15 
 
 840 
 
 720 
 
 649 
 
 800 
 
 206 
 
 205 
 
 203 
 
 20 
 
 801 
 
 695 
 
 630 
 
 850 
 
 201 
 
 199 
 
 197 
 
 25 
 
 767 
 
 673 
 
 615 
 
 900 
 
 195 
 
 193 
 
 192 
 
 30 
 
 737 
 
 653 
 
 598 
 
 950 
 
 190 
 
 189 
 
 187 
 
 35 
 
 711 
 
 634 
 
 584 
 
 1000 
 
 186 
 
 184 
 
 183 
 
 40 
 
 693 
 
 617 
 
 570 
 
 1100 
 
 177 
 
 '176 
 
 175 
 
 45 
 
 665 
 
 601 
 
 558 
 
 1200 
 
 170 
 
 169 
 
 168 
 
 50 
 
 646 
 
 586 
 
 546 
 
 1400 
 
 158 
 
 157 
 
 156 
 
 100 
 
 513 
 
 480 
 
 458 
 
 1600 
 
 148 
 
 147 
 
 146 
 
 150 
 
 439 
 
 418 
 
 403 
 
 1800 
 
 139 
 
 139 
 
 138 
 
 200 
 
 389 
 
 375 
 
 364 
 
 2000 
 
 132 
 
 132 
 
 131 
 
 250 
 
 354 
 
 345 
 
 334 
 
 2200 
 
 126 
 
 126 
 
 125 
 
 300 
 
 327 
 
 318 
 
 311 
 
 2400 
 
 120 
 
 120 
 
 120 
 
 350 
 
 304 
 
 297 
 
 292 
 
 2600 
 
 116 
 
 116 
 
 116 
 
 400 
 
 287 
 
 280 
 
 276 
 
 2800 
 
 112 
 
 112 
 
 112 
 
 450 
 
 271 
 
 266 
 
 262 
 
 3000 ,, 
 
 108 
 
 108 
 
 108 
 
 500 
 
 258 
 
 254 
 
 250 
 
 3200 
 
 105 
 
 105 
 
 104 
 
 550 
 
 247 
 
 243 
 
 240 
 
 3400 
 
 102 
 
 102 
 
 101 
 
 600 
 
 237 
 
 234 
 
 231 
 
 3600 
 
 099 
 
 099 
 
 099 
 
 We see from this table, that the effect of adding 
 to the length of the pipe is greatest next the orifice of 
 entry. The effect of a few diameters added to the 
 length in long pipes is, practically, immaterial ; but 
 in short pipes it is considerable. 
 
 As for orifices in thin plates, so also for short 
 tubes, the coefficients are found to vary according to 
 the depth of the centre below the surface of the 
 water, and to increase as the depths and diameter of 
 the tube decrease. Poleni first remarked that the 
 discharge through a short tube was greater than that 
 
OEIFICES, WEIRS, PIPES, AND RIVERS. 
 
 153 
 
 through a simple orifice, of the same diameter, in the 
 proportion of 133 to 100, or as -617 to -821. 
 
 CYLINDRICAL SHORT TUBES, A, FIG. 24. 
 
 The experiments of Bossut, as reduced by Prony, 
 give the following coefficients, at the corresponding 
 depths, for a cylindrical tube A, Fig. 24, 1 inch in 
 diameter and 2 inches long. The depths are given in 
 
 COEFFICIENTS FOE SHORT TUBES, FROM 
 
 Heads 
 in feet. 
 
 Coefficients. 
 
 Heads 
 in feet. 
 
 Coefficients. 
 
 Heads 
 in feet. 
 
 Coefficients. 
 
 1 
 
 818 
 
 6 
 
 806 
 
 11 
 
 805 
 
 2 
 
 807 
 
 7 
 
 806 
 
 12 
 
 804 
 
 3 
 
 807 
 
 8 
 
 805 
 
 13 
 
 804 
 
 4 
 
 807 
 
 9 
 
 805 
 
 14 
 
 804 
 
 5 
 
 806 
 
 10 
 
 805 
 
 15 
 
 -.803 
 
 Paris feet in the original, but the coefficients remain 
 the same, practically, for depths in English feet. 
 
 Yenturi's experiments give a coefficient *823 for a 
 short tube A, 1 J inch in diameter and 4j inches long, 
 at a depth of 2 feet 8J inches, the coefficient through 
 an orifice in a thin plate of the same diameter and at 
 the same depth being -622. We have calculated these 
 coefficients from the original experiments. The mea- 
 sures were in Paris feet and inches, from which 
 the calculations were directly made ; and as the 
 difference in the coefficient for small changes of 
 depth or dimensions is immaterial or vanishes, as 
 may be seen by the foregoing small table, and as 1 
 Paris inch or foot is equal to 1*0658 English inches 
 or feet, the former measures exceed the latter by only 
 
154 
 
 THE DISCHARGE OF WATER FROM 
 
 about iVth. We may therefore assume that the coeffi- 
 cient for any orifice, at any depth, is the same, whether 
 the dimensions be in Paris or English feet or inches. 
 This remark will be found generally useful in the con- 
 sideration of the older continental experiments, and 
 will prevent unnecessary reductions from one stand- 
 ard to another where the coefficients only have to be 
 considered. 
 
 The mean value derived from the experiments of 
 Michelotti, at depths from 3 to 20 feet, and with short 
 tubes A fromi inch to 3 inches in width, isc d zz -814. 
 Buff's experiments* give the following results for a 
 tube A of an inch wide and A of an inch long, nearly. 
 
 BUFF'S COEFFICIENTS FOE SMALL SHORT TUBES. 
 
 Head 
 in inches. 
 
 Coefficient. 
 
 Head 
 in inches. 
 
 Coefficient. 
 
 Head 
 in inches. 
 
 Coefficient. 
 
 1* 
 
 855 
 
 6 
 
 840 
 
 23 
 
 829. 
 
 ^ 
 
 861 
 
 14 
 
 840 
 
 32 
 
 826 
 
 The increase for smaller tubes and for lesser depths 
 appears by comparing these results with the foregoing, 
 and from the results in themselves, generally. Weis- 
 bach's experiments give a mean value for c d zz -815, 
 and for depths of from 9 to 24 inches the coefficients 
 
 843, -832, -821, -810 respectively, for tubes -, -, -, 
 
 10' 10' 10' 
 
 and - of an inch wide, the length of each tube being 
 three times the diameter. D'Aubuisson and CastePs 
 
 * Annalen der Physik und Chemie von Poggendorff, 1839, 
 Band 46, p. 243. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 155 
 
 experiments with a tube -61 inch diameter and 1-57 
 inch long, give -829 for the coefficient at a depth of 
 10 feet. When a pipe projects into a cistern and has 
 a sharp edge, the coefficient falls so low as -715. 
 
 We have calculated the coefficients in the two fol- 
 lowing short tables, from Rennie's experiments with 
 glass orifices and tubes, Table 7, p. 435, Philosophical 
 Transactions for 1831. The form of the orifices, or 
 length of the shorter tubes is not stated, but it is 
 probable from the result, that the arrises of the ends 
 were in some way rounded off; it is stated they were 
 " enlarged." Indeed, the discharges from the short 
 tube or orifice of J inch diameter exceed the theo- 
 retical ones in the proportion of 1-261 to l,and 1*346 
 to 1. These results could not have been derived from 
 a simple cylindrical tube, but might have arisen from 
 the arrises being more or less rounded at both ends, 
 and the orifice partaking of the nature of a compound 
 tube, which may be constructed, as we shall hereafter 
 show, so as to increase the theoretical discharge from 
 1 up to 1-553. The resulting coefficients for the | 
 
 COEFFICIENTS FOE SHORT TUBES, THE ENDS ENLARGED. 
 
 Head 
 in feet 
 
 4 inch 
 diameter. 
 
 inch 
 diameter. 
 
 finch 
 diameter. 
 
 linch 
 diameter. 
 
 1 
 
 1-231 
 
 831 
 
 766 
 
 912 
 
 2 
 
 1-261 
 
 839 
 
 820 
 
 920 
 
 3 
 
 1-346 
 
 838 
 
 821 
 
 8SO 
 
 4 
 
 1-261 
 
 831 
 
 829 
 
 991 
 
 and | inch tubes, approach very closely to those 
 obtained by other experimenters, but those for the 
 inch tube are too high, unless the arris at the ends 
 was also rounded. The coefficients derived from the 
 
156 
 
 THE DISCHAEGE OF WATEE FEOM 
 
 experiments with a cylindrical glass tube 1 foot long, 
 as here given, are very variable ; like the others they 
 
 COEFFICIENTS DERIVED FROM EXPERIMENTS WITH A GLASS TUBE ONE FOOT LONG. 
 
 Heads 
 in feet. 
 
 iinch 
 diameter. 
 
 | inch 
 diameter. 
 
 1 inch 
 diameter. 
 
 1 inch 
 diameter. 
 
 1 
 
 892 
 
 703 
 
 691 
 
 760 
 
 2 
 
 914 
 
 734 
 
 718 
 
 749 
 
 3 
 
 931 
 
 723 
 
 709 
 
 777 
 
 4 
 
 914 
 
 725 
 
 677 
 
 815 
 
 are, however, valuable, as exhibiting the uncertainty 
 attending "experiments of this nature," and the ne- 
 cessity for minutely observing and recording every 
 circumstance which tends to alter and modify them. 
 Indeed, for small tubes, a very slight difference in the 
 measurement of the diameter must alter the result a 
 good deal, particularly when it is recollected that 
 measurements are seldom taken more closely than 
 the sixteenth of an inch, unless in special cases. As 
 the author, however, states, p. 433 of the work re- 
 ferred to, that the " diameters of the tubes at their 
 extremities were carefully enlarged to prevent wire 
 edges from diminishing the sections;" this circum- 
 stance alone must have modified the discharges, and 
 would account for most of the differences. 
 
 The coefficient for rectangular short tubes differs 
 in no way materially from those given for cylindrical 
 ones, and maybe taken on an average at -814 or -815. 
 
 SHOKT TUBES WITH A ROUNDED MOUTH-PIECE, B, FIG. 24. 
 
 When the junction of a short tube with a vessel 
 takes the form of the contracted vein, Figs. 3 and 4, 
 page 48, the mean value of the coefficient c d 956, 
 
OKIFICES, WEIRS, PIPES, AND RIVERS. 
 
 157 
 
 and the actual discharge is found to be from 93 to 99 
 per cent, of the theoretical discharge. Weisbach, for 
 a tube 1| inch long and T 9 o inch diameter, rounded at 
 the junction, found at 1 foot deep c d = -958, at 5 feet 
 deep Q969, and at 10 feet deep c d 975. These 
 experiments show an increase in the coefficients, in 
 this particular case, for an increase of depth. Any 
 other form of junction than that of the contracted 
 vein, will reduce the discharge, and the coefficients 
 will vary from -715 to -814, and to -986, according to 
 the change in the junction from the cylindrical, pro- 
 jecting into the vessel, to the square and properly 
 curved forms. The coefficients derived from Ventures 
 experiments will be given hereafter. 
 
 SHORT CONICAL CONVERGENT TUBES, C, FIG. 24. 
 
 The experiments of D'Aubuisson and Castel lead to 
 the following coefficients of discharge and velocity* 
 from a conically convergent tube c at a depth of 10 
 
 COEFFICIENTS FOB CONICAL CONVEEGENT TUBES. 
 
 Converging 
 angle o. 
 
 Coefficient 
 of discharge. 
 
 Coefficient 
 of velocity. 
 
 Converging 
 angle o. 
 
 Coefficient 
 of discharge. 
 
 Coefficient 
 of velocity. 
 
 1 
 
 858 
 
 858 
 
 14 
 
 943 
 
 964 
 
 2 
 
 873 
 
 873 
 
 16 
 
 937 
 
 970 
 
 3 
 
 908 
 
 908 
 
 18 
 
 931 
 
 971 
 
 4 
 
 910 
 
 909 
 
 20 
 
 922 
 
 971 
 
 5 
 
 920 
 
 916 
 
 22 
 
 917 
 
 973 
 
 6 
 
 925 
 
 923 
 
 26 
 
 904 
 
 975 
 
 80 
 
 931 
 
 933 
 
 30 
 
 895 
 
 976 
 
 lOo 
 
 937 
 
 950 
 
 40 
 
 869 
 
 980 
 
 12 
 
 942 
 
 955 
 
 50 
 
 844 
 
 985 
 
 * Traite d'Hydraulique, Paris, p. 60. 
 
158 THE DISCHAEGE OF WATEK FROM 
 
 feet. We have interpolated the original angles and 
 coefficients so as to render the table more convenient 
 to refer to, for practical purposes, than the original. 
 The diameter of the tube at the smaller or discharging 
 orifice in the experiments was *61 inches, and the 
 length of the axis T57 inch; that is, the length was 
 2-6 times the smaller diameter of the tube. The 
 coefficient became *829 for the cylindrical tube, i.e. 
 when the angle at o was nothing. The angle of con- 
 vergence o determines, from the proportions, the 
 length of the inner and longer diameter of the tube. 
 The coefficients of discharge increase up to '943 for 
 an angle of 13 J or 14 degrees, after which they again 
 decrease ; but the coefficients of velocity increase as 
 the angle of convergence, o, increases from -829, when 
 the angle is zero up to -985 for an angle of 50 
 degrees. 
 
 When D is the discharge and A the area of the sec- 
 tion, we have, as before shown, D =C A A \/^gh; but 
 as, in conically convergent or divergent tubes, the 
 inner and outer areas (or, as they may be called, the 
 receiving and discharging sections) vary, it is clear 
 that, the discharge being the same, and also the theo- 
 retical velocity \/2 g h, the coefficient c d must vary 
 inversely with the sectional area A, and that c d XA 
 must be constant. For the coefficients tabulated, the 
 sectional area to be used is that at the smaller or 
 outside end of a convergent tube c, Fig. 24. 
 
 For a short tube c, whose length is -92 inch, lesser 
 diameter 1-21 inch, and greater diameter 1-5 inch, 
 we have found, from Venturi's experiments, that 
 c d 607 if the larger diameter be used in the calcu- 
 
OEIFICES, WEIES, PIPES, AND RIVERS. 159 
 
 lation, and c d =-934 when the lesser diameter is made 
 use of, the discharge taking place under a pressure 
 of 2 feet 8J inches. 
 
 The earlier experiments of Poleni, when reduced, 
 furnish us with the following coefficients : A tube 7-67 
 inches long, 2167 inches diameter at each end, gave 
 c d zz-854; the like tube with the inner or receiving 
 orifice increased to 2| inches, c d 903 ; increased to 
 3-5 inches, c d 898 ; increased to 5 inches, c d 888 ; 
 and increased to 9*83 inches, c d 864. The depth 
 or head was 21-33 inches, the discharging orifice 
 2-167 inches diameter, and the length 7-67 inches, in 
 each case. 
 
 In the conically divergent tube D, Fig. 24, the co- 
 efficient of discharge is larger than for the same tube 
 c, convergent, when the water fills both tubes, and 
 the smaller sections, or those at the same distances 
 from the centres o o, are made use of in the calcu- 
 lations. A tube whose angle of convergence, o, is 5 
 nearly, with a head of from 1 to 10 feet, whose axial 
 length is 3 J inches, smaller diameter 1 inch, and lar- 
 ger diameter 1-3 inch, gives, when placed as at c, -921 
 for the coefficient ; but when placed as at D, the co- 
 efficient increases to -948. In the first case the 
 smaller area, used in both calculations, being the re- 
 ceiving, and in the other the discharging, orifice. 
 The coefficient of velocity is, however, larger for the 
 tube c than for the tube D, and the discharging jet 
 of water has a greater amplitude in falling. The 
 effects of conically diverging tubes will, however, be 
 better perceived from the experiments on compound 
 short tubes. 
 
160 
 
 THE DISCHARGE OF WATER FROM 
 
 EFFECTS OF COMPOUND ADJUTAGES AND ADMISSION OF AIR 
 INTO SHORT TUBES. 
 
 If the tube A, Fig. 24, be pierced all round with 
 small holes at the distance of about half its diameter 
 from the reservoir, the discharge will be immediately 
 reduced in the proportion of '814 to *617. Venturi 
 found the reduction for a tube li inch diameter and 
 4i inches long, at a depth of 2 feet 10i inches, as 41 
 to 31, or as *823 to -622. As long as one hole re- 
 mained open, the discharge continued at the same 
 reduced rate ; but when the last hole was stopped, 
 the discharge again increased to the original quantity. 
 If a small hole be pierced in a tube 4 diameters long, 
 at the distance of li or 2 diameters at farthest from 
 the junction, the discharge will remain unaffected. 
 This shows that the contraction in the cylindrical 
 tube extends only a short distance from the junction, 
 probably li or li diameter, including the whole cur- 
 vature of the contraction. 
 
 The contraction at the entrance into a tube from a 
 reservoir accounts for the coefficients for a short tube 
 A, Fig. 24, and the short tubes, diagrams 1 and 2, 
 Fig. 25, being each the same decimal nearly, when 
 
 Fig. 25. 
 
 OR : or : : I : '8, or when or is not less than ORX '79, 
 
ORIFICES/WEIRS, PIPES, AND RIVERS. 161 
 
 O Tl 
 
 and is at the distance of nearly - - from o R. The 
 
 2 
 
 form of the junction o o r R remaining as we have de- 
 scribed it, the following coefficients will enable us to 
 judge of the discharging powers of differently formed 
 short mouth-pieces. They have been deduced and 
 calculated by us, principally, from Venturi's ex- 
 periments.* 
 
 These coefficients show very clearly that any cal- 
 culations from the mere head of water and size of the 
 orifice, without taking into consideration the form of 
 the discharging tube and its connection with the re- 
 servoir, are very uncertain ; and that the discharge 
 can only be correctly obtained when all the circum- 
 stances of the case, including the form of the dis- 
 charging orifice and its approaches, have been duly 
 considered. .oil 
 
 JfcfM 0%0 89MCfg 6rft prated ajj I {I 
 
 When a tube similar to diagram 5, Fig. 25, has the 
 junction o o r R rounded, as in Fig. 4, page 48, the 
 outer extremity s t s T, such that s t or, ss = 9 st, 
 and the diameter s T = 1-8 times the diameter ,9 , with 
 a short central cylindrical piece o r s t between, the 
 coefficient of discharge corresponding to the diameter 
 or=:rs will increase to 1-493 or 1-555 ; that is, the 
 
 1-493 1.555 
 
 discharge is =: 2-4, or zz 2-5 times as much 
 
 *622 *622 
 
 as through an orifice (whose diameter is o r) in a thin 
 plate, and - - : srjK9 times as much as through a 
 
 i * 822 ;sifj ."asifour T of s a bur, . ,.'i fl { 
 
 B*6* j.,.. . .0 .o/C ni KB sniff ift/i 
 \ * See Nicholson's translation of Venturi's Experimental IB 
 quiries, published in the Tracts on Hydraulics, London, 1836. 
 The coefficients in the table, next page, have been all calculated 
 for the first time by us. 
 
 M 
 
162 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE OF COEFFICIENTS FOR SHORT TUBES AND MOUTH-PIECES. 
 
 Description of orifice, mouth-piece, or short tube. 
 
 Coefficients 
 for the 
 diameter 
 
 OR. 
 
 Coefficients 
 for the 
 diameter 
 or. 
 
 1. An orifice 1 inch diameter in a thin plate . . 
 2. A cylindrical tube 1 inch diameter and 4 ; 
 inches long, A, Fig. 24 
 
 622 
 823 
 
 974 
 823 
 
 3. A short tube with a sharp end projecting intc 
 the cistern 
 
 715 
 
 715 
 
 4. A cylindrical tube, B, Fig. 24, having the June 
 tion rounded, as in Fig. 4, page 48 
 
 611 
 
 956 
 
 5. A short conical convergent mouth-piece, c, Fig 
 24, of the proportions of o o r R, Fig. 25 
 
 607 
 
 934 
 
 6. The like tube divergent, with the smaller dia 
 meter at the junction with the reservoir 
 length 3 inches, lesser diameter 1 inch, anc 
 greater diameter 1*3 inch . 
 
 561 
 
 948 
 
 7. The tube, o o u v r R, diagram 2, Fig. 25, when 
 o R = 1 inch, o r = 1-21 inch, u v 1-21 inch 
 and ouc=rv= 2 inches, the cylindrical por- 
 tion bein 01 shown bv dotted lines 
 
 600 
 
 923 
 
 8. The same tube when o u 11 inches 
 
 567 
 
 873 
 
 The same tube when o u 23 inches 
 
 531 
 
 817 
 
 9. The tube, oossitrn, diagram 2, Fig. 25, in 
 which OR=S=ST = I inch, from o to s 
 If inch, and ss = 3 inches, gives the same co- 
 efficient as the cylindrical tube, result No. 2 
 (see No. 19) viz 
 
 823 
 
 1-266 
 
 0. The tube, diagram 1, Fig. $5, o R = l inch. . . 
 1. The same tube, having the spaces oso and 
 r t R between the mouth-piece o o r n and the 
 cylindrical tube o s T R open to the influx of 
 
 804 
 785 
 
 1-237 
 1-209 
 
 12. The double conical tube, o o s TTR, diagram 3, 
 Fig. 25, when OR = ST = ! inch, or = l-21 
 inch, o o= '92 inches, and o s = 4-l inches . . 
 3. The like tube when, as in diagram 4, Fig. 25, 
 o o T R o s T T and o o s = 1*84 inch . . . . 
 
 928 
 823 
 
 1-428 
 1-266 
 
 4. The like tube when, sT = l-46 inch, and os 
 2'17 inches 
 
 823 
 
 1-2C6 
 
 15. The like tube when ST = 3 inches, and os = 
 
 911 
 
 1-400 
 
 16. The like tube when os = 6| inches, and ST 
 enlarged to 1*92 inch . . 
 
 1-020 
 
 1-569 
 
 17. The like tube when ST = 2| inches, and os 
 12i inches 
 
 1-215 
 
 1-855 
 
 18. A tube, diagram 5, Fig. 25, when o s = r f=3 
 inches, or=s=l-21 inch, and the tube 
 o s T r the same as described in No. 12, viz. 
 g T 1 1 inch and s s = 4'1 inches 
 
 895 
 
 1-377 
 
 9. The tube, diagram 2, Fig. 25, when s T is en- 
 larged to 1-97 inch, and s s to 7 inches, the 
 other dimensions remainin ' as in No 9. 
 
 945 
 
 1-454 
 
 20. When the junction of o s r t with s s T t, dia- 
 gram 2, Fig. 25, is improved, the other parts 
 
 850 
 
 1-309 
 
 2 ^ Another experiment gives 
 
 847 
 
 1'303 
 
 
 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 163 
 
 short cylindrical tube A, Fig. 24, whose diameter is also 
 o r. Venturi was of opinion that this discharge con- 
 tinued even when the central cylindrical portion orst 
 was of considerable length ; but this was a mistake, 
 as the maximum discharge is obtained when it is 
 reduced so that o o r a and s s t T shall join, as in 
 diagram 3, Fig. 25. We see from No. 16 of the fore- 
 
 , 1-569 . 1-569 
 
 going coefficients that - = 2-52 and 7 =1-91 are, 
 
 '622 "822 
 
 perhaps, nearer to the maximum results obtainable 
 by comparing the discharge from a compound tube 
 ooSTrR, diagram 3, Fig. 25, with those through an 
 orifice in a thin plate, and through a short cylin- 
 drical tube. When the form of 
 
 Fig\26 
 
 the tube becomes curvilineal 
 throughout, as in Fig. 26, 
 s T = 1/8 o r and o s = 9 o r, 
 the coefficient suited to the diameter o r will be 1*57 
 
 nearly, and the discharge will be - - =2 '5 2 times as 
 
 *622 
 
 much as through an orifice o r in a thin plate. 
 
 The whole of the preceding coefficients have been 
 determined from circumstances in which the co- 
 efficient for an orifice in a thin plate was -622, and 
 for a short cylindrical tube *822 or -823. When the 
 circumstances of head and approaches in the reser- 
 voir are such as to increase or decrease those 
 primary coefficients, the other coefficients for com- 
 pound adjutages will have to be increased or de- 
 creased proportionately. 
 
 After examining the foregoing results, it appears 
 sufficiently clear that the utmost effect produced by 
 
 M3 
 
164 THE DISCHABGE OF WATER FROM 
 
 the formation of the compound mouth -piece o o s T r R, 
 with the exception of No. 17, is simply a restoration 
 of the loss effected by contraction in passing through 
 the orifice o R in a thin plate, and that the coefficient 
 2 *5 applied to the contracted section at o r is simply 
 equal to the theoretical discharge* or the coefficient 
 unity, applied to the primary orifice o R ; for, as 
 orifice o R : orifice o r : : 1 : '64, very nearly, 
 when o o r R takes the form of the vena-contracta, and 
 the coefficient of discharge for an orifice o r in a 
 thin plate is -622, we get the theoretical discharge 
 through the orifice o R, to the actual discharge 
 through an orifice o r, so is 1 to -622 x '64, so is 
 1 : -39808 :: 1 : '4 very nearly ; and as -4 x 2'5 = 1, 
 it is clear that the form of the tube oosflrR^ when 
 it produces the foregoing effect, simply restores the 
 loss caused by contraction in the vena-contracta* 
 Venituri's sixteenth experiment, from which we have 
 derived the coefficients in No. 17, gives the coefficient 
 f%i5 m ior ine orifice d W E?^ 1i fMl f Hnmcai > ^SF thaf J 
 greater discharge than the theoretical, through the 
 receiving orifice, may be obtained. It is, however, 
 observable that Venturi, in his seventh proposition, 
 does not rely on this result, and Eytelwein's expe- 
 riments do not give a larger coefficient than 2-5 
 applied to the contracted orifice o r, which, we have 
 above shown, is equal to the theoretical discharge 
 through o R. 
 
 1 SHOOTS. 
 
 When the sides and under edge of an orifice or 
 notch increasie in thickness, so as to be converted 
 into a shoot or small channel, open at the top, the 
 
OEIFICES, WEIKS, PIPES, AND KIVERS. 165 
 
 coefficients reduce very considerably, and to some 
 extent beyond what the increased resistance from 
 friction, particularly for small depths, indicates. 
 Poncelet and Lesbros* found for orifices 8" x 8", that 
 the addition of a horizontal shoot 21 inches long 
 reduced the coefficient from -604 to *601, with a head 
 of about 4 feet ; but for a head of 4 J inches the 
 coefficient fell from -572 to -483. )!i 'S)W' notches 
 8" wide, with the addition of a horizontal shoot 
 P' 1,0" long, the coefficient fell from -582 to -479 for 
 a head of 8"; and from -622 to -340 for a head of I". 
 Castel also found for a notch 8" wide, with the addi- 
 tion of a shoot 8" long, inclined 4 18', the mean co- 
 efficient for heads from 2" to 4i", to be -527 nearly. 
 The effects arising from ' friction alone will be per- 
 ceived from the short table at the beginning of this 
 section, p. 1521' ( TIJ > r]y ^ 
 
 The orifice of entry into a shoot and its position 
 with reference to the sides and bottom modify the 
 discharge, the head remaining constant. Lesbrosf 
 has given the coefficients suited to different positions 
 : df "shootis both within and without a cistern, and 
 from notches and submerged Orifices ; but, however 
 valuable these ate in some respects, they are of little 
 practical use to the engineer. The general principles 
 which are involved in the modification of these coeffi- 
 cients have, however, been already pointed out by us 
 when discussing the effects 1 ' of ' ; fchfc 'position of ; toe 
 orifice, and the addition of short tubes, on the dis- 
 charge. Equation (74s. ), . 1 8$, ! -is here &ppllfcabte j 
 
 * Trait6 d'Hydraulique, pp. 46 et 94. 
 Morin's Hydrauliqne, deuxieme Edition, pp. 29 et 40. 
 
166 
 
 THE DISCHAKGE OF WATEE FKOM 
 
 SECTION YIL 
 
 LATERAL CONTACT OF THE WATER AND TUBE. ATMOSPHERIC 
 PRESSURE. HEAD MEASURED TO THE DISCHARGING ORI- 
 FICE. COEFFICIENT OF RESISTANCE. FORMULA FOR THE 
 DISCHARGE FROM A SHORT TUBE. DIAPHRAGMS. OB- 
 LIQUE JUNCTIONS. FORMULA FOR THE TIME OF THE 
 SURFACE SINKING A GIVEN DEPTH. LOCK CHAMBERS. 
 
 The contracted vein 
 o r is about 8 times 
 the diameter o R; but 
 it is found, notwith- 
 standing, that water, 
 in passing through a 
 short tube of not less 
 than 14 diameter in length, fills the whole of the 
 discharging orifice s T. This is partly effected by 
 the outflowing column of water carrying forward and 
 exhausting the air between it and the tube, and by 
 the external air then pressing on the column so as to 
 enlarge its diameter and fill the whole .tube. When 
 once the water approaches closely to the tube, or is 
 caused to approach, it is attracted and adheres with 
 some force to it. The water between the tube and 
 the vena-contracta is, however, rather in a state of 
 eddy than of forward motion, as appears from the 
 experiments, with the tube, diagram 2, Fig. 25, 
 giving the same discharge as the simple cylindrical 
 tube. If the entrance be contracted by a diaphragm, 
 as at o R, Fig. 27, the water will also generally fill 
 the tube, if it be only sufficiently long. Short cylin- 
 drical tubes do not fill when the discharge takes place 
 in an exhausted receiver ; but even diverging tubes, 
 
OKIFICES, WEIRS, PIPES, AND RIVERS. 167 
 
 V, Fig. 24, will be filled, under atmospheric pressure, 
 when the angle of divergence, o, does not exceed 7 or 8 
 degrees, and the length be not very great nor very short. 
 When a tube is fitted to the bottom or side of a 
 vessel, it is found that the discharge is that due to 
 the head measured from the surface of the water to 
 the lower or discharging extremity of the tube. It 
 must, however, be sufficiently long, and not too long, 
 to get filled throughout. G-uiglielmini first referred 
 this effect to atmospheric pressure, but the first 
 simple explanation is that given by Dr. Mathew 
 Young, in the Transactions of the Royal Irish 
 Academy, vol. vii., p. 56. Yenturi, also, in his fourth 
 proposition, gives a demonstration. 
 
 The values of the coefficients for short cylindrical 
 tubes, which we have given p. 162, have been derived 
 from experiment. Coefficients which agree pretty 
 closely with them, and which are derived from the 
 coefficients for the discharge through an orifice in a 
 thin plate, may, however, be calculated as follows : 
 Let c be the area of the approaching section, Fig. 27, 
 A the area of the discharging short tube, and a the 
 area of the orifice o R which admits the water from 
 the vessel into the tube : also put, as before, h for 
 the head measured from the surface of the water to 
 the centre of the tube, and diaphragm o R ; v for the 
 actual velocity of discharge at s T ; v & for the velocity 
 of approach in the section c towards the diaphragm 
 o R; and c c for the coefficient of contraction in 
 passing from o R to o r ; then we have c X v & = A X v, 
 the contracted section o r z= C Q x a, and consequently 
 the velocity at the contracted section is equal to 
 
168 THE DISCHAKGE OF WATEE FKOM 
 
 = - Now a theoretical head equal to 
 
 ac c ac c 
 
 ,2 
 
 : i? - - 
 
 is necessary to change the velocity v & into v by the 
 action of gravity ; but as the water at the contracted 
 
 A v 
 
 section o r, moving with a velocity , strikes against 
 
 ac c 
 
 the water between it and T s, moving, from the nature 
 of the case, with a slower velocity,* a certain loss of 
 effect takes place from the impact. If this be, sup- 
 posed, suddeD, then writers on mechanics have shown 
 that a loss of head, equal to that due to the difference 
 
 of the velocities,- - v, before and after the impact 
 must take place. This loss of head is therefore equal to 
 
 V^Au'.ii- \) y 2 
 
 -^2 * jnorli fttiw 
 
 whence we must have the whole head, ' J()1 
 
 : ewoflo] 8J3 i-)li>h 
 
 ,noi)oyay[Mj^i-'g^n:jfojlJ ad o Js 
 (bO.| ^:-r j-jo) -ofifo^oftib 
 
 from which we find for the velocity from a short tube, 
 
 '-f< ;< J \\ f ' j- 
 
 orlt Q _ ;i(M]iJ)fu^n 9jo o^inoo oil] 
 
 Now, as v2 ^ h would be the velocity of discharge 
 
 were there no resistances^ jss sustained, it^s 
 
 ( _ l _ -|| 
 evident that J ^ _ ^ , /_A_ Tx 2 [ becomes as it 
 
 ( C 2 ' \ac c ) } f'K)il gniaajsq 
 
 Sir EolJert Kane's translation of Riinhnan's book on 
 Horizontal Water Wheels, p, 49frjjnoo Olli Jfi Ydioolyv OilJ 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 169 
 
 were a coefficient of velocity. When the diameter of 
 the diaphragm o R becomes equal to the diameter s T 
 of the tube, A zz a, and as the coefficient of velocity be- 
 comes equal to the coefficient of discharge when there 
 is no contraction, we get in such case this coefficient, 
 which we shall also call c of, expressed by the formula 
 oj fcaoji>i-- [ - g 1 s 1 
 
 J ' " 1 1 ^ + ( -- 1) ( '* 
 
 i t - 111 J-ilOiJOiJj' i ! Mlkll 1 ) Y'UiHHODQJJ 
 
 and when the approaching section c is very large 
 
 compared with the area A, 
 
 '.oitfAHoaM uo ^,oa-^.ao>:ATj4ifc3a uo TKaioi'a'iaoo 
 
 (63.) '" '6'-of=\,,,i ,A' 
 
 * "T ( -- 1) i 1 ',)lTAM8iaM 
 
 If c c -z -64, we shall find from the last equation 
 c of= -872 ; if C G -601, c o/z= -833 ; if c c = -617, 
 c of= -847 ; and if ^ -62l;-^/ i !> 856. These 
 results are in excess of those derived from experi- 
 ment with cylindrical short tubes, perfectly square 
 at the ends and of uniform bore. As some loss, how- 
 ever, takes place in the eddy between o r and the 
 tube, and from the friction at the sides, not taken 
 into account in the above calculation, they will ac- 
 count for the differences of not more than from 4 to 
 
 * When the diaphragm is placed in a> tube of uniform bore, 
 then c= A, and we shall get }l 
 
 0} oi/b Bestir io4i3i<mJ fdi c B/J dud ; L- 
 
 ac c a 
 
 and the loss of head, in passing the diaphragm, becomes 
 
 /A V v* aow <ro 
 
 ^- 1 *' 
 
 i 
 
 It is evident from the equations that and c c depend mutually on 
 
 each other, and that they cannot be assumed arbitrarily. See equations 
 (66), (67)*..(1&3), (124), and (125), with the corresponding remarks. 
 
170 THE DISCHARGE OF WATER FROM 
 
 6 per cent, between the calculation and experiment. 
 If C G be assumed for calculation equal *590, then 
 cof-=. '821 ; and as this result agrees very closely 
 with the experimental one, c c should be taken of this 
 value in using the foregoing formulae, from (60) to 
 (63), for practical purposes. The thickness of the 
 diaphragm itself and the relation of that thickness to 
 the diameter, as well as the form of the orifice a, are 
 necessary elements in the consideration of this question. 
 
 COEFFICIENT OF RESISTANCE. LOSS OF MECHANICAL POWER 
 IN THE PASSAGE OF WATER THROUGH THIN PLATES AND 
 PRISMATIC TUBES. 
 
 The coefficients of contraction, velocity, and dis- 
 charge have been already denned. The coefficient of 
 resistance is the ratio of the head due to the resistance, 
 to the theoretical head due to the actual or final 
 velocity. If v be this latter velocity, the theoretical 
 
 head due to it is ^- ; and if c r be the coefficient of 
 resistance, then the head due to the resistance itself 
 is, from our definition, c r x s-. Now if c v be the 
 coefficient of velocity, the theoretical velocity of dis- 
 charge must be , and the head due to it is equal 
 
 -a 2"-; k u t as the theoretical head due to v is 
 s , we shall have 
 
 for the head due to the resistance ; and, therefore, 
 from our definition, the coefficient of resistance 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 171 
 
 
 (64) 
 
 from which we shall find the coefficient of velocity 
 
 These equations enable us to calculate the coefficient 
 of resistance from the coefficient of velocity, and 
 vice versa. If c v = 1, c r = 0, as it should be. The 
 following short table, calculated from equation (65), 
 will be of use. In short tubes, the coefficient of 
 velocity c v is equal to the coefficient of discharge c d . 
 
 COEFFICIENTS OF VELOCITY AND EESISTANCE. 
 
 Coefficient 
 of velocity. 
 
 Coefficient 
 of resistance. 
 
 Coefficient 
 of velocity. 
 
 Coefficient 
 of resistance. 
 
 Coefficient 
 of velocity. 
 
 Coefficient 
 of resistance. 
 
 990 
 
 020 
 
 910 
 
 208 
 
 830 
 
 452 
 
 970 
 
 063 
 
 890 
 
 263 
 
 820 
 
 488 
 
 950 
 
 109 
 
 870 
 
 320 
 
 814 
 
 508 
 
 930 
 
 156 
 
 850 
 
 383 
 
 810 
 
 525 
 
 The coefficient of velocity for an orifice in a thin 
 plate, or for a mouth-piece, Fig. 4, is -974 ; while 
 that for a short prismatic tube, A, Fig. 24, is -814 
 nearly. The coefficient of resistance in the former 
 case is -054, and in the latter -508 ; there is, there- 
 fore, 9-4 times as great a loss of mechanical power in 
 the passage through short prismatic tubes, as through 
 orifices in thin plates or tubes with a rounded junc- 
 tion, as in Fig. 4, the quantities of water discharged 
 and the discharging velocities being the same. 
 
 If the quantities discharged and the heads be the 
 same in both cases, then we shall have 
 2 2 
 
 equal the head ; 
 
 * See the tables of resistances, discharge, and contraction, 
 pp. 174 and 176. 
 
172 THE DISCHARGE OF WATER FROM 
 
 that is > -663 x 2 ff = -949 1 2g> r " 949 * = ' 663 v ' 
 whence we get v\ = -698 v\ and v* =. I -43 1 v* for the re- 
 lation of the discharging velocities, v , from an orifice, 
 and, Vty from a short tube. The height due to the re- 
 
 1 t' 2 
 
 sistance is therefore, ('.oi 4 2 ~~ l) o 1 for short pris- 
 
 matic tubes, and (79742 ~~l) ~o - ' for orifices in 
 
 thin plates. These are to each other as -508 to 
 054 x 1-431, or as 5-08 to -773, that is to .say, 
 the loss of mechanical power arising from the re- 
 sistance in passing through short tubes is 6 '5 7 times 
 as great as when the water passes through thin plates 
 or mouth-pieces, as in Fig. 4 ; and the discharging 
 mechanical power in plates, is to that in tubes as 
 1*431 to 1, or as 1 : *698, the heads and quantities 
 discharged being the same^ ol9V ^ j 
 
 The whole loss of mechanical power in the passage 
 is 5*4 per cent, for the plates, and about 51 per cent. 
 for short tubes. If the loss compared with the whole 
 head be sought, we get, when v is the discharging 
 
 velocity, ^j for the theoretical velocity due to the 
 
 mi& oilBflighq ho/Ia irgiJO'iJ) ouc8S./$q 9|[J 
 
 head in short tubes, and its square Tgrp = TgTjo 
 
 is as the whole head ; therefore, the whole head is to 
 
 .9iHa oil,t gniod ssiiioolsv guigidBdoaibi^dJ bn^j 
 
 the head due to the discharging velocity as ^ to v\ 
 
 or as 1 to *663 ; and as -508 is the coefficient of re- 
 sistance* for the discharging velocity, '508 x '663 
 zz '337 is the coefficient of resistance due to the 
 
 :injJaiao'i i 
 * Table, p. 171. ,jTl brm iTJ .qq 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 173 
 
 whole head ; this is equal to a loss of 34 per cent, 
 nearly, or about one-third. In like manner, we find 
 974 2 x '054 = -0512 for the coefficient when the dis- 
 charge takes place through thin plates, or 5| per 
 cent, of the whole head. 
 
 DIAPHRAGMS! '^> ^ 
 
 When a diaphragm, o R, Fig. 27, is placed at the 
 entrance of a short tube, we have shown, page 168, 
 
 A 2 
 
 janiwollirt orfKIwijsIiiofBo r (^ c ~ 1 )^ A ui _ 
 that a loss of head equal - -^ - takes place 
 
 when v is the discharging velocity, whence the co- 
 efficient of resistance is equal to ( l) ,* according 
 
 .31/roAar-fiAirt ao't aD/tATaraaa ax A ,sfo;. ^ ^c 
 
 to our definition. The coefficient of contraction c c , 
 as we have before shown, page 170, should be taken 
 equal to -590 in the application of formula (63) ; and, 
 as it must also be taken equal to about *621 when the 
 area of the tube i'ls very large compared with the 
 
 area a of the orifice o R in the diaphragm, we may 
 
 
 
 assume that when is equal to 
 
 A 
 
 0, .!. -2, -3* -4, -5, -6, -7, -8, -9, and 1 
 
 O.aoU^jTLiilO;! 10 JiTQi . , , , , , , , 
 
 successively, the coefficient c c must be taken equal to 
 
 621, -TO^HO? 48, '609, -606, -603, -600, -597, -593, and -590, 
 
 in the same order. As the approaching section c 
 
 ' * tor the sudden alteration in the velocity passing through a 
 diaphragm, we must reject the hypothesis of D'Aubuisson, 
 Traite d'Hydraulique, p. 238, and adopt that of Navier, taking 
 the loss of head to correspond to the square of the difference and 
 not to the difference of the squares of the velocities in and after 
 passing the orifice. The coefficient of contraction must, however, 
 be varied to suit the ratio of the channels, as it is in this and the 
 following pages. 
 
174 
 
 THE DISCHARGE OF WATER FROM 
 
 may be considered exceedingly large, the value of 
 the coefficient of discharge or velocity, as the tube 
 o R s T is supposed full, in equation (61), becomes 
 
 (66.) * 
 
 and the coefficient of resistance 
 
 ( 67 -) 
 
 from which equations and the above values of c c , cor- 
 responding to -, we have calculated the following 
 
 A. 
 
 values of the coefficients of discharge and resistance 
 through the tube o K s T, Fig. 27. 
 
 COEFFICIENTS OF CONTRACTION, DISCHARGE, AND RESISTANCE FOR DIAPHRAGMS. 
 
 Ratio 
 
 1 
 
 1 
 
 1 
 
 Ratio 
 
 1 
 
 "S 
 .2 
 
 1 
 
 a 
 
 6 ^ 
 
 s j 3 
 
 I* 
 
 a 
 
 3 fe 
 
 I* 
 
 ss *** 
 
 A * 
 
 1 
 
 1 
 
 1 
 
 A * 
 
 1 
 
 1 
 
 1 
 
 o-o 
 
 621 
 
 000 
 
 infinite. 
 
 0-6 
 
 603 
 
 493 
 
 3-115 
 
 0-1 
 
 618 
 
 066 
 
 231- 
 
 0-7 
 
 600 
 
 587 
 
 1-907 
 
 0-2 
 
 615 
 
 139 
 
 50-8 
 
 0-8 
 
 597 
 
 675 
 
 1-198 
 
 0-3 
 
 612 
 
 219 
 
 . 19-8 
 
 0-9 
 
 593 
 
 753 
 
 762 
 
 0-4 
 
 609 
 
 307 
 
 9-6 
 
 1-0 
 
 590 
 
 821 
 
 483 
 
 0-5 
 
 606 
 
 399 
 
 5-3 
 
 - 
 
 
 
 
 
 
 In this table c c is the coefficient of contraction, c d 
 the coefficient of discharge, suited to the larger section 
 of the pipe A, at s T ; and c r the coefficient of re- 
 sistance. The discharge is found from equation (61), 
 as c is here very large compared with A, to be 
 
 1 
 
 (67A.) D = A 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 175 
 
 A ^ 
 
 The coefficient of resistance c r is here equal ( 1) , 
 
 and the coefficient of discharge c 6 zz ^r-- -TI-* 
 
 v* T c *) 
 
 The tube must be so placed, that the water, after 
 passing the diaphragm, shall fill it ; for instance, 
 between two cisterns, when the height h must be 
 measured between the water surfaces, or when the 
 tube is sufficiently long to be filled ; in this case, 
 however, the height must be determined from the dis- 
 charging velocity , as a portion of the head is required 
 to overcome the friction, which we shall have im- 
 mediately to refer to more particularly. 
 
 The table shows that the head due to the resistance 
 is 5*3 times that due to the discharging velocity, 
 when the area of the diaphragm is half the area of 
 the tube ; that is, the whole head required is 6-3 
 times that due to the velocity, and that the coeffi- 
 cient of discharge is reduced to -399. In order to 
 find the coefficients suited to the smaller area of the 
 orifice in the diaphragm o R, when it is to be used in 
 calculations of the discharge, we have only to divide 
 
 the numbers corresponding to into those of c d , op- 
 
 A 
 
 posite to them in the table. Thus, when - zz -8, we 
 
 A 
 
 have the coefficient of discharge suited to the area a, 
 
 * For the loss sustained by contraction in the bore of a pipe 
 by a diaphragm, see equations (123), (124), and (125). The 
 actual value of c c in equation (67A) depends on the thickness of 
 the diaphragm as well as on the relation of a and A. The form 
 of the orifice a also affects the value of c c . 
 
178 
 
 THE DISCHARGE OF WATER FROM 
 
 67 
 
 Fig-,28 
 
 od i 
 
 equal -^- = -844, and so of other values of the 
 ratio -. The cc efficients in the table, page 174, are 
 
 A 
 
 for the larger orifice A in the formula D = A c d \/2 g h. 
 
 ^onuiaflKHOWi TUBES i SOBLIQUE AT THE JUNCTION. 
 
 When a tube is at- fp 
 tached obliquely, as in 
 Fig. 28, we have found 
 that if the number of 
 degrees in the angle 
 TO s, formed by the 
 direction of the tube o s, with the perpendicular o T, 
 be represented by <, then -814 -0016 <j> will give the 
 coefficient of discharge corresponding to the obliquely 
 attached short tube in the Figure. This formula is, 
 however, empirical, but it is simple, and agrees pretty 
 closely with experimental results. As the coefficient 
 
 [ . .UGG- . 1 '(. at as'fGf/Dsib lo irr^b 
 
 of resistance is equal -, 1, equation (64), we have 
 odi To -co isucma 9dKc3 Dan 
 
 here ^r ^i , __ -6oi6 V> 2 ~" ^ ; ^ rom ^ ese equations 
 we Have calculated the following table for heads 
 measured to the middle of the outside orifice ; :-rm(f 
 
 COEFFICIENTS OF DISCHARGE AND RESISTANCE FOR OBLIQUE JUNCTIONS. 
 
 in degrees. 
 
 Coefficient 
 of discharge. 
 
 Coefficient 
 of resistance. 
 
 in degrees. 
 
 Coefficient 
 of discharge. 
 
 Coefficient 
 of resistance. 
 
 
 
 814 
 
 508 
 
 35 
 
 758 
 
 740 
 
 5 
 
 806 
 
 539 
 
 40 
 
 750 
 
 778 
 
 10 
 15 
 20 
 
 "E 
 
 569 
 
 603 sr 
 
 635 (A 
 
 45 
 
 i B i ,-'742' - 
 734 
 
 81<6 [ 
 3,^87 
 
 25 
 
 . h7T.4 E '! 
 
 ingJWth-i 
 
 60 
 
 718 
 
 940;;ili 
 
 30 
 
 766 
 
 704 
 
 ilV tidj gfO 
 
 710 
 
 984 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 177 
 
 The coefficient of resistance for a tube at right angles 
 to the side, is to the like coefficient when it makes 
 an angle of 45 degrees as -508 to -816, or as 1 to 1-6 
 nearly ; and the loss of head is greater in the same 
 proportion. If the short tube be more than three or 
 four diameters in length, friction will have to be taken 
 into account. The head h is measured to the outside 
 orifice. 
 
 FORMULA FOR FINDING THE TIME THE SURFACE OF WATER 
 IN A CISTERN TAKES TO SINK A GIVEN DEPTH. DIS- 
 CHARGE FROM ONE VESSEL OR CHAMBER INTO ANOTHER. 
 LOCK CHAMBERS. 
 
 In experiments for find- 
 ing the value of the coeffi- 
 cients of discharge, one of 
 the best methods is to ob- 
 serve the time the water 
 discharged from the orifice 
 takes to sink the surface in a prismatic cistern a given 
 depth; the ratio of the observed to the theoretical 
 time will then give the coefficient sought. A formula 
 for finding the theoretical time is, therefore, of much 
 practical value. In Fig. 29, the time of falling from 
 s t to s T, in seconds, is 
 
 
 1 
 
 F 1 i *F 
 
 Fi.29 
 
 
 
 _ _ 
 
 ---i 
 
 
 1 
 
 
 
 . 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 ' 
 
 R 
 
 ...^ 
 
 
 
 
 
 in which a is the area of the orifice o R, and A the 
 area of the prismatic vessel at st or ST; this formula is 
 for measures in feet. For measures in inches, we have 
 
 EXAMPLE VII. A cylindrical vessel 5 *74 inches in 
 diameter has an orifice *2 inch in diameter at a depth 
 
178 
 
 THE DISCHARGE OF WATER FROM 
 
 0/"16 inches below the surface, measured to the centre ; 
 it is found that the water sinks 4 inches in 51 seconds ; 
 what is the coefficient of discharge ? 
 
 The theoretical time t is found from equation (69), 
 equal 
 
 5-74*X-785 6> _ ^3^ 
 
 JL ij y X *^j X io04: "OOO 
 
 1 T.fJKfJf? Q~f .Q 
 
 = -556 ~ X '5359 =31-8 seconds; hence,-gpzz-624 
 
 is the coefficient sought. When the orifice o R and 
 the horizontal section of the vessel are similar figures, 
 
 A S T 2 
 
 - is equal ^ ; and therefore, for circular cisterns 
 
 tt OR 
 
 and orifices, it is unnecessary to introduce the mul- 
 tiplier -7854. 
 
 We have given above, formulae for -the time water in 
 a prismatic vessel takes to fall a given depth, when dis- 
 
 Fig. 29<*. 
 
 charged from an orifice at the side or bottom. The time 
 
 ' the surface s T, diagram 1, Fig. 29#, takes to rise 
 
 to s t, when supplied through an orifice or tube o R, 
 
 from an upper large chamber or canal, whose surface 
 
 2 A/* 
 
 s 1 tf remains always at the same level, is "TTo^'* 
 
 * The time of rising from s to s is exactly double the time it 
 would take, if the pressure/ remained uniform, to fill the same 
 depth below R. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 179 
 
 and we thence get the time of rising from R to s for 
 measures in feet 
 
 and for measures in inches 
 
 ( 69B -) '= 
 
 in which A is the area of the horizontal section at s T ; 
 a the sectional area of the communicating channel or 
 orifice o R ; c d the coefficient of discharge suited to it, 
 and ^ and/, as shown in the diagram. 
 
 In order to find the time of filling the lower vessel 
 to the level s T, supposing it at first empty, we have 
 the contents of the portion below o R equal to A^ 2 > 
 and the time of filling it equal to 
 
 (69c.) 
 
 8-025 c d ah\ 
 
 then the time of filling up to any level s T, for mea- 
 sures in feet, is equal to the sum of (A) and (c) ; 
 that is, 
 
 8-025 c d ah$ 
 and for measures in inches 
 
 27-8 c d a 
 "When s T coincides with * t 
 
 A (2 A, + fl) 
 
 T = 8-025 c,ahi> 
 for measures in feet, and 
 
 N3 
 
180 THE DISCHARGE OF WATER FROM 
 
 A (2 fr + ft,) 
 
 27-8 c,ah\ > 
 
 for measures in inches. These equations are ex- 
 actly suited to the case of a closed lock-chamber 
 filled from an adjacent canal. 
 
 When the upper level s' T' is also variable, as in 
 Diagram 2, the time which the water in both vessels 
 takes to come to the same uniform level s' t' s t, is 
 2AQ(A 1 +/ 1 -j)* = 2AO(/+/ 1 )*_ 
 ' c d a(A+c)v/2^ c d tf(A+c)v/2#' 
 in which h + jfj h = f -f f^ is the difference of 
 levels at the beginning of the flow ; c the horizontal 
 section of the upper chamber ; and the other quanti- 
 ties as in Diagram 1. As c^ =: Af, we find 
 
 Now, in order to find the time of falling a given 
 depth d below the first level s' T', we have the head 
 above s't's t equal to^ d in the upper vessel, and 
 the depth below it in the lower vessel equal to 
 
 ^ -- ; whence the difference of levels in the two 
 
 A 
 
 vessels at the end of the fall d, is 
 
 The time of falling through d is, therefore, from 
 equation (69 H), 
 
 2AC 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 181 
 
 in which \^g = 8*025 for measures in feet, and 
 equal 27*8 for measures in inches. The whole time 
 of filling to a level the lower empty vessel, is found 
 by adding the time of filling the portion below R, de- 
 termined in a manner similar to equations (68) and 
 (69) to be 
 
 to the time of filling above R, given in equation (H), 
 when h is taken equal to zero. Equations (H), (i), 
 and (K) are applicable to the case of the upper and 
 lower chambers of a double lock, after making the 
 necessary change in the diagrams. 
 
 The above equations require further extensions 
 when water flows into the upper vessel while also 
 flowing from it into the lower ; such extensions are, 
 however, of little practical value, and we therefore 
 omit them. For sluices in flood-gates with square 
 arrises, c d may be taken at about -545, but with 
 rounded arrises, the coefficient will rise much higher. 
 See SECTIONS III. and VI. 
 
182 THE DISCHARGE OF WATER FROM 
 
 SECTION VIII. 
 
 FLOW OF WATER IN UNIFORM CHANNELS. MEAN VELOCITY. 
 MEAN RADII AND HYDRAULIC MEAN DEPTHS. BORDER. 
 TRAIN. HYDRAULIC INCLINATION. EFFECTS OF FRIC- 
 TION. FORMULA FOR CALCULATING THE MEAN VELOCITY. 
 APPLICATION OF THE FORMULA AND TABLES TO THE 
 SOLUTIONS OF THREE USEFUL PROBLEMS. 
 
 In rivers the velocity is a maximum along the 
 central line of the surface, or, more correctly, over 
 the deepest part of the channel ; and it decreases 
 thence to the sides and bottom : but when backwater 
 arises from any obstruction, either a submerged 
 weir, Fig. 22, or a contracted channel, Fig. 23, the 
 velocity in the channel approaching the obstruction 
 is a maximum at the depth of the backwater below 
 the surface, and it decreases thence to the surface, 
 sides, and bottom. When water flows in a pipe of 
 any length, the velocity at the centre is greatest, and 
 it decreases thence to the sides or circumference of 
 the pipe. If the pipe be supposed divided into two 
 portions in the direction of its length, the lower por- 
 tion or channel will be analogous to a small river 
 or stream, in which the velocity is greatest at the 
 central line of the surface, and the upper portion will 
 be simply the lower reversed. A pipe flowing full 
 may, therefore, be looked upon as a double stream, 
 and we shall soon see that the formulae for the dis- 
 charge from each kind are all but identical, though a 
 pipe may discharge full at all inclinations, while the 
 inclinations in rivers or streams, having uniform 
 motion, never exceed a few feet per mile. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 183 
 
 MEAN VELOCITY. 
 
 It is found, by experiment, that the mean velocity 
 is nearly independent of the depth or width of the 
 channel, the central or maximum velocity being the 
 same. From a number of experiments, Du Buat 
 derived empirical formulae equivalent to 
 
 v= V f=v^ v*+i, v b =(v* - I) 2 , and v=(v|+l) 2 ; 
 
 in these equations v is the mean velocity, v the max- 
 imum surface velocity, and v b the velocity at the 
 sides, or bottom, expressed in French inches. Tables 
 calculated from these formulae do not give correct 
 results for measures in English inches, though they 
 are those generally adopted. Disregarding the dif- 
 ference in the measures, which are as 1 to 1*0678, it 
 will be found that, in the generality of channels, the 
 mean velocity is not an arithmetical mean between 
 the velocity at the central surface line and that at 
 the bottom, though nearly so between the mean 
 bottom and mean surface velocities. Dr. Young,* 
 modifying Du Buat's formula, assumes for English 
 inches that v + v* zz v, and hence vzzv + i (v + 1)* 
 This gives results very nearly the same as the other 
 formula for v, but something less, particularly for 
 small surface velocities. For instance, Du Buat's 
 formula gives 5 inch for the mean velocity when 
 the central surface velocity is 1 inch, whereas Dr. 
 Young's makes it -38 inch. For large velocities both 
 formulas agree very closely, disregarding the differ- 
 ence between the measures, which is only seven per 
 
 * Philosophical Transactions, 1808, p. 487. 
 
184 THE DISCHARGE OF WATER FROM 
 
 cent. They are best suited to very small channels or 
 pipes, but unless at mean velocities of about 3 feet 
 per second, they are wholly inapplicable to rivers. 
 Prony found, from Du Buat's experiments, that 
 
 /2-37187 + vx 
 for measures in metres v (Q.I^QTO ! ) v > m which 
 
 v is also the maximum surface velocity. This, re- 
 duced for measures in English feet, becomes 
 
 and for measures in English inches, 
 93*39 
 
 For medium velocities v = -81 v. The experiments 
 from which these formulae were derived were made 
 with small channels. We have calculated the values 
 of v from that of v, equation (71), and given the 
 results in columns 3, 6, and 9, in TABLE VII. 
 Ximenes, Funk, and Briinning's experiments in 
 larger channels give the mean velocity at the centre 
 of the depth equal '914 v, when the central or 
 maximum surface velocity is v ; but as the velocity 
 also decreases in nearly the same ratio at the surface 
 from the centre to the sides of the channel, we shall 
 get the mean velocity in the whole section equal 
 
 * Francis, Lowell Experiments/p. 150, finds this formula to 
 give 1 5 per cent, less than the result found hy weir measurement 
 from the formula D = 3 '33 (I ! n h) hi, the quantity discharged 
 being about 250 cubic feet per second, and the velocity about 3 '2 
 feet. It appears, however, that Francis uses the mean surface 
 velocity, and not the maximum surface velocity required by the 
 formula : if the latter were used, the difference would be reduced 
 to 6 per cent., or thereabouts, in equation (72). 
 
^ , V*X^. fc^X^oL V^UStAjfc-^jj. 
 
 JU 
 
 2. 
 
 v v 
 
 ^ 
 
 "^<\\\, 
 
 . 
 
 0. 
 
 . 
 
 * C . 
 
 Q 
 O 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 185 
 
 914 x '914 v z= '835 v ; and hence we have, for 
 
 large channels, 
 
 (72.) v = -835 v, 
 
 in which equation v is the maximum velocity at the 
 
 surface. We have also calculated the values of v 
 
 from this formula, and given the results in columns 
 
 2, 5, and 8 of TABLE VII. This table will be found 
 
 to vary considerably from those calculated from Du 
 
 Buat's formula in French inches, hitherto generally 
 
 used in this country, and much more applicable for 
 
 all practical purposes. 
 
 MEAN KADITJS. HYDRAULIC MEAN DEPTH. BORDER. 
 COEFFICIENT OF FRICTION. 
 
 If, in the diagrams 1 and 
 2, Fig. 30, exhibiting the 
 sections of cylindrical and 
 rectangular tubes filled with 
 flowing water, the areas be 
 divided respectively by the perimeters A c B D A and 
 A B D c A, the quotients are termed " the mean radii" 
 of the tubes, diagrams 1 and 2 ; and the perimeters 
 in contact with the flowing water are termed "the 
 borders." In the diagrams 3 and 4, the surface A B 
 is not in contact with the channel, and the width of 
 the bed and sides, taken together, A c D B, becomes 
 " the border." " The mean radius " is equal to the 
 area A B D c A divided by the length of the border 
 A c D B. " The hydraulic mean depth" is the same 
 as " the mean radius" this latter term being perhaps 
 most applicable to pipes flowing full, as in diagrams 
 1 and 2 ; and the former to streams and rivers which 
 
186 THE DISCHARGE OF WATER FROM 
 
 have a surface line A B, diagrams 3 and 4. We shall, 
 throughout the following equations, designate the 
 value of the " mean radius," " hydraulic mean depth," 
 
 . area A B D c A Jt 
 or quotient, border BDCA ? b J the letter r > remarking 
 
 here that for cylindrical pipes flowing full, or rivers 
 with semicircular beds, it is always equal to half the 
 radius, or one-fourth of the diameter. 
 
 Du Buat was the first to observe that the head 
 due to the resistance of friction for water flowing in 
 a uniform channel increased directly as the length of 
 the channel /, directly as the border, and inversely 
 as the area of the cross-flowing section,! very nearly ; 
 
 that is, as -. It also increases as the square of the 
 
 velocity, nearly; therefore the head due to the re- 
 
 v* I v z I 
 
 sistance must be proportionate to - If c t X ^ 
 
 zz h { , then c f is the coefficient for the head due to the 
 resistance of friction, as h f is the head necessary to 
 overcome the friction ; c is therefore termed " the co- 
 efficient of friction" 
 
 * M. Girard has conceived it necessary to introduce the coeffi- 
 cient of correction 1-7 as a multiplier to the border for finding r, 
 to allow for the increased resistance from aquatic plants ; so that, 
 according to his reduction, 
 
 area 
 
 A* .... . , 
 
 1*7 border* 
 
 See Eennie's First Keport on Hydraulics as a Branch of En- 
 gineering ; Third Eeport of the British Association, p. 167 ; also, 
 equation (85), p. 201. 
 
 f Pitot had previously, in 1796, remarked that the diminution 
 arising from friction in pipes is, cateris paribus, inversely as the 
 diameters. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 187 
 
 HYDRAULIC INCLINATION. TRAIN. 
 
 If / be the length of a pipe or channel, and h { the 
 height due to the resistance of friction of water 
 
 flowing in it, then -/ is the hydraulic inclination. In 
 i 
 
 Fig. 31 the tubes A B, c D, of the same length /, and 
 
 Fig.31 
 
 whose discharging extremities B and D are on the 
 same horizontal plane B D, will have the same hy- 
 draulic inclination and the same discharge, no matter 
 what the actual inclinations or the depth of the en- 
 trances at A and c may be, so they be of the same 
 kind and bore ; and as the velocities in A B and c D 
 are the same, the height h due to them must be the 
 same when the circumstances of the orifices of entry 
 A and c are alike. We have the whole head H h + 
 h f (see pp. 166 and 167). The hydraulic inclination is 
 not therefore the whole head H, divided by the length 
 / of the pipe, as it is sometimes mistaken, but the 
 height h f (found by subtracting the height h, due to 
 the entrance at A or c, and the velocity in the pipe, 
 from the whole height) divided by the length /. 
 When the height h is very small compared with the 
 whole height H, as it is in very long tubes with 
 
 moderate heads, j may be substituted for j without 
 error ; but for short pipes up to 100 feet in length 
 
188 
 
 THE DISCHARGE OF WATER FROM 
 
 the latter only should be used in applying Du Buat's 
 and some other formulae ; otherwise the results will 
 be too large, and only fit to be used approximately 
 in order to determine the height h from the velocity 
 of discharge thus found. When the horizontal pipe 
 c D, Fig. 32, is equal in every way to the inclined 
 
 pipe A B, and the head at A is that due to the velocity 
 in c D, the discharge from the pipe A B will be equal 
 to that from c D ; but a peculiar property belongs 
 to the pipe A B in the position in which it is here 
 placed ; for if we cut it short at any point e, or 
 lengthen it to any extent, to E, the discharge will re- 
 main the same and equal to that through the hori- 
 zontal pipe c D. The velocity in A B at the angle of 
 inclination ABC, when A c = h { , and A B zi c D, is 
 therefore such that it remains unaffected by the 
 length A E or A e, to which it may be extended or 
 cut short ; and at this inclination the water in the 
 pipe A B is said to be " in train." In like manner a 
 river or stream is said to be "in train" when the 
 inclination of its surface bears such a relation to 
 the cross section that the mean velocity is neither 
 decreased nor increased by the length of the chan- 
 nel ; and we perceive from this that the acceleration 
 caused by the inclination is exactly counterbalanced 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 189 
 
 by the resistances to the motion when the moving water 
 in a pipe or river channel is in train. 
 
 o 
 
 As h = (1 + r ) o where c r is the coefficient of 
 the height due to the resistance at the orifice of entry 
 
 V 2 l 
 
 A or c, and h { = c { * , we therefore get 
 
 (73.) H = (l+c t ) + c f X r 
 
 and hence we find the mean velocity of discharge 
 
 as c\ zz j-y, equation (65). We have also 
 
 C T -f- 1 
 
 (74B.) v z 
 
 the values of the second member on the right hand 
 side of this equation, or of 
 
 fcM?feb$ 58S 
 
 UiP r] 
 
 are given, for different values of c , c d , and -, in the 
 small table at p. 152, and below at p. 191. 
 
 When h is small compared with A f , or, which comes 
 
 to the same thing, 1 + c t small compared with c t x - f 
 
190 THE DISCHARGE OF WATER FROM 
 
 Iv* 
 
 (75.) v = c t 
 
 and 
 
 TT 
 
 If, in the last equation, we substitute s for y, equal 
 
 the .sine of the angle of inclination A B c, we then 
 have 
 
 (77.) || . = 
 
 The average value of c f for all pipes with straight 
 channels, with velocities of about 1*5 foot per second, 
 is -0069914, from which we find equation (77) be- 
 comes, for measures in feet, 
 
 (78.) v = 96 \/rs. 
 
 As the mean value of the coefficient of resistance c y 
 for the entrance into a tube is '508, and as 
 2# z= 64-403, and c f = -0069914, equation (74), for 
 measures in feet, becomes 
 
 f 64 -403 H U 
 
 v= ~~T , or 
 
 1-508 + -0069914- 
 
 -0234r+-OOOi085/J ~(234-hl-085/ 
 
 (79.) 
 
 * \\ 
 
 58c/+l-085/J * 
 
 This, multiplied by the section, gives the discharge. 
 
 For velocities between 2 and 2J feet per second, 
 c t zz -0064403, and therefore 
 
ORIFICES, WEIRS, PIPES, AND RIVERS, 
 Hr ^1 
 
 191 
 
 UI 50 
 
 0234 r + -0001 / 
 in which d = 4 r i= diameter of a pipe. 
 
 The following table is calculated from equation 
 
 VALUES or - 
 
 3 + * X - 
 
 C 7* 
 
 N umber of diame- 
 cers in the length 
 of the pipe. 
 
 Corresponding coeffici- 
 ents of discharge. 
 
 Number of diame- 
 ters in the length 
 of the pipe. 
 
 Corresponding coeffici- 
 ents of discharge. 
 
 2 diameters 
 
 986 
 
 814 
 
 715 
 
 900 diameter 
 
 239 
 
 236 
 
 233 
 
 5 
 
 957 
 
 791 
 
 698 
 
 950 " + 
 
 234 
 
 230 
 
 227 
 
 10 
 
 919 
 
 769 
 
 683 
 
 1000 
 
 228 
 
 225 
 
 222 
 
 15 
 
 886 
 
 749 
 
 669 
 
 noo 
 
 218 
 
 215 
 
 213 
 
 20 
 
 855 
 
 731 
 
 656 
 
 1200 
 
 209 
 
 207 
 
 205 
 
 25 
 
 828 
 
 713 
 
 643 
 
 1400 
 
 194 
 
 192 
 
 191 
 
 30 
 
 804 
 
 698 
 
 632 
 
 1600 
 
 182 
 
 180 
 
 179 
 
 35 
 
 781 
 
 683 
 
 620 
 
 1800 
 
 173 
 
 171 
 
 170 
 
 40 
 
 760 
 
 668 
 
 610 
 
 2000 
 
 163 
 
 162 
 
 161 
 
 45 
 
 741 
 
 655 
 
 600 
 
 2200 
 
 156 
 
 155 
 
 154 
 
 50 
 
 723 
 
 643 
 
 590 
 
 2400 
 
 149 
 
 149 
 
 148 
 
 100 
 
 595 
 
 548 
 
 514 
 
 2600 
 
 144 
 
 143 
 
 142 
 
 150 
 
 518 
 
 485 
 
 462 
 
 2800 
 
 139 
 
 138 
 
 137 
 
 200 
 
 464 
 
 440 
 
 422 
 
 3000 
 
 134 
 
 133 
 
 133 
 
 250 
 
 424 
 
 405 
 
 391 
 
 3200 
 
 130 
 
 129 
 
 129 
 
 300 
 
 392 
 
 378 
 
 366 
 
 3400 
 
 126 
 
 125 
 
 125 
 
 350 
 
 367 
 
 356 
 
 345 
 
 3600 
 
 122 
 
 121 
 
 121 
 
 400 
 
 346 
 
 336 
 
 329 
 
 3800 
 
 119 
 
 119 
 
 118 
 
 450 
 
 329 
 
 319 
 
 314 
 
 4000 
 
 116 
 
 116 
 
 115 
 
 500 
 
 314 
 
 307 
 
 300 
 
 4200 
 
 113 
 
 113 
 
 113 
 
 550 
 
 301 
 
 295 
 
 289 
 
 4400 
 
 111 
 
 111 
 
 111 
 
 600 
 
 289 
 
 283 
 
 278 
 
 4600 
 
 108 
 
 108 
 
 108 
 
 650 
 
 279 
 
 273 
 
 269 
 
 4800 
 
 106 
 
 106 
 
 106 
 
 700 
 
 269 
 
 265 
 
 261 
 
 5000 
 
 104 
 
 104 
 
 104 
 
 750 ;-;, 
 
 261 
 
 257 
 
 253 
 
 5200 
 
 102 
 
 102 
 
 102 
 
 800 
 
 253 
 
 249 
 
 246 
 
 5400 
 
 100 
 
 100 
 
 100 
 
 850 
 
 246 
 
 242 
 
 239 
 
 5600 
 
 098 
 
 098 
 
 098 
 
192 THE DISCHARGE OF WATER FROM 
 
 (74B) for a velocity of about 20 feet per second, 
 when c f zr -004556, and for different orifices of entry, 
 in which c d varies from -986 for a rounded orifice, 
 to '715 when the pipe projects into the vessel. It 
 gives directly the coefficient, which, multiplied by 
 
 H, gives the velocity in the pipe, taking friction 
 into account. 
 
 The small table, SECTION VI., p. 152, gives the like 
 coefficients of v/2#H in equation (74B), when c f = 
 00699 suited to a velocity of about 18 inches per 
 second, and can be applied in like manner. The 
 value of \/2#H is given, in inches, in column 2, 
 TABLE II. For feet it is equal 8v/i nearly. 
 
 DU BUAT'S FORMULA. 
 
 The coefficient of friction c t is not, however, con- 
 stant, as it varies with the velocity. That which 
 we have just given answers for pipes when the 
 velocity is 20 feet per second. For pipes and 
 rivers it is found to increase as the velocity de- 
 creases ; that is, the loss of head is proportionately 
 greater for small than for large velocities. Du Buat 
 found the loss of head to be also greater for small 
 than large channels, and applied a correction accord- 
 ingly in his formula. This, expressed in French 
 inches, is 
 
 (80 .) = - - , - -3 (H - -I), 
 
 (^-hyp-iog^-M-e/ 
 
 maintaining the preceding notation, in which s j. 
 
 In this formula -1, in the numerator of the first term, 
 is deducted as a correction due to the hydraulic 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 193'*ITY 
 
 \s^U ff 
 
 mean depth, as it was found that 297 (ri 0-1) agreed 
 more exactly with experiment than 297 rf simply. The 
 
 f \ \* 
 
 second term hyp. log. (^ + I'^J, of the denominator 
 
 is also deducted to compensate for the observed loss 
 of head being greater for less velocities, and the last 
 term -3 (rl _ -1) is a deduction for a general loss of 
 velocity sustained from the unequal motions of the 
 particles of water in the cross section as they move 
 along the channel. These corrections are empirical ; 
 they were, however, determined separately, and after 
 being tested by experiment, applied, as above, to the 
 radical formula v = 297 \/r~s. 
 
 Du Buat's formula was published in his Principes 
 d'Hydraulique, in 1786. It is, as we have seen, 
 partly empirical, but deduced by an ingenious train 
 of reasoning and with considerable penetration from 
 about 125 experiments, made with pipes from the 
 19th part of an inch to 18 inches in diameter, laid 
 horizontally, inclined at various inclinations, and 
 vertical ; and also from experiments on open chan- 
 nels with sectional areas from 19 to 40,000 square 
 inches, and inclinations of from 1 in 112 to 1 in 
 36,000. The lengths of the pipes experimented with 
 varied from 1 to 3, and from 3 to 3600 feet. 
 
 In several experiments by which we have tested 
 this formula, the resulting velocities found from it 
 were from 1 to 5 per cent, too large for small pipes, 
 and too small for straight rivers in nearly the same 
 proportion. As the experiments from which it was 
 derived were made with great care, those with pipes 
 particularly so, this was to be expected. Expe- 
 
194 THE DISCHARGE OF WATEK FROM 
 
 riments with pipes of moderate or short lengths 
 should have the circumstances of the orifice of entry 
 from the reservoir duly noted ; for the close agree- 
 ment of this formula with them must depend a great 
 deal, in such pipes, on the coefficient due to the 
 height h, which must be deducted from the whole 
 
 head H before the hydraulic inclination, -j- zz s, can 
 
 be obtained ; but for very long pipes and uniform 
 channels this is not necessary. 
 
 The experiments from which Du Buat's formula 
 was constructed are given in full by the late Dr. 
 Eobinson in his able article on "rivers" in the 
 Encyclopaedia Britannica, pp. 268, 269, and 270, 
 where the calculated and observed velocities are 
 placed side by side in French inches per second. In 
 all these experiments Du Buat carefully deducted 
 the head due to the velocity and orifice of entry before 
 finding the hydraulic inclination s, and those who 
 attempt to calculate the velocity from the head and 
 length of the channel only, without making this 
 deduction, will find their calculated results very dif- 
 ferent from those there given. If there were bends, 
 curves, or contractions, deductions would have to be 
 made for these in like manner before finding s. 
 
 Under all the circumstances, and after comparing 
 the results obtained from various other formulae, we 
 have preferred calculating tables for the values of v 
 from this formula reduced for measures in English 
 inches, which is 
 
 = 
 
 
OEIFICES, WEIKS, PIPES, AND RIVERS. 195 
 
 or more simply, 
 
 , Q1 x 307 (r 1) 
 
 (81.) v = --- - - -_. 3r *_.i. 
 
 This gives the value of v a little larger than the 
 original formula, but the difference is immaterial. 
 For measures in English feet it becomes 
 (82.) = t 88-51 (r*- -03) _ _ . 084 (fi _ . 03> 
 (-)-hyp.log.(- + l-6)' 
 
 The results of equation (81) are calculated for 
 different values of s and r, and tabulated in TABLE 
 VIII. , the first eight pages of which contain the 
 velocities for values of r varying from Tsth inch to 
 6 inches ; or if pipes, diameters from | inch to 
 2 feet, and of various inclinations from horizontal to 
 vertical. The last five pages contain the velocities 
 for values of r from 6 inches to 12 feet, and with 
 falls from 6 inches to 12 feet per mile. 
 
 EXAMPLE VIII. A pipe, 1J inch diameter and 
 100 feet long, lias a constant head of 2 feet over the 
 discharging extremity ; what is the velocity of dis- 
 charge per second ? 
 
 II 3. , .,100 1 
 
 The mean radius r=- = - inches, and zz50n:-, 
 
 48 2 S 
 
 is the approximate hydraulic inclination. At page 2 
 of TABLE VIII., in the column under the mean radius 
 
 o 
 
 -, and opposite to the inclination 1 in 50, we find 
 
 8 f 
 
 30*69 inches for the velocity sought. This, however, 
 is but approximative, as the head due to the velocity 
 should be subtracted from the whole head of 2 feet, 
 before finding the true hydraulic inclination. This 
 
 o3 
 
196 THE DISCHAKGE OF WATEE FEOM 
 
 head depends on the coefficient of resistance at the 
 entrance orifice, or the coefficient of discharge for a 
 short tube. In all Du Buat's experiments this latter 
 was taken at -8125, but it will depend on the nature 
 of the junction, as, if the tube runs into the cistern, 
 it will become as small as -715 ; and, if the junction 
 be rounded into the form of the contracted vein, it 
 will rise to -974, or 1 nearly. In this case, the co- 
 efficient of discharge may be assumed -815,* from 
 which, in TABLE II., we find the head due to a velo- 
 city of 30-69 inches to be 1- = 1-87 inch nearly, 
 which is the value of h ; and hence, H h h t = 24 
 
 - 1-87 = 22-13 inches ; and| = ^^ = 54 ' 2 =p 
 the hydraulic inclination, more correctly. With this 
 
 O i ' 
 
 new inclination and the mean radius -, we find the 
 
 8 
 
 velocity by interpolating between the inclinations 
 1 in 50 and 1 in 60, given in the table to be 
 30-69 _ 1-34 ~ 29-35 inches per second. This 
 operation may be repeated until v is found to any 
 degree of accuracy according to the formula ; but it 
 is, practically, unnecessary to do so. If we now wish 
 to find the discharge per minute in cubic feet, we can 
 easily do so from TABLE IX., in which, for an inch 
 and a half pipe, we get 
 
 Inches. Cubic Feet. 
 
 For a velocity of 20-00 per second, 1-22718 per minute. 
 9-00 -55223 
 30 -01841 
 04 -00245 
 
 29-34 .1-80027 
 * See EXAMPLE 16, pp. 28, 30. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 197 
 
 The discharge found experimentally by Mr. Provis, 
 for a tube of the same length, bore, and head, was 
 1-745 cubic foot per minute. 
 
 If we suppose the coefficient of discharge due to 
 the orifice of entry and stop-cock in Mr. Provisos 
 208 experiments* with li inch lead pipes of 20, 40, 
 60, 80, and 100 feet lengths, to be '715, the results 
 calculated by the tables will agree with the experi- 
 mental results with very great accuracy, and it is 
 very probable, from the circumstances described, 
 that the ordinary coefficient -815 due to the entry 
 was reduced by the circumstances of the stop-cock 
 and fixing to about -715 ; but even with -815 for the 
 coefficient, the difference between calculation and 
 experiment is not much, the calculation being then 
 in excess in every experiment, the average being 
 about 5 per cent., and not so much in the example 
 we have given. 
 
 TABLE VIII. will give the velocity, and thence the 
 discharge, immediately, for long pipes, and TABLE X. 
 enables us to calculate the cubic feet discharged per 
 minute, with great facility. For rivers, the mean 
 velocity, and thence the discharge, is also found with 
 quickness. See also TABLES XI., XII., and XIII., 
 and the table at pp. 42 and 43. 
 
 EXAMPLE IX. A watercourse is 7 feet wide at the 
 bottom, the length of each sloping side is 6*8 feet, the 
 width at the surface is 18 feet, the depth 4 feet, and 
 the inclination of the surface 4 inches in a mile ; what 
 is the quantity flowing down per minute ? 
 
 * Transactions of the Institution of Civil Engineers, pp. 201, 
 210, vol. ii. 
 
198 THE DISCHAEGE OF WATEK FROM 
 
 /TO I I7\ v> _ 
 
 4 
 
 _ 
 
 ?. =2-4272 feet=29-126 inches 
 
 7 + 2 x 6-8 ""20-6 
 
 zzr, is the hydraulic mean depth ; and as the fall is 
 4 inches per mile, we find at the llth page of TABLE 
 VIII., the velocity v - 12-03 - -16 = 11-87 inches 
 per second ; the discharge in cubic feet per minute 
 is, therefore, 
 
 50 x X 60 = 2967-5. 
 
 12 
 
 If 94-17\/ r s = v, we have v = 94-17-Y/2 -427 + 
 
 15840 
 
 = 94 ' 17 X = =M7 feet = 14-04 inches. 
 
 Watt, in a canal of the fall and dimensions here given, 
 found the mean velocity about 13 inches per second. 
 This corresponds to a fall of 5 inches in the mile, 
 according to the formula. Du Buat's formula is less 
 by 12 J per cent, or ^th ; the common formula too 
 much by 5 per cent. 
 
 In one of the original experiments with which the 
 formula was tested on the canal of Jard, the mea- 
 surements accorded very nearly with those in this 
 
 example, viz. - = 15360, and r = 29*1 French 
 s 
 
 inches ; the observed velocity at the surface was 
 15*74, and the calculated mean velocity, from the 
 formula, 11-61 French inches.* TABLE VII. will 
 give 12-29 inches for the mean velocity, corre- 
 
 * These measures reduced to inches, give r = 31-014, v = 
 12-374; and the surface velocity 16-775 inches; reduced for mean 
 velocity 13-101 inches. 
 
ORIFICES, WEIES, PIPES, AND RIVERS. 199 
 
 spending to a superficial velocity of 15*74 inches. 
 This shows that the formula also gives too small 
 a value for v in this case, by about rrth of the 
 
 result, it being about ; part in the other. The 
 
 8'3 
 
 probable error in the formula applied to straight 
 clear rivers of about 2 feet 6 inches hydraulic mean 
 depth is nearly Ath or 8 per cent, of the tabu- 
 lated velocity, and this must be added for the more 
 correct result ; the watercourse being supposed 
 nearly straight and free from aquatic plants. 
 
 Notwithstanding the differences above remarked 
 on, we are of opinion that the results of this formula, 
 which we have calculated and tabulated, may be 
 more safely relied on as applied to general prac- 
 tical purposes than most of those others which we 
 shall proceed to lay before our readers. Eivers or 
 watercourses are seldom straight or clear from weeds, 
 and even if the sections, during any improvements, 
 be made uniform, they will seldom continue so, as 
 " the regimen" or adaptation of the velocity to the 
 tenacity of the banks, must vary with the soil and 
 bends of the channel, and can seldom continue per- 
 manent for any length of time unless protected. 
 From these causes a loss of velocity takes place, 
 difficult, if not impossible, to estimate accurately, but 
 which may be taken at from 10 to 15 per cent, of 
 that in the clear unobstructed direct channel ; but be 
 this as it may, it is safer to calculate the drainage or 
 mechanical results obtainable from a given fall and 
 river channel, from formula which give lesser, than 
 
200 THE DISCHARGE OF WATER FROM 
 
 from those which give larger velocities. This is a 
 principle engineers cannot too much observe. 
 
 We have before remarked, that for both pipes and 
 rivers the coefficient of resistance increases as the 
 velocity decreases. This is as much as to say, in the 
 simple formula for the velocity, v zz m \/r s, that m 
 must increase with v, and as some function of it. 
 This is the case in TABLE VIII., throughout which the 
 velocities increase faster than \/r, the \/s, or the 
 v/rsT In all formulae with which we are acquainted 
 but Du Buat's and Young's, the velocity found is con- 
 stant when \/r s'or r X s is constant. In Du Buat's 
 formula for r x s constant, v obtains maximum 
 values between r zz f inch and r zz 1 inch ; the dif- 
 ferences of the velocities for different values of r 
 above 1 inch, r x s being constant, are not much. 
 We may always find the maximum value, or nearly 
 so, by assuming r zz J inch, and finding the corre- 
 
 4- 7* c 
 
 spending inclination from the formula ~o~~j which is 
 equal to it. For example, if r 12 inches, and s zz 
 
 , the velocity is found equal 9*52 inches ; but 
 
 when r s is constant, the inclination s corresponding 
 
 4 X 12" 1 
 
 to r zz j- inch is zz , from which we find 
 
 3 X 10560 660' 
 
 from the table vzzlO-25 inches for the maximum 
 velocity, making a difference of fully 7 per cent. 
 
 When r zz -01 of an inch, or a pipe is Ath part 
 of an inch in diameter, Du Buat's formula fails, but 
 it gives correct results for pipes ith of an inch in 
 diameter, and two of the experiments from which it 
 
OEIFICES, WEIES, PIPES, AND EIVERS. 201 
 
 was derived were made with pipes 12 inches long and 
 only -rsth part of an inch in diameter. 
 
 COULOMB having shown that the resistance opposed 
 to a disc revolving in water increases as the function 
 a v + b v* of the velocity v, we may assume that the 
 height due to the resistance of friction in pipes and 
 rivers is also of this form ; and that 
 
 (83.) k t = (av + bv*) 1 -, 
 
 and consequently^ 
 
 (84.) r s av + bv*, and v= 
 
 G-IRARD first gave values to the coefficients a and b. 
 He assumed them equal, and each equal to -0003104 
 for measures in metres, and thence the velocity in 
 canals, 
 
 (85.) v = (3221-016 rs + -25^ -5;* 
 which reduced for measures in English feet becomes 
 
 (v = (10567-8 r s + 2-67)* 1-64, or 
 
 (v = 103 \/Ts 1-64, nearly. 
 The value of a = b = -0003104 was obtained by 
 means of twelve experiments by Du Buat and Chezy. 
 Of course the value is four times this in the original, 
 as we use the mean radius in all the formulae instead 
 of the diameter. This formula is only suited for 
 very small velocities in canals, between locks, con- 
 taining aquatic plants ; it is inapplicable to rivers 
 and channels in which the velocity exceeds an inch 
 per second. 
 
 PRONY found from thirty experiments on canals, 
 
 * See Brewster's Encyclopedia, Article Hydrodynamics, p. 259. 
 
202 THE DISCHAEGE OF WATER FROM 
 
 that a = -000044450 and b = 000309314,* for mea- 
 sures in metres, from which we find 
 (87.) v = (3232-96 r s + -005 16)* -0719 ; 
 this reduced for measures in English feet is, 
 
 rt> = (10607-02 r* + -0556)* -236 ;f or 
 
 (v = 103 \/rs -24 nearly : 
 
 the velocities did not exceed 3 feet per second in the 
 experiments from which this was derived. 
 
 For pipes, Prony found,J from fifty-one experi- 
 ments made by Du Buat, Bossut, and Couplet, with 
 pipes from 1 to 5 inches diameter, from 30 to 7,000 
 feet in length, and one pipe 19 inches diameter and 
 nearly 4,000 feet long, that a = -00001733, and b 
 zz -0003483, from which values 
 (89.) v = (2871-09 rs + -0006192)* -0249, 
 for measures in metres, and for measures in English 
 
 feet, 
 
 (v = (9419-75 rs + -00665)* '0816; or 
 ( ' (v = 97 v/ '08 nearly. 
 
 Prony also gives the following formula applicable 
 to pipes and rivers. It is derived from fifty-one 
 selected experiments with pipes, and thirty-one with 
 open channels : 
 (91.) v = (3041-47 rs + -0022065)* -0469734,g 
 
 * Reeherches Physico-Mathe'matiques sur la ThSorie des Eaux 
 Courantes. 
 
 j- For canals containing aquatic plants, reeds, &c., we must sub- 
 stitute jTy for r. See note, p. 186. 
 
 J Keeherches Physico-Mathematiques sur la The"orie du Mouve- 
 ment des Eaux Courantes, 1804. 
 
 Keeherches Physico-Mathe'inatiques sur la Theorie des Eaux 
 Oourantes. A reduction of this formula into English feet is given 
 at page 6, Article Hydrodynamics, Encyclopedia Britannica ; at 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 203 
 
 TABLE of the fifty-one Experiments referred to in Equation (89), the 
 value of g in the sixth being taken at 9-8088 metres. 
 
 It will be perceived that Prony did not take into calculation, in framing his formula, the head 
 due to the velocity in the pipe and to the orifice of entry. 
 
 Number of 
 selected expe- 
 riments. 
 
 Names of 
 Experimenters. 
 
 Heads measured 
 to the lower ori- 
 fice in metres. 
 
 n! 
 
 w 
 
 Length of the 
 pipes in metres. 
 
 Values of 
 9 r * 
 
 V 
 
 in metres. 
 
 Experimental 
 values of the 
 velocity v in 
 metres. 
 
 Calculated 
 velocity from 
 formula (89) in 
 metres. 
 
 1 
 
 Du Buat 
 
 0041 
 
 0271 
 
 19-95 
 
 000314 
 
 0430 
 
 0427 
 
 2 
 
 Couplet 
 
 1511 
 
 1353 
 
 2280-37 
 
 000404 
 
 0544 
 
 0591 
 
 3 
 
 Couplet 
 
 3068 
 
 1353 
 
 2280-37 
 
 000523 
 
 0854 
 
 0921 
 
 4 
 
 Du Buat 
 
 0135 
 
 02707 
 
 19-95 
 
 000459 
 
 0980 
 
 0926 
 
 5 
 
 Couplet 
 
 4534 
 
 1333 
 
 2280-37 
 
 000590 
 
 1117 
 
 1263 
 
 6 
 
 Couplet 
 
 6105 
 
 1333 
 
 2280-37 
 
 000638 
 
 1301 
 
 1330 
 
 7 
 
 Couplet 
 
 6497 
 
 1333 
 
 2280-37 
 
 000670 
 
 1411 
 
 1433 
 
 8 
 
 Couplet 
 
 6767 
 
 1333 
 
 2280-37 
 
 000683 
 
 1441 
 
 1467 
 
 9 
 
 Du Buat 
 
 0189 
 
 0271 
 
 3-75 
 
 001426 
 
 2352 
 
 2895 
 
 10 
 
 Du Buat 
 
 1137 
 
 0271 
 
 3-75 
 
 '001138 
 
 2826 
 
 3088 
 
 11 
 
 Du Buat 
 
 1137 
 
 0271 
 
 375 
 
 001309 
 
 2888 
 
 3088 
 
 12 
 
 Bossut 
 
 1083 
 
 0271 
 
 16-24 
 
 001337 
 
 3308 
 
 3359 
 
 13 
 
 Bossut 
 
 3248 
 
 0361 
 
 58-47 
 
 001446 
 
 3400 
 
 3553 
 
 14 
 
 Du Buat 
 
 1605 
 
 0271 
 
 19-95 
 
 001482 
 
 3604 
 
 3713 
 
 15 
 
 Bossut 
 
 3248 
 
 0361 
 
 48-75 
 
 001549 
 
 3807 
 
 3915 
 
 16 
 
 Du Buat 
 
 2106 
 
 0271 
 
 19-95 
 
 001713 
 
 4091 
 
 4287 
 
 17 
 
 Bossut 
 
 3248 
 
 0361 
 
 38-98 
 
 001687 
 
 4366 
 
 4402 
 
 18 
 
 Du Buat 
 
 2425 
 
 0271 
 
 19-95 
 
 001830 
 
 4408 
 
 4618 
 
 19 
 
 Bossut 
 
 3248 
 
 0544 
 
 58-47 
 
 001672 
 
 4433 
 
 4416 
 
 20 
 
 Du Buat 
 
 2425 
 
 0271 
 
 1995 
 
 001793 
 
 4500 
 
 4618 
 
 21 
 
 Bossut 
 
 3248 
 
 0544 
 
 48-73 
 
 001795 
 
 4955 
 
 4860 
 
 22 
 
 Bossut 
 
 6497 
 
 0361 
 
 58-47 
 
 '001922 
 
 5115 
 
 5122 
 
 23 
 
 Bossut 
 
 3248 
 
 0361 
 
 29-23 
 
 001918 
 
 5128 
 
 5122 
 
 24 
 
 Du Buat 
 
 3335 
 
 0271 
 
 19-95 
 
 002050 
 
 5411 
 
 5450 
 
 25 
 
 Bossut 
 
 3248 
 
 0544 
 
 38-98 
 
 '001981 
 
 5605 
 
 5458 
 
 26 
 
 Du But 
 
 3709 
 
 0271 
 
 19-95 
 
 '002174 
 
 5676 
 
 5766 
 
 27 
 
 Bossut 
 
 6497 
 
 0361 
 
 48-73 
 
 002073 
 
 5693 
 
 5634 
 
 28 
 
 Du Buat 
 
 3952 
 
 0271 
 
 19-95 
 
 002223 
 
 5916 
 
 5961 
 
 29 
 
 Bossut 
 
 3248 
 
 0271 
 
 26-24 
 
 002201 
 
 6032 
 
 5990 
 
 30 
 
 Bossut 
 
 3248 
 
 0361 
 
 19-49 
 
 002333 
 
 6323 
 
 6327 
 
 31 
 
 Bossut 
 
 3248 
 
 0544 
 
 29-23 
 
 002300 
 
 6444 
 
 6344 
 
 32 
 
 Bossut 
 
 6497 
 
 0361 
 
 38-98 
 
 002267 
 
 6498 
 
 6323 
 
 33 
 
 Bossut 
 
 6497 
 
 0544 
 
 58-47 
 
 002214 
 
 6695 
 
 6344 
 
 34 
 
 Bossut 
 
 6497 
 
 0544 
 
 48-73 
 
 002392 
 
 7436 
 
 6972 
 
 35 
 
 Bossut 
 
 6497 
 
 0361 
 
 29'23 
 
 002588 
 
 74 
 
 7343 
 
 36 
 
 Du Bu^t 
 
 6416 
 
 0271 
 
 19-95 
 
 002750 
 
 7761 
 
 7660 
 
 37 
 
 Bossut 
 
 3248 
 
 0544 
 
 19-49 
 
 002812 
 
 7908 
 
 7823 
 
 38 
 
 Du Butt 
 
 1624 
 
 0271 
 
 3-75 
 
 003620 
 
 7943 
 
 8930 
 
 39 
 
 Bossut 
 
 6497 
 
 0544 
 
 38-98 
 
 002656 
 
 8363 
 
 7819 
 
 40 
 
 Bossut 
 
 3248 
 
 0361 
 
 9-74 
 
 003287 
 
 8976 
 
 9048 
 
 41 
 
 Bossut 
 
 65 
 
 0361 
 
 1949 
 
 003161 
 
 9332 
 
 9048 
 
 42 
 
 Bossut 
 
 65 
 
 0544 
 
 29-23 
 
 003062 
 
 9681 
 
 9071 
 
 43 
 
 Couplet 
 
 3-9274 
 
 4873 
 
 1169-42 
 
 003785 
 
 1-0600 
 
 1-0592 
 
 44 
 
 Bossut 
 
 3248 
 
 0544 
 
 9'74 
 
 004073 
 
 1'0915 
 
 1-1164 
 
 45 
 
 Bossut 
 
 6497 
 
 0544 
 
 19-49 
 
 003821 
 
 1-1640 
 
 1-1164 
 
 46 
 
 Bossut 
 
 6497 
 
 0361 
 
 9-74 
 
 004491 
 
 1-3138 
 
 1-2896 
 
 47 
 
 Du Buat 
 
 4873 
 
 0271 
 
 3-17 
 
 006470 
 
 1-5784 
 
 1-7043 
 
 48 
 
 Du Buat 
 
 5671 
 
 0271 
 
 3-75 
 
 006307 
 
 1-5919 
 
 1-6898 
 
 49 
 
 Bossut 
 
 6497 
 
 0544 
 
 9-74 
 
 005578 
 
 1-5945 
 
 1-5890 
 
 50 
 
 Du Buat 
 
 7219 
 
 0271 
 
 3-17 
 
 007838 
 
 1-9301 
 
 2-0798 
 
 51 
 
 Du Buat 
 
 9745 
 
 0271 
 
 3-17 
 
 008882 
 
 2-2994 
 
 2-4205 
 
204 THE DISCHARGE OF WATER FROM 
 
 for measures in metres, which, reduced for measures 
 in English feet, is 
 
 v = (9978-76 r* + -02375)* -15412; or 
 
 k ' v = 100 vr s -15 nearly. 
 
 EYTELWEIN, following the method of investigation 
 pursued previously by Prony, found from a large 
 number of experiments, a = -0000242651, and b = 
 000365543 in rivers, for measures in metres ; and, 
 therefore, 
 
 (93.) v = (2735-66 rs + -001102)* -0332.* 
 This reduced for measures in English feet, is 
 
 v = (897543 r s + -0118858)* '1089 ; or 
 (94.) v = 94-5 v/rs -11 nearly = 1-3 v//> -11 
 
 = v/f-fyv -11 
 
 when f is the fall in feet per mile. He also shows,f 
 
 that irths of a mean proportional between the fall in 
 
 two English miles and the hydraulic mean depth, 
 
 gives the mean velocity very nearly. This rule for 
 
 measures in inches is equivalent to 
 
 (95.) v = 324 >/ ; 
 
 and for measures in feet 
 
 (96.) t? = 93-4\/r*. 
 
 For the velocity of water in pipes, he found,J from 
 
 the fifty-one experiments of Du Buat, Bossut, and 
 
 page 164, Third Report, British Association, by Rennie ; and at 
 pages 427 and 533, Article Hydrodynamics, Brewster's Encyclo- 
 pedia. This reduction, v = 0-1541 + (-02375 + 32806*6 rs)* 
 is entirely incorrect ; and. being the same in each of those works, 
 appears to have been copied one from the other. 
 
 * Memoires de 1'Academie de Berlin, 1814 et 1815. See 
 equation (110). 
 
 f Handbuch der Mechanik und der Hydraulik, Berlin, 1801. 
 
 J Memoires de I'Acad&nie des Sciences de Berlin, 1814 et 1815. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 205 
 
 Couplet, that a = -0000223, and b = -0002803, from 
 which we get for measures in metres, 
 (97.) v = (3567-29 r s + -00157,* -0397 ; 
 which reduced for measures in English feet becomes 
 r v = (11703-95 rs + -01698)* -1303 ; or 
 
 (98>) \v = 108 \/rs -13 nearly. 
 Another formula given by Eytelwein for pipes, which 
 includes the head due to the velocity for the orifice 
 of entry, is 
 
 in which H is the head, I the length, and d the dia- 
 meter of the pipe, all expressed in English feet. This 
 is a particular value of equation (74) suited to velo- 
 cities of about 2J feet per second. It must be here 
 mentioned, that much of the valuable information 
 presented by Eytelwein is but a modification of 
 what Du Buat had previously given, to whom only 
 for much that is attributed to the former we are 
 primarily indebted. 
 
 In the foregoing as well as in the following equa- 
 tions for the velocity, we have, unless otherwise 
 stated, maintained one class of standards. It is 
 evident if we change these standards in part, or in 
 whole, that apparently different forms of the equa- 
 tions will arise ; thus if for s, the hydraulic incli- 
 
 ryyi 
 
 nation, we substitute T^on? we shall have the fall 
 
 m in feet per mile, in place of the inclination s; so 
 that equation (94), for instance, would become 
 v = (l-7nr+-012)* -11 = (l*7mr) i -11 nearly, 
 in which v is the velocity in feet per second, m the 
 
206 THE DISCHARGE OF WATER FROM 
 
 fall in feet per mile, and r the "hydraulic mean 
 depth " in feet. In like manner equation (98) would 
 become 
 
 v = (2-2 m r + -02)* -13 = (2 -2 m r)* -13. 
 The first of these reductions, viz. : 
 
 v = (1-7 m r + -0119)* -109, 
 is given in a book of tables calculated for river 
 channels for the Commissioners of Public Works, Ire- 
 land, the original equation being Eytelwein's, and not 
 D'Aubuisson's, who merely copied it, and is suited 
 for velocities averaging about 1*3 feet per second. 
 Mr. Hawksley gives for pipes the formula 
 
 itJiMi 
 
 l,+ 1-5 
 
 in which / is the length in yards, H the head in 
 inches, d the diameter in inches, and v the velocity 
 in yards per second. For uniform feet measures, 
 for, v, d, and H, this becomes 
 
 which is only Eytelwein's equation (99) slightly 
 modified. Eytelwein's equation expressed in the 
 measures used by Mr. Hawksley would be very 
 nearly 
 
 d { ^ )i 
 
 which is far the simplest of the two ; both, however, 
 are but particular cases of the general equation 74), 
 and .only suited for velocities of about 2j feet per 
 second. 
 DR. THOMAS YOUNG* also derives his formula from 
 
 * Philosophical Transactions for 1808. 
 
OBIFICES, WEIRS, PIPES, AND RIVERS. 207 
 
 the supposition, that the head due to the resistance 
 of friction assumes the form of equation (83) ; calling 
 the diameter of a pipe d, he takes 
 
 and the whole height H = h { + j^, expressed in 
 
 inches. He found from some experiments of his 
 own, those collected by Du Buat, and some of 
 G-erstner's, that 
 
 (100.) a = -0000002 
 
 and 
 
 then as zz -00171, we get 
 
 586 
 
 / al vM a I 
 
 \2/ + -00341 dJ I '" 2 
 
 b I + -00341 3/--J 2 b I + -00341 d' 
 When the length / of the pipe is very great compared 
 with the head due to the orifice of entrance and 
 velocity, -00171 v*, we have 
 n end a 2 * a 
 
 TT 
 
 or by substituting for ^ its value s, equal the sine of 
 
 the inclination, 
 
 n (sd a 2 J a 
 
 (104.) v= -- + 
 
208 THE DISCHARGE OF WATER FROM 
 
 The values of a and b are for measures in inches. 
 For most rivers, in which d must be taken equal 4 r, 
 he finds for French inch measures, v = v/20000^ ; 
 this reduced for English inches is 
 
 (105.) ^ = 292x7^; 
 
 which again reduced for feet measures, becomes 
 
 (106.) v = 84-3v/. 
 
 These latter values, for rivers, are even smaller than 
 those found from Du Buat's formula ; less than the 
 observed velocities, and less than those found from 
 any other formula, with the exception of Girard's. 
 The values of the coefficients a and b vary in this 
 formula with the value of d = r; they are expressed 
 generally in equations (101) and (102), from which 
 we have calculated the following table for different 
 values of d and r. 
 
 An examination of this table will show that a 
 obtains a minimum value when d is between 
 10 and 11 inches ; and b when the diameter is 
 between J and f of an inch. Now, it appears from 
 
 equation (102), that v increases with <\ nearly, 
 
 or, which is the same thing, as b decreases, there 
 must, cceteris paribus^ be a maximum value of v for a 
 
 H d 
 given value of -j-, or r s, when d is between J and } 
 
 inch ; but as -~-. has a minimum value when d is 
 
 nearly 12 inches, the maximum value of v referred 
 to will be found between values of d from I inch to 
 12 inches ; in fact, when d 10 inches nearly. We 
 have already pointed out a similar peculiarity in Du 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 209 
 
 Buat's general theorem, at page 195. It will not be 
 
 a a* 
 
 necessary to take out the values* of -^-j and ^ to more 
 
 than one place of decimals. 
 The values of 77-7- are also given in the following 
 
 table, and may be used in equation (104) for finding 
 the discharge from long pipes. It is, however, neces- 
 sary to remark, that this equation is sometimes mis- 
 applied in finding the velocity from short pipes, and 
 those of moderate lengths. It is necessary to use 
 equation (102), which takes into consideration the 
 head due to the velocity and orifice of entry for such 
 pipes. 
 
 For a pipe 11 inches in diameter, the expression 
 for the velocity, equation (104), becomes for.jnch 
 measures, 
 
 and for feet measures, also substituting 4 r for d, 
 
 ( 
 
 very nearly. For a pipe -7 inch in diameter we should 
 find in a like manner for feet measures, 
 (106s.) v = 118(r*)i-5, 
 
 which is only suitable for pipes with very high velo- 
 cities. 
 
 SIR JOHN LESLIE states,* that the mean velocity 
 of a river in miles per hour, is Ifths of the mean 
 proportional between the hydraulic mean depth and 
 
 * Natural Philosophy, p. 428. 
 
210 
 
 THE DISCHAKGE OF WATEE FKOM 
 
 ft 
 
 U5 > CO 
 
 co co co 
 
 10 10 o us o 
 
 t>-<Nib-O 
 
 b- <N Oi CO O 
 
 M CM <M (M 
 
 O O 
 
 O O O 
 OJ O l> 
 
 OOOOOOiOOO 
 
 c?ot^o<Mot-oo 
 
 (MCCOO<t>OOOC? 
 
OKIFICES, WEIKS, PIPES, AND EIVEKS. 211 
 
 the fall in two miles in feet. This rule is equivalent, 
 
 for measures in feet, to 
 
 (107.) v zz 100 v/777 
 
 and is applicable to rivers with velocities of about 2i 
 
 feet per second. 
 
 D'AuBuissoN, from an examination of the results 
 obtained by Prony and Eytelwein, assumes* for 
 measures in metres that a zz -0000189, and b zz 
 0003425 for pipes, substituting these in equation (84) 
 and resolving the quadratic 
 (108.) v zz (2919-71 rs + -00074)^ _ -027 ; 
 which reduced for measures in English feet becomes 
 (v zz (9579 rs + -00813)* - -0902, or 
 \v = 98 v/_-l nearly. 
 
 For rivers he assumes with Eytelwein,f a zz 
 000024123 and b zz -0003655, for measures in 
 metres, and hence 
 
 (110.) v zz (2735-98 rs + -0011)* -033 ; 
 which for measures in English feet is 
 
 v zz (8976-5 rs + -012)' _ -109, or 
 
 v zz 94-5 N/^S '11 nearly. 
 When the velocity exceeds two feet per second, he 
 assumes, from the experiments of Couplet, a zz 0, and 
 b zz -00035875 ; these values give 
 (112.) v zz v/ 2787-46 rs, 
 
 for measures in metres, and 
 (113.) v zz 95-6 \/7s zz v/ 9145 rs 
 
 for measures in English feet. Equations (110) and 
 (111) are the same as (93) and (94), found from Eytel- 
 
 * Traite d'Hydraulique, p. 224. 
 t Traite d'Hydraulique, p. 133. See Equation (93). 
 
 p 3 
 
212 THE DISCHARGE OF WATER FROM 
 
 wein's values of a and b, and it may be remarked that 
 D'Aubuisson's equations for the velocity generally, 
 are simply those of Prony and Eytelwein. 
 
 The values which we have found to agree best 
 with experiments on clear straight rivers are a z= 
 0000035, and b = -0001150 for measures in English 
 feet, from which we find 
 
 v = (8695 -6 r s + -00023)*- '0152, or 
 
 which for an average velocity of 1J foot per second 
 will give v =. 92'3\A*s nearly, and for large velo- 
 cities vzz93'3v/rA- ; for smaller velocities than 1J foot 
 per second, the coefficients of \/r s decrease pretty 
 rapidly. This formula will be found to agree more 
 accurately with observation and experiment than any 
 other we know of this form. 
 
 WEISBACH is perhaps the only writer who has 
 modified the form of the equation r s a v + b v*. 
 In Dr. Young's formula, a and j b vary with r, but 
 
 7 j 2 
 
 Weisbach assumes that A f (a + -i)~ x ~o~ > and 
 
 finds from the fifty-one experiments of Couplet, 
 Bossut, and Du Buat, before referred to, one experi- 
 ment by Guemard, and eleven by himself, all with 
 pipes varying from an inch to five and a half inches 
 in diameter, and with velocities varying from 1J inch 
 to 15 feet per second, that a -01439, and b z= 
 0094711 for measures in metres ; hence we have for 
 the metrical standard 
 
 This reduced for the mean radius r is 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 213 
 
 (116.) 
 
 from which we find for measures in English feet 
 
 and thence 
 
 (118.) ,. = (, + 5!") *; 
 
 and by substituting for 2^7, its value 64*403, 
 (119.) ,. = (-00005585 + 
 
 /. 
 In equation (117), (-003597 + - -4~ -) = c t is the 
 
 coefficient of the head due to friction. The equation 
 does not admit of a direct solution, but the coeffi- 
 cient should be first determined for different values 
 of the velocity v and tabulated, after which the true 
 value of v can be determined by finding an approx- 
 imate value, and thence taking out the corresponding 
 coefficient from the table, which does not vary to any 
 considerable extent for small changes of velocity. 
 In the following small table we have calculated the 
 coefficients of friction, and also those of t; 2 , in equa- 
 tion (119), for different values of the velocity v. 
 
214 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE OF THE COEFFICIENTS OF FRICTION IN PIPES. 
 
 Velocity 
 in feet. 
 
 Cf 
 
 Cf 
 
 64-4 
 
 644 
 
 Cf 
 
 v/^ 4 
 
 Cf 
 
 Velocity 
 in feet. 
 
 Cf 
 
 Cf 
 
 64-4 
 
 644 
 c r 
 
 ^/m 
 
 1 
 
 017159 
 
 0002664 
 
 3078-07 
 
 55-5 
 
 2-4 
 
 006365 
 
 0000988 
 
 10121-5 
 
 100-5 
 
 2 
 
 013186 
 
 0002047 
 
 4885-2 
 
 69-9 
 
 2-5 
 
 006309 
 
 0000979 
 
 10214-5 
 
 101-0 
 
 3 
 
 011427 
 
 0001774 
 
 5636-9 
 
 75-08 
 
 2-6 
 
 006257 
 
 0000972 
 
 10288-1 
 
 101-4 
 
 4 
 
 010378 
 
 0001611 
 
 6270-3 
 
 78-8 
 
 2-7 
 
 006207 
 
 0000964 
 
 10373-4 
 
 101-8 
 
 5 
 
 009662 
 
 0001500 
 
 6666-6 
 
 81-6 
 
 2-8 
 
 006160 
 
 0000956 
 
 10460-2 
 
 102-2 
 
 6 
 
 009133 
 
 0001418 
 
 7052-2 
 
 84-0 
 
 2-9 
 
 006115 
 
 0000949 
 
 10537-4 
 
 102-6 
 
 7 
 
 008723 
 
 0001354 
 
 7385-5 
 
 85-9 
 
 3- 
 
 006073 
 
 0000943 
 
 10604-4 
 
 102-9 
 
 8 
 
 008391 
 
 0001303 
 
 7674-6 
 
 87-6 
 
 3-5 
 
 005890 
 
 0000914 
 
 10940-9 
 
 104-6 
 
 9 
 
 008117 
 
 0001260 
 
 7936-5 
 
 89-1 
 
 4- 
 
 005741 
 
 0000891 
 
 11223-3 
 
 105-9 
 
 1-0 
 
 007886 
 
 0001224 
 
 8169-9 
 
 90-4 
 
 5- 
 
 005514 
 
 0000856 
 
 11682-2 
 
 108-0 
 
 1-1 
 
 007686 
 
 0001193 
 
 8382-2 
 
 91-5 
 
 6- 
 
 005348 
 
 0000830 
 
 12048-2 
 
 109-7 
 
 1-2 
 
 007512 
 
 0001166 
 
 8576-3 
 
 92-6 
 
 7- 
 
 005218 
 
 0000810 
 
 12345-6 
 
 111-1 
 
 1-25 
 
 007433 
 
 0001154 
 
 8665-5 
 
 93-1 
 
 8- 
 
 005113 
 
 0000794 
 
 12632-2 
 
 112-4 
 
 1*8 
 
 007358 
 
 0001142 
 
 8756-5 
 
 93-5 
 
 9- 
 
 005026 
 
 0000780 
 
 12820-5 
 
 113-3 
 
 1-4 
 
 007221 
 
 0001121 
 
 8920-6 
 
 94-4 
 
 10- 
 
 004953 
 
 0000769 
 
 13003-9 
 
 114-0 
 
 1-5 
 
 007098 
 
 0001102 
 
 9074-4 
 
 95-2 
 
 15- 
 
 004704 
 
 0000730 
 
 13698-6 
 
 117-0 
 
 1-6 
 
 006987 
 
 0001085 
 
 9216-5 
 
 96-0 
 
 16- 
 
 004669 
 
 0000725 
 
 13793-1 
 
 117-4 
 
 1-7 
 
 006886 
 
 0001069 
 
 9354-5 
 
 96-7 
 
 20- 
 
 004556 
 
 0000707 
 
 14144-2 
 
 118-9 
 
 1-75 
 
 006839 
 
 0001062 
 
 9416-2 
 
 97-03 
 
 25- 
 
 004455 
 
 0000691 
 
 14471-7 
 
 120-3 
 
 1-8 
 
 006794 
 
 0001054 
 
 9487-6 
 
 97-4 
 
 30- 
 
 004380 
 
 0000680 
 
 14705-9 
 
 121-2 
 
 1-9 
 
 006715 
 
 0001042 
 
 9596-9 
 
 97-9 
 
 35- 
 
 004322 
 
 0000671 
 
 14903-1 
 
 122-0 
 
 2- 
 
 006629 
 
 0001029 
 
 9718-2 
 
 98-5 
 
 40- 
 
 004275 
 
 0000664 
 
 15060-2 
 
 122-7 
 
 2-1 
 
 006556 
 
 0001018 
 
 9823-2 
 
 99-1 
 
 45- 
 
 004236 
 
 0000658 
 
 15197-5 
 
 123-3 
 
 2-2 
 
 006488 
 
 0001007 
 
 9930-5 
 
 99-6 
 
 50- 
 
 084203 
 
 0000653 
 
 15313-8 
 
 123-7 
 
 2-3 
 
 006424 
 
 0000997 
 
 10003- 
 
 100- 
 
 100- 
 
 004208 
 
 0000625 
 
 16000-0 
 
 126-4 
 
 If the value of 
 
 the equation v = 
 
 644 
 
 here found, be substituted in 
 
 c f 
 
 rs. we shall have the value 
 
 of v. According to this table the coefficient of fric- 
 tion for a velocity of six inches is more than twice 
 that for a velocity of twenty feet, and the velocity is 
 less in the proportion of 81*6 to 118*9, or of 81*6 (rs)* 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 215 
 
 to 118*9 (rs)*. On comparing these coefficients and 
 those for pipes in the preceding formulae, with those 
 for rivers of the same hydraulic depth, we perceive 
 that the loss from friction is greatest in the latter, 
 as might have been anticipated ; but this evidently 
 arises from lesser velocities. 
 
 It has been already remarked that the coefficient 
 of friction decreases as the velocity increases. The 
 only general formula which properly meets this de- 
 fect in the common formulae is Weisbach's, but it 
 does not give the velocity v directly, as this quantity 
 is involved in both sides of his equation. As for 
 several hydraulic works it is necessary to convey 
 water through pipes to work machines under high 
 heads, and for which the common formula would 
 give results considerably under the true ones, it 
 appeared to me desirable to obtain some simple ex- 
 pression for the velocity which might be easily 
 remembered and applied, which would be equally 
 correct with other formulae for medium velocities of 
 from one to two and a half feet, and which at the 
 same time would give practically correct results for 
 lesser and greater velocities within the limits of 
 experiment. By reducing the velocity found from 
 experiment to the form v = m ^/^~ s for every case, 
 and afterwards applying a correction of the form 
 n \/^rs to meet the increasing value of m as v in- 
 creased, I discovered that the expression 
 (119 A) v= 140 (rs)* - 11 (r*)* 
 
 gave results not differing more from experiments than 
 these frequently do from each other. The following 
 table exhibits the velocities compared with those 
 
216 THE DISCHARGE OF WATER FROM 
 
 obtained from the experiments made by Du Buat, 
 Couplet, Watt, Mr. Provis, and Mr. Leslie, in the 
 Minutes of the Institution of Civil Engineers for 
 February 1855. The last experiment was furnished 
 to me by Mr. Hodson of Lincoln. Numbers 34 
 and 35 were made by myself, and give the mean 
 results of several experiments made with great care ; 
 the coefficient of the orifice of entry was found to 
 be -860.* The measures have been all reduced to 
 English feet. The results found by the same experi- 
 menters, at the same time, with the same apparatus, 
 sometimes differ by three or four per cent., as may be 
 seen by referring to Mr. Provis' experiments, (Trans- 
 actions of the Institution of Civil Engineers, vol. n., 
 p. 203,) and the difference in the experiments shown 
 in the table are apparent. The difference in the 
 velocities found from the experiments, do not exceed 
 those inseparable from practical investigations, and 
 they differ as much in themselves as from the formula, 
 which for cylindrical pipes of diameter d may be 
 thus expressed, 
 
 v zz 70 (d 5)4 6-93 (d )*, or 
 
 (119B.) 
 
 1 v = 70 (d *)4 - 7 (d s)* nearly. 
 The expression fails when 70 (d sfi is equal to or 
 less than 6-93 (d s$, but as this only happens when 
 
 f\\ \8 
 
 r s =1^40) = '000000235, and for velocities below 
 
 one inch per second, its practical value is not thereby 
 affected. The expression of Du Buat fails with a 
 
 * The coefficient for the orifice of entry was found by cutting 
 off the pipe at two diameters from the cistern at the conclusion of 
 the experiments, and finding the time of emptying. Vide p. 177. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 217 
 
 TABLE showing the Experimental Results of observed Velocities in Water 
 Channels, with the Author's general formula for Pipes and Rivers, viz. 
 v = 140 (rsj*-Il (r s)*. 
 
 c 
 
 K 
 
 Heads in 
 
 feet (H). 
 
 Lengths in 
 feet (I). 
 
 Values 
 of r. 
 
 Values 
 of s. 
 
 Values 
 of r 8. 
 
 Velocities 1 
 from 
 experiment. I 
 
 Velocities I 
 from the 1 
 formula. 1 
 
 Velocities 
 expressed in 
 the form v = 
 m \/ r s. 
 
 Experimen- 
 ters' Names. 
 
 1 
 
 08333 
 
 1086- 
 
 052083 
 
 000076 
 
 00000396 
 
 100 
 
 105 
 
 52-6 ^/T, Mr. Leslie 
 
 2 
 
 01332 
 
 65-37 
 
 022204 
 
 000116 
 
 00000434 
 
 140 
 
 113 
 
 54-0 Couwlet 
 
 8 
 
 14583 
 
 1086- 
 
 052083 
 
 000133 
 
 00000693 
 
 118 
 
 157 
 
 60-0 
 
 Mr. Leslie 
 
 4 
 
 49566 
 
 7482- 
 
 111000 
 
 000066 
 
 00000134 
 
 178 
 
 167 
 
 61-5 
 
 Couplet 
 
 6 
 
 20833 1086- 
 
 052083 
 
 000190 
 
 00000989 
 
 217 
 
 206 
 
 65-0 
 
 Mr. Leslie 
 
 6 
 
 45833 
 
 1086- 
 
 052083 
 
 000417 
 
 00002170 
 
 361 
 
 345 
 
 74-1 
 
 
 7 
 
 1-4876H 
 
 7482- 
 
 111000 
 
 000198 
 
 00002220 
 
 366 
 
 348 
 
 74-1 
 
 Couplet 
 
 8 
 g 
 
 1-448001086- 
 2-781251086- 
 
 052083 
 052083 
 
 001321 
 002538 
 
 0000688 
 0001322 
 
 715 
 
 1-085 
 
 711 
 1-050 
 
 85-7 
 91-3 
 
 Mr. Leslie 
 
 10 
 
 
 
 2-427200 
 
 000063 
 
 0001532 
 
 1-166 
 
 1-143 
 
 92-4 
 
 Watt" 
 
 11| -50000| 100- 
 
 031250 
 
 004741 
 
 0001482 
 
 1-023 
 
 1-122 
 
 92-2 
 
 Mr. Provis 
 
 12 
 
 2-78125 
 
 1086- 
 
 052083 
 
 004348 
 
 0002265 
 
 1-461 
 
 1-438 
 
 95-5 
 
 Mr. Leslie 
 
 L3 
 
 4-76042 
 
 1086- 
 
 052083 
 
 006410 
 
 0003340 
 
 1-725 
 
 1-796 
 
 98-3 
 
 
 14 
 
 1-06580 
 
 127-9 
 
 044630 
 
 007748 
 
 0003458 
 
 1-839 
 
 1-840 
 
 98-9 
 
 Bossut 
 
 15 
 
 50000 
 
 40- 
 
 031250 
 
 010810 
 
 0003378 
 
 1-711 
 
 1-816 
 
 98-7 
 
 Mr. Frovis 
 
 in 
 
 1-06580 
 
 95-92 
 
 044630 
 
 010050 
 
 0004485 
 
 2-111 
 
 2-124 
 
 100-3 
 
 Bossut 
 
 17 
 
 1-5 
 
 100- 
 
 031250 
 
 014156 
 
 0004422 
 
 2-005 
 
 2-103 
 
 100-5 
 
 Mr. Provis 
 
 18 
 
 9-9896 
 
 1086- 
 
 052083 
 
 009174 
 
 0004779 
 
 2-095 
 
 2-185 
 
 100-6 
 
 Mr. Leslie 
 
 1!) 
 
 8575 
 
 40- 
 
 031250 
 
 018042 
 
 0005638 
 
 2-380 
 
 2-414 
 
 101-7 
 
 Mr. Provis 
 
 '20 
 
 2-1316 
 
 191-9 
 
 044630 
 
 010548 
 
 0004708 
 
 2-463 
 
 2-183 
 
 100-6 
 
 Bossut 
 
 21 
 
 2-1316 
 
 159-9 
 
 044630 
 
 012524 
 
 0005589 
 
 2-440 
 
 2-404 
 
 101-7 , 
 
 
 22 
 
 
 
 ft 
 
 052083 
 
 014286 
 
 0007440 
 
 2-800 
 
 2-823 
 
 103-5 , 
 
 Mr. Leslie 
 
 2:) 
 
 2-1316 
 
 127-9 
 
 044630 
 
 015350 
 
 0006851 
 
 2-744 
 
 2-696 
 
 103-0 , 
 
 Bossut 
 
 24 
 
 n 
 
 n 
 
 044630 
 
 027921 
 
 0012465 
 
 3-819 
 
 3-760 
 
 106-5 , 
 
 
 25 
 
 n 
 
 J} 
 
 052083 
 
 025000 
 
 0013021 
 
 3-783 
 
 3-852 
 
 106-7 , 
 
 Mr. Leslie 
 
 36 
 
 3-27416 
 
 40- 
 
 031250 
 
 018040 
 
 002093 
 
 5-054 
 
 5-006 
 
 109-3 , 
 
 Mr. Provis 
 
 27 
 
 2-3684 
 
 10-39155 
 
 022204 
 
 133689 
 
 0029679 
 
 6-322 
 
 6-048 
 
 111-0 , 
 
 Du Buat 
 
 28 
 
 3-27416 
 
 20- 
 
 031250 
 
 111200 
 
 0034750 
 
 6-723 
 
 6-572 
 
 111-5 , 
 
 Mr. Provis 
 
 2!) 
 
 3-4525 
 
 20- 
 
 031250 
 
 113900 
 
 0035594 
 
 7-086 
 
 6-668 
 
 111-9 , 
 
 
 30 
 
 7-135 
 
 62-8822 
 
 029605 
 
 098861 
 
 0029268 
 
 6-157 
 
 5-999 
 
 110-9 , 
 
 Couplet 
 
 31 
 
 14-270 
 
 125-7644 
 
 029605 
 
 106151 
 
 0031426 
 
 6-151 
 
 6-239 
 
 111-3 
 
 
 :S2 
 
 21-405 
 
 188-6466 
 
 029605 
 
 108579 
 
 00321455 
 
 6-145 
 
 6-316 
 
 111-4 
 
 
 33 
 
 3-1974 
 
 10-39155 
 
 022204 
 
 176991 
 
 0039292 
 
 7-544 
 
 7-039 
 
 112-3 
 
 Du Buat 
 
 34 
 
 11-125 
 
 9-292 
 
 021250 
 
 713000 
 
 01515125 
 
 14-583 
 
 14-513 
 
 117-9 
 
 Mr. Neville 
 
 35 
 
 20-8 
 
 19-2 
 
 021250 
 
 814000 
 
 01729750 
 
 15-667 
 
 15-617 
 
 118-4 
 
 
 36 
 
 ISO- 
 
 100- 
 
 020833 
 
 1-400000 
 
 0291667 
 
 21-7 
 
 20-6 
 
 120-3^/71 
 
 Mr.Hodson 
 
 tube of one twenty-fifth part of an inch in diameter, 
 no matter what the head may be, as it then makes 
 the velocity equal to nothing, although some of the 
 experiments from which it was derived were made 
 with tubes but the eighteenth part of an inch dia- 
 
218 THE DISCHARGE OF WATER FROM 
 
 meter. The following expression is free from this 
 defect : 
 
 (119o.) v = 60 (rsfi + 120 (rs)$, 
 
 and will give results approximating very closely to 
 those found from Du Buat's formula, and, therefore, 
 with those experiments with which it most nearly 
 coincides, but agreeing much more closely with 
 Watt's and other experiments, on rivers. It gives 
 higher results than the previous formula for velo- 
 cities below six inches, but the results found by 
 different experimenters differ very much in those. 
 For higher velocities it appears to differ occasionally 
 only about one-twentieth from observation, being in 
 general less, as far as twenty feet per second, where 
 it coincides very closely with Mr. Hodson's expe- 
 riment. As the errors appear to be of an opposite 
 kind generally, in the two last expressions, we may 
 get by combining them 
 
 (119D.) v = 100 (rs)* + 60 (r s)* 5 -5 (r*)*, 
 
 an expression which, however, wants simplicity for 
 ready practical application. When the length of the 
 pipe does not exceed from 1000 to 2000 diameters, a 
 correction is due to the velocity in it, and to the 
 orifice of entry before finding the "hydraulic incli- 
 nation" (s). The coefficient used in reducing the 
 foregoing experiments for the orifice of entry was 
 
 v 2 
 815, which gives 1*508 -^ for the height due to the 
 
 joint effects of velocity and orifice. This must be 
 deducted from the head (H) before dividing it by the 
 length (/) to find the inclination (s) in our table. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 219 
 
 The following table, calculated from the formula 
 (119A), v = 140 (rs)? 11 (r s)?, gives the corre- 
 sponding values of r s and v, so that when one is 
 known the other is immediately found from inspec- 
 tion. Thus, if r s = -03125, we find 
 
 v 20-6 when rs= -029167 
 v 24-7 when r s = -041666 
 
 Difference 4-1 corresponds to -012499 
 03125 
 02917 
 
 Difference -00208 
 
 Whence -0125 : 4-1 : : -00208 : -7 nearly, and 
 20-6 4 7 z= 21-3 is the velocity sought; the same 
 practically as found in EXAMPLE 26, p. 37. If 
 allowance is to be made for the head due to the 
 orifice of entry and velocity, this head can be de- 
 termined from the velocity due to the value of r s 
 in the table next less than the given value with 
 sufficient accuracy. In this case, this velocity is 
 20-6 feet per second zz 247 inches nearly. If the 
 orifice of entry be square, the coefficient is -815, and 
 the head due to the velocity and this coefficient is, 
 TABLE II., 10 feet nearly. If r be known separately, 
 and also ,9, as well as the head H, and the length of 
 the pipe /, we had at first 
 
 H H 10 h 
 
 j zz s, and, therefore, * ~ 7 "" s ' 
 
 In EXAMPLE 26, p. 37, H = 150, and 1= 100 feet, 
 therefore, the new value of j z= =r^ is 14 ; and as 
 r must be equal '020833, rs = -02917 : the value 
 
220 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE for finding the Velocity in feet per second, from the product of 
 the hydraulic mean depths and hydraulic inclinations, and the reverse 
 calculated from the Authors formula v = 140 (rs) 4 11 (rs)*, in 
 which r, s, and v, are feet measures. 
 
 Values of r s. 
 
 Velo- 
 city V. 
 
 Values of rs. 
 
 Velo- 
 city V. 
 
 Values of r 
 
 Velo- 
 city V 
 
 Values of rs 
 
 Velo- 
 city V. 
 
 00000296 
 
 083 
 
 0001302 
 
 1-04 
 
 000689 
 
 270 
 
 003559 
 
 6-67 
 
 00000332 
 
 091 
 
 0001322 
 
 1-05 
 
 000710 
 
 2-75 
 
 003599 
 
 6-71 
 
 00000395 
 
 104 
 
 0001420 
 
 1-09 
 
 000744 
 
 2-83 
 
 003630 
 
 6-74 
 
 00000427 
 
 111 
 
 0001482 
 
 1-12 
 
 000758 
 
 2-85 
 
 003788 
 
 6-90 
 
 00000543 
 
 133 
 
 0001532 
 
 1-14 
 
 000789 
 
 2-91 
 
 003929 
 
 7-04 
 
 00000592 
 
 142 
 
 0001578 
 
 1-16 
 
 000805 
 
 2'94 
 
 003946 
 
 7-05 
 
 00000690 
 
 158 
 
 0001610 
 
 1-17 
 
 000833 
 
 3-00 
 
 003977 
 
 7-08 
 
 00000734 
 
 167 
 
 0001657 
 
 1-19 
 
 000852 
 
 304 
 
 004104 
 
 7-20 
 
 00000947 
 
 198 
 
 0001736 
 
 1-21 
 
 000900 
 
 3-13 
 
 004167 
 
 7-27 
 
 00000989 
 
 206 
 
 0001776 
 
 1-24 
 
 000947 
 
 3'22 
 
 004356 
 
 7-44 
 
 00001184 
 
 231 
 
 0001815 
 
 1-26 
 
 001042 
 
 340 
 
 004546 
 
 762 
 
 00001263 
 
 241 
 
 0001894 
 
 1-30 
 
 001105 
 
 3-51 
 
 004630 
 
 7-69 
 
 00001420 
 
 261 
 
 0002052 
 
 1 35 
 
 001136 
 
 3-57 
 
 004735 
 
 7-78 
 
 00001578 
 
 280 
 
 0002131 
 
 1-38 
 
 001231 
 
 3-73 
 
 005556 
 
 8-49 
 
 00001677 
 
 292 
 
 0002265 
 
 1-43 
 
 001246 
 
 3'76 
 
 006944 
 
 9-61 
 
 00001894 
 
 316 
 
 0002367 
 
 1-47 
 
 001263 
 
 3'78 
 
 007576 
 
 10-0 
 
 00001973 
 
 325 
 
 0002552 
 
 1-50 
 
 001302 
 
 3-85 
 
 008333 
 
 10'5 
 
 00002170 
 
 345 
 
 0002604 
 
 1-55 
 
 001326 
 
 3-89 
 
 009259 
 
 11 1 
 
 00002367 
 
 365 
 
 0002652 
 
 1-57 
 
 001420 
 
 4-04 
 
 010417 
 
 118 
 
 00002565 
 
 385 
 
 0002778 
 
 1-61 
 
 001515 
 
 4'18 
 
 011905 
 
 12-7 
 
 00002841 
 
 411 
 
 0002841 
 
 1-63 
 
 001576 
 
 4-28 
 
 013889 
 
 13-8 
 
 00003255 
 
 448 
 
 0003030 
 
 1-69 
 
 001610 
 
 4-32 
 
 015151 
 
 14-5 
 
 00003354 
 
 457 
 
 0003157 
 
 1-73 
 
 001667 
 
 4-41 
 
 016667 
 
 15-3 
 
 00003551 
 
 473 
 
 0003220 
 
 1 75 
 
 001705 
 
 4'46 
 
 017297 
 
 156 
 
 00003748 
 
 489 
 
 0003314 
 
 1-79 
 
 001735 
 
 4'51 
 
 020833 
 
 17-1 
 
 00003946 
 
 505 
 
 0003378 
 
 1-80 
 
 001799 
 
 4'60 
 
 027778 
 
 20-2 
 
 OOC04143 
 
 521 
 
 0003409 
 
 1-81 
 
 001894 
 
 4-73 
 
 029167 
 
 20-6 
 
 00004340 
 
 536 
 
 0003551 
 
 1-85 
 
 001989 
 
 4-87 
 
 041666 
 
 24-7 
 
 00004632 
 
 558 
 
 0003630 
 
 1-89 
 
 002052 
 
 4-94 
 
 055556 
 
 28-8 
 
 00005130 
 
 594 
 
 0003706 
 
 1-90 
 
 002083 
 
 4-98 
 
 062500 
 
 30-6 
 
 00005327 
 
 608 
 
 0003788 
 
 1-92 
 
 002093 
 
 5-00 
 
 072916 
 
 33-2 
 
 00005524 
 
 622 
 
 0003946 
 
 1-98 
 
 002178 
 
 5-10 
 
 083333 
 
 35-6 
 
 00005919 
 
 648 
 
 0004022 
 
 1-10 
 
 002210 
 
 5-14 
 
 104167 
 
 40-0 
 
 00006314 
 
 674 
 
 0004103 
 
 2-02 
 
 002273 
 
 5-22 
 
 125 
 
 43-9 
 
 00006708 
 
 699 
 
 0004261 
 
 36 
 
 002375 
 
 5*35 
 
 145583 
 
 47-6 
 
 0000688 
 
 711 
 
 0004419 
 
 2-10 
 
 002462 
 
 5-46 
 
 166667 
 
 51-1 
 
 00007102 
 
 724 
 
 0004485 
 
 2-12 
 
 002533 
 
 553 
 
 208333 
 
 57-3 
 
 00007694 
 
 760 
 
 0004546 
 
 2-14 
 
 002652 
 
 56-8 
 
 229167 
 
 60-2 
 
 00008049 
 
 781 
 
 0004708 
 
 2-18 
 
 002683 
 
 5-72 
 
 250000 
 
 63-0 
 
 00008523 
 
 808 
 
 0004735 
 
 2-18 
 
 002841 
 
 5'90 
 
 270833 
 
 65-7 
 
 00008681 
 
 828 
 
 0004893 
 
 2-23 
 
 002968 
 
 6-05 
 
 312500 
 
 70-7 
 
 00009270 
 
 849 
 
 0005051 
 
 2-27 
 
 002999 
 
 6-08 
 
 333333 
 
 73-2 
 
 00009470 
 
 861 
 
 0005208 
 
 2-31 
 
 003030 
 
 6-11 
 
 354167 
 
 75-5 
 
 00010259 
 
 903 
 
 0005303 
 
 2-33 
 
 003143 
 
 6-23 
 
 375000 
 
 77-7 
 
 00010654 
 
 923 
 
 0005638 
 
 2-41 
 
 003157 
 
 6-25 
 
 395833 
 
 80-0 
 
 00011048 
 
 945 
 
 0006061 
 
 2-52 
 
 003214 
 
 6-31 
 
 416667 
 
 62-1 
 
 00011364 
 
 960 
 
 0006155 
 
 2-54 
 
 003220 
 
 6-32 
 
 437500 
 
 84-2 
 
 00011837 
 
 983 
 
 0006313 
 
 2-57 
 
 003314 
 
 6-42 
 
 458333 
 
 86-2 
 
 00012232 
 
 1-00 
 
 0006440 
 
 2-60 
 
 003409 
 
 6-51 
 
 479166 
 
 88-3 
 
 00012627 
 
 1-02 
 
 0006629 
 
 2-64 
 
 003475 
 
 6-58 
 
 500000 
 
 90-2 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 221 
 
 corresponding to which, in our table, is 20*6, the 
 velocity when allowance is made for the head due 
 to the velocity and orifice of entry. *+ 
 
 In general, by taking the value of v for the next 
 less value of r s in the table, we shall find the 
 velocity with sufficient accuracy, and also the value 
 of r s from that of v by taking it as the next 
 greater. If we had taken r s -0008523, the table 
 would give v = 3-04 feet, the same practically as 
 already found in EXAMPLE 27, p. 38. 
 
 The value of r s, when known, determines the value 
 of v. If r be assumed of any convenient dimen- 
 sions, s is determined ; and, in like manner, any 
 suitable value of s determines r ; thus : 
 
 r s _ r s 
 
 =s, and z= r. 
 r s 
 
 It is well to remark, here again, that for pipes the 
 value of r is the fourth part of the diameter J, and 
 that 
 
 r zz j, and 4 r = d. 
 
 In 1857, M. Darcy, inspecteur des ponts et 
 chausees, published his Eecherches experimentales 
 relatives au Mouvement de 1'Eau dans les Tuyaux,* 
 the result of 198 experiments, in which the velocities 
 varied from '03 to 5 or 6 metres per second, or from H 
 inch to 16 or 19 feet, and with pipes varying from 
 J inch to 20 inches diameter. The formula by which 
 he presents the results is in metres, 
 (a.) R j = x u 2 , 
 
 in which R is the radius of the pipe, j the hydraulic 
 * Morin's Hydraulique, deuxieme edition, Paris, p. 164. 
 
222 
 
 THE DISCHARGE OF WATER FROM 
 
 inclination, b-^ a variable coefficient dependent on the 
 circumstances, and u the velocity per second. For 
 wrought and cast iron pipes of the same state of 
 bore, the value of b is expressed by M. Darcy, by 
 the equation 
 
 (ft.) Z, = -000507 + 0000 647 , ' 
 
 R 
 
 the agreement between which and experiment is 
 shown in the following table. ;. 
 
 Diameters 
 in English 
 inches. 
 
 Diameters in 
 metres. 
 
 Values of ^ 
 
 from experi- 
 ments. 
 
 Values of b^ 
 
 by the 
 formula. 
 
 Kemarks. 
 
 5 
 
 0122 
 
 001'673 
 
 001568 
 
 
 1- 
 
 0266 
 
 000918 
 
 000993 
 
 
 1-5 
 
 0395 
 
 000785 
 
 000835 
 
 
 3-2 
 
 0819 
 
 000695 
 
 000665 
 
 
 5-4 
 
 1370 
 
 000553 
 
 000601 
 
 Well polished bore. 
 
 7.4 
 11-7 
 
 1880 
 2970 
 
 000584 
 000612 
 
 000576 
 000551 
 
 f Pipe already in use, 
 \ but the bore cleaned. 
 
 19-7 
 
 5000 
 
 000509 
 
 000532 
 
 
 For iron coated with bitumen, the value of b in 
 a pipe *196 metres in diameter was -0004334 ; for 
 a newly cast pipe of -188 metres, b was -000584; 
 and for a pipe -2432 metres in diameter, ^ was 
 001168 ; the relative proportions of b in these 
 three instances, being as 
 
 1-1 to 1-5 and to 3 ; 
 
 and, therefore, the velocities, or discharges, would 
 be inversely as the square roots of these, or as 
 95 to -82 and to -58. 
 
 By substituting our notation for that of M. Darcy, 
 we shall have in metres, from equations (a) and (b), 
 
ORIFICES, WEIRS, PIPES, AND RIVERS, - 223 
 
 which for feet measures becomes (as 1 metre z= 
 3-281 feet) 
 
 r " 3-281 x '0000016175! v 2 
 
 rs= {'0002535 + - ^- ~| x 3^281 : 
 
 hence we get 
 
 rs 
 
 **= ^-00007726 + miQ ' 2 
 
 and, therefore, 
 
 -00007726 
 
 For all half-inch pipes this becomes 
 
 -00023278 
 for all inch pipes, 
 
 t f* C \ TV 
 
 v = 1-000155021 = 
 for all two-inch pipes, 
 
 for all four-inch pipes, 
 
 c =1-0000967) = 
 
 for all six-inch pipes, 
 
 v = 1-000090221 = 105 ' 3 
 
224 THE DISCHARGE OF WATER FROM 
 
 for all nine-inch pipes, 
 
 = 107-8 v 
 
 for all twelve-inch pipes, 
 
 *= 1- 
 
 -00008374 
 for all eighteen-inch pipes, 
 
 * =1- 
 
 00008158) 
 for all twenty-four-inch pipes, 
 rs 
 
 ={: 
 
 0000805) 
 
 = 111-5 
 
 and when r is large, as for very large pipes and 
 channels, we get the velocity 
 
 f 
 
 v 
 
 -00007726J 
 
 = 113-8 vW 
 
 There is evidently, on an examination of these 
 results, a great error in the formula of M. Darcy. 
 As long as the diameter of a long pipe continues 
 constant, the velocity is always represented by a 
 given fixed multiple of \/~rs, or of the square root 
 of the product of the hydraulic inclination and 
 hydraulic mean depth, no matter how small or great 
 the velocity In the pipe may be. For an inch pipe 
 this multiplier for feet measures is 80-3. Now with 
 a lead pipe I have found, from several experiments, 
 that for a velocity of about 15 feet per second, the 
 multiplier to be 117 or 118 ; and for a velocity of 
 about 22 feet per second, Mr. Hodson's experiment 
 gives a multiplier of about 120. Taking the other 
 
ORIFICES, WEIRS, PIPES, AND RIVERS, 225 
 
 extreme for large pipes, the multiplier derived from 
 M. Darcy' s formula is 113*8, no matter how small 
 the velocity may be. Now we have experiments in 
 abundance to prove that for velocities of about 12 or 
 13 inches per second, the multiplier cannot exceed 
 95. We, therefore, look upon these researches of 
 M. Darcy as partial and defective, and his formula 
 as a representation, at best, of a limited range of 
 velocities, in which those at either side are omitted 
 or not perceived. 
 
 For small pipes, any obstruction arising from de- 
 fective bore, decomposition, encrustation, or from 
 diminished bore, affects the discharge much more 
 considerably than the same obstructions in a large 
 pipe. In order to compare correctly the effects of 
 the state of the bore on the discharge, we must use 
 pipes of exactly the same diameter, and determine 
 the value of b : from experiments in which the velo- 
 city is the same, otherwise the results, as deduced 
 by M. Darcy and given by Morin, cannot be de- 
 pended upon. 
 
 COEFFICIENTS DUE TO THE ORIFICE OF ENTRY. PROBLEMS. 
 
 Unless where otherwise expressed, the head due 
 to the velocity and orifice of entry is , not considered 
 in the preceding equations. In equation (74), where 
 it is taken into calculation generally, 
 
 Cf X - 
 
 /1\ 2 
 in which 1 + c r is equal to ( ) , c r being the coefficient 
 
226 THE DISCHARGE OF WATER FROM 
 
 of resistance clue to the orifice of entry, and c v the 
 coefficient of velocity or discharge from a short tube. 
 If the tube project into the reservoir, and be of small 
 thickness, c v will be equal -715 nearly, and therefore 
 c r zz -956 ; if the tube be square at the junction, the 
 mean value of c v will be -814, and therefore c r = -508; 
 and if the junction be rounded in the form of the 
 contracted vein, c v is equal to unity very nearly, and 
 C T z= 0. For other forms of junction the coefficients 
 of discharge and resistance will vary between these 
 limits, and particular attention must be paid to their 
 values in finding the discharge from shorter tubes 
 and those of moderate lengths ; but in very long tubes 
 
 1 + c r becomes very small compared with c f x , 
 
 and maybe neglected without practical error. These 
 remarks are necessary to prevent the misapplication 
 of the tables and formulae, as the height due to the 
 velocity and orifice of entry is an important element 
 in all calculations for short tubes. 
 
 We have considered it unnecessary to give any 
 formulae for finding the discharge itself, because, the 
 mean velocity once determined, the calculation of the 
 discharge from -the area of the section is one of 
 simple mensuration ; and the introduction of this 
 element into the three problems to which this por- 
 tion of hydraulic engineering applies itself, renders 
 the equations of solution complex, though easily 
 derived ; and presents them with an appearance of 
 difficulty and want of simplicity which excludes 
 them, nearly altogether, from practical application. 
 The three problems are as follows : 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 227 
 
 I. Given the fall, length, and diameter of a pipe 
 or hydraulic mean depth of any channel, to find the 
 discharge. 
 
 Here all that is necessary is to find the mean 
 velocity of discharge, which, multiplied by the area 
 of the section (equal d 2 x *7854 in a cylindrical 
 pipe), gives the discharge sought. TABLE VIII. 
 gives the velocity at once for long channels, accord- 
 ing to Du Buat, or it can be found from equation 
 (119A) by calculation. TABLE IX. gives the dis- 
 charge in cubic feet per minute for different diameters 
 of pipes, and velocities in inches per second, when 
 found from TABLE VIII. or formula (119A). See 
 also TABLES XI. and XII. For a pipe 6 inches in 
 diameter, the velocity per second is practically equal 
 to the discharge in cubic feet per minute. See 
 also the tables, pp. 42, 43, 252, and 253. 
 
 II. Given the discharge and cross section of a 
 channel, to find the fall or hydraulic inclination. 
 
 If the cross section be circular, as in most pipes, 
 the hydraulic mean depth is one-fourth of the dia- 
 meter ; in other channels it is found by dividing the 
 water and channel line of the section, wetted peri- 
 meter, or border, into the area. The velocity is 
 found by dividing the area into the discharge, and 
 reducing it to inches per second ; then in TABLE 
 VIII., under the hydraulic mean depth, find the 
 velocity, corresponding to which the fall per mile 
 will be found in the first column, and the hydraulic 
 inclination in the second. This result can be cor- 
 rected by trial and error to accord with formula 
 (11 BA), and the table for the values of r s and v, p. 220, 
 
 Q3 
 
228 THE DISCHARGE OF WATER FROM 
 
 calculated from it. See also the tables, pp. 42, 43, 
 252, and 253. 
 
 III. Given the discharge, length, and fall, to find 
 the diameter of a pipe, or hydraulic mean depth and 
 dimensions of a channel. 
 
 This is the most useful problem of the three. 
 Assume any mean radius r a , and find the discharge D a 
 by Problem I. We shall then have for cylindrical 
 pipes 
 
 rf : 7*5 : : D a : D : : 1 : ; 
 
 and as r a , D, and D a are known, r% becomes also 
 known, and thence r. TABLE XIII. will enable us to 
 find r with great facility. Thus, if we had assumed 
 r & =. 1 and found D a = 15, D being 33, we then have 
 
 1 : r*:: 1 : - :: 1 : 2*2, therefore r* = 22 ; 
 
 15 
 
 and thence by TABLE XIII., r = 1*37, the mean radius 
 required, four times which is the diameter of the 
 pipe. For other channels, the quantity thus found 
 must be the hydraulic mean depth ; and all channels, 
 however varied in the cross section, will have the 
 same velocity of discharge, when the fall, length, and 
 hydraulic mean depth are constant. If r a be as- 
 sumed equal to 1J inch, the velocity found from 
 TABLE VIII. will then be the discharge in cubic feet 
 per minute nearly, and this "mean radius" can 
 always be assumed for the first term of the pro- 
 portion. See also the tables, pp. 42, 43, 252, and 253. 
 In order to find the dimensions of any polygonal 
 channel whatever, which will give a discharge equal 
 to D, we may assume any channel similar to that 
 
OEIFICES, WEIES, PIPES, AND RIVERS. 229 
 
 proposed, one of whose known sides is s a , and find 
 the corresponding discharge, D a , by Problem L, or 
 from TABLES XI. and XII. ; then, if we call the like 
 side of the required channel, s, we shall have 
 
 sn:s a ( j, and thence the numerical value from 
 
 TABLE XIII. The result, as before, can be corrected 
 to accord with any of our formulae by the method of 
 trial and error. 
 
 As it frequently happens that deposits in and en- 
 crustations on a pipe take place from time to time, 
 which diminish the flowing section considerably, it 
 is always prudent, when calculating the necessary 
 diameter, to take the largest coefficient of friction, c f ,or 
 to double its mean value, particularly for small pipes, 
 when calculating the diameter from any of the for- 
 mulae. Some engineers, as D'Aubuisson, increase 
 the quantity of water by one-half to find the dia- 
 meter ; but much must depend on the peculiar cir- 
 cumstances of each case, as sometimes less may be 
 sufficient, or more necessary. Tne discharge increases 
 in similar figures, nearly as r% or as d? 9 that is, as the 
 square root of the fifth power of the diameter, and 
 the corresponding increase in the diameter for any 
 given or allowed increase in the discharge can be 
 easily found by means of TABLE XIII., as shown 
 above. If we increase the dimensions by one-sixth, 
 the discharge will be increased by one-half nearly, 
 and by doubling them the discharge is increased in 
 the proportion of 5 1 to 1. 
 
 For shorter pipes, we have to take into considera- 
 tion the head due to the velocity and orifice of entry. 
 
230 THE DISCHARGE OF WATER FROM 
 
 Taking the mean coefficient of velocity or discharge, 
 we find from TABLE II. the head due to the velocity, 
 if it be known ; this subtracted from the whole head, 
 H, leaves the head, h iy due to the hydraulic inclina- 
 tion, which is that we must make use of in the table. 
 If the velocity be not given, we can find it approxi- 
 mately ; the head found for this velocity, due to the 
 orifice of entry, when deducted, as before, will give a 
 close value of h t , from which the velocity may be 
 determined with greater accuracy, and so on to any 
 degree of approximation. In general, one approxi- 
 mation to h t will be sufficient, unless the pipes be 
 very short, in which case it is best to use equation 
 (74). EXAMPLE VIII., p. 195, and the explanation 
 of the use of the tables, SECTION I., may be usefully 
 referred to. 
 
 TABLES XI., XII., and XIII. enable us to solve 
 with considerable facility all questions connected 
 with discharge, dimensions of channel, and the ordi- 
 nary surface inclinations of rivers. The discharge 
 corresponding to any intermediate channels or falls 
 to those given in TABLES XI. or XII., will be found 
 with abundant accuracy, by inspection and simple 
 interpolation ; and in the same manner the channels 
 from the discharges. Rivers have seldom greater 
 falls than those given in TABLE XII., but in such an 
 event we have only to divide the fall by 4, then 
 twice the corresponding discharge will be that re- 
 quired. TABLE XIII. gives the comparative dis- 
 charging powers of all similar channels, whether 
 pipes or rivers, and the comparative dimensions from 
 the discharges. We perceive from it, that an increase 
 
OEIFICES, WEIKS, PIPES, AND RIVERS. 231 
 
 of one-third in the dimensions doubles, and a de- 
 crease of one-fourth reduces the discharge to one- 
 half. By means of this table, we can determine by 
 a simple proportion, the dimensions of any given 
 form of channel when the discharge is known. See 
 EXAMPLE 17, p. 30. See also the tables pp. 42, 
 43, 252 and 253. 
 
 The mean widths in TABLES XI. and XII. are cal- 
 culated for rectangular channels, and those having 
 side slopes of 1J- to 1. Both these tables are, how- 
 ever, practically, equally applicable to any side 
 slopes from to 1 up to 2 to 1, or even higher, when 
 the mean widths are taken and not those at top or 
 bottom. A semihexagon of all trapezoidal channels 
 of equal area has the greatest discharging power, and 
 the semisquare and all rectangles exactly the same 
 as channels of equal areas and depths with side 
 slopes of H to 1. The maximum discharge is ob- 
 tained between these for the semihexagon with side 
 slopes, of nearly \ to 1, but for equal areas and 
 depths the discharge decreases afterwards as the slope 
 flattens. The question of "HOW MUCH?" is here, 
 however, a very important one ; for, as we have 
 already pointed out in equations (28) and (31), the 
 differences for any practical purposes may be imma- 
 terial. This is particularly so in the case of chan- 
 nels with different side slopes, if, instead of the top 
 or bottom, we make use of the mean width to calcu- 
 late from. We then have only to subtract the ratio 
 of the slope multiplied by the depth to find the 
 bottom, and add it to find the top. If the mean 
 width be 50 feet, the depth 5 feet, and the side slopes 
 
232 THE DISCHARGE OF WATER FROM 
 
 2 to 1, we get 50 (2 x 5) z= 40 for the bottom, and 
 50 + (2 x 5) = 60 for the top width. 
 
 Side slopes of 2 to 1 present a greater difference 
 from the mean slope of H to 1, than any others in 
 general practice when new cuts are to be made. A 
 triangular channel having slopes of 2 to 1, and bot- 
 tom equal to zero, differs more in its discharging 
 power from the half square, equal to it in depth and 
 area, than if the bottom in each was equally in- 
 creased, yet even here it is easy to show that this 
 maximum difference is only 5 per cent. If the 
 bottom be increased so as to equal the depth, it is 
 only 4J per cent. ; when equal to twice the depth, 3 -8 
 per cent. ; and when equal to four times the depth, to 
 2 per cent. ; while the differences in the dimensions 
 taken in the same order are only 2-2, 1-8, 1-5, and 
 0-8 per cent. For greater bottoms in proportion to 
 the depth the differences become of no comparative 
 value. It therefore appears pretty evident, that 
 TABLES XL and XII. will be found equally applicable 
 to all side slopes from to\upto1to\,by taking the 
 mean widths. When new cuts are to be made, we see 
 no reason whatever in starting from bottom rather 
 than mean widths, to calculate the other dimensions ; 
 indeed, the necessary extra tables and calculations 
 involved ought entirely to preclude us from doing 
 so. Besides, the formulas for finding the discharge 
 vary in themselves, and for different velocities the 
 coefficient of friction also varies.* Added to which 
 
 * The coefficient m in the formula v = m (r s)* in rivers for 
 velocities from 3 inches to 3 feet per second, varies from about 
 72 to 103 ; yet, strange to say, most tables are calculated from 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 233 
 
 the inequalities in every river channel, caused by 
 bends and unequal regimen, preclude altogether any 
 regularity in the working slopes and bottom, though 
 the mean width would continue pretty uniform under 
 all circumstances. 
 
 The quantities in TABLE XII. are calculated, from 
 the velocities found from TABLE VIII., to correspond 
 to a channel 70 feet wide and of different depths, the 
 equivalents to which are given in TABLE XI. In 
 order to apply these tables generally to all open 
 channels, the latter are to be reduced to rectangular 
 ones of the same depth and mean width, or the re- 
 verse, as already pointed out. If the dimensions of 
 the given channel be not within the limits of TABLE 
 XI., divide the dimensions of the larger channels by 
 4, and multiply the corresponding discharge found 
 in TABLE XII. by 32 ; for smaller channels, multiply 
 the dimensions by 4, and divide by 32. In like 
 manner, if the discharge be given and exceed any to 
 be found in TABLE XIII., divide by 32, and multiply 
 the dimensions of the suitable equivalent channel 
 found in TABLE XI. by 4. If we wish to find equiva- 
 lent channels of less widths than 10 feet for small 
 discharges, multiply the discharge by 32, and divide 
 the dimensions of the corresponding equivalent by 4, 
 Many other multipliers and divisors as well as 4 and 
 32 may be found from TABLE XIII., such as 3 and 
 
 one coefficient alone ; or, rather, from a formula equivalent to 
 94-17 (r s)$, which gives results suited only to a velocity of 16 
 inches. Dimensions of channels calculated by means of this 
 formula are too small in one case, and too large in the other. In 
 pipes the variation of the coefficients is shown in the small tables, 
 pp. 214 and 217. 
 
234 THE DISCHARGE OF WATER FROM 
 
 15-6, 6 and 88-2, 7 and 130, 9 and 243, 10 and 316, 
 12 and 499, &c. The differences indicated at pages 
 198 and 199, must be expected in the application of 
 these rules, which will give, however, dimensions for 
 new channels which can be depended on for doing 
 duty. 
 
 It will be seen from TABLE XIII. that a very 
 small increase in the dimensions increases the dis- 
 charging power very considerably. TABLE XII. 
 also shows that a small increase in the depth alone 
 adds very much to the discharge. If we express in 
 
 this latter case a small increase in the depth, d, by - , 
 
 7i 
 
 then it is easy to prove that the corresponding in- 
 crease in the velocity, v, will be ; and that in the 
 
 A 1 
 
 discharge D, o~, if the surface inclination continue 
 
 unchanged ; but as it is always observable in rivers 
 that the surface inclinations increase with floods, the 
 differences in practice will be found greater than 
 these expressions make it. As in a large river the 
 surface inclination must be very small, four times 
 the fall will add very little to the sectional area ; yet 
 this increase of fall will double the discharge, and 
 we thence perceive how tributaries can be absorbed 
 into the main channel without any great increase to 
 its depth. 
 
OKIFICES, WEIRS, PIPES, AND EIVERS. 
 
 235 
 
 SECTION IX. 
 
 BEST FORMS OF THE CHANNEL. REGIMEN. VELOCITY. 
 EQUALLY DISCHARGING CHANNELS. 
 
 We have seen above, that the determination of the 
 hydraulic mean depth does not necessarily determine 
 the section of the channel. If the form be a circle, 
 the diameter is four times the mean radius ; but, 
 though this form be almost always adopted for pipes, 
 the beds of rivers take almost every curvilineal and 
 trapezoidal shape. Other things being the same, 
 that form of a river channel, in which the area 
 divided by the border is a maximum, is the best. 
 This is a semicircle having the diameter for the sur- 
 face line, and in the same manner, half the regular 
 figures, an octagon, hex- 
 agon, and square, in Fig. 
 33, are better forms for 
 the channel, the areas 
 and side slopes being 
 constant, than any others 
 of the same number of 
 sides. Of all rectangular channels, Diagram 4, in 
 which A B c D is half a square, is the best cross section ; 
 and in Diagram 3, A c D B, half a hexagon, is the best 
 trapezoidal form of cross section. When the width 
 of the bottom, c D, Diagram 3, is given, and the slope 
 
 Aa 
 
 = n, then, in order that the discharge may be the 
 
 greatest possible, we must have 
 
236 THE DISCHARGE OF WATER FROM 
 
 A ** 
 
 -12 
 
 in which A is the given area of the channel. As, 
 however, we have never known a river in which the 
 slope of the natural banks continued uniform, even 
 though made so for any improvements, we consider it 
 almost unnecessary to give tables for different values 
 of n. If, notwithstanding, we put $ for the inclina- 
 tion of the slope A c, equal angle c A a, we shall find, 
 
 1 
 
 as cot. < = n, and vri 2 + 1 = - - , that the fore- 
 
 sin. <f> 
 
 going equations become 
 
 ( A sin. <fr | * _ CJD 
 
 12 cos. *! = 2 {(n* + l)*-n}' 
 
 and 
 
 (121.) CD = ca x cot. <,* 
 
 C cl 
 
 which will give the best dimensions for the channel 
 when the angle of the slope for the banks is known. 
 When the discharge from a channel of a given 
 area, with given side slopes, is a maximum, it is easy 
 to show that THE HYDRAULIC MEAN DEPTH MUST BE HALF 
 
 OF THE CENTRAL OR GREATEST DEPTH. This Simple 
 
 principle enables us to construct the best form of 
 channel with great facility. Describe any circle on 
 the drawing-board ; draw the diameter and produce 
 
 * When c D = o. The channel is triangular ; we get A = c a 2 
 
OKIFICES, WEIES, PIPES, AND RIVERS. 237 
 
 it on both sides, outside the circle ; draw a tangent to 
 the lower circumference parallel to this diameter, and 
 draw the side slopes at the given inclinations, touching 
 the circumference also on each side and terminating 
 on the parallel lines : the trapezoid thus formed will 
 be the best form of channel, and the width at the 
 surface will be equal to the sum of the two side 
 slopes. It is easy to perceive that this construction 
 may be, simply, extended for finding the best form 
 of a channel having any polygonal border whatever 
 of more sides than three and of given inclinations. 
 
 Commencing with the best discharging form of chan- 
 nel, which in practice will have the mean width, about 
 double the depth ; an equally discharging section of 
 double the width of the first will have the contents one- 
 eleventh greater, and the depth less in the proportion 
 of 1 to 1*85. A channel of double the mean width 
 of the second must have the sectional area further 
 increased by about one-fifth, and a further decrease 
 in the depth from 1*67 to 1 nearly. The greater 
 expense of the excavation at greater depths will, 
 in general, more than counterbalance these differences 
 in the contents of the channel. When the banks rise 
 above the flood line, and are unequal in their section, 
 the wider channel involves further upper extra cut- 
 ting, but there is greater capacity to discharge extra 
 and extraordinary flooding, the banks are less liable 
 to slip or give way, the slopes may be less, and the 
 velocity being also less, the regimen will, in general, 
 be better preserved. The table of equally discharg- 
 ing channels, p. 2 5 2, will afford the means of calculating 
 the difference of cubical contents. 
 
238 
 
 THE DISCHARGE OF WATER FROM 
 
 If" 
 
 % % % ^ % 
 
 uo i i <?q T-I o 
 
 O <M - t- 00 
 
 rH CN? T^ O J>- 
 
 C{ CH Cq rH 
 
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 COt^i-tt-^Ht-OOO 0^3,- 
 
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 TS S K * i 
 
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 J-i O 
 
 1 1 
 
 CL-fi'Z " 
 
OKIFICES, WEIES, PIPES, AND RIVERS. 239 
 
 When the sectional area is given, the above table 
 shows that the semicircle is the best discharging 
 channel, and the complete circle the worst; the 
 latter is so, however, only compared with the open 
 channels given in the table, it being the very best 
 form for an enclosed channel flowing full. T/ie best 
 form of channel is particularly suited for new cuts in 
 flat, marsh, callow, and fen lands, in which it is also 
 often advisable to cut them with a level bed, up from 
 the discharging point, in order to increase the hy- 
 draulic mean depth, and consequently the velocity 
 and discharge. 
 
 As the quantity of water coming down a river 
 channel in a season varies very considerably, we 
 have observed it in one case to vary from one to 
 thirty, and occasionally in the same channel from 
 one to seventy-five, the proportion of the water 
 section to the channel itself must also vary, and 
 those relations of the depth, sides, and width to each 
 other, above referred to, cease to hold good and be 
 the best under such circumstances. If the object be 
 to construct a mill-race, temporary drain for un- 
 watering a river, or other small channel, in which 
 the depth remains nearly constant, channels of the 
 form of a half hexagon, diagram 3, Fig. 33, will be, 
 perhaps, the best, if the tenacity of the banks per- 
 mit the slope ; but rivers, in which the quantity of 
 water varies considerably, require wider channels in 
 proportion to the depth ; and also, that the velocity 
 be so proportioned to the tenacity of the soil, or as 
 it is termed " the regimen" that the banks and bed 
 
Fig. 34 
 
 240 THE DISCHAKGE OF WATER FROM 
 
 shall not vary from time to time to any injurious 
 extent, and that any deposits made during their 
 summer state, and during light freshes, shall be 
 carried off periodically by floods. Another circum- 
 stance, also, modifies the effects of the water on the 
 banks. It is this, that at curves, and turns, the 
 current acts wit,h greatest effect against the bank, 
 concave to the di- 
 rection in which 
 it is moving; deep- 
 ening the channel 
 there ; undermin- 
 ing also the bank, 
 as at A, Fig. 34 ; and raising the bed towards the 
 opposite side B. The reflexion of the current to 
 the opposite bank from A acts also in a similar man- 
 ner, lower down, upon it ; and this natural operation 
 proceeds, until the number of turns, increased length 
 of channel, and loss of head from reflexion and 
 unequal depths, bring the currents into regimen with 
 the bed and banks. At all bends it is, therefore, 
 prudent to widen the channel on the convex side B, 
 Fig. 34, in order to reduce the velocity of approach ; 
 and if the bed be here also sunk below its natural 
 inclination, as we see it in most rivers at bends, the 
 velocity will be farther reduced, and the permanence 
 of the bed better established. 
 
 The circumstances to be considered in deciding on 
 the dimensions and fall of a new river course, after 
 the depth to which the surface of the water is to be 
 brought has been decided on, are the following : 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 241 
 
 The mean velocity must not be too slow, or 
 aquatic plants will grow, and deposits take place, 
 reducing the sectional area until a new and smaller 
 channel is formed within the first with just sufficient 
 velocity to keep itself clear. This velocity should 
 not in general be less than from ten to fourteen 
 inches per second. The velocity in a canal or river 
 is increased very considerably by cutting or re- 
 moving reeds and aquatic plants growing on the 
 sides or bottom.* 
 
 The mean velocity must not be too quick, and 
 should be so determined as to suit the tenacity 
 and resistance of the channel, otherwise the bed and 
 banks will change continually, unless artificially 
 protected ; it should not exceed 
 
 25 feet per minute in soft alluvial deposits. 
 
 40 clayey beds. 
 
 60 sandy and silty beds. 
 
 120 gravelly. 
 
 180 strong gravelly shingle. 
 
 * " M. Girard a fait observer, avec raison, que les plantes aqua- 
 tiques, qui croissent toujours sur le fond et sur les berges des 
 canaux, augmentent conside"rablement le perimetre mouille, et par 
 suite la resistance ; il a rapelle* que Du Buat, ayant mesure la 
 vitesse de 1'eau dans le canal du Jard, avant et apres la coupe des 
 roseaux dont il e"tait garni, avait trouve un resultat bien moindre 
 avant qu'apres. En consequence, il a presque double la pente 
 donne"e par le calcul . . ." Traite d'Hydraulique, p. 135. 
 When the fall does not exceed a few inches per mile, the velocity, 
 as determined from the inclination, is very uncertain, and for this 
 reason it is always prudent to increase the depths and sectional 
 areas of channels in flat lands, as far as the regimen will permit. 
 In such cases the section of the channel should approximate 
 towards the best form. See p. 238. 
 
 R 
 
242 
 
 THE DISCHARGE OF WATER FROM 
 
 240 feet per minute in shingly. 
 
 300 ., shingly and rocky. 
 
 400 and upwards in rocky and shingly.* 
 
 A velocity of 180 feet per minute will remove an- 
 gular stones the size of an egg. Mr. Phillips, under 
 the Metropolitan Commissioners of Sewers, states 
 that 2| feet per second, or 150 feet per minute, 
 is sufficient to prevent soil depositing in sewers. 
 
 The fall per mile should decrease as the hydraulic 
 mean depth increases, and both be so proportioned 
 that floods may have sufficient power to carry off the 
 deposits, if any, periodically. The proportion of the 
 width to the depth of the channel should not be 
 derived, for new cuts or river courses, from any 
 formula, but taken from such portions of the old 
 channel as approximate in depth and in the inclina- 
 tion of the surface to that proposed. When the 
 depth is nearly half the width, the formula shows, 
 cceteris paribus, that the discharge will be a maxi- 
 mum ; but as (altogether apart from the question 
 
 * TABLE OF VELOCITIES OP SOME MOVING BODIES COMPARED WITH THOSE OF RIVERS. 
 
 Objects in motion. 
 
 Miles 
 per 
 hour. 
 
 Feet 
 per 
 second. 
 
 Objects in motion. 
 
 Miles 
 per 
 hour. 
 
 Feet 
 per 
 second. 
 
 Current of slow rivers . . 
 
 Currents of ordinary rivers, 
 up to 
 
 A 
 ii 
 
 1 
 
 Railway trains, French . . 
 German . 
 Sound when atmosphere 
 is at 32 Fahr 
 
 27 
 24 
 
 743 
 
 89* 
 
 53i 
 
 1 090 
 
 Currents of rapid rivers . 
 
 7 
 
 lof 
 
 Ditto 60 Fahr 
 
 765 
 
 1,122 
 
 Man walking 
 
 3 
 
 4? 
 
 
 850 
 
 1 247 
 
 Horse trotting 
 
 7 
 
 10i 
 
 TVtt 'h fh h t ' 
 
 
 
 Swiftest race-horse . . . 
 Moderate winds .... 
 
 60 
 7 
 36 
 
 88 
 101 
 52$ 
 
 stands at 30 inches . . 
 Common musket-ball . . 
 Rifle-ball 
 
 917 
 
 850 
 1000 
 
 1,344 
 1,247 
 1/67 
 
 
 80 
 
 117t 
 
 
 1 091 
 
 1 600 
 
 Swift English steamboats 
 navigating the channels . 
 Swift American Kiver 
 steamers 
 
 14 
 18 
 
 **t 
 
 20| 
 26^ 
 
 Bullet discharged from air- 
 gun, air being compress- 
 ed into the hundredth 
 
 477 
 
 700 
 
 Fast sailing vessels . . . 
 Railway trains, English . 
 American . 
 Belgian . 
 
 12 
 
 32 
 18 
 25 
 
 Hi 
 
 47 
 26* 
 36f 
 
 A point on earth's surface 
 at the equator moving 
 round the axis .... 
 Earth moving round sun . 
 
 1,040 
 68,182 
 
 1,525 
 100,000 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 243 
 
 of expense) the quantity of water discharged daily, 
 at different seasons, may vary from one to seventy, 
 or more, and " the regimen" has to be maintained, 
 the best proportion between the width and depth 
 of a new cut should be obtained, as we have stated, 
 from some selected portion of the old channel, whose 
 general circumstances and surface inclination ap- 
 proximate to those of the one proposed ; and the 
 side slopes of the banks must be such as are best 
 suited to the soil. The resistance of the banks to 
 the current being in general less than that of the 
 beds, which get covered with gravel, and the neces- 
 sary provision required for floods, appears to be 
 the principal reason why rivers are in general so 
 very much wider than about twice the depth, the 
 relation which gives the minimum of friction. 
 
 The following table is given by Bennie, as an 
 approximation, generally, to the actual state of 
 rivers.* The surface inclinations, however, given 
 in this table for the first and second classes, are 
 very considerable for large rivers, and would give 
 velocities which would effectually scour them. For 
 a hydraulic mean depth of 12 feet, the velocity, with 
 
 a fall of 10000' would be 2 feet 8 inches per second 
 
 by Du Buat's formula ; and 3-3 feet per second by 
 our formula. The description, therefore, can only 
 apply to small channels. In fact, 4 inches to a 
 
 mile, or , is a considerable inclination for a 
 
 large river. 
 
 * Eeport to the British Association 1834. 
 
 R3 
 
244 
 
 THE DISCHARGE OF WATER PHOM 
 
 
 i* 2 
 
 2 a S 
 
 tjj 
 
 ||| 
 
 *8J 
 
 i?l 
 
 'S'o 
 
 DISTINCTIVE ATTBEBUTE8 
 
 **f 
 
 - 'o "3 
 
 6 
 
 | ** 
 
 * 
 
 00 *H rfl '> 
 
 09 THE 
 
 f 
 "o > &o 
 
 i.'|| 
 
 o" 
 
 a! 2 
 
 1 .& 
 
 I-H 
 
 VABIOUS KINDS OF BIVEBS. 
 
 
 g | fl 
 
 'S'wo 
 fi "3 * 
 
 o-g 
 
 
 B& 13 
 
 
 1*1 
 
 u - a 
 
 "! a 
 
 ill 
 
 III 
 
 t$l 
 
 -||o 
 
 Channels wherein the resist- 
 
 
 
 
 
 
 
 ance from the bed, and other 
 
 
 
 
 
 
 
 obstacles, equal the quantity of 
 
 
 
 
 
 
 
 current acquired from the de- 
 clivity ; so that the waters would 
 
 1st. 
 
 
 
 0" 
 
 
 
 12000 
 
 14 
 
 stagnate therein, were it not for 
 
 
 
 
 
 
 
 the compression and impulsion 
 
 
 
 
 
 
 
 of the upper and back waters . 
 
 
 
 
 
 
 
 Artificial canals in the Dutch i 
 and Austrian Netherlands . . J 
 
 2nd. 
 
 1 
 
 180 
 
 
 
 1000 
 
 8 
 
 Rivers in low flat countries, "] 
 
 
 
 
 
 
 
 full of turns and windings, and 
 
 
 
 
 
 
 
 of a very slow current, subject 
 to frequent and lasting inunda- 
 
 - 3rd. 
 
 1 
 
 120 
 
 10 
 
 5200 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Rivers in most countries that \ 
 
 
 
 
 
 
 
 are a mean between flat and 
 
 
 
 
 
 
 
 hilly, which have good currents, 
 but are subject to overflow ; also 
 
 4th. 
 
 li 
 
 80 
 
 15 
 
 Wtro 
 
 4| 
 
 the upper parts of rivers in flat 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Rivers in hilly countries with\ 
 
 
 
 
 
 
 
 a strong current, and seldom 
 
 
 
 
 
 
 
 subject to inundations ; also all 1 
 rivers near their sources have ( 
 
 5th. 
 
 *i 
 
 55 
 
 21* 
 
 T2"0(7 
 
 3f 
 
 this declivity and velocity, and 
 
 
 
 
 
 
 
 often much more J 
 
 
 
 
 
 
 
 Rivers in mountainous coun- \ 
 
 
 
 
 
 
 
 tries having a rapid current and ( 
 straight course, and very rarely [ 
 
 6th. 
 
 3 
 
 40 
 
 30 
 
 2600 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Rivers in their descent from\ 
 
 
 
 
 
 
 
 among mountains down into the 1 
 plains below, in which plains [ 
 
 7th. 
 
 5 
 
 24 
 
 50 
 
 "2000 
 
 2J 
 
 they run torrent- wise. . . . j 
 
 
 
 
 
 
 
 Absolute torrents among i 
 
 8th. 
 
 8 
 
 15 
 
 80 
 
 x 
 
 2 
 
 mountains . . i 
 
 
 
 
 
 1 7 
 
 
 
 
 
 
 
 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 245 
 
 The following information with reference to the sur- 
 face inclinations of the Thames, is from Rennie's Report 
 on Hydraulics,* as a branch of engineering science. 
 
 Names of places. 
 
 Length. 
 
 Fall 
 
 Fall in feet 
 per mile. 
 
 Ratio of 
 inclinations. 
 
 From Lechdale at St. John's 
 Bridge to Oxford at FoUy 
 
 Miles. Fur. 
 
 28 
 
 Feet. In. 
 47 
 
 1-68 
 
 TTTTS 
 
 From Oxford to Abingdon 
 
 9 
 
 13 11 
 
 1-73 
 
 1 
 
 From Abingdon to Walling- 
 ford Bridge .... 
 
 14 
 
 27 4 
 
 1-95 
 
 
 From Wallingford to Read- 
 in s Bridsre 
 
 18 
 
 24 1 
 
 1-81 
 
 
 From Reading to Henley 
 Bridge 
 
 9 
 
 19 3 
 
 2-14 
 
 TT A & 7 
 
 From Henley to Mario w 
 Bridge 
 
 9 
 
 12 2 
 
 1-35 
 
 
 From Marlow to Maiden- 
 head Bridge 
 
 8 
 
 15 1 
 
 1-86 
 
 
 From Maidenhead Bridge to 
 Windsor Bridge .... 
 From Windsor 'to Staines 
 Bridge 
 
 7 
 8 
 
 13 6 
 
 15 8 
 
 1-93 
 1-96 
 
 2 1*8" 3 ^ 
 ff,T3TF 
 
 From Staines to Chertsey 
 Bridge .... 
 
 4 6 
 
 6 6 
 
 1-44 
 
 
 From Chertsey to Tedding- 
 ton-Lock . 
 
 13 6 
 
 19 8 
 
 1-45 
 
 s-T^rr 
 
 From Teddington-Lock to 
 London Bridge .... 
 From London to Yanlet 
 Creek 
 
 19 
 40 
 
 2 9 
 2 1 
 
 145 
 052 
 
 3.64 1 
 TnTi4'14 
 TO 1637 
 
 
 
 
 
 
 From Lechdale to Yanlet 
 Creek 
 
 186 4 
 
 218 
 
 
 
 Deduct . . . 
 
 40 
 
 
 
 .- 
 
 From Lechdale to London . 
 
 146 4 
 
 
 
 
 For enclosed channels, the circular form of sewer 
 will have the largest scouring power, at a given 
 hydraulic inclination. For the area of the sections 
 being the same, the velocity in the circular channel 
 will be a maximum. When the supply is intermit- 
 tent, and the channel too large, the egg-shaped form 
 
 * Report, for 1834, of the British Association. 
 
246 THE DISCHARGE OF WATER FROM 
 
 with the smaller end for the bottom, or the sides 
 vertical with an inverted ridge-tile or V bottom for 
 drains, will have a hydrostatic flushing power to 
 remove soil and obstructions, which a cylindrical 
 channel, only partly fall, does not possess ; because 
 a given quantity of water rises higher against the 
 same obstruction, or obstacle, to the flow in the 
 pipe. It must be confessed, however, that for small 
 drains and house-sewage, this gain is immaterial, and 
 is at best but effected by a sacrifice of space, mate- 
 rial, and friction in the upper part of drains, from 
 6 to 12 inches in diameter. Besides this, the mere 
 hydrostatic pressure is only intermittent, and during 
 an ordinary, or heavy, fall of rain, the hydrodynamic 
 power is always more efficient in scouring properly- 
 proportioned cylindrical drains ; and the workman- 
 ship in the form and joints is less imperfect than for 
 more compound forms, as those with egg-shaped and 
 inverted tile bottoms. The moulds and joints of 
 cylindrical stone- ware drains, exceeding 12 inches 
 in diameter, are seldom, however, in large quantities 
 perfect ; and the expense will exceed that of brick, 
 stone, or other sufficient drains in most localities. 
 
 As to the increased discharging power which it is 
 asserted by some, stone-ware cylindrical drains pos- 
 sess over other ordinary drains, no doubt it is true 
 for small sizes, because the form, jointing, and sur- 
 face are in general more smooth and circular ; and 
 for sewage matter* the friction and adherence to the 
 sides and bottom is less ; any advantage from these 
 causes becomes, however, immaterial for the larger 
 
 * Weisbach found the coefficient of resistance 1'75 times as 
 great for small wooden as for metallic pipes. All permeable pipes 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 247 
 
 sizes, as these can be constructed of brick or stone 
 abundantly perfect to any form, and sufficiently 
 smooth for all practical purposes, for in the larger 
 properly-proportioned sizes the same amount of sur- 
 face roughness opposed to the sewage matter is, com- 
 paratively, of no effect. The judicious inclination 
 and form of the bottom, and properly curved junc- 
 tions, are the principal points to be attended to. 
 Smaller drains tile-bottomed, with brick or stone 
 sides, and flat-covered, have one great advantage 
 over circular pipes,* They can be opened up, for 
 examination and repairs at any time with facility, 
 and at the smallest expense ; but greater certainty 
 must be attached to the working of small stone-ware 
 drains than to equally-sized small brick or stone 
 drains, and they will be found, in general, also 
 cheaper. This, however will depend on the locality. 
 It may be observed in numerous experiments, 
 that water flowing from a pipe does not entirely fill 
 the orifice of exit, when the velocities are not con- 
 siderable, and yet the results are found to be but 
 slightly affected if a little more than three-fourths of 
 the circumference be full. It is easy to demonstrate 
 that the full circle does not give the maximum dis- 
 charging velocity as has been generally believed, but 
 
 present greater resistance than impermeable ones ; hence the prin- 
 cipal advantage derived from glazing. 
 
 * Half-socket joints at bottom would remedy this imperfection 
 in small pipes, and they could be better laid and cemented. A 
 semicircular flange laid on at top would effectually protect the 
 joint on the upper side. Latterly Doulton has cut off an upper 
 segment from the pipe, which can be removed for cleaning. And 
 it may be demonstrated, that when this is a segment of 78J 
 degrees, the lower portion will discharge more than a full pipe at 
 the same inclination. 
 
248 
 
 THE DISCHARGE OF WATER FROM 
 
 when filled to the height 
 of the chord ac of arc 
 a e c of 784 degrees, and 
 where the velocity is 9i 
 per cent, over that due 
 to the full circle, for then 
 
 ,1 area ad c 
 
 the =- is a maxi- 
 
 arc aae 
 
 mum, and the length of 
 the arc adc is equal to the tangent of the supple- 
 mental arc a e c, as may be without difficulty demon- 
 strated. The hydraulic mean depths of the circle 
 and larger segment are to each other as -5 to -6, and 
 their square roots, which are as the velocities or 
 scouring powers, are as 1 to 1-095. The discharging 
 powers are to each other as 1x3*1416 to 1-095 
 X 2*946, or as 1 to 1-026, which shews that the seg- 
 ment adc has also a greater discharging power than 
 the whole circle of nearly three per cent. These facts, 
 which were first pointed out by the author, are not 
 unimportant in matters connected with pipe-drains 
 and sewerage. The effects of greater velocity and 
 discharge here pointed out, are sometimes increased, 
 in short pipes, from the fall between the surface a c, and 
 the surface from which the head is measured, being 
 greater than the fall to the top of the pipe at e, or from 
 the inclination of the surface of the water in the pipe 
 being greater than the inclination of the pipe itself. 
 
 EQUALLY DISCHARGING CHANNELS. 
 
 In order that different channels should have the 
 same discharging power, the inclination of the sur- 
 face being the same, the areas must be inversely as 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 249 
 
 the square roots of the hydraulic mean depths. The 
 channel a dcv, Fig. 35, will have the same discharge 
 as the channel A D c B if they be to each other 
 as f ADCB ,i to , 
 
 I A D 4- DC 4- C B J ( 
 
 AD + DC + CB 
 
 ad + dc + CB 
 
 and hence the square root of the cube of the channel 
 area, divided by the border, must be constant. With 
 a fall of one or more feet to a mile, two channels, one 
 70 feet wide and 1 foot deep, and the other 20 feet 
 wide and 2i feet deep, will have the same dis- 
 charge. If we put w for the width and d for 
 the depth of any rectangular channel, then 
 
 w 
 
 w 
 
 y 
 + 2 d) 
 
 m ; we therefore have the cubic equation 
 
 (122.) d--d = 
 
 w 3 w 2 
 
 for finding the depth, d of any other rectangular 
 channel whose width is w, of the same discharging 
 power. We have calculated the depths d for different 
 widths of channel from this equation, assuming a 
 width of 70 feet and different depths to find m from. 
 The results are given in TABLE XI., which will be 
 found sufficiently accurate for all practical purposes, 
 when the banks are sloped, by taking the mean width. 
 This table is equally applicable to any measures 
 whatever, to their multiples, and sub-multiples. 
 
 If the hydraulic inclinations vary, then the \/rs 
 must be inversely as the areas of the channels when 
 \/rs x channel or the discharge is constant ; and if 
 
250 THE DISCHARGE OF WATER FROM 
 
 the area of the channel and discharge be each con- 
 stant, r must vary inversely as s; and r s be also 
 constant. For instance, a channel which has a fall 
 of four feet per mile, and a hydraulic mean depth of 
 one foot, will have the same discharge as another chan- 
 nel of equal area, having a hydraulic mean depth of 
 four feet, and a fall per mile of only one foot. If in 
 TABLE XII. we take the same discharge from the 
 columns for different inclinations, we shall get the 
 mean rectangular dimensions corresponding to them 
 in the first column, and thereby be enabled to select 
 an equally discharging channel from TABLE XL, suited 
 to an increase or decrease of the hydraulic inclina- 
 tion.* 
 
 We have, however, calculated for this edition the 
 table at p. 252, of equally discharging river channels, 
 with a primary channel having a mean width of 100, 
 instead of 70, as in TABLE XI. ; and in the table at 
 p. 253 we have given the discharges at different in- 
 clinations from this new primary channel, to find 
 those from its equivalents. The tables at pp. 42, 43, 
 253, and TABLE VIII., have been calculated from 
 Du Buat's formula. For slow velocity of only a few 
 inches per second, the dimensions should be increased 
 by about one-sixth, and the discharges by about one- 
 half. 
 
 With reference to pipes, it is apparent that a given 
 depth of roughness or contraction arising from any 
 
 * Tables similar to numbers XL, XII., and XIII., but on a 
 much more extended scale, have been printed and published by 
 MB. WEALE, on a separate sheet for office use, and may be had 
 from him. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 251 
 
 cause will have a greater effect the smaller the dia- 
 meter becomes. Now in practice, it is necessary to 
 increase the diameter beyond what is found by calcu- 
 lation. For small service pipes half-an-inch is the 
 smallest diameter in general use. For mains and 
 sub-mains the value of c t in equation (74B), or at p. 
 214, should at least be doubled, or the discharge taken 
 at one and a half times its amount to find the dia- 
 meter. By enlarging the diameter by one-seventh, one- 
 half the amount will be added to the discharge, very 
 nearly; and by increasing the diameter by one-third, the; 
 discharge will be doubled. In a broad and practical 
 sense, and considering the losses arising from depo- 
 sitions,* pipes under two inches should have one- 
 third or more added to their calculated dimensions, 
 and larger pipes from one-third to one- seven th even 
 after making allowance for junctions, bends, and con- 
 tractions. For large conduits or channels the allow- 
 ance need not be so large, if the maximum quantity 
 to be conveyed be duly estimated. 
 
 * Mr. Bateman lately in giving evidence says :^ " He wished 
 to mention a circumstance which might he useful with regard to 
 the spongillse found in the Dublin water pipes. At Manchester, 
 before the introduction of soft water, the city was supplied with 
 hard water, which favoured the growth of a small fresh- water 
 mussel, which thickly line the reservoirs and pipes. There were 
 myriads of them, and they lay in the pipes as thick as paving 
 stones. These were caused by the large quantity of lime in the 
 water. He was curious to see what would be the effect of passing 
 water without lime. This was done ten or eleven years ago, and 
 the result was that these mussels had entirely disappeared. There 
 was no longer anything from which they could make their shells, 
 and for years, on their discharge, the small pipes were found 
 choked with them. If soft water were supplied to Dublin in 
 place of the present hard water, which probably favoured the 
 growth of spongillaD, they would probably disappear." 
 
TABLE of mean widtJis and depths of equally discharging trapezoidal River-channels, or 
 Sewers, with side slopes up to 2 J to 1. Practically all river-channels may be reduced to 
 rectangular sections of equal areas and depths to find the discharge. 
 
 Primary 
 Channel 
 
 Mean rectangular dimensions of equally discharging water-channels or sewers, in any 
 measures whatever, inches, feet, yards, fathoms, or their aliquot parts, or multiples. 
 
 Primary 
 Channel 
 
 Mean 
 width 
 100 
 
 Mean 
 width 
 90 
 
 Mean 
 width 
 80 
 
 Mean 
 width 
 70 
 
 Mean 
 width 
 60 
 
 Mean 
 width 
 50 
 
 Mean 
 width 
 40 
 
 Mean 
 width 
 30 
 
 Mean 
 width 
 20 
 
 Mean 
 width 
 15 
 
 Mean 
 width 
 10 
 
 Mean 
 width 
 100 
 
 1 
 
 11 
 
 12 
 
 13 
 
 14 
 
 16 
 
 18 
 
 22 
 
 29 
 
 35 
 
 47 
 
 1 
 
 125 
 
 13 
 
 14 
 
 16 
 
 17 
 
 20 
 
 23 
 
 28 
 
 37 
 
 45 
 
 60 
 
 125 
 
 2 
 
 21 
 
 23 
 
 25 
 
 28 
 
 32 
 
 37 
 
 45 
 
 60 
 
 73 
 
 98 
 
 2 
 
 25 
 
 27 
 
 29 
 
 32 
 
 35 
 
 40 
 
 46 
 
 56 
 
 75 
 
 92 
 
 1-25 
 
 25 
 
 3 
 
 32 
 
 35 
 
 38 
 
 42 
 
 48 
 
 56 
 
 68 
 
 90 
 
 1-11 
 
 1-52 
 
 3 
 
 375 
 
 40 
 
 44 
 
 .48 
 
 53 
 
 60 
 
 70 
 
 85 
 
 1-13 
 
 1-40 
 
 1-94 
 
 375 
 
 4 
 
 43 
 
 46 
 
 51 
 
 56 
 
 64 
 
 74 
 
 91 
 
 1-21 
 
 1-50 
 
 2-08 
 
 4 
 
 5 
 
 54 
 
 58 
 
 64 
 
 71 
 
 80 
 
 93 
 
 14 
 
 1-53 
 
 1-90 
 
 2-67 
 
 5 
 
 6 
 
 64 
 
 70 
 
 76 
 
 85 
 
 96 
 
 1-12 
 
 37 
 
 1-85 
 
 2-31 
 
 3-28 
 
 6 
 
 625 
 
 67 
 
 73 
 
 79 
 
 88 
 
 1-00 
 
 1-16 
 
 43 
 
 1-93 
 
 2-42 
 
 3-44 
 
 625 
 
 7 
 
 75 
 
 81 
 
 89 
 
 99 
 
 1-12 
 
 1-31 
 
 61 
 
 2-17 
 
 2-73 
 
 3-92 
 
 7 
 
 75 
 
 80 
 
 87 
 
 95 
 
 06 
 
 1-20 
 
 1-41 
 
 73 
 
 2-34 
 
 2-95 
 
 425 
 
 75 
 
 8 
 
 86 
 
 93 
 
 1-02 
 
 13 
 
 1-29 
 
 1-50 
 
 85 
 
 251 
 
 3-17 
 
 4-59 
 
 8 
 
 875 
 
 94 
 
 02 
 
 1-12 
 
 24 
 
 1-40 
 
 1-64 
 
 2-02 
 
 2-76 
 
 3-50 
 
 5-10* 
 
 875 
 
 9 
 
 97 
 
 05 
 
 1-15 
 
 27 
 
 1-45 
 
 1-69 
 
 2-08 
 
 2-84 
 
 3-61 
 
 5-28 
 
 9 
 
 1-0 
 
 1-07 
 
 16 
 
 1-27 
 
 42 
 
 1-61 
 
 1-88 
 
 2-32 
 
 3-18 
 
 4-07 
 
 5-99 
 
 1-0 
 
 1-125 
 
 1-21 
 
 31 
 
 1-43 
 
 60 
 
 1-81 
 
 2-13 
 
 2-63 
 
 3-62 
 
 4-64 
 
 6-92 
 
 1-25 
 
 1-2 
 
 1-29 
 
 40 
 
 1-53 
 
 70 
 
 1-94 
 
 2-27 
 
 2-81 
 
 3-88 
 
 5-00 
 
 7-50 
 
 1-2 
 
 1-25 
 
 1-35 
 
 46 
 
 1-60 
 
 78 
 
 2-02 
 
 2-37 
 
 2-94 
 
 4-06 
 
 5-24 
 
 7-89 
 
 1-125 
 
 1-3 
 
 1-40 
 
 51 
 
 1-66 
 
 85 
 
 2-10 
 
 2-47 
 
 3-06 
 
 4-24 
 
 5-48 
 
 8-29 
 
 1-3 
 
 1-375 
 
 1-48 
 
 60 
 
 1-76 
 
 1-96 
 
 2-23 
 
 2-62 
 
 3-25 
 
 4-51 
 
 5-85 
 
 8-89 
 
 1-375 
 
 1-4 
 
 1-50 
 
 63 
 
 1-79 
 
 1-99 
 
 2-27 
 
 2-66 
 
 3-31 
 
 4-60 
 
 5-97 
 
 9-10 
 
 1-4 
 
 1-5 
 
 1-61 
 
 75 
 
 1-92 
 
 2-14 
 
 2-43 
 
 2-86 
 
 3-56 
 
 4-97 
 
 6-47 
 
 9-92 
 
 1-5 
 
 1-6 
 
 1-72 
 
 1-86 
 
 2-05 
 
 2-28 
 
 2-60 
 
 3-06 
 
 3-81 
 
 5-34 
 
 6-U8 
 
 10-78 
 
 16 
 
 1-625 
 
 1-75 
 
 1-89 
 
 2-08 
 
 2-32 
 
 2-64 
 
 3-11 
 
 3-87 
 
 5-43 
 
 7-11 
 
 11-00 
 
 1-625 
 
 1-7 
 
 1-83 
 
 1-98 
 
 2-17 
 
 2-43 
 
 2-76 
 
 3-26 
 
 4-06 
 
 5-72 
 
 7-50* 
 
 11-66 
 
 1-7 
 
 1-75 
 
 1-88 
 
 2-04 
 
 2-24 
 
 2-50 
 
 2-85 
 
 3-36 
 
 4-19 
 
 5-91 
 
 7-77 
 
 12-10 
 
 175 
 
 1-8 
 
 1"93 
 
 2-10 
 
 2-30 
 
 2-57 
 
 2-93 
 
 3-45 
 
 4-32 
 
 6-09 
 
 8-03 
 
 12-54 
 
 1-8 
 
 1-875 
 
 2-02 
 
 2-19 
 
 2-40 
 
 2-68 
 
 3-05 
 
 3-60 
 
 4-51 
 
 6-38 
 
 8-43 
 
 13-23 
 
 . 1-875 
 
 1-9 
 
 2-04 
 
 2-22 
 
 2-43 
 
 2-71 
 
 3-10 
 
 3-65 
 
 4-57 
 
 6-48 
 
 8-57 
 
 13-46 
 
 1-9 
 
 2-0 
 
 2-15 
 
 2-33 
 
 2-56 
 
 2-86 
 
 3-26 
 
 3-86 
 
 4-83 
 
 6-87 
 
 9-11 
 
 14-39 
 
 2-0 
 
 2-1 
 
 2-26 
 
 2-45 
 
 2-69 
 
 3-01 
 
 3-43 
 
 4-06 
 
 5-09 
 
 7-27 
 
 9-67 
 
 15-35 
 
 2-1 
 
 2-2 
 
 2-37 
 
 2-57 
 
 2-82 
 
 3-15 
 
 3-60 
 
 4-26 
 
 5-36 
 
 7-66 
 
 10-23 
 
 1632 
 
 2-2 
 
 2-3 
 
 2-47 
 
 2-69 
 
 2-95 
 
 3-30 
 
 3-77 
 
 4-4'J 
 
 5-62 
 
 8-07 
 
 10-80 
 
 17-31 
 
 2-3 
 
 2-4 
 
 2-58 
 
 2-80 
 
 3-08 
 
 3-44 
 
 3-94 
 
 4-67 
 
 5-89 
 
 8-48 
 
 11-38 
 
 18-33 
 
 2-4 
 
 2-5 
 
 2-69 
 
 2-92 
 
 3-21 
 
 3-59 
 
 4-11 
 
 4-87 
 
 6-16 
 
 8-79 
 
 11-97 
 
 19-35 
 
 2-5 
 
 2-6 
 
 2-80 
 
 3-04 
 
 3-34 
 
 3-74 
 
 4-28 
 
 5-08 
 
 6-42 
 
 9-31 
 
 12-57 
 
 20-40 
 
 2-6 
 
 2-7 
 
 2-91 
 
 3-16 
 
 3-47 
 
 3-88 
 
 4-55 
 
 5-28 
 
 6-69 
 
 9-73 
 
 13-17 
 
 21-46 
 
 2-7 
 
 2-8 
 
 3-01 
 
 3-27 
 
 3-60 
 
 4-03 
 
 4-62 
 
 5-49 
 
 6-97 
 
 10-16* 
 
 13-78 
 
 22-52 
 
 2-8 
 
 2-9 
 
 3-12 
 
 3-39 
 
 3-73 
 
 4-18 
 
 4-79 
 
 5-70 
 
 7-24 
 
 10-59 
 
 14-40 
 
 23-63 
 
 2-9 
 
 3-0 
 
 3-23 
 
 3-51 
 
 3-86 
 
 4-42 
 
 4-96 
 
 5-91 
 
 7-52 
 
 11-02 
 
 15-03 
 
 24-75 
 
 3-0 
 
 3-1 
 
 3-34 
 
 3-63 
 
 3-99 
 
 4-47 
 
 5-13 
 
 6-12 
 
 7-79 
 
 11-46 
 
 15-68 
 
 
 3-1 
 
 3-2 
 
 3-45 
 
 3-75 
 
 4-13 
 
 4-62 
 
 5-30 
 
 6-33 
 
 8-07 
 
 11-90 
 
 16-32 
 
 el 
 
 3-2 
 
 3-3 
 
 3-55 
 
 3-86 
 
 4-26 
 
 4-77 
 
 5-48 
 
 6-54 
 
 8-35 
 
 12-35 
 
 16-97 
 
 ,. 
 
 3-3 
 
 3-4 
 
 3-66 
 
 3-98 
 
 4-39 
 
 4-92 
 
 5-65 
 
 6-75 
 
 8-64 
 
 12-80 
 
 17-63 
 
 || 
 
 3-4 
 
 3-5 
 
 3-77 
 
 4-10 
 
 4-52 
 
 5-06 
 
 5-82 
 
 6-96 
 
 8-92 
 
 13-26 
 
 18-29 
 
 PL 
 
 3-5 
 
 3-6 
 
 3-88 
 
 4-22 
 
 4-65 
 
 5-21 
 
 6-00 
 
 7-18 
 
 9-21 
 
 13-71 
 
 18-96 
 
 in 
 
 3-6 
 
 3-7 
 
 3-99 
 
 4-34 
 
 4-78 
 
 5-36 
 
 6-17 
 
 7-39 
 
 9-49 
 
 14-18 
 
 1965 
 
 gs 
 
 3-7 
 
 3-8 
 
 4-09 
 
 4-46 
 
 4-91 
 
 5-51 
 
 6-35 
 
 7-60 
 
 9-78 
 
 14-65 
 
 20-34 
 
 ^ 
 
 3-8 
 
 3-9 
 
 4-20 
 
 4-58 
 
 5-05 
 
 5-66 
 
 6-52 
 
 7-82 
 
 10-07 
 
 15-12 
 
 *j "5-^ 
 
 3-9 
 
 4-0 
 
 4-31 
 
 4-69 
 
 5-18 
 
 5-81 
 
 6-70 
 
 8-04 
 
 10-36 
 
 15-59 
 
 ilif-si 
 
 4-0 
 
 4-1 
 
 4-42 
 
 4-81 
 
 5-31 
 
 5-96 
 
 6-87 
 
 8-25 
 
 10-66 
 
 16-07 
 
 "t* o3 H^'n fl> 
 
 Ss^g-s^ 
 
 4-1 
 
 4-2 
 
 4-53 
 
 4-93 
 
 5-44 
 
 6-11 
 
 7-05 
 
 8-47 
 
 10-95 
 
 16-55 
 
 ^llt = l 
 
 4-2 
 
 4-3 
 
 4-64 
 
 5-05 
 
 5-57 
 
 6-26 
 
 7-23 
 
 8-69 
 
 11-25 
 
 17-04 
 
 2**lid 
 
 4-3 
 
 4-4 
 
 4-74 
 
 5-17 
 
 5-71 
 
 6-41 
 
 7-40 
 
 8-91 
 
 11-55 
 
 17-53 
 
 KfS-&8 
 
 4-4 
 
 4-5 
 
 4-85 
 
 5-29 
 
 5-84 
 
 6-56 
 
 7-58 
 
 9-13 
 
 11-85 
 
 18-02 
 
 IHi^s 
 
 4-5 
 
 4-6 
 
 4-96 
 
 5-47 
 
 5-97 
 
 6-72 
 
 7-76 
 
 935 
 
 12-15 
 
 18-52 
 
 !*uHi 
 
 4-6 
 
 4-7 
 
 5-07 
 
 5-53 
 
 6-10 
 
 6-87 
 
 7-94 
 
 9-57 
 
 12-45 
 
 19-02 
 
 ftSI&:J 
 
 4-7 
 
 4-8 
 
 5-18 
 
 5-64 
 
 6-24 
 
 7-02 
 
 8-12 
 
 9-79 
 
 12-75 
 
 19-53 
 
 li*tgS1 
 
 4-8 
 
 4-9 
 
 5-29 
 
 5-76 
 
 6-37 
 
 7-17 
 
 8-29 
 
 10-02 
 
 13-06 
 
 20-04 
 
 S|.sB*-8 
 
 4-9 
 
 5-0 
 
 5-40 
 
 5-88 
 
 6-50 
 
 7-32 
 
 8-47 
 
 10-24 
 
 13-37 
 
 
 ?iliii* 
 
 5-0 
 
TABLE of the Discharges in cubic feet per minute from the primary Channel in the opposite 
 page, taken in feet; and from the corresponding equivalent Channels, also taken in feet. 
 
 Depths of the prln.a- 1 
 ry channel in oppo- 1 
 site table, having a 1 
 mean width of 100 ; 1 
 In feet. 
 
 Discharges In cubic feet per minute. Interpolate for Intermediate falls or Inclinations: divide 
 greater falls or Inclinations by 4, and donble the corresponding discharges. If the dimensions be 
 in Inches, divide the discharges here given by 500 ; if in tenths, by 316 ; If in eighths, by ibl ; if in 
 sixths, by 88; if in fifths, by 56; if in quarters, l>y 32; if in thirds, by 15-6; and if in halves, dy 
 5-66. Reverse the operation and multiply for like multiples. 
 
 Depths of the prima- 
 ry channel In oppo- 
 site table, having a 
 mean width of lOo; 
 in feet. 
 
 4 inches 
 per mile 
 lin 
 15840. 
 
 1 6 inches 
 per mile 
 lin 
 10560. 
 
 9 inches 
 per mile 
 lin 
 7040. 
 
 ;12inches 
 per mile 
 lin 
 5280. 
 
 15inchef 
 per mile 
 lin 
 4224. 
 
 18inche 
 per mile 
 lin 
 3520. 
 
 21 inches 
 per mile, 
 lin 
 3017-1. 
 
 24 inches 
 per mile, 
 lin 
 2640. 
 
 80 inches 
 per mile, 
 lin 
 2112. 
 
 36inche 
 per mile 
 lin 
 1760. 
 
 1 
 
 IK 
 
 140 
 
 176 
 
 207 
 
 235 
 
 260 
 
 284 
 
 305 
 
 345 
 
 383 
 
 1 
 
 125 
 
 157 
 
 198 
 
 250 
 
 294 
 
 332 
 
 373 
 
 402 
 
 433 
 
 490 
 
 543 
 
 125 
 
 2 
 
 325 
 
 409 
 
 515 
 
 606 
 
 6*6 
 
 760 
 
 828 
 
 b91 
 
 1,009 
 
 1,117 
 
 2 
 
 25 
 
 455 
 
 675 
 
 725 
 
 853 
 
 966 
 
 1,070 
 
 1,166 
 
 1,256 
 
 1,422 
 
 1,574 
 
 25 
 
 3 
 
 601 
 
 760 
 
 957 
 
 1,125 
 
 1,275 
 
 1,412 
 
 1,539 
 
 1,658 
 
 1,876 
 
 2,076 
 
 3 
 
 375 
 
 844 
 
 1,068 
 
 1,344 
 
 1,582 
 
 1,793 
 
 1,985 
 
 2,162 
 
 2.329 
 
 2,637 
 
 2,919 
 
 375 
 
 4 
 
 931 
 
 1,178 
 
 1,484 
 
 1,745 
 
 1,977 
 
 2,189 
 
 2,386 
 
 2,569 
 
 2,900 
 
 3,220 
 
 4 
 
 5 
 
 1,308 
 
 1,653 
 
 2,()81 
 
 2,447 
 
 2,775 
 
 3.071 
 
 3,347 
 
 3.6<6 
 
 4,0*3 
 
 4,513 
 
 5 
 
 6 
 
 1,721 
 
 2,178 
 
 2.743 
 
 3,227 
 
 3,657 
 
 4,047 
 
 4,410 
 
 4,752 
 
 5.401 
 
 5,956 
 
 6 
 
 625 
 
 1,830 
 
 2,316 
 
 2,917 
 
 3,431 
 
 3,887 
 
 4,303 
 
 4,690 
 
 5,053 
 
 5,795 
 
 6,332 
 
 625 
 
 7 
 
 2,177 
 
 2,750 
 
 3,463 
 
 4,0'2 
 
 4,614 
 
 5,109 
 
 5.568 
 
 5,999 
 
 6,936 
 
 7,516 
 
 7 
 
 75 
 
 2,414 
 
 3,029 
 
 3,844 
 
 4,520 
 
 5,123 
 
 5.674 
 
 6,180 
 
 6.660 
 
 7,567 
 
 8,342 
 
 75 
 
 8 
 
 2,660 
 
 3,363 
 
 4,236 
 
 4,982 
 
 5,646 
 
 6,253 
 
 6,811 
 
 7,340 
 
 8,309 
 
 9,194 
 
 8 
 
 875 
 
 3,044 
 
 3,850 
 
 4,846 
 
 5,703 
 
 6,463 
 
 7,157 
 
 7,770 
 
 8,401 
 
 9,513 
 
 10,527 
 
 875 
 
 9 
 
 3,175 
 
 4,017 
 
 5,060 
 
 5,951 
 
 6,743 
 
 7,467 
 
 8,082 
 
 8,765 
 
 9,926 
 
 10,984 
 
 9 
 
 1-0 
 
 3,731 
 
 4,711 
 
 5,933 
 
 6,973 
 
 7,903 
 
 8.750 
 
 9,513 
 
 10,273 
 
 11,634 
 
 12,877 
 
 1-0 
 
 1-125 
 
 4,441 
 
 5,614 
 
 7,071 
 
 8,313 
 
 9,421 
 
 10,430 
 
 11,369 
 
 12,216 
 
 13,867 
 
 15,347 
 
 1-125 
 
 1-2 
 
 4,88!) 
 
 6,186 
 
 7,791 
 
 9,163 
 
 10,381 
 
 11,494 
 
 12,521 
 
 13,494 
 
 15,280 
 
 16.914 
 
 1-2 
 
 1-25 
 
 5,207 
 
 6,582 
 
 8,291 
 
 9,752 
 
 11,048 
 
 12,232 
 
 13,336 
 
 14,361 
 
 16,261 
 
 18,000 
 
 1-25 
 
 1-3 
 
 5,529 
 
 6,981 
 
 8,793 
 
 10,357 
 
 11,718 
 
 12,974 
 
 14,146 
 
 15,234 
 
 17,246 
 
 19,091 
 
 1-3 
 
 1-375 
 
 6,004 
 
 7,591 
 
 9,561 
 
 11,245 
 
 12,734 
 
 14,107 
 
 15,386 
 
 16,576 
 
 18,752 
 
 20,756 
 
 1-375 
 
 1-4 
 
 6,167 
 
 7,797 
 
 9,821 
 
 11,544 
 
 13,087 
 
 14,491 
 
 15,794 
 
 17,031 
 
 19,262 
 
 21,318 
 
 1-4 
 
 1-5 
 
 6,844 
 
 8,653 
 
 10,^98 
 
 12,818 
 
 14,524 
 
 16,081 
 
 17,523 
 
 18,917 
 
 21,376 
 
 23,658 
 
 1-5 
 
 1-6 
 
 7,538 
 
 9,520 
 
 12,002 
 
 14,115 
 
 15,994 
 
 17,709 
 
 19,296 
 
 20,829 
 
 23,539 
 
 26,053 
 
 1-6 
 
 1-625 
 
 7,705 
 
 9,741 
 
 12,272 
 
 14.428 
 
 16,348 
 
 18,102 
 
 19,724 
 
 21,286 
 
 24,061 
 
 26,631 
 
 1-625 
 
 1-7 
 
 8.252 
 
 10,432 
 
 3,139 
 
 15,432 
 
 17,509 
 
 19,360 
 
 21,126 
 
 22,780 
 
 25,769 
 
 28,523 
 
 1-7 
 
 1-75 
 
 8,617 
 
 10,893 
 
 3,719 
 
 16,134 
 
 18,282 20,241 
 
 22,060 
 
 23,776 
 
 26,907 
 
 29,784 
 
 1-75 
 
 1-8 
 1-875 
 
 8,993 
 9,561 
 
 11,369 
 
 12,088 
 
 4,318 
 15,226 
 
 16,851 19,079 
 17,905 |20,287 
 
 21,124 
 22,463 
 
 23,024 
 24,476 
 
 24,821 
 26,372 
 
 28,081 
 29,860 
 
 31,085 
 33,052 
 
 1-8 
 
 1-875 
 
 1-9 
 
 9,741 
 
 12,316)15,515 
 
 18,245 120,672 
 
 22.890 
 
 24,946 
 
 26,872 
 
 30,426 
 
 33,682 
 
 1-9 
 
 2-0 
 
 10,515 
 
 13,297 
 
 16,753 
 
 19,702 
 
 22.320 
 
 24.718 
 
 26,935 
 
 29,019 
 
 32,852 
 
 36,358 
 
 2-0 
 
 2-1 
 
 11,307 
 
 14,300 
 
 18,020 21,192 
 
 23,991 
 
 26,561 
 
 29,074 
 
 31,213 
 
 35,334 
 
 39,106 
 
 2-1 
 
 2-2 
 
 12,110 15,314 19,297 22,689 
 
 25,708 
 
 28,467 
 
 31,024 
 
 33,424 
 
 37,838 
 
 41,878 
 
 2-2 
 
 2-3 
 
 12,935 16,357 ! 20,608 24,235 
 
 27,456 
 
 30,407 
 
 33,134 
 
 35,694 
 
 40,410 
 
 44,724 
 
 2'3 
 
 2-4 
 
 13,781 
 
 17,425 121,954 25,816 
 
 29,250 
 
 32,32 
 
 35,299 
 
 38,022 
 
 43,048 
 
 47,643 
 
 2-4 
 
 2-5 
 
 14,647 
 
 18,520 123,332 ! 27,436 31,087 
 
 34,425 
 
 37,516 
 
 40,407 
 
 45,750 
 
 50,634 
 
 2-5 
 
 2-6 
 
 15,538 
 
 19,645 '24,747 129,100 132,974 
 
 36,514 
 
 39,794 
 
 42,856 
 
 48,526 
 
 53,706 
 
 2-6 
 
 2-7 
 
 16,430 
 
 20,773 26,167 30,770 34,867 
 
 38,610 
 
 42,078 
 
 45,316 
 
 51.311 
 
 56,789 
 
 2-7 
 
 2-8 
 
 17,333 
 
 21,915 
 
 27,605 32,462 36,784 
 
 40,733 
 
 44.390 
 
 47,809 
 
 54,131 
 
 59,913 
 
 2-8 
 
 2-9 
 
 18,257 
 
 23,084 
 
 29,076 ;34,193 38,744 
 
 42,905 
 
 46,755 
 
 50,359 
 
 57,017 
 
 63,110 
 
 2-9 
 
 3-0 
 
 19,203 
 
 24,280 30,581 35,963 (40,750 
 
 45,127 
 
 49,175 
 
 52,968 
 
 59,968 
 
 66,379 
 
 3-0 
 
 3-1 
 
 20,167 25,498 32,120 
 
 37,767 (42.794 
 
 47,392 
 
 51,640 
 
 55,634 
 
 62,986 
 
 69,709 
 
 3-1 
 
 3-2 
 
 21,146 26,737 33,673 
 
 39,600 144,871 
 
 49,692 
 
 54,148 
 
 58,327 
 
 66,033 
 
 73,097 
 
 3-2 
 
 3-3 
 
 22,118 
 
 27,969 35,225 
 
 41,425 
 
 46,939 
 
 51,978 
 
 56,640 
 
 61,017 
 
 69,077 
 
 76,465 
 
 3-3 
 
 3-4 
 
 23,106 
 
 29,220 36,798 
 
 13,275 
 
 49,036 
 
 54,302 
 
 59,171 
 
 63,745 
 
 72,164 
 
 79,879 
 
 3-4 
 
 3-5 
 
 24,115 
 
 30,497 
 
 38,407 
 
 45,166 
 
 51,1*0 
 
 56,675 
 
 61,758 
 
 66,534 
 
 75.322 
 
 83,371 
 
 3-5 
 
 3-6 
 
 25,139|31,795 40,040 
 
 47,086 
 
 53,356 
 
 59,084 
 
 64,384 
 
 69,366 
 
 78,526 
 
 86,915 
 
 3-6 
 
 3-7 
 
 26,18233.116 
 
 41,702 
 
 49,041 
 
 55,572 
 
 61,532 
 
 67,058 
 
 72,249 
 
 81,789 
 
 90,524 
 
 3-7 
 
 3-8 
 
 27,23:334,446 
 
 43,379 
 
 51,013 
 
 57,807 
 
 64,009 
 
 6:>,753 
 
 75,158 
 
 85,078 
 
 94,162 
 
 3-8 
 
 3-9 
 
 28-287 
 
 35,777 
 
 45,060 
 
 52,989 
 
 50,046 
 
 66.489 
 
 72,455 
 
 78.061 
 
 88.371 
 
 97,810 
 
 3-9 
 
 4-0 
 
 29,356 
 
 37,128 
 
 46,766 
 
 54.9M 
 
 52,318 
 
 69,006 
 
 75,197 
 
 81,012 
 
 91,710 
 
 101,512 
 
 4*0 
 
 4-1 
 
 30,438 
 
 38,495 
 
 48,492 
 
 57,024 64,616 
 
 n,553 
 
 77,973 
 
 83, '.*99 
 
 95,093 
 
 105,259 
 
 4-1 
 
 4-2 
 
 31,538 
 
 39.H84 
 
 50,246 59,0*6 66.960 
 
 74.141 
 
 80,7^3 
 
 87,033 
 
 98,535 
 
 109,065 
 
 4-2 
 
 4-3 
 
 32,654 
 
 41,294 
 
 "2,027 161,180 6.327 
 
 76,769 
 
 83,655 
 
 90,116 
 
 102,025 
 
 112,93" 
 
 4-3 
 
 4-4 
 
 33,776 
 
 +2,712 
 
 53,816 63,283 
 
 71,709 
 
 79,406 
 
 86,529 
 
 93.209 
 
 105,531 
 
 116,811 
 
 4-4 
 
 4-5 
 
 34,908 
 
 44,138 55,613 !6r>,394 ' 
 
 "4.100 
 
 32,052 
 
 89.413 
 
 96,318 
 
 L09.054 
 
 I2<),709 
 
 4-5 
 
 4-6 
 
 36,041 
 
 45,579 ;57,429 67,527 76,500 J84,725 
 
 92,327 
 
 99,460 
 
 Ll26i4 
 
 124,647 
 
 4-6 
 
 4-7 
 
 37,193 
 
 47,034 59,2(i2 6.*, 682 78,955 (87.426 
 
 95,271 
 
 102,632 
 
 116,209 ; 
 
 128,625 
 
 4-7 
 
 4-8 
 
 38,36:3 
 
 18,514 61,128 71,874 |81,438 |90,173 
 
 98,266 
 
 105,860 
 
 119,866 132,672 
 
 4-8 
 
 4-9 
 
 39,544 
 
 50,009 
 
 ^3,011 74.087 83,944 < 
 
 )2,946 
 
 01,289 : 
 
 L09.119 
 
 123,559 ] 
 
 L36.758 
 
 4-9 
 
 5-0 
 
 40,725 
 
 51,507 
 
 54,895 7 6,298 |86,450 < 
 
 >5,720 
 
 04,313 : 
 
 112,376 
 
 127,248 ] 
 
 L40,841 
 
 5-0 
 
254 THE DISCHARGE OF WATER FROM 
 
 SECTION X. 
 
 EFFECTS OF ENLARGEMENTS AND CONTRACTIONS. BACKWATER 
 WEIR CASE, LONG AND SHORT WEIRS. 
 
 When the flowing section in pipes or rivers expands 
 or contracts suddenly, a loss of head always ensues ; 
 this is probably expended in forming eddies at the 
 sides, or in giving the water its new section. A side 
 current, moving slowly upwards, may be frequently 
 observed in the wide parts of rivers, when the channel 
 is unequal, though the downward current, at the 
 centre, be pretty rapid ; and though we may assume 
 generally that the velocities are inversely as the sec- 
 tions, when the channels are uniform, we cannot 
 properly do so when they are not, and the motions 
 so uncertain as those referred to. When a pipe is 
 contracted by a diaphragm at 
 
 Fig. 3 6 
 
 the orifice of entry, Fig. 27, 
 we have seen (equation 60), 
 that the loss of head is, 
 
 (123.) h = 
 
 When the diaphragm is placed in a uniform pipe, 
 Fig. 36, then A zz c, and we get the loss of head 
 
 (124.) h= - 27 > 
 
 and the coefficient of resistance 
 
 as in equation (67). The coefficient of discharge c d is 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 255 
 
 here equal to the coefficient of contraction c c , o * very 
 nearly. Now we have shown in equation (45), and 
 the remarks following it, that the value of the co* 
 efficient of discharge, c d , varies according to the ratio 
 
 of the sections, A * and in TABLE V. we have calcu- 
 a 
 
 lated the new coefficients for different values of the 
 ratios, and different values of the primary coefficient 
 c d . If we assume c d , when A is very large compared 
 with a, to be '628, we then find by attending to the 
 remarks at pp. 109 and 128, that the different values of 
 
 c d corresponding to -807 x^, taken from TABLE V., 
 
 a 
 
 are those in columns Nos. 2 and 5 of the next small 
 
 TABLE OF COEFFICIENTS FOR CONTRACTION, BY A DIAPHRAGM IN A PIPE, 
 
 a 
 
 A 
 
 Pd 
 
 C r 
 
 a 
 A 
 
 CA 
 
 c r 
 
 
 
 628 
 
 infinite 
 
 6 
 
 713 
 
 1-790 
 
 1 
 
 630 
 
 221-2 
 
 7 
 
 753 
 
 807 
 
 2 
 
 636 
 
 47-1 
 
 8 
 
 807 
 
 301 
 
 3 
 
 647 
 
 17-2 
 
 85 
 
 845 
 
 154 
 
 4 
 
 661 
 
 7-7 
 
 9 
 
 890 
 
 062 
 
 5 
 
 683 
 
 3-7 
 
 1 
 
 1-000 
 
 000 
 
 * The general value of c c , as given by Professor Eankine, is 
 
 618 
 c c = ~ ^-7, which is equal to unity when = A, as it 
 
 should be ; and equal to -618, when a is very small, compared with 
 A, as it also should be when the diaphragm is a thin plate, but 
 not otherwise. If the thickness of the diaphragm be twice the dia- 
 meter of the orifice a, the coefficient of discharge would be -815 ; 
 and if the higher arris be rounded, this would be increased to 1, in 
 which cases the expression would clearly fail ; the thickness of the 
 diaphragm and the form of the aperture a must also be considered. 
 
256 THE DISCHARGE OF WATER FROM 
 
 table, the values of the coefficient of resistance, in 
 columns 3 and 6, being calculated from equation (125) 
 for the respective new values of the coefficient of 
 discharge thus found. The table shows that when 
 the aperture in a diaphragm is Aths of the section of 
 the pipe, that 47 times the head due to the velocity 
 is lost thereby. If the aperture in the diaphragm be 
 rounded at the arrises, the loss will not be so great, as 
 the primary coefficient c d will then be greater than 
 that due to an orifice in a thin plate : see coefficients, 
 p. 174 
 
 When there are a number of diaphragms in a tube, 
 the loss of head for each must be found separately, 
 and all added together for the total loss. If the 
 diaphragms, however, approach each other, so that 
 the water issuing from one of the orifices a, Fig. 36, 
 shall pass into the next before it again takes the 
 velocity due to the diameter of the pipe, the loss will 
 not be so great as when the distance is sufficient to 
 allow this change to take place. This view is fully 
 borne out by the experiments of Eytelwein with tubes 
 1-03 inch in diameter, having apertures in the diaph- 
 ragms of -51 inch in diameter. 
 
 Venturi's twenty-fourth experiment, with tubes 
 varying from -75 inch to -934 inch in diameter at the 
 junction with the cistern, so as to take the form of 
 the contracted vein, and expanding and contracting 
 along the length from -75 to 2 inches and from 2 
 inches to *75 inch alternately, shows the great loss of 
 head sustained by successive enlargements and con- 
 tractions of a channel, even when the junction of the 
 parts is gradual. Calling the coefficient for the short 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 257 
 
 tube, with a junction of nearly the form of the con- 
 tracted vein, 1, then the following coefficients are 
 derivable from the experiment : 
 
 Short tube with rounded junction . . , . 1- 
 
 One enlargement ......... "741 
 
 Three enlargements ........ -569 
 
 Five enlargements , -454 
 
 Simple tube with a rounded junction of the 
 same length, 36 inches, as the tube with 
 
 the five enlarged parts . . . . , , . ? 786 
 
 The head, in the experiment, was 32 J inches. Venturi 
 states that no observable differences occurred in the 
 times of discharge when the enlarged portions were 
 lengthened from 3i to 6^ inches. See tables, pp. 152 
 and 191. 
 
 With reference to this experiment, so often quoted, 
 it is necessary to remark that the diameters of the 
 enlarged portions were 2 inches each, while the lengths 
 varied only from 3 to 6-J- inches, and consequently 
 were at most only 3 times the diameter. Now with 
 such a large ratio of the 
 
 width to the length of the H 
 enlarged portions, a A B b, 
 Fig. 37, it is pretty clear 
 that a good deal of the head 
 is lost by the impact of the moving water on the 
 shoulders A and B. That this is so is evident from 
 the fact, stated by the experimenter, of the time of 
 discharge remaining the same when a A, in five dif- 
 ferent enlargements, was increased from 3i to 6i 
 inches ; though this must have lengthened the whole 
 
258 THE DISCHARGE OF WATER FROM 
 
 tube from 36 to 50 inches,* thereby increasing the 
 loss from friction proportionately, but which happened 
 to be compensated for by the reduction in the resist- 
 ances from impact at A and B, and in the eddies, by 
 doubling the lengths from a to A. 
 
 If, however, the length from a to A be very large 
 compared with the diameter, and the junctions at a, 
 A, B, and by be well grafted, less loss will arise from 
 the enlargement than if the smaller diameter con- 
 tinued all along uniform. The explanation is clear, 
 as the resistance from friction is inversely as the 
 square roots of the mean radii ; and the length being 
 the same, the loss must be less with a large than a 
 small diameter, 
 
 These remarks, mutatis mutandis, apply equally to 
 rivers as to pipes. We have already, pp. 140 and 147, 
 pointed out the effects of submerged weirs and con- 
 tracted river channels, and given formulae for calcu- 
 lating them. 
 
 BACKWATER FROM CONTRACTIONS IN RIVERS. 
 
 A river may be contracted in width or depth, by 
 jetties or by weirs ; and when the quantity to be dis- 
 charged is known, we have given, in formulae (9), (55), 
 and (57), equations from which the increase of head 
 may be determined. The effect of a weir, jetty, or 
 contracted channel of any kind, is to increase the 
 depth of water above ; and this is sometimes neces- 
 sary for navigation purposes, or to obtain a head for 
 mill power. When a weir is to rise over the surface, 
 we can easily find, from the discharge and length, the 
 discharge per minute over each foot of length, with 
 which, and the coefficient due to the ratio of the 
 
 * The dimensions throughout this experiment are given as in 
 the original, viz. in French inches. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 259 
 
 sections, on and above the weir, found from TABLE V., 
 we can find the head from TABLE VI. For submerged 
 weirs and contracted widths of channel, the head can 
 be best calculated, by approximation, from the equa- 
 tions above referred to. 
 
 The head once determined, the extent of the back- 
 water is a question of some importance. If F c o D, 
 Fig. 38, be the original surface of a river, and a A B p 
 the raised surface by backwater from the weir at a, then 
 the extent a F of this backwater, in a regular channel, 
 will be from 1-5 to 1*9 times ac drawn parallel to the 
 horizon to meet the original surface in c. This rule 
 
 Fig-. 38 
 
 will be found useful for practical purposes ; but in 
 order to determine more accurately the rise for a 
 given length, B : B 2 or B! B, of the channel, it is neces- 
 sary to commence at the weir and calculate the 
 heights from A to B, B to B I? and from BJ to B 2 sepa- 
 rately, the distance from A to B 2 being supposed 
 divided into some convenient number of equal parts, 
 so that the lengths A B, BB^ &c., may be considered 
 free from curvature. Now, as the head A D is known, 
 or may be calculated by some of the preceding for- 
 mulae, the section of the channel at the head of the 
 weir also becomes known, and thence the mean velo- 
 city in it, by means of the discharge over the weir. 
 Putting A for the area of the channel at A n, d for its 
 depth A H, and v for the mean velocity ; also A X for 
 
 s3 
 
260 THE DISCHARGE OF WATER FROM 
 
 the area of the channel at B i, d 1 for its depth, and v l 
 for its mean velocity ; b m the mean border between 
 the sections at AH and BI; r m the mean hydraulic 
 
 V -j V-i 
 
 depth ; ' the mean velocity ; AD A; B o zz ^ ; 
 the sine of angle o D E zz s ; and the length A B zz D o 
 nearly zz /; we get A x v = A X x v i and r m zz ' , -; 
 
 ^ ^m 
 
 but as, in passing from B to A, the velocity changes 
 from v l to v, there is a loss of head equal -^ ? 
 and if c t be the coefficient of friction, there is a loss 
 
 of head from this cause equal c f X -- X 
 
 ^*m 
 
 hence the whole change of head in passing from B to A 
 is equal to c { x X Q -^ . But this 
 
 * 
 
 change of head is equal to BE ADZZBO + OE AD 
 zz h^ + Is h, whence we get 
 
 (126.) 
 
 AV A + A! , , 
 
 or as v l = . and r m zz ( ', . we get, by a few re- 
 A! ZO m 
 
 ductions and change of signs, 
 
 (127.) A- A! = *- * x ft m X X 
 
 and therefore we get 
 
ORIFICES, WEIES, PIPES, AND RIVERS. 261 
 
 _A 2 -A* 
 
 , 6 m x (A + A,) v ^ 
 
 ~ Ct ' 2 A? v 2# 
 
 from which we can calculate the length / corre- 
 sponding to any assumed change of level between A 
 and B. Then, by a simple proportion we can find 
 the change of level for any smaller length. To find 
 the change of level directly from a given length does 
 not admit of a direct solution, for the value of h k 
 in equation (127) involves A 1? which depends again 
 on h h ly and further reduction leads to an equation 
 of a higher order ; but the length corresponding to 
 a given rise, k ly is found directly by equation (128). 
 When the width of the channel, w, is constant, and 
 the section equal to w x d nearly, the above equa- 
 tions admit of a further reduction for A X = d l w and 
 A = dw, by substituting these values in equation 
 (127) it becomes, after a few reductions, 
 (129.) h h l =dd 1 
 
 dxd l t 
 
 or, as it may be further reduced, 
 
 J^ 
 s ~~ Ct X X 
 
 (130.) A-A, zz 
 
 
 d\ f *g 
 
 Now, we may take in this equation for all practical 
 purposes, 
 
 approximately, b being the border of the section at 
 
262 THE DISCHARGE OF WATER FROM 
 
 AH; and also, , a l = -?, approximately; there- 
 ! a 
 
 fore we shall have 
 
 f dw 2 a 
 (131.) h-h 1 = - -3 ^r--X/5 
 
 and 
 
 f 2 - 
 (132.) /= b d j" 9 
 
 Now, as -T = -, 2# r 64-4, and the mean value of 
 
 the coefficient of friction for small velocities c f -z 
 0078, we shall get 
 
 64 -4 ds- -0078 ^ 2 
 
 (133.) *, = A 64-4 rf- 2 t/ X /5 " / 
 
 and 
 
 64-4^5- -0078 - r v* 
 
 very nearly* Having by means of these equations 
 found A B from B o or B E, and B o from A B, we can in 
 the same manner proceed up the channel and calcu- 
 late B! c, B 2 G!, &c>, until the points B, B!, Ba in the 
 curve of the backwater shall have been determined, 
 and until the last nearly coincides with the original 
 surface of the river. When A x = 0, we shall have 
 
 64-4 tfs -0078- v 2 
 r 
 
ORIFICES, WEIRS, PIPES, AND RIVE 
 
 ^ 
 
 If we examine equation (134) it appears that w 
 64-4 d -=L 2 v 2 , /must be equal to zero; or. when 
 
 d v 2 
 
 2 == gr-ij equal the height due to the velocity v. 
 
 When / is infinite, 64 -4 d must exceed 2# 2 , and 64 
 equal to -0078 ^ 2 ; 
 64*4 7*5 
 
 > .Q078 
 
 This is the velocity clue to friction in a channel of 
 the depth d, hydraulic mean depth r, and inclination 
 5 ; and, as in wide rivers r d nearly, v m 90'9 
 \/d~s, but when the numerator was zero we had from 
 it v m \/32-2 6/ ; equating these values of v, we get 
 
 5 ='0039 n --nearly : see p. 139. Now, the larger 
 
 the fraction s is, the larger will the velocity v become ; 
 and the larger v becomes, the more nearly, in all 
 practical cases, will the terms 
 
 64-4 d -2 v 2 and 64-4tfs - -0078 ^ 2 , 
 
 in the numerator and denominator of equation (134), 
 approach zero ; when 64-4 d 2 v 2 becomes zero first, 
 
 / = ; when 64'4 ds -0078 - v 2 becomes zero first, 
 / equals infinity ; and when they both become zero 
 at the same time, l = h h l9 and 5 m , see p. 139 ; 
 
 if 5 be larger than this fraction, the numerator in 
 equation (134) will generally become zero before the 
 denominator, or negative, in which cases / will also 
 be zero, or negative ; and the backwater will take the 
 
264 THE DISCHARGE OF WATEE FROM 
 
 form F c 2 b 2 j b a 1 a, Fig. 38, with a hollow at C* 
 Bidone first observed a hollow, as F c 2 b 2 , when the 
 
 inclination s was -. When the inclination of a river 
 
 30 
 
 channel changes from greater to less, the velocity is 
 obstructed, and a hollow similar to F c 2 b 2 sometimes 
 occurs ; when the difference of velocity is consider- 
 able, the upper water at b 2 falls backwards towards 
 c 2 and F, and forms a bore, a splendid instance of 
 which is the pororoca, on the Amazon, which takes 
 place where the inclination of the surface changes 
 from 6 inches to ^th of an inch per mile, and the 
 velocity from about 22 feet to 4J feet per second. 
 
 WEIR CASE, LONG AND SHORT WEIRS. 
 
 When a channel is of very unequal widths, above 
 a weir, we have found the following simple method of 
 calculating the backwater sufficiently accurate, and 
 the results to agree with observation. Having as- 
 certained the surf ace fall due to friction in the channel 
 at a uniform mean section, add to this fall the height 
 which the whole quantity of water flowing down would 
 rise on a weir having its crest on the same level as the 
 lower weir, and of the same length as the width of the 
 channel in the contracted pass. The sum will be the 
 head of water at some distance above such pass very 
 nearly. A weir was recently constructed on the 
 river Blackwater, at the bounds of the counties 
 Armagh and Tyrone, half a mile below certain mills, 
 which, it was asserted, were injuriously affected by 
 backwater thrown into the wheel-pits. The crest of 
 the weir, 220 feet long, was 2 feet 6 inches below the 
 
ORIFICES, WEIRS* PIPES, AND RIVERS. 265 
 
 pit; the river channel between varied from 50 and 
 57 feet to 123 feet in width, from 1 foot to 14 feet 
 deep ; and the fall of the surface, with 3 inches of 
 water passing over the weir and the sluices down> 
 was nearly 4 inches in the length of half a mile. 
 Having seen the river in this state in summer, the 
 writer had to calculate the backwater produced by 
 different depths passing over the weir in autumn and 
 winter, which in some cases of extraordinary floods 
 were known to rise to 3 feet. The width of the 
 channel about 60 feet above the weir averaged 120 
 feet; The width, 2050 feet above the weir and 550 
 feet below the mills, was narrowed by a slip in an 
 adjacent canal bank, to 45 feet at the level of the top 
 of the weir, the average width at this place as the 
 water rose being 55 feet. The channel above and 
 below the slip widened to 80 and 123 feet. Between 
 the mills and the weir there were, therefore^ two 
 passes ; one at the slip, averaging 55 feet wide ; 
 another above the weir, about 120 feet wide. As- 
 suming as above, that the water rises to the heights 
 due to weirs 55 and 120 feet long, at these passes, 
 we get, by an easy calculation, or by means of TABLE 
 X., the heads in columns two and four of the follow- 
 ing table, corresponding to the assumed ones on the 
 weir, given in the first column. 
 
 As the length of the river was short, and the 
 hydraulic mean depth pretty large, the fall due to 
 friction for 60 feet above the weir was very small, 
 and therefore no allowance was made for it ; even the 
 distance to the slip was comparatively short, being 
 less than half a mile, and as the water approached it 
 
266 
 
 THE DISCHARGE 0V WATER FROM 
 
 TABLE OF CALCULATED AND OBSERVED HEIGHTS ABOVE M'KEAN's 
 WEIR ON THE RIVER BLACKWATER. 
 
 Heights at 
 M'Kean's 
 weir 220 feet 
 long, in 
 inches. 
 
 Heights 60 feet above the 
 Weir channel 120 feet wide. 
 
 Heights 2050 fef>t above the 
 weir channel 55 feet wide ; 
 average. 
 
 Calculated 
 inches. 
 
 Observed 
 inches. 
 
 Calculated 
 inches. 
 
 Observed 
 inches. 
 
 1* 
 
 2J 
 
 2J 
 
 4J 
 
 5* 
 
 2 
 
 
 :. 
 
 
 ^ii ?'!< 
 
 3 
 
 4* 
 
 i . 
 
 7i 
 
 7 
 
 4 
 
 6 
 
 .. 
 
 10 
 
 9 
 
 5 
 
 7 * 
 
 
 . 12* 
 
 llj 
 
 6 
 
 9 
 
 9 
 
 15 
 
 !% 
 
 7 
 
 10* 
 
 10* 
 
 17* 
 
 18| 
 
 8 
 
 12 
 
 '" t?::v 
 
 20 
 
 20* 
 
 9 
 
 13* 
 
 12* 
 
 22* 
 
 20| 
 
 10 
 
 15 
 
 . . 
 
 24J 
 
 20 
 
 11 
 
 16* 
 
 . . 
 
 27| 
 
 24 
 
 12 
 
 18 
 
 17 
 
 30 
 
 31 
 
 13 
 
 19* 
 
 18* 
 
 32| 
 
 33 
 
 15 
 
 22* 
 
 21 
 
 37| 
 
 40 
 
 18 
 
 27 
 
 25 
 
 45J 
 
 46 
 
 21 
 
 31* 
 
 29| 
 
 53 
 
 54 
 
 24 
 
 36 
 
 34 
 
 604 
 
 62 
 
 with considerable velocity, this was conceived, as the 
 observations afterwards showed, to be a sufficient 
 compensation for the loss of head below by friction. 
 The observations were made by a separate party, 
 over whom the writer had no control, and it is neces- 
 sary to remark, that with the same head of water on 
 the weir, they often differed more from each other 
 than from the calculation. This, probably, arose from 
 the different directions of the wind, and the water 
 rising during one observation, and falling during 
 another. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 267 
 
 The true principle for determining the head at #, 
 Fig. 39, apart from that due to friction, is that 
 pointed out at pages 142 and 147 ; but when the passes 
 are very near each other, or the depth d 2 , Fig. 23, is 
 small, the effect of the discharge through d 2 is incon- 
 siderable in reducing the head, as the contraction and 
 loss of vis-viva are then large, and the head d^ be- 
 comes, that due to a weir of the width of the con- 
 tracted channel at A, nearly. The reduction in the 
 extent of the backwater, by lowering the head on 
 a longer weir, is found by taking the difference of the 
 amplitudes due to the heads at g, Fig. 39, in both 
 cases, as determined from equations (56), (128), etseq. 
 This will seldom exceed a mile up the river, as the 
 surface inclination is found to be considerably greater 
 than that due to mere friction and velocity, and hence 
 the general failure of drainage works designed on the 
 assumption that the lowering of the head below, by 
 means of long weirs, extends its effects all the way 
 up a channel. We must nearly treble the length of 
 a weir before the head passing over can be reduced 
 by one-half, TABLE X., even supposing the circum- 
 stances of approach to be the same : surely several 
 engineering appliances for shorter weirs, during 
 periods of flood, would be found more effective and 
 far less expensive than this alternative, with its extra 
 sinking and weir basin for drainage purposes. 
 
 The advocates for the necessity of weirs longer 
 than the width of the channel, for drainage purposes, 
 must show that the reduction of the head and extent 
 of backwater above ^, Fig. 39, is not small, and 
 that the effects extend .the whole way up the channel, 
 
268 
 
 THE DISCHAKCiE OF WATEK FBOM 
 
 or at least as far as the district to be benefited. 
 Practice has heretofore shown, that long weirs have 
 failed (unless after the introduction of sluices or 
 other appliances) in producing the expected arterial 
 drainage results, notwithstanding the increased leak- 
 age from increased length, which must accompany 
 their construction. 
 
 The deepening in the weir basin a b B E A is mostly 
 of use in reducing the surface inclination between 
 a b and A B by increasing the hydraulic mean depth ; 
 but, thereby, the velocity of approach is lessened, 
 and therefore the head at E increased. When the 
 length of a weir basin a E exceeds that point where 
 these two opposite effects balance each other, there 
 will be a gain by the difference of the surface in- 
 clinations in favor of the long weir : but unless a E 
 exceeds half a mile, this difference cannot amount to 
 more than 3 or 4 inches, unless the river be very 
 small indeed ; and if the channel be sunk for the long 
 weirs B A or b a ly it should also be sunk to at least the 
 same depth and extent for the short weirs B e, b a, 
 otherwise there is no fair comparison of their separate 
 merits. The effect of the widening between a b and 
 A B, the depth being the same, is also to reduce the 
 
OKIFICES, WEIES, PIPES, AND EIVEES. 269 
 
 surface inclination from a toE ; but, as before, unless a D 
 be of considerable length, this gain will also be small. 
 Now A B, at best, is but a weir the direct width of 
 the new channel at A B, and if the length a E be 
 considerable, we have an entirely new river channel 
 with a direct weir at the lower end, and the saving of 
 head effected arises entirely from the larger channel, 
 with a direct transverse weir at the lower end. 
 
 By referring to TABLE VIII., it will be found that 
 for a hydraulic mean depth of 5 feet a fall of 74 
 inches per mile will give a velocity of 2 feet per 
 second ; if we double the depth, a fall of 4 inches 
 will give the same velocity ; and for a depth of only 
 2 feet 6 inches, a fall of 121 inches is necessary, 
 This is a velocity much larger than we have ever 
 observed in a weir basin, yet we easily perceive 
 that the difference in the inclinations for a short 
 distance, E a of a few hundred feet, must be small. 
 If one section be double the other, the hydraulic 
 mean depth remaining constant, the velocity must be 
 one-half, and the fall per mile one-four-th, nearly. 
 This would leave 7| 2 = 5| inches per mile, or 1 
 inch per 1000 feet nearly, as the gain with a hydrau- 
 lic mean depth of 5 feet for a double water channel. 
 For greater depths the gain would be less, and the 
 contrary for lesser depths. 
 
 Is the saving of head and amplitude of backwater 
 we have calculated worth the increased cost of long 
 weirs and the consequent necessity and expense of 
 sinking and widening the channels for such long 
 distances ? We think not ; indeed, the sinking in the 
 basin immediately at the weir is absolutely injurious, 
 
270 THE DISCHAEGE OF WATER FROM 
 
 by destroying the velocity of approach, and increasing 
 the contraction. The gradual approach of the bottom 
 towards the crest, shown by the upper dotted line 
 b E in the section, Fig. 39, and a sudden overfall, will 
 be found more effective in reducing the head, unless 
 so far as leakage takes place, than any depth of 
 sinking for nearly 80 or 100 feet above long weirs. 
 
 In most instances, the extra head will be only per- 
 ceived by an increased surface inclination, which 
 may extend for a mile or more up the channel, 
 according to the sinking and widening. 
 
 It is a general rule that, for shorter weirs, the co- 
 efficients of discharge decrease ; this arises from the 
 greater amount of lateral contraction, and is more 
 marked in notches or Poncelet weirs, than for weirs 
 extending from side to side of the channel ; but for 
 weirs exceeding 10 feet in length the decrease in the 
 coefficients from this cause is immaterial, unless the 
 head passing over bear a large ratio to the length ; 
 and we , even see from the coefficients, page 80, de- 
 rived from Mr. Blackwell's experiments, that with 
 10 inches head passing over a 2-inch plank, the co- 
 efficient for a length of 3 feet is *614 ; for a length of 
 6 feet *539 ; and for a length of 10 feet -534 ; show- 
 ing a decrease as the weir lengthens, but which may, 
 in the particular cases, be accounted for. We have 
 before referred to other circumstances which modify 
 the coefficients, yet we may assume generally, without 
 any error of practical value, that the coefficients are 
 the same for different weirs extending from side to 
 side of a river. If, then, we put w and w l for the 
 lengths of two such weirs, we shall have the relation 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 271 
 
 of the heads d and d l for the same quantity of water 
 passing over, as in the following proportion : 
 
 d : d l ; : w^ : w 3 ; 
 and therefore 
 
 (135.) #=(-)* X d, 
 
 M0i' 
 
 By means of this equation we have calculated 
 
 TABLE X,, the ratio being given in columns 1, 3, 
 w l 
 
 l 
 
 5 and 7, and the value of ( IL Y, or the coefficient by 
 
 \Wif 
 
 which d is to be multiplied, to find d l in columns 2, 
 4, 6 and 8. It appears also, that if we take the heads 
 passing over any weir in a river in an arithmetical 
 progression, the heads then passing over any other weir 
 in the same river must also be in arithmetical progres- 
 sion, unless the quantity flowing down varies from 
 erogation or supply, such as drawing off by millraces, 
 &;c. If c d be the coefficient for a direct weir, *94 d 
 will answer for an obliquity of 45, and -91 c d for an 
 angle of 65. 
 
 SECTION XL 
 
 BENDS AND CURVES. BRANCH PIPES. DIFFERENT LOSSES OF 
 HEAD. GENERAL EQUATION FOR FINDING THE VELOCITY. 
 HYDROSTATIC AND HYDRAULIC PRESSURE. PIEZOMETER. 
 CATCHMENT BASINS. RAIN-FALL PER ANNUM. 
 
 The resistance or loss of head due to bends and 
 curves has now to be considered. If we fix a bent 
 pipe, F B c D E G, Fig. 40, between two cisterns, so as 
 
272 
 
 THE DISCHARGE OF WATER FROM 
 
 to be capable of re- 
 volving round in col- 
 lars at F and G, we 
 shall find the time the 
 water takes to sink a 
 given distance from/ 
 to F in the upper cistern the same, whether the tube 
 occupy the position shown in the figure or the hori- 
 zontal position shown by the dotted line YbcdeG. 
 I This shows that the resistances due to friction and to 
 \ bends are independent of the pressure. If the tube 
 were straight, the discharge would depend on the 
 length, diameter, and difference of level between 
 f and a, and may be determined from the mean 
 velocity of discharge, found from TABLE VIII. or 
 equation (79). Here, however, we h^ve to take into 
 consideration the logs sustained at the bends and 
 curves, and our illustration shows that it is unaffected 
 by the pressure. 
 
 / The experiments of Bossut, Du Buat, and others, 
 show that the loss of head from bends and curves 
 like that from friction increases as the square of the 
 velocity ; but when the curves have large radii, and 
 the bends are very obtuse, the loss is very small. 
 With a head of nearly 3 feet, Venturi's twenty- third 
 experiment, when reduced, gives for a short straight 
 tube 15 inches long, and 1^ inch in diameter, having 
 the junction of the form of the contracted vein, very 
 nearly *873 for the coefficient of discharge. When of 
 the same length and diameter, but bent as in Diagram 
 I, Fig. 40, the coefficient is reduced to -785 ; and 
 when bent at a right angle as at H, Fig. 40, the co- 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 273 
 
 efficient is further reduced to -560. In these respec- 
 tive cases we have therefore* 
 
 1. 
 2. 
 
 , _ 
 v = -873 \/^g~h, and h = 1-312 x 3-; 
 
 -2 
 
 vzz-785 
 
 , and k = 1-623 x 
 
 _ 
 3. w = -560 v/2 # A, and A = 3-188 x j- ; 
 
 showing that the loss of head in the tube H, Fig. 40, 
 
 v 2 
 from the bend, is 1-876 x o~> or nearly double the 
 
 theoretical head due to the velocity in the tube. The 
 
 v 2 
 loss of head by the circular bend is only -3 11s, 
 
 or not quite one-sixth of the other. 
 
 Du Buat deduced, from about twenty-five experi- 
 ments, that the head due to the resistance in any 
 bent tube ABCDEFGH, diagram 1, Fig. 41, depends 
 
 IT O 
 
 on the number of deflections between the entrance at 
 A and the departure at H ; that it increases at each 
 
 * It is stated that the time necessary for the discharge of a 
 given quantity of water through a straight pipe being 1, the time 
 for an equal quantity through a curve of 90 would be 1-11, with 
 a right angle 1-57; two right angles would increase the time to 
 2'464, and two curved junctions to only 1*23. Vide REPORT ON 
 THE SUPPLY OF WATER TO THE METROPOLIS, p. 337, APPENDIX 
 No. 3. 
 
274 THE DISCHARGE OF WATER FROM 
 
 reflection as the square of the sine of the deflected 
 angle, A B R for instance, and as the square of the 
 velocity; and that if <, < 1? < 2 , < 3 , &c., be the number 
 of degrees in the angles of deflection at B, c, D, E, &c., 
 then for measures in French inches the height A b , due 
 to the resistances from curves, is 
 
 v 2 (sin. 2 4> + sin. 2 ^ + sin. 2 4> 2 + sin. 2 4> 3 + &c.) 
 *t>.) /**= 3000 
 
 which for measures in English inches becomes 
 
 ^ 2 (sin. 2 < + sin. 2 ^ + sin. 2 < 2 + sin. 2 
 (137.) h*- 3197 
 
 and for measures in English feet, 
 
 58.) K- 266-4 
 
 or, as it may be more generally expressed for all 
 measures, 
 
 (139.) ^ = (sin/^+sin. 2 ^ + sin. 2 < 2 + sin. 2 4> 3 + &c.) X 
 
 
 2 
 
 , in which 8>27 zz ^grj = '00375 v 2 in feet. 
 
 The angle of deflection, in the experiments from 
 which equation (136) was derived, did not exceed 
 36. We have already shown the loss of head from 
 the circular bend in diagram L Fig. 40, where the 
 
 angle of deflection is nearly 45, to be *311 K = 
 
 00483 v\ but as the sin. 45 = -707 ; sin. 2 45 = -5 
 we get -00483 v* = -00966 v* x sin. 2 45, or more than 
 two and a half times as much as Du Buat's formula 
 would give; and if we compare it with Rennie's 
 experiments,* with a pipe 15 feet long, J inch dia- 
 meter, bent into fifteen curves, each 3 inches radius, 
 
 * Philosophical Transactions for 1831, p. 438. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 275 
 
 we should find the formula gives a loss of head not 
 much more than one half of that which may be 
 derived from the observed change, -419 to -370 cubic 
 feet per minute in the discharge. See p. 278. 
 
 Dr. Young* first perceived the necessity of taking 
 into consideration the length of the curve and the 
 radius of curvature. In the twenty-five experiments 
 made by Du Buat, he rejected ten in framing his 
 formula, and the remaining fifteen agreed with it very 
 closely. Dr. Young finds 
 
 (140.) ^OOOOM^x*. 
 
 where </> is the number of degrees in the curve N P, 
 diagram 2, Fig. 41, equal the angle N o P ; P = o N 
 the radius of curvature of the axis ; h b the head due 
 to the resistance of the curve, and v the velocity, all 
 expressed in French inches. This formula reduced 
 for measures in English inches is 
 (141.) ^ = -000004*^ X*. 
 
 P 
 
 and for measures in English feet, 
 (U2 .) , b 
 
 Equation (140) agrees to ^ n of the whole with 
 twenty of Du Buat's experiments, his own formula 
 agreeing so closely with only fifteen of them. The 
 resistance must evidently increase with the number 
 of bends or curves ; but when they come close upon, 
 and are grafted into each other, as in diagram 1, 
 Fig. 41, and in the tube F B c D E G, Fig. 40, the 
 motion in one bend or curve immediately affects those 
 
 * Philosophical Transactions for 1808, pp. 173 175. 
 
 T3 
 
276 THE DISCHARGE OF WATER FROM 
 
 in the adjacent bends or curves, and this law does 
 not hold. 
 
 Neither Du Buat nor Young took any notice of 
 the relation that must exist between the resistance 
 and the ratio of the radius of curvature to the radius 
 of the pipe. Weisbach does, and combining Du Buat's 
 experiments with some of his own, finds for circular 
 tubes, 
 
 (143.) Ab = Tf0 X {-131 + 1-847 (^J) x 
 
 $*;. s 2^ 
 
 and for quadrangular tubes, 
 
 d_ 7 
 
 p- 
 
 (144) A b=T f X ("124 4 3-104(^7) x ; 
 
 in which < is equal the angle N o P = N i R, diagram 2, 
 Fig. 41 ; d the mean diameter of the tube, and p the 
 
 radius N o of the axis. When - exceeds -2, the value 
 of -131 + 1-847 * exceeds -124 + 3-104 
 
 and the resistance due to the quadrangular tube ex- 
 ceeds that due to the circular one. We have ar- 
 ranged and calculated the following table of the 
 numerical values of these two expressions for the 
 more easy application of equations (143) and (144). 
 This table will be found of considerable use in cal- 
 culating the values of equations (143) and (144), as 
 the second and fifth columns contain the values of 
 
 131 + 1-847 o~ 5 and the third and sixth columns 
 the values of -124 + 3-104 (:;)> corresponding to 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 277 
 
 
 different values of ~- ; and it is carried to twice the 
 
 & P 
 
 extent of those given by Weisbach. 
 
 TABLE OF THE VALUES OF THE EXPEESSIONS 
 
 131 + 1-847 (-\* and -124 + 3-104 ( - 
 
 d 
 2p ' 
 
 Circular 
 tubes. 
 
 Quadrangular 
 tubes. 
 
 d 
 
 V 
 
 Circular 
 tubes. 
 
 Quadrangular 
 tubes. 
 
 1 
 
 131 
 
 124 
 
 6 
 
 440 
 
 643 
 
 15 
 
 133 
 
 128 
 
 65 
 
 540 
 
 811 
 
 2 
 
 1S8 
 
 135 
 
 7 
 
 661 
 
 1-015 
 
 25 
 
 145 
 
 148 
 
 75 
 
 806 
 
 1-258 
 
 3 
 
 158 
 
 170 
 
 8 
 
 977 
 
 1-545 
 
 35 
 
 178 
 
 203 
 
 85 
 
 1-177 
 
 1-881 
 
 4 
 
 206 
 
 250 
 
 9 
 
 1-408 
 
 2-271 
 
 45 
 
 244 
 
 314 
 
 95 
 
 1-674 
 
 2-718 
 
 5* 
 
 294 
 
 308 
 
 1-00 
 
 1-978 
 
 3-228 
 
 For bent tubes, diagrams 3, 4, and 5, Fig. 41, the 
 loss of head is considerably greater than for rounded 
 tubes. If, as before, we put the angle N i R = <, i R 
 being at right angles to i o the line bisecting the 
 angle or bend, we shall find, by decomposing the 
 
 V* V 2 
 
 motion, that the head ^ becomes ^ x cos. 2 $ from 
 the change of direction ; and that a loss of head 
 (145.) A b = (1 - cos. 2 2 $) ^- = sin. 2 2 < f- 
 
 must take place. When the angle is a right angle, 
 
 v 2 
 
 as in diagram 4, cos. 2 <#> = 0, and k b z= ^- ; that 
 
 ^9 
 
 is to say, the loss of head is exactly equal to the 
 
 * The values corresponding to Q =-55 are -350 and *507 for 
 circular and quadrangular tubes. 
 
278 THE DISCHARGE OF WATER FROM 
 
 theoretical head. When the angle or bend is acute, 
 
 v 2 
 as in diagram 5, the loss of head is (1 + cos. 2 2 <j> ~- 9 
 
 for then cos. 2 < becomes negative. Weisbach does 
 not find the loss of head in a right angular bend 
 
 v 2 
 greater than -984 - ; while Venturi's twenty-third 
 
 experiment, made with extreme care, p. 273, shows 
 the loss to be 1-876 s When the pipes are long, 
 
 however, the value of o - is in general small, and 
 
 this correction does not affect the final results in any 
 material degree. 
 
 Rennie's experiments,* with a pipe 15 feet long, J 
 inch in diameter, and with 4 feet head, give the dis- 
 charge per second 
 
 Cubic feet. 
 
 1. Straight, see table, p. 152 . . -00699 
 
 2. Fifteen semicircular bends . . -00617 
 
 3. One bend, a right angle 8J inches 
 
 from the end of the pipe . . -00556 
 
 4. Twenty-four right angles . . -00253 
 From these data we find consecutively, the theoreti- 
 cal discharge being -021885 cubic feet per second, 
 
 Q 
 
 and the theoretical head H s , that 
 
 1. v = -319 \/2#H, and therefore H = 9-82 x j- ; 
 
 2. v = -282 \X2^H, H = 12-58 x^-; 
 
 * Philosophical Transactions for 1831, p. 438. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 279 
 ^2 
 
 3. v n -254 \/2#H,and therefore H = 15*50 x 
 
 ^2 
 
 4. v = -116 v/2#H, HIZ 74-34 x g-- 
 The loss of head, therefore, by the introduction of 
 15 semicircular bends, is 2*76 ; by the intro- 
 
 2 
 
 duction of one right angle, 5-68 n ; and by the 
 
 2 
 
 introduction of 24 right angles, 64-52 -, or about 
 
 12 times the loss due to one right angle. This 
 shows that the resistance does not increase as the 
 number of bends, as we before remarked, p. 256, 
 when they are close to each other. The loss of head 
 
 from one right angle, 5*68 o , is more than double 
 
 the loss from 15 semicircular bends, or 2-76 ^~. 
 
 The loss of head for a right angular bend, determined 
 
 v 2 
 from Venturi's experiment, is 1-876 s-; formula 
 
 v 2 
 (145) makes it ^- ; and Weisbach's empirical for- 
 
 v 2 
 mula, (-9457 sin. < + 2-047 sin. 4 <) H-, makes it only 
 
 v 2 
 984 s . The formulae now in use give, therefore, 
 
 results considerably under the truth. It appears 
 to us, that the velocity of the water moving 
 directly towards the bend must be taken into 
 consideration, and also the loss of mechanical ef- 
 fect from contraction, and eddies at the bend, as 
 
280 
 
 THE DISCHAEGE OF WATEE FEOM 
 
 well as the loss arising from the mere change of 
 
 direction. 
 
 BRANCH PIPES. 
 
 When a pipe is joined to another, the quantity of 
 water flowing below the junction B, diagram 1, 
 Fig. 42, must be equal to the sum of the quantities 
 
 Fig*. 42 
 
 flowing in the upper branches in the case of supply ; 
 and when the branch pipe draws off a portion of the 
 water, as in diagram 2, the quantity flowing above 
 the junction must be equal to the quantities flowing 
 in the lower branches. Both cases differ only in the 
 motion being from or to the branches, which, in 
 pipes, are generally grafted at right angles to the 
 main, for practical convenience, as shown at bb, and 
 then carried on in any given direction. The loss of 
 head arising from change of direction, equation (145), 
 
 2 
 
 is sin. 2 2 < n~, in which 2 < zz angle ABO; but as in 
 
 general 2 <#> is a right angle for branches to mains, 
 
 v 2 
 this source of loss becomes then simply s . I n 
 
 addition to this, a loss of head is sustained at the 
 junction, from a certain amount of force required to 
 unite or separate the water in the new channel. In 
 the case of drawing off, diagram 2, this loss was 
 estimated by D'Aubuisson, from experiments by 
 Genieys, to be about twice the theoretical head due 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 281 
 
 2 v 2 
 to the velocity in the branch, or o , so that the 
 
 whole loss of head arising from the junction is 
 
 v 2 2v 2 3 v 2 
 
 o + ~ - =: ^ , or three times the theoretical head 
 
 2# n 1g 2g> 
 
 due to the velocity. In the case of supply, the loss 
 is probably nearly the same. The actual loss is, 
 however, very uncertain; but, as was before ob- 
 served when discussing the loss of head occasioned 
 
 v 2 
 
 by bends, two or three times ^ is in general so 
 
 comparatively small, that its omission does not ma- 
 terially affect the final results. A loss also arises 
 from contraction, &c. See pp. 175, 176. 
 
 The calculations for mains and branches become 
 often very troublesome, but they may always be 
 simplified by rejecting at first any minor corrections 
 for contraction at orifice of entry, bends, junctions, 
 or curves. If, in diagram 2, Fig. 42, we put h for 
 the head at B, or height of the surface of the 
 reservoir over it ; h & for the fall from B to A ; A d for 
 the fall from B to D ; / equal the length of pipe from 
 B to the reservoir ; / a equal the length B A ; 4 equal 
 the length B D ; r equal the mean radius of the pipe 
 B c ; r a the mean radius of the pipe B A ; r d the mean 
 radius of B D ; v the mean velocity in B c ; v a the 
 velocity in B A ; and v d the velocity in B D, we then 
 find, by means of equation (73), the fall from the 
 reservoir to A equal to 
 
 (146.) A+A> 
 
 the fall from the reservoir to D equal to 
 
282 THE DISCHARGE OF WATER FROM 
 
 (147.) k + ht 
 
 and, as the quantity of water passing from c to B is 
 
 equal to the sum of the quantities passing from B to 
 
 A and from B to D, 
 
 (148.) vr 2 = v a rl + v d rl 
 
 By means of these three equations we can find any 
 
 three of the quantities A, A a , A d , r, r a , r d , b, b a , b A , 
 
 the others being given. Equations (146) and (147) 
 
 may be simplified by neglecting c r , the coefficient due 
 
 to the orifice of entry from the reservoir, and 1, the 
 
 coefficient of velocity. They will then become 
 
 (148A.) h + A. = c t x - X + X 
 and 
 
 (149.) A + A d = c f 
 
 The mean value of c { for a velocity of 4 feet per 
 
 second is -005741. and of ^, -0000891. The values 
 
 *9 
 
 for any other velocities may be had from the table of 
 coefficients of friction given at p. 214. When /, h, 
 and r are given, the velocity v can be had from 
 
 the equation, v = ( x -y) , or more immediately 
 from TABLE VIII. 
 
 GENERAL EQUATION FOR MEAN VELOCITY. 
 
 We are now enabled to give a general equation for 
 finding the whole head H, and the mean velocity v, in 
 any channel ; and to extend equations (73) and (74) 
 so as to comprehend the corrections due to bends, 
 
ORIFICES, WEIKS, PIPES, AND EIVERS. 283 
 
 curves, &c. Designating, as before, the height due 
 to the resistance at the orifice of entry by 
 
 h r , and the corresponding coefficient by c r ; 
 
 h { the head due to friction, and c ( the coefficient of friction ; 
 
 7i b the head due to hends, and c b the coefficient of hends ; 
 
 h c the head due to curves, and c c the coefficient of curves ; 
 
 h e the head due to erogation, and c e the coefficient of erogation ; 
 
 /i x the head due to other resistances, and c x their mean coefficient ; 
 
 then we get 
 
 (150.) n = h t + h f + h l) + h c +h e + ^ + ~; 
 
 that is to say, by substituting for h r , h f , &c., their 
 values as previously found, 
 
 v* I v 2 v 2 
 
 V 2 V 2 V 2 
 
 X + Ce X + Cx X 
 
 or, more briefly, 
 
 7 
 
 (151.) H = 
 
 from which we find 
 
 (152.) v= 
 
 c t x - 
 
 It is to be observed here, that for very long 
 uniform channels, the value of the mean velocity 
 
 will be found in general equal to I ^ rH | a s the 
 
 \ Cf I ) 
 
 other resistances and the head due to the velocity 
 are all trifling compared with the friction, and may 
 be rejected without error ; but, as we before stated, 
 it is advisable in practice, when determining the 
 diameter of pipes, p. 229, to increase the value of c 
 
284 THE DISCHARGE OF WATER FROM 
 
 table, p. 214, or to increase the diameter found 
 from the formula by one-sixth, which will increase 
 the discharging power by one half. (See TABLE XIII.) 
 In equations (74) and (151), the coefficient of fric- 
 tion c { depends on the velocity v, and its value can 
 be found from an approximate value of that velocity 
 from the small table, p. 214. If, however, we use 
 both powers of the velocity, as in equation (83), we 
 shall get, when H is the whole head, and h the head 
 from the surface to the orifice of entry 
 
 I v 2 
 
 (av + bv 2 )- + (I + ;)*- + h=n, 
 
 a quadratic equation from which we find 
 
 f (H-fe)2<7r / gal 
 
 1(1 + c T )r + 2gbl 
 
 (1 + c r )r 
 
 for a more general value of the velocity than that 
 given in equation (74). If now we put c s = c r + c b 
 + c c + c e + c x in equation (151) we shall find 
 
 gal 
 
 Zgbl 
 
 for a more general expression of equation (152), when 
 the simple power of the velocity, as in equation (83), 
 is taken into consideration. For measures in English 
 feet, we may take a -0000223 and b = -0000854, 
 which correspond to those of Eytelwein, in equation 
 (97). The value of a is the same in English as in 
 French measures, but the value of b in equation (83), 
 for measures in metres, must be divided by 3-2809 
 to find its corresponding value for measures in 
 
 v 2 
 English feet. In considering the head ^- c n due to 
 
 contraction at the orifice of entry as not implicitly 
 comprised in the primary values of a and b, equation 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 285 
 
 (83), Eytelwein is certainly more correct than D'Au- 
 buisson, Traite d'Hydraulique, pp. 223 et 224, as 
 this head varies with the nature of the junction, and 
 should be considered in connection with the head 
 due to the velocity, or separately. It can never be 
 correctly considered as a portion of the head due to 
 friction. In all Du Buat's experiments, this head 
 was considered as a portion of that due to the velo- 
 
 v 2 
 city, and the whole head, (1 + c r ) 9 ? deducted to 
 
 find the head due to friction and thence the hydraulic 
 inclination. 
 
 VALUES OF a AND I FOR MEASURES IN ENGLISH FEET. 
 
 Equation (88.) 
 (90.) 
 (94.) 
 
 (98.; 
 
 (109.) 
 
 (HI.) 
 (114.) 
 
 Mean values for all 
 
 straight channels, 
 
 pipes, or rivers 
 These mean values of a and b give the equation 
 
 r s = -0001040 v 2 + -0000221 v, 
 from which we find 
 
 9615 r s =v 2 + -21 v, 
 and thence 
 
 (153.) v =(9615r* + -Oil)* -105 = 98 v/~ 1, 
 very nearly, suited to velocities of about 2 feet, p. 217. 
 
 
 a. 
 
 b. 
 
 ) 
 
 0000445 
 
 0000944 
 
 ) 
 
 0000173 
 
 0001061 
 
 ) 
 
 0000243 
 
 0001114 
 
 ) 
 
 0000223 
 
 0000854 
 
 ) 
 
 0000189 
 
 0001044 
 
 ) 
 
 0000241 
 
 0001114 
 
 ) 
 
 0000035 
 
 0001150 
 
 
 0000221 
 
 0001040 
 
286 
 
 THE DISCHARGE OF WATER FROM 
 
 HYDROSTATIC AND HYDRAULIC PRESSURE. PIEZOMETER. 
 
 When water is at rest in any vessel or channel, the 
 pressure on a unit of surface is proportionate to the 
 head at its centre,* measured to the surface, and is 
 expressed in Ibs. for measures in feet, by 624 H s, in 
 which H is the head, and s the surface exposed to the 
 pressure, both in feet measures. This is the hydro- 
 static pressure. In the pipe A B c D F E, Fig. 43, the 
 pressure at the points B, c, D, F, and E, on the sides of 
 the tube will be respectively as the heads B b, c c, D d, 
 F/J and E e, if all motion in the tube be prevented by 
 
 stopping the discharging orifice at E. In this case the 
 pressure is a maximum and hydrostatic ; but if the 
 discharging orifice at E be partially or entirely open, 
 a portion of each pressure at B, c, D, F, &c., is ab- 
 sorbed in overcoming the different resistances of 
 friction, bends, &c., between it and the orifice of entry 
 at A, and also by the velocity in the tube, and the 
 difference is the hydraulic pressure. 
 
 * This is only correct when the surface is small in depth com- 
 pared with the head. If H he the depth of a rectangular surface 
 in feet, and also the head of water measured to the lower hori- 
 zontal edge, then the pressure in Ibs. is expressed by 31 H 2 ; and 
 the centre of pressure is at f rds- of the depth. 
 
OEIFICES, WEIES, PIPES, AND EIVEES. 287 
 
 Bernoulli first showed that the head due to the 
 pressure at any point, in any tube, is equal to the 
 effective head at that point, minus the head due to the 
 velocity. When the resistances in a tube vanish, the 
 effective head becomes the hydrostatic head, and by 
 representing the former byA ef we shall have, adopting 
 the notation in equation (150), 
 
 and consequently the head due to the hydraulic pres- 
 sure equal 
 
 v * X 
 
 If small tubes be inserted, as shown in Fig. 43, at 
 the points B, c, D, and F, the heights B b 1 , c c 1 , D d 1 , F/ 1 , 
 which the water rises to, will be represented by the 
 corresponding values of A p in the preceding equation; 
 and the difference between the heights c c 1 , F/ 1 , at 
 c and F, for instance, added to the fall from c to F 
 will, evidently, express the head due to all the re- 
 sistances between c and F. When H = E e, and the 
 orifice at E is open, we have, from equation (150), 
 
 , and therefore Ap=0, 
 
 that is, the pressure at the discharging orifice is 
 nothing. 
 
 The vertical tubes at B, c, D, F, when properly 
 graduated, are termed piezometers or pressure gauges; 
 they not only show the actual pressure at the points 
 where placed, but also the difference between any 
 two ; D d 1 B b 1 , for instance, added to the difference 
 of head between D and B, or D d 2 will give D d 1 sb l 
 + D d 2 for the head or pressure due to the resistances 
 
288 THE DISCHARGE OF WATER FROM 
 
 between B and D. This instrument affords, perhaps, 
 the very best means of determining the loss of head 
 due to bends, curves, diaphragms, &c. The loss of 
 head due to friction, bend, diaphragms, &c., between 
 K and L, Fig. 43, is equal to K k L / + K v. If M 
 be the same distance from L as K is, L / M m will be 
 the height due to the friction (L and M being on the 
 same level) ; therefore K k L / + KV L / + M m m 
 K k + K v + M m 2 L / is the head due to the 
 diaphragm and bend both together. If the diaphragm 
 be absent, we get the head due to the bend, and if 
 the bend be absent, the head due to the diaphragm in 
 like manner. 
 
 When the discharging orifice, as at E, is quite open, 
 we have seen that the pressure there is zero; but 
 when, as at G, it is only partly open, this is no longer 
 the case, and the hydraulic pressure increases from 
 zero to hydrostatic pressure, as the orifice decreases 
 from the full section to one indefinitely small com- 
 pared with it. A piezometer, placed a short distance 
 inside G, will give this pressure ; and the difference 
 between it and the whole head will be the head due to 
 the resistances and velocity in the pipe : from which, 
 and also the length and diameter, the discharge may 
 be calculated as before shown. Again, by means of 
 the head M m 1 , and that due to the velocity of ap- 
 proach, we can also find the discharge through the 
 diaphragm G ; see equation (45) and the remarks fol- 
 lowing it. This result must be equal to the other, 
 and we may in this way test the formulae anew or 
 correct them by the practical results. 
 
 The velocity of discharge of the tube A c D E, may 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 289 
 
 be calculated by means of any piezometric height 
 c c 1 ; for by putting the whole fall from c 1 to E equal 
 
 (2flrrH c lH 
 
 to H C I, we get, disregarding bends, v = \ - \ , j n 
 
 1 Me 1 J 
 
 which I C \G E. This is evident from equation (152), 
 as we have supposed that no part of the head is ab- 
 sorbed in generating velocity, or in overcoming the 
 resistance of bends. If the bend at D were taken 
 
 into consideration, then v = ' 
 
 -fX^r + 
 
 SECTION XII. 
 
 RAIN-FALL CATCHMENT BASINS. DISCHARGE INTO CHANNELS. 
 
 DISCHARGE FROM SEWERS. LOSS FROM EVAPORATION, ETC. 
 
 A catchment basin is a district which drains itself 
 into a river and its tributaries. It is bounded gene- 
 rally by the summits of the neighbouring hills, ridges, 
 or high lands forming the water-shed boundary ; and 
 may vary in extent from a few square miles to many 
 thousands ; that of the Shannon is 4,544 square 
 miles. The average quantity of water which dis- 
 charges itself into a river will, cceteris paribus, depend 
 on the extent of its catchment basin, and the whole 
 quantity of rain discharged on the area of the catch- 
 ment basin, including lakes and rivers. 
 
 The quantity of rain which falls annually varies 
 with the district and the year ; and it also varies 
 at different parts of the same district. The average 
 quantity in Ireland may be taken at about 34 inches 
 
290 
 
 TABLE of some Catchment Basins in Ireland. 
 
 Names of Drainage-districts, 
 or Rivers. 
 
 Counties or Towns. 
 
 Area of 
 Catchment 
 in acres. 
 
 Area of 
 Catchment in 
 square miles. 
 
 Avonmore . . 
 
 Wicklow and Wexford 
 
 128 000 
 
 200- 
 
 
 Wicklow . . 
 
 179,840 
 
 281- 
 
 Ballinasloe . 
 
 Mayo 
 
 70000 
 
 110- 
 
 Barrow, Nore, and Suir . . . 
 
 Waterford 
 
 2,176,000 
 
 3400- 
 
 Blackwater and Boyne .... 
 
 Meath, &c 
 
 695,040 
 
 1086- 
 
 Blackwater .... 
 
 Waterford Youghal 
 
 780,160 
 
 1219- 
 
 Blackwater 
 
 Armagh 
 
 336,640 
 
 526- 
 
 Blackwater 
 
 Meath and Kildare 
 
 50,000 
 
 78-1 
 
 Bandon River 
 
 Cork 
 
 145,920 
 
 228- 
 
 Bann, Upper and Lower, and 
 the Main 
 
 
 810,240 
 
 1266- 
 
 Boyne 
 
 Meath, Westmeath Kildare 
 
 
 
 
 and King's 
 
 304,139 
 
 478-2 
 
 Brusna (Ferbane) 
 
 King's . . . 
 
 389,120 
 
 608- 
 
 
 Wexford 
 
 26,752 
 
 41-8 
 
 Ballinamore and Bally connel 
 
 Cavan, Fermanagh, Leitrim, 
 and Roscommon 
 
 101,455 
 
 158-5 
 
 Breeogue 
 
 Sligo . . 
 
 180,408 
 
 282- 
 
 
 Cork 
 
 23,500 
 
 36-7 
 
 Cappagh 
 
 
 34,856 
 
 54-4 
 
 
 Sliffo 
 
 90744 
 
 141-8 
 
 Camoge 
 
 Limerick . . . 
 
 61,184 
 
 95-6 
 
 
 Galway, Mayo, and Roscom- 
 
 
 
 
 mon . . . 
 
 96,161 
 
 150-2 
 
 Dodder 
 
 Dublin 
 
 35,200 
 
 55- 
 
 Deel 
 
 Meath and Westmeath 
 
 64,000 
 
 100- 
 
 Dee 
 
 Louth and Meath 
 
 78,000 
 
 121-9 
 
 Erne 
 
 Belturbet Enniskillen 
 
 1,014,400 
 
 1585- 
 
 Fovle 
 
 Londonderry 
 
 944,640 
 
 1476- 
 
 Fergus 
 
 Clare and Galway 
 
 134,400 
 
 210- 
 
 Fane 
 
 Louth 
 
 87,400 
 
 136-6 
 
 Glyde 
 
 Louth, Meath Monaghan 
 
 
 
 
 and Cavan 
 
 176,813 
 
 276-3 
 
 Inny . . 
 
 Meath Westmeath Long- 
 
 
 
 
 ford, and Cavan 
 
 231,116 
 
 361-1 
 
 Kilbeggan . . 
 
 Westmeath and King's 
 
 88,030 
 
 137-5 
 
 lifi'ey and Tolka 
 
 Dublin, &c 
 
 328,320 
 
 513-0 
 
 Lee ... 
 
 Cork 
 
 470,400 
 
 735- 
 
 Lough Gara and Mantua . . 
 Loughs Oughter and Gowna 
 and River Erne 
 Lough Neagh 
 
 Roscommon, Mayo, and Sligo 
 
 Cavan, Leitrim, and Longford 
 Londonderry Antrim Down 
 
 128,000 
 260,480 
 
 200- 
 407- 
 
 
 and Armagh 
 
 1,411,320 
 
 2205-2 
 
 Lough Mask and River Robe 
 
 Mayo and Galway 
 
 225,000 
 
 351-5 
 
 Loughs Corrib, Mask, and 
 Carra . 
 
 Galwav and Mayo 
 
 780,000 
 
 1218-7 
 
 Lonford 
 
 Longford . 
 
 72320 
 
 113-0 
 
 Moy . 
 
 Mayo, Ballina 
 
 661,120 
 
 1033- 
 
 r ? 
 
 Main . , 
 
 Antrim 
 
 37,600 
 
 90- 
 
 
 
 54,000 
 
 84-4 
 
 Ma^hera . 
 
 Down 
 
 19000 
 
 29-7 
 
 Nobber 
 
 Meath 
 
 40,000 
 
 62-3 
 
 Quoile 
 
 Down 
 
 57000 
 
 89-1 
 
 Rinn and Black River .... 
 Strokestown 
 
 Leitrim and Longford .... 
 Roscommon 
 
 74,000 
 70000 
 
 115-6 
 109-4 
 
 
 Different Counties, Towns of 
 
 
 
 
 Athlone, Limerick 
 
 2,908 160 
 
 4544- 
 
 Slaney . . 
 
 Wexford 
 
 521 600 
 
 815- 
 
 
 
 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 291 
 
 deep, that which falls in Dublin being 27 inches, and 
 that in Cork 41 inches nearly. The average yearly 
 fall in Dublin for seven years, ending with 1849, was 
 26*407 inches ; and the maximum fall in any month 
 took place in April 1846, being 5-082 inches. "The 
 average fall in inches per month for seven years, 
 ending with 1849, was as follows : October, 3 -060 ; 
 August, 2-936 ; January, 2-544 ; April, 2-503 ; No- 
 vember, 2-300 ; July, 2-116 ; June, 2-005 ; Decem- 
 ber, 1*938; September, 1-860 ; May, 1-814; March, 
 1-739 ; February, 1-534."* A gauge at London- 
 derry, 1795 to 1801, gives 31 inches average ; one 
 at Belfast, from 1836 to 1841, gives 35 inches; at 
 Mountjoy, Phoenix Park, 182 feet above low water, 
 1839 and 1840, there is an average of 33 inches ; 
 and at the College of Surgeons, 52 feet over low 
 water, the average is 30 inches for the same two 
 years. Sir Robert Kane assumes that 36 inches is 
 the average fall in Ireland, and that out of that depth 
 12 inches, or one-third, passes on to the sea, two- 
 thirds being evaporated and taken up by plants. 
 The quantity varies a good deal with the altitude of 
 the district. In parts of Westmoreland it rises 
 sometimes to 140 inches ; in London, an average 
 of 20 years' observations, gives a fall of nearly 25 
 inches. 
 
 Forty years' observation at Greenwich, Kent, at 
 155 feet above the level of the sea, gives the following 
 results : 
 
 * Proceedings of the Koyal Irish Academy, vol. v., p. 18. 
 
 TJ3 
 
292 
 
 THE DISCHAEGE OF WATER FROM 
 
 Description of fall. 
 
 Winter. 
 
 Spring. 
 
 Summer 
 
 Entire 
 Years. 
 
 Mean annual fall . . . 
 
 inches. 
 7-86 
 
 inches. 
 7-25 
 
 inches. 
 10-47 
 
 inches, 
 f 25-48 
 
 Maximum fall; being a mean of 
 five of the wettest years during 
 forty years 
 
 11-05 
 
 10-86 
 
 14-96 
 
 { 25-58 
 ( 34-00 
 
 Minimum fall ; being a mean of 
 five of the driest years during 
 forty years . . 
 
 5-22 
 
 4-05 
 
 6-80 
 
 { 36-87 
 ( 18-40 
 
 
 
 
 
 (16-07 
 
 In this table Winter comprises November, De- 
 cember, January, and February ; Spring, the next 
 four months ; and Summer, the months of July, 
 August, September, and October. The last column 
 contains means of two classes of years : the first 
 figures showing the ordinary years from January to 
 December, and the second, under the first, years 
 from November to October.* We see here that the 
 mean maximum is fully double of the mean minimum, 
 and about one-and-a-half times the mean annual fall, 
 and therefore the necessity for calculating from the 
 minimum fall for all water works in which it is 
 an element, and from the maximum for sewerage 
 works where it is not intended to pass off a portion 
 on the surface or through other available channels. 
 
 In the district surrounding the Bann reservoirs in 
 the County Down, the average fall has been so high 
 as 72 inches. In Keswick, the average fall is said to 
 be 67| inches, and in Upminster, Essex, only 19| 
 inches. Indeed, it is requisite to obtain the fall 
 from observation for any particular district, when it 
 
 * See Mr. James Simpson in the Metropolitan Main Drainage 
 Report, 1857, p. 115. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 293 
 
 is necessary to apply the results to scientific pur- 
 poses ; and not the mean average fall alone, but also 
 the maximums and minimums in a series of years and 
 months in each year. 
 
 Mr. Symons gives (see Builder for I860, p. 230) 
 the following heavy falls of rain during 1859 : 
 Wandsworth, June 12th, 217 inches in two hours ; 
 Manchester, August 7th ? 1*849 inches in twenty-four 
 hours ; Southampton, September 26th, 2-05 inches 
 in two-and-a-quarter hours ; Truro, October 25th, 
 during the day, 2-4 inches. The mean falls in the 
 South Western Counties were 39-1 inches ; in the 
 South Eastern Counties, 30-2 inches ; in the West 
 Midland Counties, 28 inches ; in the Eastern Coun- 
 ties, 2 5 *4 inches ; in the North Midland Counties, 
 24 inches ; in the North Western Counties, 39 
 inches ; in the Northern Counties, 55 inches ; and 
 the average of all England, 31-857 inches. 
 
 As an instance of extraordinary rain-fall, in 
 connexion with the sewage question, it is stated that 
 4 inches of rain fell in one hour in the Holborn and 
 Finsbury sewers' district, on the 1st of August, 
 1846 ; at Highgate, 3-5 to 3 -3 inches ; and at 
 Greenwich, 0-95 inches.* 
 
 In the upland districts about Manchester, Mr. 
 Homershamf gives the result of observations at 
 Fairfield, Bolton, Rocksdale, Marple, Comlis re- 
 servoir, Belmont, Chapel-en-le-Frith, and Whiteholme 
 reservoir, for four years. These give a maximum 
 fall of 61-4 inches at Belmont Sharpies in 1847, and 
 
 * Metropolitan Main Drainage Keport, p. 16. 
 
 f Report on the Supply of Water to Manchester. WEALE. 
 
294 THE DISCHARGE OF WATER FROM 
 
 a minimum of 24-8 at Whiteholme reservoir in 1844. 
 The general average for the four years being 42-49 
 inches. 
 
 April is the driest month, and October, or about 
 it, the wettest month, and the average fall during 
 the year varies sometimes as two to one. 
 
 The proportion between the quantity which falls, 
 and that which passes from a catchment basin into 
 its river, also varies very considerably. When the 
 sides of a catchment basin are steep, and the water 
 passes off rapidly into the adjacent river or tri- 
 butaries, there is less loss by evaporation and perco- 
 lation than when they are nearly flat. The soil, 
 subsoil, and stratification, have also considerable 
 effect on the proportion. Reservoirs being generally 
 constructed adjacent to steep side falls, give a much 
 larger proportion of the quantity fallen than can be 
 obtained from rivers in flatter districts ; besides, the 
 quantity of rain which falls on the high summits, 
 near reservoirs, almost always considerably exceeds 
 the average fall. As 640 acres is equal to one 
 square mile, and one acre is equal to 43,560 square 
 feet, a fall of one inch of rain is equal to 3,630 cubic 
 feet per acre, and to 3,630 x 640 = 2,323,200 cubic 
 feet per square mile : the proportion of this fall, for 
 each acre, or square mile of the catchment basin, 
 which enters the river, must depend entirely on the 
 district and local circumstances, the full or maximum 
 quantity being retained on lakes. A stream de- 
 livering 53 cubic feet per minute supplies an equiva- 
 lent to 12 inches of rain-fall collected per square 
 mile per annum. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 295 
 
 It is too often taken for granted that the discharge 
 from a catchment basin takes place, into the con- 
 veying channels, in nearly the same time that a 
 given quantity of rain falls. Perhaps the largest 
 registry on record in Great Britain is a fall of four 
 inches in an hour. The maximum fall in any hour of 
 any year seldom exceeds half of this amount, and 
 then perhaps only once in several years. The quan- 
 tity which falls will not be discharged into the 
 channels in the same time. The quantity discharged, 
 
 QUANTITY PER ACRE FOR A GIVEN DEPTH OF FALL. 
 
 Fall in 
 inches. 
 
 Cubic feet 
 per acre. 
 
 Fall in 
 inches. 
 
 Cubic feet 
 per acre. 
 
 Fall in 
 inches. 
 
 Cubic feet 
 per acre. 
 
 Fall in 
 inches. 
 
 Cubic feet 
 per acre. 
 
 2 
 
 7260 
 
 i 
 
 1815 
 
 1 
 
 454 
 
 JL_ 
 
 181 
 
 If 
 
 6352 
 
 -i 
 
 1361 
 
 | 
 
 403 
 
 TO- 
 
 121 
 
 ii 
 
 5445 
 
 
 907 
 
 1 
 
 363 
 
 1 
 
 91 
 
 2 
 
 
 4 
 
 
 TO 
 
 
 4: 
 
 
 U 
 
 4537 
 
 i 
 
 726 
 
 A 
 
 302 
 
 50 
 
 73 
 
 i 
 
 3630 
 
 i 
 
 605 
 
 yV 
 
 259 
 
 "g"0" 
 
 61 
 
 I 
 
 2723 
 
 i 
 
 T 
 
 519 
 
 TV 
 
 227 
 
 TV 
 
 52 
 
 and time, will depend a good deal on the season and 
 district. The arterial channel receives the supply at 
 different places and from different distances, and the 
 water in passing into and from it does not encounter 
 the same amount of resistance as if it all passed 
 first into the upper end. Less sectional area is 
 therefore necessary than if the whole discharge had 
 to pass through the whole length of the channel and 
 during the time of fall. The relation of the quantity 
 of rain -fall to the portion which flows into the main 
 channel, as well as the time which it takes to arrive 
 at it, and the places of arrival, must be known 
 
296 THE DISCHARGE OF WATER FROM 
 
 before the proper size of a new channel can be 
 determined, particularly sewers in urban districts. 
 A pipe sufficient to discharge the water from 200 
 acres need not be 20 times the discharging power of 
 one exactly suited to 10 acres of the same district, 
 for the discharge from the outlying 190 acres will 
 not arrive at the main in the same time as that from 
 the adjacent 10 acres. 
 
 The following table of rain-fall, at Athlone, central 
 in Ireland, was furnished to the Royal Irish Academy 
 by General Sir H. D. Jones, and is printed in the 
 Proceedings.* The average for four years, gives 29 
 inches, and the effect on the Upper and Lower Sills 
 of the Lock as affecting the rise and fall of the 
 Shannon, affords valuable data, although not analysed. 
 The rise and fall on the sills is the sum of the 
 monthly risings and fallings for each year, and must 
 be divided by 12 to get the average monthly rise and 
 fall. In 1845 the greatest rise was in January, 
 2 feet 9 inches at the upper sill, and 3 feet 1 1| inches 
 at the lower sill. In 1846 the greatest rise was 
 2 feet 5 inches in October, at the upper sill ; and 
 5 feet 6J inches on the lower sill, in August. 
 
 Upper SiU. Lower Sill. 
 
 Maximum rise in Maximum rise in 
 
 one month. one month. 
 
 1845 .... 2 ft. 9 in. January . . 3 ft. !! in. January. 
 
 1846 . . . . 2 ft. 5 in. October . . 5 ft. 6 in. January. 
 
 1847 .... 3ft. 1 in. November. .4ft. 6 in. May. 
 
 1848 .... 3 ft. 3 in. February . . 4ft. 11 in. February. 
 
 The sum of the risings and fallings for each month, 
 taken as a mean of four years, is nearly the same 
 
 * Vol. iv. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 297 
 
 
 
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 M O? CO Ol CO 
 
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 rH 
 
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 Amount for \ 
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 P 
 
298 
 
 THE DISCHARGE OF WATER FROM 
 
 at either sill. The general average of the rise and 
 fall for the upper sill, is about 1 foot 3i inches each 
 way, and 1 foot 101 inches at the lower sill. These 
 would give 2 feet 7 inches for the average difference 
 of level in the Shannon above, and 3 feet 9 1 inches 
 for that in the Shannon below. In Lough Allen 
 catchment of 146 square miles, the maximum rise 
 was sometimes 6 inches in 24 hours, calculated at 
 568 inch of depth of rain, over the catchment area. 
 Above Killaloe, the catchment is 3,611 square miles, 
 and the floods about once a year rose 6 inches in 
 24 hours, or *296 inch in depth of rain over the 
 catchment. Once, in 1840, it is reported to have 
 risen 12 inches, or *6 inch of rain over the catchment 
 in one day. 
 
 MAXIMUM DISCHARGES OF THE SHANNON AND ERNE, AND A TRIBUTARY OF THE LATTER, 
 THE WOODFORD RIVER. 
 
 BIVERS IN IRELAND. 
 
 Extent of 
 catchment, 
 statute acres. 
 
 Square 
 miles. 
 
 Maximum 
 discharge 
 per minute 
 in cubic feet. 
 
 Cubic feet 
 per minute 
 from each 
 acre. 
 
 Cubic feet 
 per minute 
 from each 
 square mile 
 
 Shannon, at Killaloe, measured 
 
 
 
 
 
 
 previous to the commence- 
 
 
 
 
 
 
 ment of Shannon Works, 
 
 
 
 
 
 
 
 3,000,000 
 
 4687'5 
 
 1,000,000 
 
 0-33 
 
 211 
 
 Lower Erne, measured during 
 
 
 
 
 
 
 the very high floods of Jan. 
 
 
 
 
 
 
 1851, at BeUeek 
 
 974,000 
 
 1521-9 
 
 657,511 
 
 0-67 
 
 429- 
 
 Upper Erne, measured during 
 the very high floods of Jan. 
 
 
 
 
 
 
 1851, at Belturhet 
 
 309,000 
 
 482-8 
 
 257,771 
 
 0-83 
 
 531 
 
 Woodford River, Counties of 
 
 
 
 
 
 
 Lei trim and Cavan, measured 
 
 
 
 
 
 
 during the very high floods 
 
 
 
 
 
 
 of Jan. 1851, at Ballyconnell 
 
 90,000 
 
 140-6 
 
 101,035 
 
 1-12 
 
 717 
 
 Yellow River, or upper portion 
 
 
 
 
 
 
 of the Woodford River, mea- 
 
 
 
 
 
 
 sured during the very high 
 
 
 
 
 
 
 floods of Jan. 1851, Co. Leitrim 
 
 5,000 
 
 7-8 
 
 52,125 
 
 10-43 
 
 6675- 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 299 
 
 These results show how difficult it is to draw any 
 inference from discharge and area of catchment 
 alone, as the discharge, per minute per acre, must 
 vary with the contour and elevation of the district 
 in the same course ; and with the climate also, in 
 different countries. We have observed ourselves 
 the maximum discharges to vary up to 6 cubic feet 
 per minute per acre, the lesser maximums being due 
 to broad flat districts, and the greater maximums to 
 higher and steeper districts, near the sources. In 
 the Proceedings of the Institution of Civil Engi- 
 neers, Ireland, vol. iv., from which we have collected 
 and arranged some of the foregoing information, it 
 is stated, p. 96, that the ratio of the discharge to the 
 rain-fall, on a catchment on the Glyde, of 79,433 
 acres, for three months, ending March 13th, 1851, 
 was 1*49 to 1 up to January 13th ; 1*39 to 1 up to 
 February 13th ; and 3-86 to 1 up to March 13th, 
 making a general average of 1*59 to 1 ; the whole 
 rain-fall for the three months being only 5 '89 inches, 
 while the discharge was 9-35 inches ! We fancy 
 there is a mistake here. The whole catchment of 
 the Glyde is 176,813 acres, and there is no data to 
 show the discharge previous to or after the rain-fall 
 from which to calculate the difference due to it per 
 se for the three months; nor is the place or method 
 of gauging stated. The supply from springs and 
 the actual discharge before and after rain-fall must 
 be correctly gauged before the proportion passing 
 into the main channel in a given time, can be pro- 
 perly estimated ; the results just stated clearly con- 
 tradict themselves. The following anomalous results 
 
300 
 
 THE DISCHARGE OF WATER FROM 
 
 from p. 47 of the same work are also worthy of 
 note. In five different districts the discharge is 
 gauged, or estimated, greater than the fall, as shown 
 in the following table. It is not stated, however, if 
 
 District. 
 
 River. 
 
 Catchment 
 in Acres. 
 
 DECEMBER 1850. 
 
 JANUARY 1851. 
 
 Total fall 
 of Bain by 
 gauge in 
 inches. 
 
 Total 
 depth of 
 discharge 
 off catch- 
 ment in 
 inches. 
 
 Total fall 
 of Rain by 
 Castlebar 
 gauge in 
 inches. 
 
 Total 
 depth of 
 discharge 
 off catch- 
 ment in 
 inches. 
 
 Saleen . . . 
 
 Saleen 
 
 2,625 
 20,640 
 33,500 
 70,000 
 
 3,200 
 32,000 
 
 . 3-55 , 
 J4-00 
 
 6-26 
 5-46 
 5-46 
 
 6527 
 5-705 
 
 . 6-33 . 
 
 9-20 
 
 $55 
 8-18 
 7-39 
 
 Ijannagh . . . 
 
 Castlebar . . 
 Manulla . . . 
 Robe 
 
 Balla 
 
 Mask and Robe . 
 DaUa . . . . I 
 
 Dalla 
 
 Owenmore . 
 
 I 
 
 the depths passed off, estimated over the catchments, 
 include the flow before the commencement of the 
 rain. If so the results are so far useless ; and if 
 they do not include it, there must be an error 
 somewhere. Indeed, in the Eobe we have evidence 
 that not more than 58 per cent, passed from the 
 catchment to the river, from Mr. Betagh's paper, 
 the results of which are arranged below. Also, in 
 July 1850, it is shown that in the Lannagh district 
 only -53 inch in depth passed off the catchment 
 from a fall of 1'83 inches, or about one-third of the 
 depth. The method of determining this was un- 
 objectionable. Where such discrepancies as above 
 exhibited exist, it is important that the method of 
 gauging, and the whole calculation, should be shown, 
 in order that other engineers should be able to judge 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 301 
 
 of their accuracy ; otherwise the results should be 
 rejected, no matter under whose authority they may 
 be published. 
 
 The following information has been collected and 
 arranged by us from a paper by Mr. Betagh, in the 
 Proceedings of the Institution of Civil Engineers, 
 Ireland, vol. iv. In January 1851, 3*41 inches of 
 rain fell in seven days, producing the maximum dis- 
 charge of 85,836 cubic feet; while in December 
 1852, 3-17 inches, also falling in seven days, pro- 
 duced 115,656 for the maximum. At the beginning 
 of the first fall there was flowing 2 6, 640 feet, leaving 
 the effects of the seven days' rain 85,836 - 26,640 
 zz 5 9,1 96 cubic feet, while in the second year the 
 quantity flowing at first was 75,360 cubic feet, leaving 
 the effects of the seven days' rain-fall equal to 
 115,656 - 75,360 zz 40,296 cubic feet. The effect 
 of the previous state of the weather on the catch- 
 ment must always modify, to a considerable extent, 
 the discharge from a given rain-fall, and this has 
 more to do with the results than the effect of arterial 
 drainage itself, unless so far as one is a result of the 
 other. Taking the mean of 1851 and 1852, it ap- 
 pears that the evaporation in the Ballinrobe catch- 
 ment was to the rain-fall as 41 '6 to 98*7, or about 42 
 per cent. This is certainly, from the nature of the 
 catchment, less than the average through Ireland, 
 which cannot be less than 60 per cent. In high, 
 steep districts, fully three-fourths or 75 per cent, of 
 the rain-fall can be collected, and at times, when the 
 catchment is saturated, nearly the whole ; even in 
 some few limited cases, when springs or hidden 
 
302 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLES showing in detail, for the years 1851 and 1852, the Monthly Fall of Rain and 
 the corresponding Discharge of the River Robe, at Ballinrobe, County Mayo ; the 
 catchment basin being 70,000 acres, or 110 square miles ; the lower end 100 feet, 
 the upper end 336 feet ; and the average height of the surface about 180 feet above 
 the level of the sea. The average fall of the river, not including the rapids, is 
 from one to two feet per mile ; the catchment is about 20 miles long, about one- 
 tenth of the area bog or low marsh, and nine-tenths clayey and gravelly. The 
 river is about 33 miles long. 
 
 OBSERVATIONS IN 1851. 
 
 MONTHS. 
 
 Eain-fall each 
 Month 
 in inches. 
 
 Discharge each 
 Month 
 of rain-fall in 
 inches. 
 
 Discharge in cubic feet per mi- 
 nute, from a catchment of 70,000 
 acres, for each month. 
 
 Discharge in cubic feet per 
 minute, per acre, for each 
 month. 
 
 Maximum 
 
 minimum. 
 
 Average. 
 
 Maximum. 
 
 Minium m . 
 
 Average. 
 
 January 
 
 9.2 
 
 7.4 
 
 85,836 
 
 20,133 
 
 43,373 
 
 1-158 
 
 287 
 
 620 
 
 February . 
 
 6-8 
 
 4.7 
 
 72,448 
 
 18,420 
 
 30,410 
 
 1-034 
 
 263 
 
 434 
 
 March . . 
 
 4.4 
 
 3-6 
 
 49,137 
 
 10,860 
 
 20,945 
 
 702 
 
 155 
 
 300 
 
 April . . 
 
 3-4 
 
 2-5 
 
 24,200 
 
 5,760 
 
 14,355 
 
 345 
 
 082 
 
 205 
 
 May . . . 
 
 1-0 
 
 8 
 
 5,820 
 
 4,125 
 
 5,001 
 
 083 
 
 059 
 
 071 
 
 June . . . 
 
 3-8 
 
 8 
 
 7,040 
 
 1,114 
 
 4,230 
 
 100 
 
 016 
 
 060 
 
 July. . . 
 
 3-8 
 
 5 
 
 4,920 
 
 1,500 
 
 2,558 
 
 070 
 
 021 
 
 036 
 
 August . . 
 
 2-4 
 
 0-9 
 
 17,055 
 
 1,240 
 
 4,866 
 
 243 
 
 017 
 
 069 
 
 September . 
 
 1-9 
 
 0-5 
 
 4,746 
 
 1,200 
 
 2,854 
 
 067 
 
 017 
 
 040 
 
 October . . 
 
 5-0 
 
 1-6 
 
 23,980 
 
 6,940 
 
 12,588 
 
 342 
 
 099 
 
 179 
 
 November . 
 
 1-3 
 
 1-2 
 
 12,852 
 
 6,000 
 
 7,827 
 
 183 
 
 085 
 
 111 
 
 December . 
 Total 
 
 2-6 
 
 2-5 
 
 44,715 
 
 6,210 
 
 14,373 
 
 638 
 
 088 
 
 205 
 
 45-6 
 
 27- 
 
 352,749 
 
 83,502 
 
 163,380 
 
 4-965 
 
 1-189 
 
 2-33 
 
 supplies are re-tapped, a larger discharge may take 
 place than that due to the catchment and rain-fall ; 
 but these do not affect the general question. 
 
 " The future population of the suburbs of London 
 is calculated at 30,000 inhabitants per square mile. 
 According to the following data, some of the densest 
 portions of our large towns have a population of 
 220 persons to an acre. The population on the 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 303 
 
 RIVER ROBE OBSERVATIONS IN 1852. Continued from last page. 
 
 MONTHS. 
 
 Eain-f all each 
 Month 
 in inches. 
 
 Discharge each 
 Month 
 of rain-fall in 
 inches. 
 
 Discharge in cubic feet per mi- 
 nute, from a catchment of 70,000 
 acres, for each month. 
 
 Discharge in cubic feet per 
 minute, per acre, for each 
 month. 
 
 Maximum. 
 
 Minimum . 
 
 Average. 
 
 Maximum. 
 
 Minimum. 
 
 Average. 
 
 January. . 
 
 7-5 
 
 54 
 
 41,600 
 
 12,852 
 
 28,730 
 
 594 
 
 183 
 
 410 
 
 February . 
 
 4-8 
 
 4-3 
 
 56,400 
 
 8,190 
 
 25,296 
 
 805 
 
 117 
 
 361 
 
 March . . 
 
 1-0 
 
 0-7 
 
 9,600 
 
 2,737 
 
 6,702 
 
 137 
 
 039 
 
 095 
 
 April . ' *. 
 
 1-1 
 
 0-5 
 
 3,931 
 
 1,468 
 
 2,477 
 
 056 
 
 020 
 
 035 
 
 May. .' ; . 
 
 1-9 
 
 0-4 
 
 3,931 
 
 1,050 
 
 1,861 
 
 056 
 
 015 
 
 026 
 
 June . . . 
 
 6-6 
 
 1-2 
 
 22,764 
 
 1,400 
 
 6,547 
 
 325 
 
 020 
 
 093 
 
 July. . .. 
 
 2-5 
 
 1-0 
 
 15,439 
 
 3,172 
 
 6,057 
 
 220 
 
 045 
 
 087 
 
 August . . 
 
 4-5 
 
 0-6 
 
 3,856 
 
 2,236 
 
 3,070 
 
 055 
 
 032 
 
 043 
 
 September . 
 
 1-8 
 
 0-5 
 
 3,427 
 
 2,642 
 
 2,874 
 
 048 
 
 037 
 
 041 
 
 October . . 
 
 3-9 
 
 1-0 
 
 32,040 
 
 1,114 
 
 5,932 
 
 457 
 
 016 
 
 084 
 
 November . 
 
 5-5 
 
 5-2 
 
 45,360 
 
 17,000 
 
 30,742 
 
 648 
 
 242 
 
 439 
 
 December . 
 Total 
 
 12-0 
 
 9-5 
 
 115,656 
 
 23,232 
 
 54,846 
 
 1-657 
 
 331 
 
 783 
 
 53-1 
 
 30-1 
 
 354,004 
 
 77,093 
 
 175,134 
 
 5-058 
 
 1-097 
 
 2-497 
 
 north side of the Thames is about 75 persons per 
 acre, and on the south side 28 persons per acre. 
 Taking the average density of population in our 
 twenty-one principal towns, there appear to be 5045 
 inhabitants to the square mile ; but, from the fol- 
 lowing table, extracted from Dr. Duncan's report 
 on Liverpool, it will be seen that if we select five 
 of our most populous cities, the average in these is 
 much greater, while in others, it is equally certain 
 that the crowding is far less than the general stand- 
 ard to which we have referred : 
 
304 THE DISCHARGE OF WATER FROM 
 
 Inhabitants to a Square Mile. 
 
 Towns. Total Area. Builded Area. 
 
 Leeds 20,892 . . 87,256 
 
 London .... 27,423 . . 50,000 
 
 Birmingham . . 33,669 . . 40,000 
 
 Manchester . . . 83,224 . . 100,000 
 
 Liverpool . . . 100,899 . . 138,224 
 
 Dr. Duncan, however, states that there is a district 
 in Liverpool containing 12,000 inhabitants crowded 
 together on a surface of only 105,000 square yards, 
 which gives a ratio of 460,000 inhabitants to the 
 geographical square mile. In the East and West 
 London Unions, Mr. Farr has estimated that there 
 are nearly 243,000 inhabitants to a geographical 
 square mile ; but, great as this overcrowding is, the 
 maximum density of Liverpool exceeds that of the 
 metropolis by nearly double."* 
 
 The amount of sewage from each person is cal- 
 culated about FIVE CUBIC FEET PER PERSON, including 
 the supply from manufactories, breweries, distilleries, 
 &c. SEVEN FEET PER HEAD has been recommended 
 as data to calculate from by Captain Galton, Messrs. 
 Simpson and Blackwell, in their Eeport on the 
 Main Drainage, and it has been found that about 
 half of the estimated quantity of sewage would pass 
 off in six or eight hours. 
 
 In calculating the size of sewers, however, the 
 rain-fall must be provided for, in addition to the 
 sewage matter from houses and public establish- 
 ments. Mr. Bazalgette calculated this for the Lon- 
 don sewerage at ith of an inch fall in 24 hours in 
 the urban districts, and ith of an inch for the 
 suburban districts. Captain Galton and the Messrs. 
 
 * Illustrated News, September 8th, 1855. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 305 
 
 Simpson and Blackwell assumed fths of an inch fall 
 during eight hours' maximum flow. This would be 
 ]452 feet per acre. Assuming the highest data, we 
 shall have to provide sewers to discharge in eight 
 hours 
 
 1,452 cubic feet of rain water per acre, 
 
 3J cubic feet of sewage nearly per person. 
 Assuming a population of 80 persons per acre, then 
 these figures would become 
 
 in eight hours, or 
 1,452 cubic feet for ram, ' 
 
 OOA i,- f 4. f \ at)OUt 3 i cubic f eet 
 
 280 cubic feet for sewage. 
 
 per minute, per acre, 
 
 which shows that the sewage is not more than -Hh 
 of the rain water ; and that, in calculations for the 
 size of sewers, the surface water is the most im- 
 portant element to be considered. If we had as- 
 sumed a larger fall of rain, the difference between 
 sewage and rain would be greater. On the 20th 
 June, 1857, the day after heavy rain, the referees on 
 the Metropolitan Drainage question found the Nor- 
 folk-street sewer to discharge 3 feet ; the Essex- 
 street sewer 5| feet; the Northumberland-street 
 sewer 3J feet ; and the Savoy-street sewer 20J feet 
 per minute per acre ; but the last result has been 
 controverted. 
 
 It appears that the daily amount of sewage 
 varies from 4*8 cubic feet per head in the more 
 thickly inhabited portions of London, occupied by 
 a larger portion of the poorer classes, to 8 cubic feet 
 per head in the western districts, where the value of 
 water is more appreciated, and the cost less a matter 
 of consideration; and the average of the whole 
 
306 THE DISCHARGE OF WATER FROM 
 
 metropolitan districts appears to be 5-8 cubic feet per 
 head per diem. If the day be divided into three periods 
 of eight hours each, the amount of the maximum flow 
 is between nine A.M. and five P.M. and 49 per cent. 
 of the whole, whilst only 18 per cent, flows during 
 the eight hours of minimum flow, which occur be- 
 tween eleven P.M. and seven A.M.* The advantage 
 of storm flows in flushing is shown by the heavy 
 rain which occurred on the 20th of June, causing a 
 flow in the Savoy-street sewer .which was equivalent 
 to 20 times the ordinary flow at the time. This was 
 six times the maximum flow, and although the sewer 
 had been scoured, to a considerable extent, by a heavy 
 fall of rain on the previous night, the sample con- 
 tained more than double the amount of total impurity 
 contained in specimens of ordinary sewage. 
 
 In a town district, such as that drained by the 
 Savoy and Northumberland-street sewers, the quan- 
 tity running off into sewers, within six hours after 
 the fall, varies from 10 to 60 per cent, of the quan- 
 tity fallen. Of the rain during the storm of the 
 20th June, 1857, nearly one inch-and-a-quarter in an 
 hour, 65 per cent., ran off within 15 hours of the 
 fall, viz. : 
 
 46 per cent, in 45 minutes after the rain ceased, 
 14 in the next 6| hours, 
 5 in the next 7i hours. 
 
 In a suburban locality, such as the Counter Creek 
 sewer drain, the quantity reaching the sewers would 
 
 * Metropolitan Main Drainage Keport, pp. 15, 17. 
 
OEIFICES, WEIKS, PIPES, AND KIVEES. 307 
 
 vary from to 30 or 40 per cent, in 24 hours after 
 the rain.* 
 
 In the Holborn and Finsbury divisions Mr. Roe 
 calculated that an 18-inch cylindrical pipe, laid at 
 an inclination of 1 in 80, is sufficient for 20 acres of 
 house-sewage, while a 5-inch pipe, laid at an inclina- 
 tion of 1 in 20, is necessary for 1 acre, and a 3-inch 
 pipe, laid also at I in 20, for j acre. A pipe 30" 
 in diameter, laid with an inclination of 1 in 200, 
 would discharge 1700 cubic feet per minute, and per- 
 fectly drain 200 acres of urban land covered with 
 houses to the extent of 4000 or upwards, and each 
 house having a water supply of 150 gallons per 
 diem. In each of these cases, however, the dis- 
 charge must depend on the head and length of the 
 pipe as well as the inclination at which it is laid. 
 Assuming the inclination of those pipes to corre- 
 spond with the hydraulic inclination, we have calcu- 
 lated their discharging powers with water to be 
 respectively 807, 72, 20, and 1700 cubic feet per 
 minute, the areas to be drained being 20, 1, J, and 
 200 acres. In all calculations of this kind it is 
 necessary , for accuracy, to ascertain not only the max- 
 imum rain-fall per hour, but also the proportions dis- 
 charged per hour, according to the season and district, 
 into the main channel, as well as the junctions or 
 places of arrival. In urban districts, 1500, 2100, 
 and sometimes 3600 cubic feet per hour per acre, 
 have to be discharged after extraordinary rain-falls. 
 These may be taken as maximum results. The 
 
 * Metropolitan Main Drainage Keport, pp. 75, 76. 
 
 x3 
 
308 THE DISCHARGE OF WATER FROM 
 
 gaugings of the Westminster sewers in summer 
 give 53 cubic feet per hour for the urban, and 17 
 feet for the suburban, according to Mr. Hawkins. 
 
 In urban districts, however, a much larger quantity 
 of water is conveyed more rapidly, cceteris paribus, 
 to the mains, than in suburban districts and catch- 
 ment basins generally, in which the maximum dis- 
 charge per acre per hour, even in the steeper and higher 
 districts, seldom exceeds 700 cubic feet, and varies 
 from about 20 cubic feet for the larger and flatter 
 districts upwards. This arises from the impervious 
 nature of the surfaces it falls upon in towns, and the 
 lesser waste in passing to the drains, as well as a 
 large portion of the supply being often artificial. 
 From 70 to 90 cubic feet* per acre per hour, is 
 generally taken for the maximum discharge from the 
 average number of catchment basins ; this is nearly 
 equal to a supply of one-fiftieth part of an inch in 
 depth from the whole area. Thorough-drainage in- 
 creases the supply and discharge. Every catchment 
 basin has, however, its own peculiar data, and a 
 knowledge of these is necessary before we can draw 
 any correct conclusions for new waterworks in connec- 
 tion with it. It may be remarked, however, that any 
 conclusions drawn from experiments on the supply 
 of tributaries, particularly in high districts, are 
 wholly inapplicable to the main channel into which 
 they flow. The flow into tributaries and mountain 
 streams, or rivers, is always more rapid than into 
 
 * Some interesting observations on rain-fall and flood dis- 
 charges are given in the Transactions of the Institution of Civil 
 Engineers, Ireland, for 1851, pp. 19-33, and pp. 44-52. 
 
OBIFICES, WEIES, PIPES, AND RIVERS. 
 
 309 
 
 main channels and rivers in flat districts, and the 
 supply from springs often forms a large portion of 
 the water flowing in them. 
 
 TABLE showing Summer Discharges of some English Rivers, as collected from 
 various authorities, re-arranged, showing to some extent the effect of 
 Springs in supplying Channels in different places. 
 
 NAMES OF RIVEKS. 
 
 1 
 
 Valley. Hill. 
 
 Catchment in 1 
 square miles. 
 
 1 
 B 
 
 11 
 
 Discharge per 1 
 square mile in I 
 cubic feet per 1 
 minute. 
 
 Representing 
 inches of rain-fall I 
 per annum. 
 
 Total average 1 
 rain-fall in inches 1 
 per annum. 
 
 Gade, at Hunton 
 
 
 
 
 
 
 
 Bridge, chalk . . 
 
 150 to 500 
 
 69-5 
 
 2,500 
 
 36-2 
 
 8-19 
 
 
 
 Lea, at Lea Bridge, 
 
 
 
 
 
 
 
 chalk. (Rennie, 
 
 
 
 
 
 
 
 April 1796) . . . 
 
 30 to 600 
 
 570-0 
 
 8,880 
 
 15-58 
 
 3-53 
 
 - 
 
 Loddon, (Feb. 1850,) 
 
 
 
 
 
 
 
 green sand . . . 
 
 110 to 700 
 
 221-8 
 
 3,000 
 
 13-53 
 
 3-01 
 
 25-4 
 
 Medway, driest sea- 
 
 
 
 
 
 
 
 sons,(Rennie!787,) 
 
 
 
 
 
 
 
 clay 
 
 
 481-5 
 
 2,209 
 
 4-59 
 
 1-04 
 
 
 Mimram, atPanshan- 
 
 
 
 
 
 
 
 ger, chalk . . . 
 
 200 to 500 
 
 29-2 
 
 1,500 
 
 51-4 
 
 11-58 
 
 26-6 
 
 Medway, ordinary 
 
 
 
 
 
 
 
 summer run, (Ren- 
 
 
 
 
 
 
 
 nie, 1787,) clay . . 
 
 
 
 481-5 
 
 2,520 
 
 5-23 
 
 2-19 
 
 
 Nene, at Peterbo- 
 
 
 
 
 
 
 
 rough, oolites, Ox- 
 
 
 
 
 
 
 
 ford clay, and lias . 
 
 10 to 600 
 
 620-0 
 
 5,000 
 
 8-45 
 
 1-88 
 
 23-1 
 
 Plym, at Sheepstor, 
 
 
 
 
 
 
 
 granite .... 
 
 800 to 1,500 
 
 7-6 
 
 500 
 
 71-4 
 
 15-10 
 
 45-0 
 
 Severn, at Stone- 
 
 
 
 
 
 
 
 bench, silurian . . 
 
 400 to 2,600 
 
 3,900 
 
 33,111 
 
 8-49 
 
 1-98 
 
 
 
 Thames, at Staines, 
 
 
 
 
 
 
 
 chalk, green sand, 
 
 
 
 
 
 
 
 Oxford clay, oolites, 
 
 
 
 
 
 
 
 &c 
 
 40 to 700 
 
 3,086 
 
 40,000 
 
 12-98 
 
 2-93 
 
 24-5 
 
 Verulam, at Bushey 
 
 
 
 
 
 
 
 Hall, chalk . . . 
 
 150 to 500 
 
 120-8 
 
 1,800 
 
 14-9 
 
 3-37 
 
 
 
 Wandle, below Car- 
 
 
 
 
 
 
 
 shalton, chalk .;-.. 
 
 70 to 350 
 
 41-0 
 
 1,800 
 
 43-9 
 
 9-93 
 
 24-0 
 
 Trent, at its mouth, 
 
 
 
 
 
 
 
 oolites and Oxford 
 
 
 
 
 
 
 
 
 100 to 600 
 
 3,921 
 
 
 
 
 
 
 
 
 
 
 
 
310 THE DISCHARGE OF WATER FROM 
 
 The above information has- been obtained from 
 Mr. Beardmore and Mr. Hughes' books, and from 
 Rennie's reports. The effect of the geology and 
 fissures in the chalk and mountain limestone forma- 
 tions on the springs of a catchment basin, and on 
 maintaining the summer discharge, should be carefully 
 noted as one of the elements entering into catchment 
 basin statistics. Indeed, the maximum and minimum 
 discharges from catchments are of as much impor- 
 tance to the engineer as the averages, and, for many 
 purposes, more important. There were abundant 
 opportunities of acquiring this information for all 
 our Irish rivers, but we are not aware if these were 
 turned to account. 
 
 The effects of evaporation are very variable ; some- 
 times 58 or 60 per cent, of the annual fall is carried 
 off in this way from ordinary flat tillage soils, and 
 other estimates are much higher ; much, however, 
 depends on the soil, subsoil, inclination, stratification, 
 and season. The evaporation from water surfaces 
 exceeds the annual fall in these countries by 
 about one-third ; and that from flat, marsh, and 
 callow lands exceeds the evaporation from ordinary 
 tillage, porous, and high lands. When the flat lands 
 along the banks of rivers extend considerably on 
 both sides, an extra fall is necessary into the main 
 channel, along the normal drains, otherwise such 
 lands must suffer from excessive evaporation as well 
 as floods. Evaporation also varies with the climate, 
 and in this country we may assume that one-third of 
 the whole rain-fall passes on to the sea. 
 
 In a paper in the Journal of the Royal Agricultural 
 
OKIF1CES, WEIRS, PIPES, AND RIVERS. 
 
 311 
 
 Society of England, vol. v, part 1, 1844, Mr. Josiah 
 Parkes shows, that 42 1 per cent, of the whole annual 
 rain of England filters through the soil, and 57| per 
 cent, evaporated, being the mean results of eight 
 years' observations, from 1836 to 1843, both included. 
 The mean evaporation and filtration for each month 
 during this period is shown and arranged by us in 
 the following table : 
 
 MONTHS. 
 
 Total 
 falling. 
 
 Evaporated. 
 
 Remaining. 
 
 Deposited in Tons 
 and Cubic feet per 
 acre. 
 
 Inches. 
 
 Inches. 
 
 per cent. 
 
 Inches. 
 
 per cent. 
 
 Cubic feet 
 
 Tons. 
 
 January .... 
 
 1-847 
 
 540 
 
 29-3 
 
 1-307 
 
 70-7 
 
 4,744 
 
 132 
 
 February . . . 
 
 1-971 
 
 424 
 
 21-6 
 
 1-547 
 
 78-4 
 
 5,616 
 
 156 
 
 March .... 
 April 
 
 1-617 
 1-456 
 1-856 
 2-213 
 
 540 
 1-150 
 1-748 
 2-174 
 
 33-4 
 79-0 
 94-2 
 98-3 
 
 1-077 
 0-306 
 0-108 
 0-039 
 
 66-6 
 21-0 
 
 5-8 
 1-7 
 
 3,910 
 1,111 
 392 
 142 
 
 109 
 39 
 11 
 4 
 
 ]\lav . 
 
 June 
 
 July 
 
 2-287 
 
 2-245 
 
 982 
 
 0-024 
 
 1-8 
 
 87 
 
 2-4 
 
 August .... 
 
 2-427 
 
 2-391 
 
 98-6 
 
 0-036 
 
 1-4 
 
 131 
 
 3-6 
 
 September . . . 
 
 2-639 
 
 2-270 
 
 80-1 
 
 0-369 
 
 13-9 
 
 1,339 
 
 37 
 
 October .... 
 
 2-823 
 
 1-423 
 
 50-5 
 
 1-400 
 
 49-5 
 
 5,082 
 
 141 
 
 November . . . 
 
 3-837 
 
 0-579 
 
 15-1 
 
 3-258 
 
 84-9 
 
 11,826 
 
 328 
 
 December . . . 
 
 1-641 
 
 0-164 
 
 00-0 
 
 1-805 
 
 ico-o 
 
 6,552 
 
 182 
 
 Yearly averages . 
 
 26-614 
 
 15-320 
 
 57-6 
 
 11-294 
 
 42-4 
 
 40,932 
 
 1145 
 
 The maximum quantity, 32-10 inches, fell in 184.1, 
 and the minimum in 1837, 21-10 inches. The maxi- 
 mum quantity which fell in January was 3-95 inches, 
 and the minimum -31 inch; in February 2*85 and 
 1-02 inches; in March 3-65 and 0*34 inches; in 
 April, 2-57 and -34 inches; in May 5-00 and -70 
 
312 
 
 THE DISCHARGE OF WATER FROM 
 
 inches ; in June 3-31 and 1*33 inches ; in July 4-36 
 and 1-30 inches; in August 3-65 and 0-95 inches; in 
 September 4-50 and 0*63 inches ; in October 4-82 
 and 1*41 inches ; in November 5*77 and 2*05 inches; 
 and in December 3-02 and *40 inches. The greatest 
 quantities fall in September, October, and November; 
 and the least in February, March, and April. The 
 general mean fall for England is said to be 3 H inches, 
 and near London 25 inches. 
 
 The amount of rain varies, not only at different 
 places and different elevations, but also at different 
 elevations in the same place. The following table 
 shows the amount of rain collected in each month in 
 1855 at Greenwich observatory, at different elevations: 
 
 MONTH IN 1855. 
 
 Osier's anemo- 
 meter gauge, 
 inches. 
 
 On the roof 
 of the 
 library. 
 
 Cylinder 
 partly sunk in 
 the ground. 
 
 January 
 
 0-2 
 
 1*0 
 
 1 5 
 
 February 
 
 0'2 
 
 1-4 
 
 1-0 
 
 
 0*5 
 
 1'3 
 
 2-0 
 
 AprU... 
 
 0-1 
 
 0-1 
 
 0-1 
 
 May ., 
 
 0-5 
 
 1-5 
 
 1-8 
 
 
 0'5 
 
 0*7 
 
 0'9 
 
 July 
 
 3*1 
 
 4'8 
 
 5.3 
 
 August 
 
 0-6 
 
 0*8 
 
 1-4 
 
 September 
 
 0'8 
 
 1-1 
 
 2-0 
 
 
 2'6 
 
 4*5 
 
 6'2 
 
 
 0'5 
 
 ri 
 
 1*5 
 
 
 0*4 
 
 0-9 
 
 1-1 
 
 
 
 
 
 Totals 
 
 10-0 
 
 19-2 
 
 23-8 
 
 The cylinder gauge was placed 155 feet above the 
 
OKIFICES, WEIES, PIPES, AND EIVERS. 313 
 
 level of the sea; the gauge on the roof of the 
 library 22 feet over the cylinder gauge, and Osier's 
 anemometer gauge 28 feet higher than the gauge on 
 the roof of the library. In the valleys in the lake 
 districts, Westmoreland and Cumberland, the annual 
 fall varies occasionally from 50 to 100 inches, and 
 the maximum fall is said to obtain at about 2000 feet 
 above the level of the sea on high catchments. 
 
 At Ballinrobe, a gauge placed on the church 
 tower, 60 feet above the ground, indicated 42 per cent, 
 less rain than one on the ground ; and another experi- 
 ment with a change of gauges, gave 68 per cent, less 
 at the greater elevation. 
 
 At Kinfauns Castle, Scotland, a gauge 600 feet 
 high on a hill, gave 41 J inches, while one at the base, 
 580 feet lower, gave only 25 \ inches. In Keswick, 
 the fall is 65! inches, and in Carlisle only 30 inches. 
 At Kendall the fall is 60 inches ; at Manchester 33 
 inches ; at Lancaster 45 inches ; at Liverpool 34 inches. 
 
 From the 23rd of February to the 6th of June, 1 860, 
 the rain at Dublin was 8 inches. At the Leefin Moun- 
 tain, which is 2000 feet high, the rain was 13-1 inches. 
 From the 23rd of February to the 9th of July, the 
 rain at Dublin was 10-674 inches ; and at the same 
 time, on the Leefin Mountains (over Ballysmutten), 
 181 inches ; that is, an increase of nearly 80 per cent, 
 in that time. From the 23rd February to the 21st 
 August, inclusive, the rain-fall at Dublin was 17 inches ; 
 at Blessington 21 inches ; at Ballysmutten, on the site 
 of a proposed reservoir, 27 inches. This showed an 
 increase over Dublin of 10 inches. It would appear 
 that from 50 to nearly 80 per cent, more rain fell at 
 Ballysmutten than at Dublin. 
 
314 THE DISCHARGE OF WATER FROM 
 
 Experiments were made at York in 1832, 1833, and 
 1834, for the British Association, with three gauges, 
 the first placed on a large grass plot in the grounds 
 of the Yorkshire Museum ; the second at a higher 
 elevation, 43 feet 8 inches, on the roof of the Mu- 
 seum; and the third on a pole 9 feet above the 
 battlements of the great tower of the Minster, at an 
 elevation over the gauge on the ground of 212 feet 
 10i inches. The quantities received were as follows : 
 
 Average depth for 
 
 Depth for three years. one year. 
 
 First gauge . . 64-430 inches . . 91-477 inches 
 Second gauge. . 52-169 \.-.. 17-389 
 Third gauge . . 38-972 . . 12-991 
 
 Professor Phillips gives the following formula for 
 calculating the difference between the ratios of rain 
 falling on the ground and at any height h in the same 
 place 1 the temperature of the season, and c a co- 
 efficient dependent upon it ; then the difference d is 
 
 > - ' d==ch Tw' 
 
 The mean height at which rain begins to be formed 
 by this formula is 1,747 feet over the ground; and at 
 356 feet high, the depth which falls is one-half of 
 what falls on the ground.* 
 
 A discussion of the mean temperature in connexion 
 with the fall of rain, has been made at Greenwich for 
 the years 1852, 1853, and 1854 ; and at Oxford for 
 the years 1855, 1856, and 1857. The result shows an 
 average of 160-3 rainy days at Greenwich for each 
 year, and 146-6 at Oxford. The difference of the mean 
 temperatures of the day of rain and the day before 
 is less than that of the day of rain and the day after. 
 
 * Vide Civil Engineer and Architect's Journal for 1860, p. 167. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 315 
 
 Mean tempera- Mean tempera- Mean tempera- 
 ture, day ture, day of ture, day 
 before rain. rain. after rain. 
 
 Greenwich observations 49-25 . 49-27 . 48-98 
 Oxford do. 49-50 . 49-63 -V 49-44 
 
 Dividing the winds into two groups, northerly 
 and southerly, the Oxford observations give the di- 
 rection for 218'5 days' fair weather. The wind was 
 northerly for 131-5 days, and southerly for 87 days. 
 For the remaining 146-5 rainy days, the wind was 
 northerly for 64-5 days, and southerly for 82 days. 
 
 SECTION XIII. 
 
 WATER SUPPLY FOR TOWNS. STRENGTH OF PIPES. SEWER- 
 AGE ESTIMATES AND COST. THOROUGH-DRAINAGE. 
 ARTERIAL DRAINAGE. 
 
 SUPPLY. QUALITY. 
 
 The supply of water to towns has become latterly 
 a subject of considerable importance. Three points 
 have to be considered, first, a sufficient supply 
 at high pressure, when it can be obtained within a 
 reasonable expenditure ; secondly, the quality ; and, 
 thirdly, the cost. The advantages in towns of high 
 pressure are now apparent to all in overcoming fire ; 
 fronts of houses and pavements may also be cleaned, 
 and streets watered if the supply be abundant. The 
 highest apartments ean be supplied, and even mecha- 
 nical power can be obtained for many purposes, as 
 grinding coffee, at a reasonable cost. Mr. Glynn 
 says,* " In many parts of London water is supplied 
 at 4< for 1000 gallons, at a pressure of 150 feet : 
 a gallon of water weighs 10 Ibs., so that 1000 
 * Power of Water. WEALE. 
 
316 THE DISCHARGE OF WATER FROM 
 
 gallons of water falling 150 feet, are equal to 
 1,500,000 Ibs. falling one foot ; and if 1500 gal- 
 lons of water be used in one hour, they are equal 
 to 37,500 Ibs. falling one foot in one minute, or 
 somewhat more than a horse's power, which is 33,000; 
 therefore, it may be assumed, that the cost of a horse's 
 power for an hour in such cases, is only 6<" 
 
 The number of gallons of water required for the 
 supply of each person, including all collateral uses, 
 has been differently estimated, and varies in almost 
 every town, and even in the same city London, for 
 instance, when supplied by different companies and 
 under different systems. 44 gallons per head, per 
 diem, were supplied by the several companies of 
 London in 1853, while evidence has been given to 
 show that the actual average consumption for all 
 purposes did not exceed 10 gallons per head, per 
 diem ; the remainder having been wasted under an 
 imperfect system of distribution. It is asserted that 
 when the supply is 25 gallons per head, per diem, 
 that 5 gallons of it are used for purposes requiring 
 nitration, 10 gallons for purposes not requiring fil- 
 tration, and 10 gallons wasted, or two-fifths of the 
 supply. As there must be a considerable, loss under 
 even the best system of supply, we may assume, 
 with the Board of Health, that a minimum supply of 
 75 gallons per house, per diem, or 15 gallons per 
 person, per diem, is necessary. 
 
 The following is an abstract of the average num- 
 ber of gallons of water furnished per diem, by 
 different water companies in London, during the 
 year 1853, to each house, including manufactories 
 and public establishments as houses : 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 317 
 
 New River Company 
 
 Gallons. 
 
 Per House. 
 
 Per Person. 
 
 193 
 
 38-3-5 
 
 East London Water Works . . , . 
 
 187 
 
 37-4-5 
 
 West Middlesex Water Works 
 
 204 
 
 40-5-5 
 
 Grand Junction Water Works 
 
 f319 
 (336 
 
 f 63-4-5 
 1 67-1-5 
 
 Southwark and Vauxhall Companies' Houses 
 
 175 
 
 35 
 
 Ditto average houses, manufactories, public 
 
 209 
 227 
 
 41-4-5 
 45-2-5 
 
 Chelsea Water Works . . . . ..- 
 
 Hampstead Water Works .... 
 
 111 
 
 22-1-5 
 
 Kent Water Works 
 Mean Values . 
 
 270 
 
 55 
 
 2233 
 
 446-3-5 
 
 223-3-10 
 
 44-3-5 
 
 These quantities have been calculated from the par- 
 liamentary returns made in 1854; and if there be 
 any truth in the calculations and returns of the 
 quantities actually consumed per person said to be 
 10 gallons we get the proportion, as 10 is to 34 so is 
 the quantity consumed to the quantity wasted. But, 
 even assuming the quantity consumed to be 20 gal- 
 lons per head, what an immense loss is here exhibited 
 from want of a suitable system of distribution. 
 
 For large towns it is safe to provide for many 
 purposes, besides mere personal or house wants ; and 
 it is safer, where it can be done without much cost, 
 to provide for a supply of 40 gallons to each inha- 
 bitant, even if this quantity shall not be used or 
 
318 THE DISCHARGE OF WATER FROM 
 
 raised. For high pressure, the supply required will 
 generally vary from 15 to 42 gallons, or from 3 to 
 7 cubic feet to each inhabitant, or an average of 
 about 30 gallons, including the supply to stables, 
 offices, manufactories, and breweries. 
 
 The quality of water for drinking, washing, or 
 cooking, is also an important element in selecting 
 a source of supply. Hardness is measured by the 
 number of grains of chalk or carbonate of lime to 
 a gallon of water, each called a degree. The average 
 hardness of spring water is about 26, that is, 26 
 grains of carbonate of lime to one gallon of water. 
 Eivers and brooks have an average hardness of 13, 
 and water derived from surface drainage 5 ; hence 
 the great advantage of the latter kinds of water in 
 washing. The average hardness of the London pipe 
 waters is from 10 to 16. The following report 
 and analyses furnished to me, in 1855, by Professor 
 Sullivan, of the Museum of Irish Industry, Dublin, 
 will show what is generally required on this head : 
 
 " On the annexed page you will find the numerical 
 results of my analyses of the four samples of water 
 which you left with me for examination. From the 
 table you will perceive that the water of the Mattock 
 Eiver appears to be the purest, so far as the nature 
 and the amount of the foreign substances held dis- 
 solved in it is concerned. The water of the Boyne 
 comes next in quality to that of the Mattock River, 
 the pump water being in every sense the worst, so far as 
 amount of ingredients can be taken as a test of the 
 quality of a water ; in this respect, indeed, it resembles 
 the water of the deep wells of London and elsewhere. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 319 
 
 " As the ordinary mode in which the quality of 
 a water, for drinking and for culinary and like pur- 
 poses, is judged of is, by the comparative amount of 
 organic matter, the total amount of dissolved matter, 
 and its hardness, according to the 'soap test,' I 
 shall give in the following table the numbers repre- 
 senting each of these qualities : 
 
 TABLE showing the number of grains of Organic Matter, and the number of 
 grains of Solid Matter, in an imperial gallon of 
 
 Water from 
 
 Number of 
 Grains of 
 Organic Matter, 
 per 
 Imperial Gal. 
 
 Number of 
 Grains of 
 Solid Matter, 
 per 
 Imperial Gal. 
 
 Degree of 
 Hardness 
 according to 
 the 
 Soap Test. 
 
 No. 1. Tullyescar . . . 
 
 8-975 grs. 
 
 31,175 
 
 15 8-10ths. 
 
 2. Eiver Mattock . 
 
 2- (about) 
 
 15,360 
 
 9 l-10th. 
 
 3. Kiver Boyne . . 
 
 3-250 
 
 22,700 
 
 14 9-10ths. 
 
 4. Burn's Pump . . 
 
 7-100 
 
 76,850 
 
 34 4-10ths. 
 
 " In order to render this table more instructive, it 
 may be well to subjoin a few of the results obtained 
 from the analyses of the waters of other localities. 
 
 TABLE showing the number of grains of Solid Matter contained in one gal- 
 lon of the following Water : 
 
 Thames, at Greenwich . *' i*-> . V*? 27-9 grains. 
 
 London .,.*,, . .. . : . .... , . 5 ., .28-0 
 
 Westminster ...._;"..;* ..= * f . 24-4 
 
 Twickenham . . . 22-4 
 
 Teddington .... 17-4 
 New River (London) . . . . 19-2 
 
 Lea ... ;'- 23-7 
 
 Trafalgar Square Fountain, Deep Well . 68-9 
 Well in St. Giles', Holborn . . . 105-0 
 Artesian Well at Grenelle (Paris) . T"; 9-86 
 
 " The following are some of the results obtained 
 from an examination of the waters in the neighbour- 
 
320 
 
 THE DISCHARGE OF WATER FROM 
 
 hood of Dublin, or which have been proposed as a 
 source of supply :* 
 
 Locality from whence Water 
 was obtained. 
 
 Total Number 
 of Grains 
 per Imperial 
 Gallon. 
 
 Total Number 
 of Grains 
 of Organic 
 Matter. 
 
 Degree of Hard- 
 ness according 
 to the Soap 
 
 Test. 
 
 Royal Canal (12th Lock) . . 
 
 21-0 
 
 2-80 
 
 degs. 
 14-0 
 
 Grand Canal (7th Lock) . . 
 
 16-300 
 
 2-30 
 
 10 3-4ths. 
 
 River Liffey, at Kippure . . 
 
 3-525 
 
 1-90 
 
 2-10ths. 
 
 at Phoulaphouca 
 
 5-125 
 
 1-50 
 
 2-10ths. 
 
 Lough Dan, Co. Wicklow . . 
 
 2*800 
 
 1-225 
 
 8-10ths. 
 
 River Dodder, at City Weir . 
 Lough Owel , 
 
 8-350 
 10-225 
 
 1-625 
 1-550 
 
 1 8-10ths. 
 6 7-10ths. 
 
 
 
 
 
 " The quality of a water for drinking purposes 
 depends in a great degree upon the condition in 
 which the organic matter is found, much more than 
 upon its quantity. This is, however, a question 
 outside of the domain of chemistry, and can only 
 be solved by the aid of the microscope. I may, 
 however, venture to remark that the organic matter 
 contained in the water of the Boyne and the Mat- 
 
 * While these pages were passing through the press, Dr. Apjohn 
 gave the following analyses : 
 
 * Total matter Organic 
 
 dissolved, matter. Hardness. 
 
 Grand Canal mean of seven analyses . 20-78 -95 15-9 
 
 Royal Canal mean of five analyses . 20-76 1-64 14-1 
 
 Liffey mean of eleven analyses . . 8*62 1-77 6*1 
 
 Analysis of the deposition on pipes from the Portobello basin : 
 
 Water . . . "..;.' ... ' . 2-20 
 
 Organic Matter '; . ' . ; . , . 9-71 
 
 Sand . . . . 10-20 
 
 Per Oxide of Iron and Alumina . 3-50 
 
 Carbonate of Lime . .... . 74-20 
 
 Carbonate of Magnesia . '..' . -19 
 
 100 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 321 
 
 tock River is of vegetable origin, and would not, 
 so far as I believe, be injurious to health. 
 
 " As a general rule, I believe that the water of 
 clear flowing rivers, even though it may contain a 
 large amount of solid matter, and even of organic 
 matter, will be found wholesomer than well water, 
 especially in towns. 
 
 u For certain manufacturing purposes, and for 
 culinary purposes, too large an amount of lime is 
 injurious, but I believe that a certain quantity pre- 
 sent in water, is not only not injurious, but in my 
 opinion is of the greatest utility, and renders the 
 waters wholesome. I think the rage for extracting 
 pure water containing only one grain of solid matter 
 to the gallon, or thereabouts, for supplying towns, is 
 carried too far. Such water is, no doubt, the best 
 on a hill side ; but, I question whether it is equally 
 well adapted for resting in basins, tanks, pipes, &c., 
 with that containing some lime. The Eiver Dodder 
 and Lough Owel waters appear to me the best 
 adapted for city and town supplies. The River 
 Mattock contains rather more than either, but it 
 is decidedly better than the water of either of the 
 canals from which our Dublin supply is drawn. 
 
 " Drogheda is rather badly situated for a supply 
 of very soft water, as almost the whole drainage 
 basin of the Boyne is either situated upon limestone, 
 or the feeders of that river rise through the cal- 
 careous drift gravel which covers so much of the 
 country. The water of the Boyne appears to be 
 an excellent water for most purposes, and perhaps 
 the difference between it and the Mattock River 
 
322 
 
 THE DISCHARGE OF WATER FROM 
 
 Tabular Results of the Special Analyses of Four Samples of Water 
 from the neighbourhood of Drogheda. 
 
 Nature of dissolved matter. 
 
 No. 1. 
 Tullyescar. 
 
 No. 2. 
 Mattock 
 River. 
 
 No. 8. 
 Boyne 
 River. 
 
 No. 4. 
 Burns's 
 pump water. 
 
 Observations. 
 
 Carbonate of lime . . . 
 Carbonate of magnesia . 
 
 9-350 
 0-429 
 
 7-302 
 0-510 
 
 11-648 
 0-888 
 
 21-475^ 
 0-585J 
 
 (^Inclusive of a very 
 small quantity of 
 -j phosphate of lime 
 and iron not sepa- 
 t^rated from the lime. 
 
 Sulphate of lime . . . 
 
 9-043 
 
 2-514 
 
 4-459 
 
 4-568 
 
 
 Chloride of magnesium . 
 
 0-743 
 
 1-258 
 
 1-685 
 
 8-445 
 
 
 Chloride of calcium . . 
 
 .. 
 
 
 
 9-524 
 
 
 Chloride of soldium . . 
 
 .. 
 
 0-991 
 
 .. 
 
 
 
 Magnesia existing as cre- 
 nate, &c., in the water . 
 
 0*464 
 
 .. 
 
 .. 
 
 .. 
 
 
 Lime do. do. . . 
 
 
 
 
 0-548 
 
 
 Silica do. do. . . 
 
 0-627 
 
 
 0-322 
 
 2-212 
 
 
 Potash and eoda existing 
 in water, as nitrates, 
 crenates, and other or- 
 ganic salts ..... 
 
 1-644 
 
 2-785 
 
 0-448 
 
 22-393 
 
 
 Organic matter . . . . 
 
 Total number of grains 
 per Imperial gallon . . 
 
 8-975 
 
 
 
 8-250 
 
 7-100 
 
 
 31-175 
 
 15-360 
 
 22-700 
 
 76-850 
 
 may in part be accounted for by its being taken 
 near the banks, or more probably, perhaps, because 
 it was above and close to where some small stream 
 entered. 
 
 " The quantity of solid matter in it, however, was 
 not more than I would expect considering the nature 
 of the locality. I did not draw attention in my 
 Report to a point of some importance namely, the 
 proportion of lime and magnesia existing as car- 
 bonates, and as sulphates, and chlorides. The whole 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 323 
 
 of the lime and magnesia existing as carbonates, and 
 as sulphates, and chlorides, is precipitated by boil- 
 ing, the water being thus proportionably rendered 
 less hard ; lime and magnesia existing as sulphates 
 or chlorides, on the other hand, are not precipitated. 
 This difference is of great consequence in culinary 
 operations, as where boiled water is used, the car- 
 bonates of lime and magnesia are not injurious, and 
 if no sulphates or chlorides be present, the water 
 may be soft after boiling. The same observation 
 applies to water applied to washing clothes when 
 boiled. And lastly, sulphate of lime forms one of 
 the worst elements of fur or deposits upon steam 
 boilers." 
 
 The saving in soap effected by a reduction of 10 
 degrees in hardness, is found to be over 50 per cent. 
 Some of the metropolitan waters analyzed by Dr. 
 Eobert Dundas Thomson, F.R.S., were found, in 
 May 1860, much more impure than others, the sam- 
 ples of which had been taken at the beginning of 
 the month, before the impurities conveyed by the 
 rains had contaminated them. The supply afforded 
 by large and small rivers, as in London, in this 
 table, contrasts most unfavourably with that afforded 
 by the drainage of mountain ridges, as at Glasgow 
 and Manchester. The specimens of water from the 
 two latter cities were taken by the instructions of 
 Mr. Bateman, F.R.S., the engineer, from the main 
 pipes during the month. It should be the object of 
 the London Companies to avoid pumping the water 
 in its most impure state, and to store it when in the 
 condition of the greatest purity. 
 
 Y3 
 
324 
 
 THE DISCHARGE OP WATER FROM 
 
 
 Total 
 Impurity 
 per gallon. 
 
 Organic 
 Impurity 
 per gallon. 
 
 
 Grs., or . 
 
 o-o 
 
 Grs., or . 
 
 o-o 
 
 Loch Katrine water, new supply to Glasgow 
 
 3-16 
 
 0-96 
 
 Manchester water supply .... 
 
 4-32 
 
 0-64 
 
 THAMES COMPANIES : Chelsea . 
 
 17-84 
 17-08 
 
 1-48 
 1-64 
 
 Grand Junction 
 
 20-72 
 
 2-00 
 
 West Middlesex 
 
 20-08 
 20-80 
 
 2-08 
 2'40 
 
 OTHER COMPANIES": New Kiver 
 
 18-52 
 23-64 
 
 1-56 
 3-20 
 
 
 21-68 
 
 2-96 
 
 
 
 
 The table is read thus : Loch Katrine water con- 
 tains in the gallon 3-16 degrees or grains of foreign 
 matter in solution, of which *96 degrees or grains 
 are of vegetable or animal origin. 
 
 Professor Apjohn gives the following analyses of 
 waters furnished to the city of Dublin in 1860. It 
 shows how necessary it is to distinguish the time of 
 taking specimens for analysis, and the previous state 
 of the weather as affecting the foreign matters in the 
 water. The specimens were collected on the 5th and 
 19th of May, 1860. The quantity operated upon in 
 each instance was an imperial gallon, or 277-273 
 cubic inches : 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 325 
 
 Carbonate of lime . . . 
 Carbonate of magnesia . . j 
 Sulphate of lime and chlo- 
 rides of sodium and mag- 
 
 CITY WATER COURSE, DODDER. 
 5th May. 
 
 1 4-056 
 
 19th May. 
 
 7-308^1 
 
 2-269 
 
 nesium 
 
 Silex 0-166 
 
 Organic matter 1*811 
 
 8-302 
 
 0-700 
 
 2-171 
 
 0-526 
 
 Specific gravity of 
 specimen (5th May) 
 
 1-00011. 
 
 Specific gravity of 
 specimen (19th 
 May) 1-00014. 
 
 11-086 
 
 PORTOBELLO BASIN. 
 
 7-687 
 
 Carbonate of lime 
 Carbonate of magnesia 
 Sulphate of lime and chlo- ) 
 rides of sodium and mag- j- 4-058 
 
 nesium j 
 
 Silex 0-073 
 
 Organic matter 3-308 
 
 15-126 
 
 11-660- 
 0-764 
 
 3-751 
 
 0-194 
 
 2-289 
 
 18-658 
 
 Specific gravity of 
 specimen (5th May) 
 
 1-00023. 
 
 Specific gravity of 
 specimen (19th 
 May) 1-00031. 
 
 It will be observed that the quantities of saline 
 and other ingredients found in specimens of same 
 water collected at the two separate periods above 
 mentioned are materially different; those obtained 
 at the later date (May 19) containing the larger 
 portion of foreign matters. The extent of this 
 variation is very considerable, and it appears to Dr. 
 Apjohn to have been the consequence of a very con- 
 siderable fall of rain, which took place in the in- 
 terval between the periods at which the specimens 
 were taken up for analysis. 
 
 When the means of the preceding analyses are 
 taken, we obtain the following results : 
 
 City Water Course. Portobello Basin. 
 
 Mean amount of saline matter . 8-598 14-094 
 
 organic matter . 1-456 2*798 
 
326 THE DISCHARGE OF WATER FROM 
 
 SOURCES AND GATHERING GROUNDS. 
 
 The sources from which a water supply for towns 
 may be derived are lakes, rivers, and streams, 
 springs, wells, and gathering grounds. Of the latter 
 it may be said that, however ably put forward under 
 the auspices of the Board of Health, it is far safer 
 to resort to good river waters than trust to what 
 has been termed, with some satirical truth, "new 
 fangled schemes of pot-piped gathering grounds." 
 Springs and wells afford, at best, but a partial sup- 
 ply unless for villages or manufactories; and we 
 must almost always trust to lakes, rivers, or streams, 
 with sometimes reservoirs, for stowage, for a suffi- 
 cient supply for large towns. The Grot on aqueduct, 
 conveying water with an average of three degrees of 
 hardness, to New York, is perhaps the noblest work 
 for water supply of modern times. The length of 
 the aqueduct is about 44 miles, with a channel in- 
 clination of about 15 inches per mile. The receiving 
 reservoir is about two miles higher up the channel 
 than the distributing reservoir, which latter is 115 
 feet over the level of the sea, and commands the 
 highest buildings of the city. In the driest weather 
 the supply is equal to 28,000,000 gallons.* The cost 
 of the work, including the purchase of land and 
 water rights, was 8,575,000 dollars, or 8 per lineal 
 foot nearly. The cost of distributing pipes was 
 1,800,000 dollars. Latterly we have had the Loch 
 Katrine and Glasgow aqueduct, also a noble work, 
 
 * Schrainke's Croton Aqueduct, New York. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 327 
 
 constructed after this model by Mr. Bateman, not- 
 withstanding the previous supply of that city, or a 
 portion of it, the G-orbals, from gathering grounds at 
 a high level. It is, however, sometimes necessary 
 to make use of such grounds, particularly when 
 
 TABLE showing the Quantities of Gathering Ground and Reservoir Room to 
 supply a given population with 15, 30, and 40 gallons of water per head 
 per diem. The reservoir room is calculated to hold 12 inches in depth 
 of rain-fall per mile as a guide for lesser depths. For 4 inches the 
 results are to be divided by 3; and for 6 inches by 2. 
 
 ttd 
 
 *4 
 
 SB 
 
 S.g-1 
 
 1 
 
 *|^* S 
 
 f*J 
 
 JH 
 
 si 
 
 it 
 
 *i 
 
 11 a 
 
 li 
 
 Ifftl 
 
 |P| 
 
 <B 
 
 If 
 
 ft 
 
 |4* 
 
 i 
 
 flijfj 
 
 flf| 
 
 II 
 
 It 
 
 il 
 
 ||f|l 
 
 If 
 
 ||^|| 
 
 l||!-' 
 
 
 
 
 9 
 
 o m 
 
 -w S i 
 
 ^"2 
 
 tc'3 it, *< 
 
 ^*" <H 
 
 It 
 
 si 
 ti 
 
 ft 
 
 k!l 
 
 || 
 
 f|f|| 
 
 ||*t 
 
 2a 
 
 (SS 
 
 &$ 
 
 SJflt 
 
 1" 
 
 llsls 
 
 iiii 
 
 2,500 
 
 1,250 
 
 937 
 
 4-179 
 
 0375 
 
 0789 
 
 2-196 
 
 5,000 
 
 2,500 
 
 1,875 
 
 8-358 
 
 075 
 
 1577 
 
 4-393 
 
 7,500 
 
 3,750 
 
 2,812 
 
 12-536 
 
 1125 
 
 2366 
 
 6-589 
 
 10.000 
 
 5,000 
 
 3,750 
 
 16-715 
 
 16 
 
 3154 
 
 8-786 
 
 12,500 
 
 6,250 
 
 4,687 
 
 20-894 
 
 1875 
 
 3942 
 
 10-982 
 
 15,000 
 
 7,500 
 
 5,625 
 
 25-072 
 
 225 
 
 4731 
 
 13-179 
 
 17,500 
 
 8,750 
 
 6,562 
 
 29-251 
 
 2625 
 
 5519 
 
 15-375 
 
 20,000 
 
 10,000 
 
 7,500 
 
 33-430 
 
 800 
 
 6308 
 
 17-572 
 
 25,000 
 
 12,500 
 
 9,375 
 
 41-788 
 
 375 
 
 7885 
 
 21-965 
 
 30,000 
 
 15,000 
 
 11,250 
 
 60-145 
 
 45 
 
 9462 
 
 26-358 
 
 35,000 
 
 17,500 
 
 13,125 
 
 58-5 
 
 525 
 
 1-1039 
 
 30-75 
 
 40,000 
 
 20,000 
 
 15,000 
 
 66-9 
 
 6 
 
 1-2616 
 
 35-144 
 
 45,000 
 
 22,500 
 
 16,875 
 
 75-217 
 
 675 
 
 1-4193 
 
 39-537 
 
 50,000 
 
 25,000 
 
 18,750 
 
 83-57 
 
 75 
 
 1-577 
 
 43-93 
 
 55,000 
 
 27,500 
 
 20,625 
 
 91-932 
 
 825 
 
 1-734 
 
 48-32 
 
 60,000 
 
 30,000 
 
 22,500 
 
 100-29 
 
 9 
 
 1-8924 
 
 52-716 
 
 65,000 
 
 32,500 
 
 24,375 
 
 108-65 
 
 975 
 
 2-0501 
 
 57-109 
 
 70,000 
 
 35,000 
 
 26,250 
 
 117- 
 
 1-05 
 
 2-2078 
 
 61-502 
 
 75,000 
 
 37,500 
 
 28,125 
 
 125-36 
 
 1-125 
 
 2-3655 
 
 65-895 
 
 80,000 
 
 40,000 
 
 30,000 
 
 133-72 
 
 1-2 
 
 2-5232 
 
 70-288 
 
 85,000 
 
 42,500 
 
 31,875 
 
 142-1 
 
 1-275 
 
 2-6809 
 
 74-681 
 
 90,000 
 
 45,000 
 
 33,750 
 
 150-435 
 
 1-36 
 
 2-8386 
 
 79-074 
 
 95,000 
 
 47,500 
 
 35,625 
 
 158-8 
 
 1-425 
 
 2-970 
 
 83-467 
 
 100,000 
 
 50,000 
 
 37,500 
 
 167-16 
 
 1-5 
 
 3-154 
 
 87-86 
 
 105,000 
 
 52,500 
 
 39,375 
 
 175-5 
 
 1-57 
 
 3-311 
 
 92-25 
 
 110,000 
 
 55,000 
 
 41,250 
 
 183-86 
 
 1-65 
 
 3-469 
 
 96-64 
 
 115,000 
 
 57,500 
 
 43,125 
 
 192-22 
 
 1-72 
 
 3-62 
 
 01-10 
 
 120,000 
 
 60,000 
 
 46,000 
 
 200-58 
 
 1-8 
 
 3-785 
 
 05-43 
 
328 THE DISCHARGE OF WATER FROM 
 
 flanking or lying above glens where an embankment 
 may be easily thrown across, and the supply stored 
 for use, which would otherwise pass quickly off. 
 The table, page 327, gives the areas of reservoirs 
 and gathering grounds according to a collection of 
 one foot in depth from the catchment ; it can be 
 easily modified when the storeage or required supply 
 exceeds or falls short of this depth. One acre of 
 gathering ground with a collection of twelve inches of 
 rain-fall from it annually will give a daily supply of 
 Jive cubic feet per head to twenty-four inhabitants. 
 
 The next table will be of use in showing the 
 actual quantities which have been collected, or could 
 have been collected, for storeage. Hbmersham, 
 Hughes, and Beardmore's books have been consulted 
 in arranging it. 
 
 The various methods employed for purification may 
 be classed under three heads : mechanical, by filtering 
 or straining ; chemical, or antiseptic media, such as 
 peat and animal charcoal, and precipitation by the 
 use of lime water ; and the natural precipitation of 
 impurities when the water is at rest, as well as the 
 purification which takes place from oxidation and 
 neutralization on thorough exposure by the ozone of 
 the atmosphere. This latter plan has, however, been 
 tried, and signally failed. Filter beds may be con- 
 structed to have a surface area of one square yard 
 to every 1,000 gallons to be filtered in twenty-four 
 hours. For executed works the proportions vary 
 from 1 in 460 to 1 in 1140. The cost of filtering, 
 in capital, may be said to vary from 30 to 70 for 
 each million of gallons, or 30*. to 70s. annually. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 329 
 
 TABLE showing information with reference to size of Reservoirs, Catchment Areas, 
 (&c., collected and arranged from various authorities. The first, fifth, and sixth 
 columns contain information with reference to reservoirs and the collecting areas; 
 the second, third, and fourth, show for different districts the whole rain-fall, and 
 the portions or per centage flowing off and available. 
 
 Names of Drainage Areas 
 and names of Reservoirs. 
 
 a 
 '* 
 1 
 
 If 
 
 Rain-fall in 
 inches per 
 annum. 
 
 IP 
 
 *ss 
 
 M 
 
 rS g S 
 ||| 
 
 P.Sto 
 
 ! 
 
 933 . 
 
 tJD > CJ 
 
 1a!l 
 
 Si 1 
 
 Reservoir room 
 per square mile 
 in millions of cu- 
 bic feet of water. 
 
 Contents of 
 Reservoir in mil- 1 
 lions of cubic 1 
 feet. 
 
 Ashton 
 
 59 
 
 40-0 
 
 15-5 
 
 39 
 
 21-0 
 
 12 
 
 Albany Works, U.S 
 
 29 
 
 
 
 
 1-1 
 
 32 
 
 Ballinrobe, Ireland 
 
 11-0 
 
 49-3 
 
 28-5 
 
 58 
 
 
 
 Belmont (moorland, mean 
 
 
 
 
 
 
 
 of four years) 
 
 2-81 
 
 54-5 
 
 39-6 
 
 72 
 
 26-8 
 
 75 
 
 Bolton 
 
 80 
 
 
 
 
 25-6 
 
 20 
 
 Bute (low country) 
 
 
 45-4 
 
 23-9 
 
 53 
 
 
 
 Bateman's Evidence on the 
 
 
 
 
 
 
 
 drainage area of Long- 
 
 
 
 
 
 
 
 dendale : 
 
 
 
 
 
 
 
 First half of 1845, very dry 
 
 . . 
 
 21-2 
 
 13-5 
 
 64 
 
 .. 
 
 .. 
 
 Second half of 1845 
 
 
 38-6 
 
 27-25 
 
 71 
 
 
 
 First half of 1846 
 
 
 22-5 
 
 17-5 
 
 78 
 
 
 
 Oct., Nov., and Dec., 1846 
 
 
 
 10-2 
 
 8-67 
 
 85 
 
 
 
 
 
 Bann Reservoir (moorland 
 
 .. 
 
 72- 
 
 48-0 
 
 66 
 
 
 
 Drainage areas on south 
 side of Longridge Fell, 
 near Preston, May 1852, 
 
 i :: 
 
 54-' 
 
 15-5 
 18-0 
 oo.ft 
 
 29 
 33 
 40 
 
 
 
 
 
 to April 1853 
 
 / 
 
 
 
 A/Si U 
 
 4bO 
 
 r 
 
 
 
 DilworthReservoir of Pres 
 
 
 
 
 
 
 
 ton Works, Lancashire 
 
 092 
 
 
 
 .. 
 
 .. 
 
 54-0 
 
 5 
 
 
 6-00 
 
 37-0 
 
 22-3 
 
 60 
 
 7-66 
 
 46 
 
 Grreenock 
 
 7-88 
 
 60-0 
 
 41-0 
 
 68 
 
 38* 
 
 300 
 
 Homersham's estimate of 
 
 
 
 
 
 
 
 24,000 cubic feet of Re- 
 
 
 
 
 
 
 
 servoir to each acre oi 
 
 
 
 
 
 
 
 drainage 
 
 1 
 
 
 
 
 15-36 
 
 15-36 
 
 Longdendale 
 
 23-8 
 
 
 
 
 12'3 
 
 292 
 
 Proposed Reservoir for 
 
 
 
 
 
 
 
 Wolverhampton Works 
 
 22- 
 
 
 
 
 
 .. 
 
 7 
 
 16 
 
 Rivington Pike 
 
 16-25 
 
 55-5 
 
 24'25 
 
 44 
 
 29-6 
 
 481 
 
 Sheffield 
 
 1-42 
 
 
 
 
 36"5 
 
 52 
 
 Turton and Entwistle . 
 
 3-18 
 
 46-2 
 
 41-0 
 
 89 
 
 31-43 
 
 100 
 
330 
 
 THE DISCHARGE OF WATER FROM 
 
 COST. 
 
 With reference to cost, the following tables, ar- 
 ranged by us from various sources, will afford in- 
 formation from works executed. 
 
 The actual cost of all works for house service 
 varies very much in different towns, and with the 
 quantities supplied, from a general average of Id. 
 per house per week, to 2d., and from an annual 
 rate of 9d. in the pound to Is. 6<, and higher. 
 The cost of raising and supplying 1000 gallons 
 from a height of 135 feet in Nottingham, is said to 
 be 3d., and the charge for house service to vary 
 from 5s. to 60s. annually. In Eugby, the average 
 cost per house is 19s. per year, 4&d. per week, or an 
 annual charge of 3s. 3d. per year, or ^d. per week 
 per head of the population, and for a bare supply 
 of 13 gallons. In Croydon, for a supply of only 
 14 gallons per head, the cost of works varied from 
 lid. to 2%d. per house per week. The parliamentary 
 returns, showing the number of houses supplied, 
 and cost of supply, by different water companies of 
 London, in 1834, give the following results : 
 
 COMPANIES. 
 
 Number 
 of 
 Houses. 
 
 Daily 
 average 
 Supply in 
 Gallons. 
 
 Height of 
 Supply 
 over 
 Thames. 
 
 Amount of 
 charge per 
 Company. 
 
 
 73,212 
 
 241 
 
 145 
 
 s. d. 
 166 
 
 Chelsea . 
 
 13891 
 
 168 
 
 135 
 
 1 13 3 
 
 West Middlesex 
 
 16000 
 
 185 
 
 155 
 
 2 16 10 
 
 Grand. Junction 
 
 11,140 
 
 350 
 
 152 
 
 286 
 
 
 46,421 
 
 120 
 
 107 
 
 129 
 
 South London 
 
 12,046 
 
 100 
 
 80 
 
 15 
 
 Lambeth 
 
 16,682 
 
 124 
 
 185 
 
 17 
 
 
 7,100 
 
 156 
 
 60 
 
 113 
 
 
 
 
 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 331 
 
 Cost of house apparatus for private supply from 
 street mains, as averaged by the Board of Health, 
 for first-rate houses, is 3 13s. Id. ; second-rate 
 houses, 2 18s. 6d. ; third-rate, 2 3s. 3d. ; fourth- 
 rate and cottages, 17s. 5d. ; average cost for houses 
 and cottages, 2 8s. Id. 
 
 The actual cost of private works to take water 
 from mains for the supply of cottages is shown in 
 the following table : 
 
 Work 
 executed in 
 
 NAME OF PLACE. 
 
 Mean Expense 
 of Private 
 Works for 
 each Cottage. 
 
 Annual Value 
 of 
 each Cottage. 
 
 Jan. 1852 
 Mar.1852 
 1852 
 Aug.1852 
 Mean valu< 
 
 Rugby, mean of 6 Cottages 
 
 s. d. 
 1 12 11 
 
 200 
 1 18 li 
 2 11 10 
 
 s. d. 
 5 10 
 
 400 
 326 
 10 
 
 
 Barnard Castle 11 
 
 Tottenham .... 6 .... 
 
 3s for each Cottage 
 
 209 
 
 5 13 1J 
 
 
 The water rate charged by the Local Board at 
 Tottenham, is given as follows : 
 
 
 In the Special District Rate 
 Assessment. 
 
 Water Bate 
 per week. 
 
 Water Rate 
 per annum. 
 
 Above 
 
 And not 
 exceeding 
 
 On Premises 
 assessed. 
 
 s. d. 
 
 
 
 8. d. 
 
 10 
 
 8. d. 
 
 
 8. d. 
 026 
 
 
 
 10 
 
 15 
 
 n 
 
 039 
 
 
 
 15 
 
 20 
 
 H 
 
 050 
 
 n 
 
 20 
 
 25 
 
 n 
 
 063 
 
 tt 
 
 25 
 
 30 
 
 n 
 
 080 
 
 n 
 
 30 
 
 40 
 
 H 
 
 11 
 
 
 
 40 
 
 50 
 
 
 
 14 
 
 and 3s. for every additional rate of 10. 
 
332 
 
 THE DISCHARGE OF WATER FROM 
 
 p ^ 
 
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 H 
 
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 papuadxg ^ 
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 ^ ^ 
 
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 saiqB, S 
 
 03 
 
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 rH 
 
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 ^ 
 
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 rH 
 
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 sasnoH 
 
 BSBIQ pug 
 
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 <* * 
 
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 rH 
 
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 rH 
 
 
 
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 g 
 
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 sraooa 91 
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 rH 
 
 
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 00 
 
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 SUTUIB^UOO 
 
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 ^ 
 
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 ^ <* 
 
 
 
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 sraoog OT 
 Smure^uoo 
 
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 sraoojj g 
 sasnoH 
 
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 rH 
 
 Surarajuoo 
 
 898UOH 
 
 l>i O O 
 
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 rH 
 ^ rH rH 
 
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 rH 
 
 
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 rH 
 
 rH 1-1 rH 
 
 
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 B98UOH 
 
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 ^ rH rH 
 
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 0* O 
 
 rH rH 
 
 rH TH rH 
 
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 00 
 
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 sraooa e 
 
 CO 
 
 rH 
 
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 00 
 l-H t- O 
 
 co 
 
 Smuro^uoo 
 
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 fc. CO ' ' 
 
 rH 
 
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 m 
 1 
 
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 d 
 
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 lal Charge 
 
 COMPAQ 
 
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 Southwark & 
 
 1 
 
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 rl 6 M W 
 
 Average Annt 
 
 ON | 
 
 rH (M 
 
 CO 
 
 2 
 
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ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 333 
 
 PUBLIC WORKS OF WATER SUPPLY, PRESTON. 
 
 Yards. Cost of Pipes. s. d. 
 
 44 of 2-in. iron pipes, including valves, 
 fire-plugs, outlet-pipes, and all 
 
 appurtenances, at Is. 7^. ... 398 
 
 1,496 of 3-in. ditto, at 3s. 4<Z 249 6 8 
 
 321 of 4-in. ditto, at 4s. 9d 76 4 9 
 
 625 of 5-in. ditto, at 6s 187 10 
 
 30 of 9-in. ditto, at 
 
 14 5 
 
 2,516 
 
 530 16 1 
 
 Water Supply and its Cost for some Cities and Towns, from a Paper read to 
 the British Association at Leeds, in 1858, by Dr. Strang, of Glasgow. 
 Vide Builder, for 1858, p. 653. 
 
 TOWNS. 
 
 Population 
 within 
 bounds of 
 Supply. 
 
 Daily Supply. 
 
 Daily 
 Supply 
 'or each 
 inhabi- 
 tant. 
 
 Cost of 
 undertaking. 
 
 Daily 
 Supply 
 for every 
 1 ex- 
 pended. 
 
 Prospective 
 Supply daily 
 in addition. 
 
 London .... 
 Paris 
 
 2,667,917 
 1,100,000 
 
 Gallons. 
 81,025,842 
 
 26,350,000 
 
 Gallons. 
 
 30-3 
 
 24- 
 
 
 
 7,102,823 
 
 800,000 
 
 Gallons. 
 11-4 
 
 33- 
 
 Gallons. 
 
 20,000,000 
 
 Hamburgh . 
 
 160,000 
 
 5,000,000 
 
 31-25 
 
 170,000 
 
 29-50 
 
 
 New York . . 
 
 713,000 
 
 28,000,000 
 
 39-27 
 
 1,800,000 
 
 15-5 
 
 .. 
 
 Manchester . 
 
 500,000 
 
 11,000,000 
 
 22- 
 
 1,300,000 
 
 8-5 
 
 14,000,000 
 
 Liverpool . . 
 Leeds 
 
 500,000 
 153,000 
 
 11,000,000 
 1,850,000 
 
 22- 
 12- 
 
 1,640,000 
 
 283,871 
 
 7- 
 7- 
 
 
 
 Edinburgh . 
 
 215,000 
 
 4,800,000 
 
 22-3 
 
 456,000 
 
 10-5 
 
 2,000,000 
 
 Aberdeen . . 
 
 65,000 
 
 1,200,000 
 
 18-4 
 
 50,000 
 
 24- 
 
 .. 
 
 Dundee .... 
 
 96,000 
 
 1,750,000 
 
 18-2 
 
 139,000 
 
 12-5 
 
 .. 
 
 Greenock . . 
 
 40,000 
 
 2,112,500 
 
 52-8 
 
 90,000 
 
 23-4 
 
 .. 
 
 Paisley 
 
 48,000 
 
 1,021,452 
 
 21- 
 
 60,000 
 
 17- 
 
 
 
 Glasgow 
 
 420,000 
 
 16,710,000 
 
 39-8 
 
 651,199 
 
 26- 
 
 20,000,000 
 
 The cost of pumping varies with circumstances ; 
 we believe that pumping engines cannot be put down at 
 less than from 60 to 100 per horse power, dependent 
 
334 
 
 THE DISCHARGE OF WATER FROM 
 
 on the size of the engine, although the Board of 
 Health adopted a standard of .50 per horse power. 
 For the town of Drogheda we estimated for two 
 engines at 75 per horse power. The following in- 
 formation has been furnished to us respecting the 
 cost of the Waterworks, Cork, by Sir John Benson 
 the engineer, who designed and carried out the works. 
 
 CORK WATER WORKS. 
 
 Steam engine 100-horse 
 power. 
 
 Iwo 50-horse power Tur- 
 bines. 
 
 Reservoirs 
 One of 3,500,000 gallons. 
 One of 563,000 gallons. 
 
 Cost per head. 
 
 Valuation standard per 
 pound on the valuation. 
 
 Yearly cost per five in- 
 habitants. 
 
 Water supplied. 
 
 Direct acting Cornish Engine with 
 three cylindrical flue boilers, includ- 
 ing engine and boiler house, setting 
 boilers, chimnies, &c., &c., per horse 
 power 
 
 Two Turbines completed with four 11 
 in. ram pumps on each, including 
 buildings, cisterns, sluices, gates, 
 screens, per horse power 
 
 One reservoir on a levelof 186 feet over 
 
 weir 
 One reservoir on a level of 360 feet over 
 
 weir 
 
 The inhabitants in 1851, 86,000. , 
 The inhabitants in 1861, 100,000. 
 City valuation, 112,000 , 
 
 Distribution per house of every five 
 persons 
 
 Quantity supplied, including manufac- 
 tories, to one person per day 
 
 s. d. 
 55 
 
 44 
 4,900 
 
 15 3 
 13 
 11 7 
 
 050 
 
 30 gallons. 
 
 The total estimated cost of engines, including 
 pumps, engine houses, wells, &c., for raising the 
 London sewage, is 70 per horse power, and the 
 annual cost 20 per horse power. * 
 
 When coals are 10*. per ton, the cost of an engine 
 
 * Main Drainage Eeport, 1857, p. 29. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 335 
 
 exceeding 100-horse power, single acting Cornish, 
 working night and day, will be 10 per horse power; 
 when coals are 15s. per ton, the cost would be 13 
 per horse power ; when coals are 20s. per ton, the 
 cost would be 16 per horse power; when coals are 
 25,9. per ton, the cost would be 19 per horse power. 
 These estimates have been given by Mr. Hughes, and 
 include every expense of coals, wages, oil, tallow, 
 materials for packing, cleaning, &c., but none for 
 interest of capital or depreciation of machinery.* 
 
 At Ely the cost of pumping is stated by a writer 
 in the Builder to be as follows : 
 
 To pump one million gallons 140 feet high, the old engine 
 consumes s. d. 
 
 Four tons of coal, at 16s. per ton 340 
 
 Oil, tallow, and packing 12 
 
 090 
 
 Total cost of pumping one million gallons 4 5 
 
 which gives Id. per 1,000 gallons pumped 140 feet 
 high (not a very high price). 
 The new engine requires 
 
 Five and a half tons of coal at 16s 480 
 
 Oil, tallow, and packing 1 10 
 
 Wages 120 
 
 Total cost of pumping one million gallons 140 feet high 700 
 which is 65 per cent, more money than the old 
 engine requires. 
 
 While another writer in the same periodical states, 
 that the cost of pumping 1,000,000 gallons with the 
 old engine was 4 18s. 8|^., and with the new 
 engine, 4 10s. Id. In the following table, arranged 
 from information in Mr. Hughes' book,f the estimated 
 cost of pumping engines for various works, English 
 and American, is given : 
 * Main Drainage Keport, 1857, p. 447. f WEALE, London. 
 
336 
 
 THE DISCHARGE OF WATER FROM 
 
 ESTIMATED COST OF ENGINES FOR PUMPING, AND OTHER INFORMATION. 
 
 Description of Works, and Engineers' estimates, 
 with short description of engines. 
 
 Millions of Gal- 1 
 Ions daily. 
 
 Height to be 
 raised in feet. 1 
 
 "o <u -u 
 
 cc- ^ 
 
 J3.a 
 11- 
 
 Estimated cost in 
 s. 
 
 Work to be done 1 
 in units of 33,000 Ibs.j 
 raised 1 ft. high perl 
 minute, or horse I 
 power. 
 
 Cost for each unit 
 of 33,000 Ibs. raised 
 1 foot high per mi- 
 nute, or horse 
 power, in a. 
 
 Daglish's estimate for Liverpool Works, exclusive 
 of buildings one engine and pumps 
 
 2 
 
 800 
 
 600 
 
 8470 
 
 126-26 
 
 67-1 
 
 Ditto Two engines and pumps 
 
 2 
 
 300 
 
 600 
 
 9850 
 
 126-26 
 
 78-0 
 
 Ditto For one engine and pump 
 
 1 
 
 75 
 
 75 
 
 3400 
 
 15-78 
 
 215-4 
 
 Homersham, for Watford supply 
 
 8 
 
 174 
 
 1392 
 
 85,357 
 
 292-92 
 
 120-7 
 
 Forester & Co.'s estimate for Liverpool engines, 
 one or two, exclusive of buildings 
 
 2 
 
 300 
 
 600 
 
 9700 
 
 126-26 
 
 76-8 
 
 Ditto ditto ditto 
 
 1 
 
 75 
 
 75 
 
 3200 
 
 15-78 
 
 202-8 
 
 Hurry and Bateman's estimate, Wolverhampton 
 Water Works 
 
 w 
 
 Hit 
 
 314\ 
 264) 
 
 471) 
 1188 J 
 
 85,400 
 
 349-11 
 
 101-4 
 
 Harvey and West's for ditto, exclusive of buildings, 
 84 inch cylinder engine 
 
 11 
 
 306 
 
 459 
 
 11000 
 
 96-59 
 
 113-9 
 
 Ditto ditto 64 inch cylinder engine 
 Ditto ditto 58 inch cylinder engine 
 Messrs. Hawthorne for ditto, double power expan- 
 sive condensing beam engines, exclusive of 
 
 8 
 
 14 
 
 n 
 
 180 
 140 
 
 814 
 
 270 
 210 
 
 471 
 
 7000 
 6500 
 
 6055 
 
 56-82 
 44-19 
 
 99-11 
 
 123-2 
 147-1 
 
 61-1 
 
 Ditto 
 
 if 
 
 163 
 
 244 
 
 2520 
 
 51-45 
 
 48-9 
 
 Messrs. Hawthorne's high-pressure double acting 
 horizontal engines, to be raised in 6 hours 
 Ditto to be raised in 9 hours 
 
 It 
 
 22 
 22 
 
 132 
 
 108 
 
 1170 
 
 780 
 
 27-78 
 22-73 
 
 42-1 
 34-3 
 
 
 is 
 
 22 
 
 66 
 
 570 
 
 13-89 
 
 41-0 
 
 Mr. Hocking's estimate for Wolverhampton, single 
 
 14 
 
 500 
 
 750 
 
 20800 
 
 157-83 
 
 131-8 
 
 
 5J 
 
 500 
 
 750 
 
 13200 
 
 157-83 
 
 83-6 
 
 Ditto to be raised in 12 hours, by two 60 inch , 
 two 45 inch, and one 36 inch cylinders 
 Ditto exclusive of buildings 
 
 14 
 
 Is 
 
 500 
 500 
 
 1500 
 1500 
 
 32,600 
 21800 
 
 315-65 
 315-65 
 
 103-3 
 69-0 
 
 Mr. Quick's estimate, for Grand Junction Works, 
 one 64, or two 45 inch cylinder, single acting 
 Cornish 
 
 5 
 
 46 
 
 230 
 
 7000 
 
 48-40 
 
 144-6 
 
 Ditto for Southwark and Vauxhall 
 
 8 
 
 40 
 
 320 
 
 7000 
 
 67-34 
 
 103-9 
 
 Ditto for West Middlesex 
 
 5 
 
 46 
 
 230 
 
 10000 
 
 48*40 
 
 206-6 
 
 Sandys, Vivian & Co., two engines for Liverpool, 
 
 2 
 
 300 
 
 600 
 
 8000 
 
 126-26 
 
 63-4 
 
 Ditto one engine, exclusive of buildings 
 Seaward and Capel's, for Wolverhampton, exclusive 
 
 1 
 11 
 
 75 
 566 
 
 75 
 849 
 
 1800 
 
 r 22,000^1 
 
 1 * < 
 
 15-78 
 178-65 
 
 114-1 
 f 123-2 \ 
 
 West's estimate, 65 inch cylinder, exclusive of 
 
 Is 
 
 814 
 
 471 
 
 I 25,000 J 
 6000 
 
 99-11 
 
 1 139-9 j 
 60-5 
 
 Ditto 50 inch cylinder, including buildings . . 
 Ditto exclusive of buildings 
 
 U 
 
 1 
 
 163 
 163 
 
 244 
 244 
 
 7100 
 5000 
 
 51-45 
 51-45 
 
 138-0 
 97-2 
 
 Ditto raised in 6 hours, by a 35 inch cylinder 
 
 
 22 
 
 132 
 
 4600 
 
 27-78 
 
 165-5 
 
 Ditto exclusive of buildings 
 
 1A 
 
 22 
 
 132 
 
 8100 
 
 27-78 
 
 111-6 
 
 
 {i 
 
 566 
 
 849 
 
 38200 
 
 158-66 
 
 240-8 
 
 Ditto 65 inch cylinder engine, including 
 buildings (?) 
 
 
 814 
 
 471 
 
 8400 
 
 99-11 
 
 84-8 
 
 AMERICAN ESTIMATES AND WORKS. 
 
 Mr. McAlpine's, for Brooklyn, double acting, expan- 
 sive, high-pressure, condensing engine, 72 inch 
 cylinder engine, to be raised in 12 hours, ex- 
 
 5 
 
 190 
 
 1950 
 
 18,000 
 
 399-83 
 
 45-0 
 
 Ditto non-condensing 30 inch cylinder engine , 
 raised in 24 hours, exclusive of buildings 
 Ditto in 12 hours, exclusive of buildings 
 Ditto exclusive of buildings 
 
 5 
 10 
 20 
 
 190 
 190 
 190 
 
 1035 
 8800 
 7600 
 
 8000 
 39,000 
 53,000 
 
 199-92 
 799-66 
 1599-23 
 
 15-0 
 
 48-7 
 83-1 
 
 Ditto ditto 
 
 30 
 
 190 
 
 11,400 
 
 85,000 
 
 2398-99 
 
 35-4 
 
 Ditto Albany Water Works, 58 inch condens- 
 ing beam engine, and one duplicate not con- 
 densing, working for day only 
 Ditto Chicago Works, as in last, including 
 buildings, 46 inch cylinder condensing engine 
 
 }*{ 
 
 3 
 
 1561 
 
 238J 
 
 107 
 
 624"! 
 476J 
 
 642 
 
 13,320 
 11,258 
 
 231-48 
 135-10 
 
 67'5 
 83-3 
 
 
 3 
 
 116 
 
 696 
 
 11,642 
 
 146-46 
 
 79-5 
 
 
 
 
 
 
 
 
OEIFICES, WEIKS, PIPES, AND EIVEES. 337 
 
 In EXAMPLE 28, pages 39 and 40, we have pointed 
 out the method of calculating the increase of horse 
 power required in raising water through pipes from 
 friction, and also the great increase of this extra 
 head if the velocity increases ; the increase being 
 nearly as the square of the velocity. In addition to - 
 this, an allowance of horse power must be made for 
 bends, curves, junctions, and other obstructions, for 
 the effects of which see SECTION XI. The more 
 slowly the water is pumped, the less will the loss be 
 from these causes through the same pipe. It is 
 therefore, so far, advisable to give as large a diameter 
 to the pipes supplying a reservoir from a pumping 
 engine as other aspects of the question, cost, and 
 engine power, will admit. 
 
 THICKNESS OF PIPES FOR WATER WORKS. 
 
 It is evident that the thickness of a pipe should be 
 at least sufficient to bear the pressure of the atmos- 
 phere, and therefore the whole pressure in a pipe is 
 best expressed by a determinate number of pressures, 
 each equal to that of a column of water 33 feet high. 
 If n be the number of such pressures, or the number 
 of units each equal 33 feet high, d the diameter of 
 the pipe in inches, and t the thickness, also in inches, 
 we shall have for 
 
 (A). 
 
 ( 1. Iron pipes, plate t = -0009 nd + -13. 
 
 2. Iron pipes cast horizontally . t = -0024 nd + -33. 
 
 3. Iron pipes cast vertically . . t = 'OOlQnd+ -32. 
 
 4. Copper pipes, plate .... Z = '0015nd + -16. 
 
 5. Lead pipes * = -0024nd+ -19. 
 
 6. Zinc pipes t-='005lnd+ -16. 
 
 7. Artificial stone * = -0054 nd -f 1-60. 
 
 z 
 
338 THE DISCHARGE OF WATER FROM 
 
 For cast-iron pipes the engineer of the Paris water 
 works, M. Dupuis, adopted in his practice a formula 
 which is equivalent to 
 (B.) * 0016rad + -32 + -013d 
 
 in the foregoing measures. This formula may also be 
 expressed as follows : 
 
 (0.) t = (-0016/1 + -013) d + -32. 
 
 If d be 12 inches, and n = 9, corresponding to a pres- 
 sure of 297 feet, we shall find from the last equation, 
 
 All pipes should however be proved with ten atmos- 
 pheres, or 330 feet, and in practically applying the 
 above formulae in equation (A), for finding the 
 thickness of pipes, the value of n should always have 
 10 added to it. Hence, applying formula (A), No. 3, 
 to our example, we get Jzz-OOlGx 19 x 12 + -32=: 
 6848 inch, which is the same practically as found 
 from equation (C). 
 
 SEWERAGE COST. 
 
 As for water works, the minimum rain-fall of a 
 district should be calculated upon, so the maximum 
 fall must be considered in sewerage and drainage. 
 We have already shown, page 305, that for a popu- 
 lation of 80 persons per statute acre, and a dis- 
 charge of two-fifths of an inch in eight hours, 
 sewers should be calculated to discharge about 31 
 cubic feet per minute, the rain supply being about 
 seven times the house supply, or sewage, including 
 house water supply. Instances are quoted in which 
 the discharge, after a heavy rain-fall, amounted to 
 
ORIFICES, WEIRS, PIPES, AND RIVERS, 339 
 
 201 cubic feet per minute per acre, as in the Savoy- 
 street sewer, which of course was principally sur- 
 face water, as the sewage of 80 persons at 7 cubic 
 feet per person, one-half of which, if discharged in 
 
 80 x 7 
 eight hours, would only be g x o = ^ cubic feet 
 
 35 
 per hour, or TTQ = *59 feet nearly per minute, which 
 
 is only about the thirty-third part of 201 feet. In 
 other words, the storm waters were thirty-three times 
 the amount of house sewage. It would be waste to 
 provide drainage for so much surface water con- 
 sidered in itself, where it can be passed off from 
 the surface channels. But sewage is not water, 
 and it is essential, in the greater number of cases, 
 that sewers should be flushed occasionally. It is 
 absurd to calculate the size of sewers, as if the 
 sewage matter were thoroughly diluted or passed 
 off like water. In fact, the sewage in part lies at 
 the bottom of the sewer, or is deposited there in 
 nine cases out of ten, while the house supply of 
 water passes on and escapes over it, removing only 
 diluted and detached portions. It is, therefore, of 
 importance, where artificial flushing and cleansing 
 out are not provided, that storm waters should oc- 
 casionally pass through and flush a system of sewers, 
 particularly the main or arterial lines. An engineer 
 must be guided, in calculating the dimensions, &c., 
 of main sewers, by the circumstances of each case. 
 The inclinations to be obtained, the form of the 
 bottom or invert, the rain-fall, the amount of sewage 
 which will not affect the size to any considerable 
 
 z 3 
 
340 THE DISCHARGE OF WATER FROM 
 
 extent, the material and the cost consistent with 
 permanency. 
 
 The discharging power of a water channel is more 
 than doubled by increasing its dimensions by one- 
 third; and it is increased in the proportion of 5*7 to 
 1 by doubling the dimensions. By giving four times 
 the fall, the same channel will only double the 
 discharge. Now a pipe 2 feet in diameter with a 
 fall of 1 in 200, would discharge fully 1000 cubic 
 feet of water flowing full with a velocity of 5*4 feet 
 per second : at 3 j cubic feet per minute per acre, for 
 a population of 80 to the acre, the thoroughly diluted 
 sewage of 280 acres would be passed off by one such 
 pipe ; that is, the sewage from 20,400 persons, on 
 280 acres, and also two-fifths of an inch of rain 
 falling for eight hours, can be conveyed by a 2 
 feet pipe, with a fall of 1 in 200. But as this rain 
 supply is about seven times the house supply, passing 
 2^ feet off per person in eight hours, made up of 
 faeces and used-up water supply. It is apparent 
 that such a pipe would convey about eight times the 
 sewage alone of the district, if flowing as water ; but, 
 under any circumstances, would be abundantly large 
 for the duty, even when assuming the whole quantity 
 to pass in at the upper end. For a fall of 1 in 800, 
 two such pipes would be required, or one pipe 32 
 inches in diameter ; for a fall of 1 in 3200, four 2 
 feet pipes would be required, or one pipe 3 feet 6 
 inches. 
 
 ' House drains should not be less than 6 inches in 
 diameter, and should have facilities for being cleaned, 
 either by using half-flange joints, or by having a 
 
OKIFICES, WEIES, PIPES, AND KIVEKS. 
 
 341 
 
 moveable upper segment. The inclination for these 
 drains should be uniform, but the amount is not so 
 important as some appear to think, if proper provi- 
 sion be made for cleaning. Where flushing is used, 
 cast-iron pipes are the best, but they are also the 
 most expensive. House drains of brick with a v 
 tile bottom covered with flags or bricks are perhaps 
 the best, as the capacity can be considerably aug- 
 mented by adding to the height of the sides, and they 
 can be at all times easily opened and cleaned. If 
 inclinations from 1 in 50 to 1 in 20 can be had, so 
 mucn the better. The following items as to cost 
 have been selected from the Builder : 
 
 COST OF SEWERS, NEWPORT, MONMOUTHSHIRE. 
 
 Total 
 lengths. 
 
 Average 
 depths. 
 
 Sizes of sewers. 
 
 Thick- 
 nesses. 
 
 Cost 
 per foot 
 lineaL 
 
 1,322 ft. 
 
 15 ft. 6 in. 
 
 ft. in. ft. in. 
 4 6 by 3 6 
 
 9 in. 
 
 8. d. 
 11 8 
 
 2,217 ft. 
 
 13 ft. in. 
 
 4 6 by 3 
 
 9 in. 
 
 10 1 
 
 6,1 10 ft. 
 
 12 ft. in. 
 
 3 by 2 2 
 
 9 in. 
 
 7 7i 
 
 12,354 ft. 
 
 11 ft. 8 in. 
 
 3 by 2 2 
 
 6 in. 
 
 5 3| 
 
 1,953 ft. 
 
 9 ft. 3 in. 
 
 2 6 by 1 10 
 
 6 in. 
 
 4 7 
 
 9,663 ft. 
 
 10 ft. in. 
 
 2 6 by 1 10 
 
 4in. 
 
 3 8J 
 
 690 ft. 
 
 10 ft. 2 in. 
 
 2 3 by 1 9 
 
 4in. 
 
 3 6J 
 
 3,264 ft. 
 
 8 ft. 6 in. 
 
 1 2 diameter 
 
 4^ in. 
 
 2 4| 
 
 COST OF SEWERS AND PIPES IN PRESTON. 
 
 The following extract from the recently published 
 summary of public works executed during the year 
 ending April 30th, 1859, contains some useful infor- 
 mation : 
 
342 
 
 THE DISCHARGE OF WATER FROM 
 
 Yards. s. d. 8. d. 
 60 of Brick Sewers, 2ft. 6in. dia- 
 meter, at 7s 21 
 
 538 3ft. by 2ft., at 17s. 6d 470 14 3 
 
 294 3ft. 6in. by 2ft. 4in., at 28s 412 12 11 
 
 372 3ft. 9in. by 2ft. 6in., at 28s.... 520 16 
 
 250 4ft. 3in. by 2ft. lOin., at 41s. U. . 521 16 8 
 
 56 4ft. 6in. by 3ft., at 75s. Id 211 13 4 
 
 66 4ft. 6in. diameter, at 40s. 9d 134 9 6 
 
 2,293 2 8 
 
 1,678 
 
 42 of Cast-iron Sewer, 2ft. diameter, 
 
 at 36s 75 12 
 
 22 of Earthenware Pipe Sewer, 6in. 
 
 diameter, at 4s 480 
 
 1,129 9in. diameter, at 7s. 5d ... 418 13 5 
 
 565 12in. diameter, at 8s. 9d 247 2 9 
 
 88 15in. diameter, at 11s. 3d 49 10 
 
 98 1 Sin. diameter, at 13s 63 14 
 
 145 21in. diameter, at 18s. Qd 134 2 6 
 
 917 10 8 
 
 2,047 
 
 Total, including superintendence, also man-boles, 
 
 street gullies, and all appurtenances 3,286 5 4 
 
 TABLE showing the prices of Tubular Drains as made by the Board of Health 
 in 1852, fifty per cent, being added for profit, &c. ; and the sale prices in 
 the market. 
 
 Diame- 
 ter in 
 inches. 
 
 Lengths. 
 
 For 1,000 feet 
 For 1,000 feet 
 For 1,000 feet 
 
 Red earthen- 
 ware pipes 
 made by the 
 Board. 
 
 s. d. 
 
 6 15 
 
 9 14 
 
 15 1 6 
 
 Red pipes 
 
 at 
 Sale prices. 
 
 S. d. 
 
 20 16 8 
 
 25 
 
 37 10 
 
 Stoneware 
 
 glazed at 
 
 Sale prices. 
 
 S. d. 
 
 25 
 
 29 3 4 
 
 50 
 
 Assumed gain. 
 
 On 
 
 red ware 
 pipes. 
 
 S. d. 
 
 14 1 8 
 
 15 6 
 
 22 8 6 
 
 Over glazed 
 
 stoneware 
 
 pipes. 
 
 s. d. 
 
 18 5 
 
 19 9 4 
 34 18 6 
 
 Did the Board of Health here add the cost of their own 
 establishment and staff to the cost of production ? The manufac- 
 turer must surely live. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 343 
 
 1 
 
 8i 
 
 R 
 
 H rt 
 
 73' OOOOOi-lrHOtM 
 
 ! 
 
 ,3 *00 >0 CO *<N<M 
 
 2||j 
 
 73' 00001>t>COCOCO 
 
 i 
 
 73* *O <M -* <M rH 
 
 co <N <N rH rH rH 
 
 "ft^ 
 
 
 iW 
 
 73' OOCOOlOrHOO 
 rH 
 
 73 CO CO O <M rH 
 oi CO CO <M rH O 
 
 f 
 > 
 
 If 
 Si 
 
 I 
 
 H rt 
 
 78 rH 
 
 ST. THOMAS'S, EXETEE. 
 Pipes laid 8 to 7 feet deep. 
 
 73 CO CD CO 00 CO ~ O O 00 
 rH 
 
 <* *CO " O 00 10 CO * <M !N ,H rH 
 i 1 rH 
 
 Illl 
 
 73 OCOCO^COdd 
 oi lOdddfMdd 
 
 Ts'COCO CO CO O CO CO O O O 
 CO CO " d d <N rH ' rH rH rH rH 
 
 ol^l 
 
 -14 
 
 73 CSCO'OCOOCOrH 
 
 73' _ O OOooco ^OOOO 
 
 ,5 O 00 CO CO (M -HrHOO 
 
 1 SOUTHAMPTON. 
 Pipes laid 9 feet deep. 
 
 1 s * 
 
 ' Oi fOlO^ ' 
 
 1 TOTTENHAM. 
 Pipes laid 5 to 13 feet :deep. 
 
 rH rH 
 co- CO J> Tfl CO O rH 
 
 nil 
 
 73' t- 00 
 
 ^ co co co d 
 
 73 10 00 O O rH CO 
 drHrHrHOO* 
 
 iW 
 
 73* .0 co co * 
 
 oj CO CO d rH 
 
 73 <* IO O rH 00 OJ 
 =5 ' t~ IQ CO 0? rH rH 
 
 |3 
 
 .s. 
 
 j!h 
 
 73* O O O t-H t- 
 co ' CO CO d d 
 
 f 
 
 73 
 
 O {O 
 
 73 00 d rH t> IO 
 
 co O -^ d d <M 
 
 III! 
 
 73 O O 00 00 00 
 
 73* 05 CO CO CO CO 
 
 * * 
 
 --a?. 
 
 
 rH rH rH rH rH 
 
 il4 
 
 73* O <N <M CO rH 
 4 *U5 CON rH O 
 
 73 rH O 5 rH rH 
 
 
 tS CO d rH I 1 O 
 
 p. 
 
 1 
 
 g 3 
 
 s 
 
 H - 1 
 
 >eco co d J> -i os o 
 
 rfCO ' rH CO CO <*< CO CO 
 
 1 
 
 1 
 
 73 d rH rH CO O d 00 CO 
 
 aj rHO5tO^ COdd 
 rH 
 
 if 
 
 73'CO CO d ^ CO d 
 
 73 COOJCOCOO OO5OS 
 
 iW 
 
 73O O O CO d CO O 
 
 aSCO OOCOCOdrHO 
 rH 
 
 ^ * "oOtU5COd rHOO 
 
 sadid |o ja^eureid 
 
 gddrHrHiH <N d d rH rH rH 
 
344 
 
 THE DISCHARGE OF WATER FROM 
 
 ESTIMATE FOR SEWERS AT BRIGHTON. 
 
 DESCRIPTION OF SEWEBS. 
 
 Length in yards. 
 
 .Price 
 per 
 yard. 
 
 Amount. 
 
 Brick Sewers: 
 Diameter. 
 6ft 
 
 4,850 
 350 
 3,990 
 1,890 
 
 2,820 
 8,580 
 
 22,480 
 53,896 
 2,010 
 
 8. 
 
 3 10 
 2 10 
 2 8 
 2 2 
 1 16 
 18 
 
 8. 
 
 13 6 
 10 
 
 8. 
 
 7 
 3 
 
 20 
 4 
 3 10 
 
 8. 
 
 16,975 
 875 
 9,600 
 3,969 
 5,076 
 7,722 
 
 6,389 11 
 22,215 
 
 5,250 
 3,780 
 
 12,000 
 2,400 
 10,500 
 
 5,000 
 
 11,178 9 
 122,930 
 
 4ft. Gin 
 
 4ft. 6in. by 3ft 
 
 3ft. 9in. by 2ft. Gin 
 
 3ft. by 2ft 
 
 2ft. 3in. by 1ft. Gin. . 
 
 Total brick sewers . 
 
 Earthenware Pipe Sewers : 
 15 inches diameter . 
 
 9,466 
 44,430 
 
 750 
 1,260 
 
 12 
 
 Total earthenware pipe sewers . 
 Cast-iron Pipe Sewers : 
 
 1ft. Gin. 
 
 Total cast-iron pipe sewers .... 
 Total length of sewers 
 
 
 78,386 
 
 Number 
 600 
 600 
 3,000 
 
 Or 44 miles 956 yards. 
 Man-holes and ventilating shafts 
 
 Gullies 
 
 Outlet works, overflows, and ex- 
 tra work on steep gradients, 
 &c 
 
 Contingencies, in eluding repairs, 
 &c. of existing sewers, 10 per 
 cent 
 
 Total. 
 
 
 
 The following estimates have been made for lay- 
 ing pipes at Tottenham, not including their cost : 
 
 Depth 
 6 feet. 
 
 Depth 
 8 feet. 
 
 lid. 
 
 Depth 
 10 feet. 
 
 Diameter 
 
 of pipe 
 
 in inches. 
 
 6 
 
 12 
 
 The cost of laying alone at St. Thomas's, Exeter, 
 was 
 
OEIFICES, WEIRS, PIPES, AND RIVERS. 345 
 
 6 inch pipes 5d. per foot lineal 3 to 4 feet deep. 
 9 5d. 3 to 4 feet deep. 
 
 12 Sd. 5 feet deep. 
 
 15 9d. 5 feet deep. 
 
 18 lid. 5 feet deep. 
 
 2d. per foot lineal for relaying pitching ; 4d. for macadamized 
 roads ; and Qd. for pavements. 
 
 The author has constructed a large quantity 
 of main sewers, from 18 inches to 2 feet and 2 
 feet 6 inches wide, and 4 feet 6 inches high; the 
 side walls built with rubble masonry, 9-inch seg- 
 ment inverts laid with 4i-inch courses in cement ; 
 the top sometimes flagged, when flags of sufficient 
 length could be procured, and sometimes arched 
 with rough rubble arches. The invert was laid 
 on, well bedded, well rammed, rubble to prevent 
 subsidence, and preserve the bottom inclination 
 uniform. The cost, at an average depth of about 
 9 feet, was 9s. per running foot, the side walls 
 being about 18 inches thick. Upright side walls, 
 where rubble is cheap, have many advantages 
 in giving a considerable increase of capacity for a 
 small outlay. The tenement and house drains were 
 of earthenware pipes. Cast-iron gully grates and 
 traps, weighing 3 cwt., cost 30s. each; the grate 
 fastened by a wrought-iron chain. 
 
 The following regulations have been laid down 
 for Cambridge and Carlisle : 
 
 STIPULATIONS FOR CAMBRIDGE DRAINAGE. 
 
 " Water from the rear of premises should not be conveyed to 
 the front under the hasement floor. 
 
 " Rain-water from the roofs should not be conveyed into the 
 basement, but conducted into the sewer by shallow drains. 
 
346 THE DISCHARGE OF WATER FROM 
 
 " Cast-iron pipes may be used for basement drains in some 
 instances. 
 
 " The scullery sink should be kept as high as possible, and 
 approached by a step. A flap trap should be fixed between the 
 sink and sewer. 
 
 " There should be no water-closet on the basement floor ; if it 
 cannot be arranged elsewhere, the soil-pipe should have a flap 
 trap, or similar contrivance, to prevent the influx of sewage 
 water." 
 
 FOE CARLISLE DRAINAGE. 
 
 " STIPULATION 1. If water-closets are to be generally used, 
 the description of such to be sanctioned by the Board, the same 
 to be fixed to the satisfaction of the Surveyor. 
 
 " 2. All down-spouts to be connected with the sewers where 
 it may be proper to connect the same ; in all cases where they 
 are not connected with the sewer they are to be connected with 
 the channel. 
 
 " 3. All stench traps to be similar to samples furnished by 
 the Surveyor, or others approved by him, and properly fixed to 
 his satisfaction. 
 
 " 4. All sewers to water-closets not to be less than six inches 
 diameter. 
 
 " 5. All sewers to yards, stables, kitchens, and sculleries, not 
 to be less than four inches diameter. 
 
 " 6. In every case the whole of the fall to be made available 
 from the junction with the main sewer to the end of the private 
 drain, that is to say, only one inclination to be used from the 
 junction with the public sewer to the end of the private drain ; 
 and all branches from the private drain to sinks, water-closets, 
 &c., to have one inclination from the junction of such drain. 
 None of the above instructions to be departed from without the 
 express sanction of the Surveyor. 
 
 " 7. In no case must a private drain be put in with a less fall 
 than one in fifty, without the sanction of the Surveyor. 
 
 " 8. No pipes, water-closets, stench traps, gullies, kitchen 
 sinks, bends, junction or tapering pipes, to be used without 
 being approved by the Surveyor. 
 
 " 9. All ash pits and dung depots to be raised to the level of 
 the adjoining ground, to be properly paved and drained as the 
 Surveyor may direct. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 347 
 
 " 10. All buildings, outhouses, &c., to be properly spouted, 
 and the water conveyed into the sewers where approved of by the 
 Surveyor." 
 
 THOROUGH LAND DRAINAGE. 
 
 The following instructions and general specifica- 
 tions, have been prepared by the Commissioners of 
 Public Works in Ireland, for the use of the District 
 Inspectors, and persons reporting on thorough- 
 drainage. The drains are made in general parallel, 
 and to suit the fall of the ground. The depths must 
 alter in order that the bottoms should have an un- 
 interrupted fall, and may vary from 2 feet to 4 feet 
 6 inches in practice, averaging, say about 3 feet 
 6 inches, but dependent on circumstances. The 
 portions printed in italics are from specifications 
 prepared by officers of the Board, and are varied 
 according to each particular case : 
 
 GENERAL OBSERVATIONS. 
 
 " No drainage works should be undertaken until it has been 
 clearly ascertained that the surface level of the maximum floods 
 in the main drain can be discharged at a level that will admit of 
 the submain drains venting the waters from the lowest point of the 
 lands proposed to be thorough-drained, at a level sufficiently below 
 the surface of such land, that the highest floods shall not prevent 
 the free discharge of such submain. 
 
 " When sufficient out-fall can be obtained, no open main drain 
 should be of a less depth than five feet, and in all cases a greater 
 depth is desirable, in order to insure a permanent and efficient 
 drainage, and at the same time to prevent cattle, &c., from 
 crossing. 
 
 "As it has been found by practical experiments on different 
 varieties of soils, that deep drains, say from four to five feet deep, 
 are more effective than shallow ones : they should always be 
 estimated for, when the open main drains admit of their being 
 
348 THE DISCHARGE OF WATER FROM 
 
 cut to that depth, or when, hy a moderate outlay per acre, the 
 main drains can be cut to a sufficient depth ; the distance be- 
 tween the parallel drains must necessarily vary with the texture 
 of the soil, forty feet may be taken as a general rule. 
 
 OPEN MAIN DRAINS. 
 
 " Main drains should have gradients of such inclination, and 
 be sunk to a depth that will admit of the above stipulations, as 
 to the discharge of the submain drains being carried out. They 
 should have such width at bottom and side slopes as may be 
 necessary ; and be free of sharp angles, projecting stones, and 
 other impediments to the quick discharge of the waters. 
 
 " The spoil or material raised in sinking and improving the 
 drains, where not available for filling up useless holes or drains, 
 should be removed to a proper distance from the edge of the 
 main drains, and dressed off in a workmanlike manner. 
 
 " The abutments and piers of such bridges as have sufficient 
 breadth of water-way, should, if necessary, be carefully under- 
 pinned; and those bridges which are insufficient to discharge 
 floods, should be taken down and rebuilt of suitable dimensions. 
 
 COVERED MAIN DRAINS. 
 
 " Whenever, from the nature of the lands, the extent of the 
 district under drainage, and the quantity of water to be voided, 
 it may be necessary to form covered main drains to receive the 
 water discharged from the submains, their dimensions must be 
 proportional to the amount of water to be voided, well flagged or 
 paved at bottom, the sides built of stone or brick, and covered 
 with a flag or arch at top. 
 
 SUBMAINS. 
 
 " The submains to be of such depth and width at top and bot- 
 tom as may be necessary. The fall in each to be as great as the 
 above-described main drainage of the district will allow, and not 
 to be allowed to run beyond a suitable length without discharging 
 itself into a covered or open main drain. 
 
OEIFICES, WEIES, PIPES, AND KIVERS. 349 
 
 THE MINOR DRAINS 
 
 " To be of such depth, width at top and bottom, and at such 
 distance apart, as will secure the perfect drainage of the land, to 
 be run in a straight direction parallel to each other, directly up 
 and down the declivity, unless where the declivity happens to be 
 very steep, and then to be carried across the fall at such an angle 
 as to secure a free discharge for the water. The fall in each 
 minor drain to be as great as the main drainage and submain 
 drainage, previously described, will admit. 
 
 " In filling in the stones, great care should be taken that the 
 bottom of the drain be clean, and that no clay or dirt be put in 
 with them ; a sod, grass side down, or a few inches of tough clay, 
 to be placed on the surface of the stones, and trodden firmly. 
 The drain should then be filled up with the stuff previously sho- 
 velled out, observing to keep the active soil for the top. The 
 putting in of the stones to be commenced at the highest part or 
 head of the drain. 
 
 " In using draining pipes or other tiles, care should be taken 
 that they be laid firmly on the bottom for their entire length, so 
 as to prevent them being deranged by the filling of the drain, and 
 that the points be fitted as closely together as possible. 
 
 " In cases of unfavourable ground, caused by running sand or 
 otherwise, whereby the level of the conduit might be deranged, 
 collared pipe tiles offer considerable advantages in the way of 
 remedy. 
 
 " When gripes may be necessary on the sides of farm roads, 
 they should be on the field side of the fences." 
 
 SPECIFICATION FOR MAIN DRAINAGE. 
 
 OPEN MAIN DRAINS. 
 
 " The deepening and improving of the main drain, No. , 
 
 is to be commenced at the point on the accompanying map, 
 
 and from thence a gradient carried up to the point , having 
 
 an inclination of at least feet per statute mile, and sunk to 
 
 the depth of feet. It shall be feet wide at bottom, and 
 
 the side slopes shall average at least, unless in rock cutting, 
 
 when the side slopes may be diminished to six inches to one foot; 
 all sharp angles, projecting stones, and other impediments to the 
 
350 THE DISCHAEGE OF WATER FROM 
 
 free discharge of the water, must be carefully removed. The 
 spoil or material raised in sinking and improving the drain, when 
 not immediately used for top-dressing the adjoining lands, or for 
 filling useless holes or drains, is to he removed to a distance of 
 
 feet from the edge of the main drain, and dressed off in a 
 
 workmanlike manner. 
 
 " The bridge marked at the point on the accompanying 
 
 map to be . 
 
 " The whole to be executed in a proper and workmanlike manner, 
 and the works to be maintained in good .order for so long as any 
 interest shall be payable for the money advanced on account of its 
 execution." 
 
 SPECIFICATION FOR THOROUGH-DRAINAGE 
 (WITH TILES). 
 
 COVERED MAIN DRAINS. 
 
 " These shall be cut jiffy-four inches deep, thirty-six inches wide 
 at top, twenty-four inches wide at bottom ; the materials used in 
 them shall be double row of three-inch pipe tiles. 
 
 " The side walls shall be inches in height, inches 
 
 thick, and well at bottom. They shall be covered with a 
 
 flag not less than in thickness. 
 
 SUBMAINS. 
 
 " These shall be cut fifty inches deep, thirty inches wide at top, 
 eighteen inches wide at bottom. They shall be carried along the 
 low side of the fields, or portions of land to be drained, at a dis- 
 tance from the fence of fifteen feet, and through natural hollows 
 where necessary. No submain to be allowed to run beyond a 
 length of two hundred yards without discharging itself into a 
 covered or open main drain. 
 
 MINOR DRAINS. 
 
 " These shall be cut forty-eight inches deep, sixteen inches wide at 
 top, five inches wide at bottom, and at a distance of forty feet 
 apart. They shall be run in a straight direction, parallel to each 
 other, directly up and down the declivity (when possible). No 
 minor drain to be allowed to run beyond a length of two hundred 
 yards without discharging itself into a submain. 
 
OKIFICES, WEIES, PIPES, AND RIVERS. 351 
 
 FILLING IN. 
 
 "All the drains (or a large number of them) having been opened 
 and cut in a workmanlike manner, and it being ascertained that 
 no water is standing in any of them, the filling in may be com- 
 menced. 
 
 MINOR DRAINS. 
 
 " Into each minor drain shall be put pipe tiles twelve inches in 
 length, one-and-a-half inch in the ope, for one hundred yards, com- 
 mencing from the upper end of the drain, and pipe tiles twelve 
 inches in length, one-and-three-quarter inch in the ope, in con- 
 tinuation from thence to the submains. 
 
 SUBMAINS. 
 
 " Into each submain shall be put pipe tiles twelve inches in length, 
 two inches in the ope, for one hundred yards, commencing from the 
 upper end of the drain, and pipe tiles twelve inches in length, three 
 inches in the ope, in continuation to the end or point where they 
 discharge themselves. 
 
 GENERAL RULES. 
 
 " All tiles to be of good sound material, and well burned. The 
 tiles shall be laid firmly on the bottoms of the drains for their 
 entire length ; the joints fitted as closely as possible, they shall 
 be carefully covered with a thin grassy sod or screen. The stuff pre- 
 viously taken out of the drains shall then be returned, observing 
 to keep the active soil uppermost. 
 
 " The mouths of the covered main or submain drains shall be 
 built about with solid masonry set in mortar, carried up with the 
 same slope as the sides of the open main drain, into which they 
 discharge themselves. 
 
 " Before laying the tiles, great care must be taken that the 
 bottom of the drains be clean. The putting in of the tiles to be 
 commenced at the highest point or head of the drains. 
 
 " In case of an entire field being thorough-drained, a drain 
 shall be cut at the top of it, parallel to the fence, and running at 
 a distance from it equal to one-half of the distance between each 
 of the minor drains, into one or more of which (as may be neces- 
 sary) it shall discharge itself. The remainder of the minor 
 drains to be discontinued at a distance from this drain equal to 
 
352 THE DISCHARGE OF WATER FROM 
 
 one-half the entire distance between each of the minor drains ; 
 this drain to be of the same dimensions, and filled with the same 
 materials, and in like manner, as the above described. 
 
 " No open drain shall run into a closed one. 
 
 " In passing through unfavourable ground, caused by running 
 sand or otherwise, whereby the level of the conduit might be de- 
 ranged, and where pipe tiles are the materials used for forming 
 the conduit, collars must be used, so as to connect the ends of the 
 tiles, and they must be fitted as closely as possible. 
 
 " Soles must, in all cases, be used when laying single D tiles, 
 and they must be so laid that the ends of the tiles shall rest 
 equally on them ; when inverted D tiles are used, they shall also 
 be connected from end to end by placing one-half of the upper 
 tiles on one-half of the adjoining tiles below them. 
 
 " The whole to be executed in a proper and workmanlike 
 manner ; and the work to be maintained in like good order as 
 when approved of at its completion, for so long as any interest 
 shall be payable for the money advanced on account of its 
 execution." [Collars for up to k-inch pipes can be had at the Florence 
 Court Tilery.] 
 
 SPECIFICATION FOR THOROUGH-DRAINAGE (WITH 
 BROKEN STONES). 
 
 COVERED MAIN DRAINS. 
 
 " These shall be cut forty >-tivo inches deep, thirty inches wide 
 at top, twenty-four inches wide at bottom ; the materials used in 
 them shall be . 
 
 " The side walls in them shall be twelve inches in height, six 
 
 inches thick, and well at bottom. They shall be covered 
 
 with . 
 
 SUBMAINS. 
 
 " These shall be cut forty-two inches deep, eighteen inches 
 wide &t top, fourteen inches wide at bottom. They shall be car- 
 ried along the low side of the fields, or portions of land to be 
 drained, at a distance from the fences of thirteen feet, and 
 through natural hollows, where necessary. No submain to be 
 allowed to run beyond the length of one hundred and fifty yards, 
 without discharging itself into a covered or open main drain. 
 
/ x<s iegs. 
 
 ORIFICES, WEIRS, PIPES, AND RIVERS. 353 
 
 ft VNJYE115S1TY | 
 
 MINOE DRAINS. 
 
 " These shall be cut thirty-six inches deep, fifteen inches wide 
 at top, four inches wide at bottom, and at a distance of twenty- 
 six feet apart. They shall be run in a straight direction, parallel 
 to each other, directly up and down the declivity (when possible). 
 No minor drain to be allowed to run beyond a length of two 
 hundred yards without discharging itself into a submain. 
 
 FILLING IN. 
 
 " All the drains (or a large number of them) having been 
 opened and cut in a workmanlike manner, and it being ascertained 
 that no water is standing in any of them, the filling in may be 
 commenced. 
 
 MINOR DRAINS. 
 
 " Into each minor drain shall be put ten inches of broken 
 stones in depth, the stones having been broken to a size not ex- 
 ceeding two-and-a-half inches in diameter. Great care should be 
 taken that the bottom of the drain be clean, and that no clay or 
 dirt be put in along with the stones ; a sod (or clay, as may be 
 convenient) three inches thick shall be placed carefully on top, 
 and the whole trampled upon or rammed hard. The drain shall 
 then be filled up with the stuff previously shovelled out, ob- 
 serving to keep the active soil for covering the top. The putting 
 in of the stones shall invariably be commenced at the highest 
 part or head of the drain. 
 
 FILLING IN SUBMAINS. 
 
 " In each submain a conduit shall be formed of six inches in 
 height, four inches wide, and the filling in completed as above 
 described. 
 
 GENERAL RULES. 
 
 " The mouths of the covered main or submain drains shall be 
 built about with solid masonry set in mortar, carried up with the 
 same slope as the sides of the open main drain into which they 
 discharge themselves. 
 
 " Before filling in the stones, great care must be taken that the 
 bottom of the drains be clean, and that no clay or dirt be put in 
 
 A A 
 
354 THE DISCHARGE OF WATER FROM 
 
 along with them. The putting in of the stones to be commenced 
 at the highest part or head of the "drains. 
 
 " In case of an entire field being thorough-drained, a drain 
 shall be cut at the top of it, parallel to the fence, and running 
 at a distance from it equal to one-half the distance between each 
 of the minor drains, into one or more of which (as may be 
 necessary) it shall discharge itself. The remainder of the minor 
 drains to be discontinued at a distance from this drain, equal to 
 one-half the entire distance between each of the minor drains ; 
 this drain to be of the same dimensions, to be filled with the 
 same material, and in like manner, as the above described. 
 
 " No open drain shall run into a closed one. 
 
 " The whole to be executed in a proper and workmanlike man- 
 ner, and the work to be maintained in like good order as when 
 approved of at its completion, for so long as any interest shall be 
 payable for the money advanced on account of its execution." 
 
 One of the officers of the Commissioners of Public 
 Works, Ireland, the Inspector of Drainage for Ros- 
 common, a gentleman residing in that county, writes 
 to me as follows, with reference to tile and broken 
 stone drains on the carboniferous formation : 
 
 " With respect to tile drainage my experience has 
 not been very extensive, as the proprietors of the dis- 
 trict, with scarcely any exception, give a decided 
 preference to broken stones ; but from what I have 
 seen, I am very much inclined to prefer good well- 
 burnt pipes to any other draining material, provided 
 that collars be used, but not otherwise. As to the 
 best diameters, I have found the li" collared pipes 
 of the Clonbrock Tile Works (now closed) very satis- 
 factory ; but when the length of minor drains ex- 
 ceeded 100 yards, I should like an increase to li or 
 1|. For submains (say 150 or 180 yards long) I 
 have recommended pipes of 2 inches, 2i, and 3 
 inches in succession, all of which were to be had 
 
OKIFICES, WEIKS, PIPES, AND KIVEES. 355 
 
 with collars: if 4-inch pipes were to be had with 
 collars, I should have recommended longer submains. 
 The larger-sized pipes are not provided with collars 
 in our present tileries, and on this account I generally 
 put a note on the margin of the printed form, sug- 
 gesting that a stone duct of the ordinary size of 
 submain, say 6 inches in height and 4 inches wide, 
 be substituted for the tile filling. 
 
 " I decidedly prefer an open duct to broken stone 
 filling ; and in nine-tenths of my own drainage I 
 have made the minor drains on the same plan as the 
 submain, with an open stone conduit ; the only differ- 
 ence being, that the minor drains are a few inches 
 shallower, with a smaller duct. The increase of 
 expense is a mere trifle, and when the substratum (as 
 very frequently occurs here) is a fine calcareous gravel, 
 containing 40 to 60 per cent, carbonate of lime, the 
 additional spoil is a very cheap fertilizer for the land. 
 
 " With respect to depths and distances apart, the 
 two most commonly used in my specification are 
 3i feet deep, 33 feet apart, and 4 feet deep, 42 
 feet apart. These arrangements will not suit all 
 cases, and I vary accordingly. Thus, in one case 
 of exceedingly retentive land of peculiar texture, 
 4 feet drains, 27 feet apart, produced the required 
 result, while in another, 3J feet drains, 66 feet apart, 
 effected all that was required. In the latter case 
 there was a mixed soil, which might be described 
 as " half wet ;" yet the water lingered sufficiently 
 long to make the land unsound for sheep, and greatly 
 to injure the crops in quality as well as quantity." 
 
 Mr. Josiah Parkes says, in 1843 : " Experiment 
 
 AA3 
 
356 THE DISCHARGE OF WATER FROM 
 
 and experience have rapidly induced the adoption 
 of a system of parallel drains, considerably deeper, 
 and less frequent, than those commonly advocated 
 by professed drainers, or in general use. I gave 
 several instances of this practice in Kent in the 
 Report of last year, 1843, already alluded to, and 
 it is rapidly extending. Mr. Hammond stated to 
 you that he drained < stiff clays 2 feet deep, and 24 
 feet between the drains, at 3 4s. 3d. per acre, 
 and porous soils 3 feet deep, 33J feet asunder, at 
 2 5s. 2d. per acre.' I now find him continuing 
 his drainage at 4 feet deep, wherever he can obtain 
 the outfall, from a conviction founded on the ex- 
 perience of a cautious progressive practice as to the 
 depth and distance, that depth consists with economy 
 of outlay as well as with superior effect. He has 
 found 4-feet drains to be efficient, at 50 feet asunder, 
 in soils of varied texture not uniform clays and 
 executes them at a cost of about 2 5s. per acre, 
 being 185. 4d. for 871 pipes, and 1 6s. 6d. for 
 53 rods of digging. Communications have been 
 recently made to me by several respectable Kentish 
 farmers, of the satisfactory performance of drains 
 deeply laid in the Weald clays, at distances ranging 
 from 30 to 40 feet, but I have not had the oppor- 
 tunity of personally inspecting these drainages. 
 
 " The following little table shows the actual and 
 respective cost of the above three cases of under- 
 draining, calculated on the effects really produced, 
 that is, on the masses of earth effectively relieved 
 of their surplus water at an equal expense. I con- 
 ceive this to be the true expression of the work 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 357 
 
 done, as a mere statement of the cost of drainage 
 per acre of surface conveys but an imperfect, in- 
 deed a very erroneous, idea of the substantive and 
 useful expenditure on any particular system. This 
 will be apparent on reference to the two last columns 
 of the table, which give the cost in cubic yards and 
 square yards of soil drained for one penny, at the 
 above mentioned prices, depths, and distances. 
 
 Depth of drains, 
 in feet. 
 
 Distance 
 between the 
 drains, in feet. 
 
 Mass of soil 
 drained per acre, 
 in cubic yards. 
 
 Mass of soil 
 drained for one 
 penny, in cubic 
 yards. 
 
 Surface of soil 
 drained for one 
 penny, in square 
 . f yards. 
 
 2 
 
 24 
 
 3226 * 
 
 4-1 
 
 6'27 
 
 3 
 
 33* 
 
 4840 
 
 8-93 
 
 8-93 
 
 4 
 
 50 
 
 6453 
 
 12-00 
 
 8-96 
 
 " I may here observe, that Mr. Hammond, when 
 draining tenacious clays, chooses the month of 
 February for the work, when he lays his pipes 
 (just covering them with clay to prevent crumbs 
 from getting in), and leaves the trenches open 
 through March, if it be drying weather, by which 
 means he finds the cracking of the soil much accele- 
 rated, and the complete action of the drains ad- 
 vanced a full season. The process of cracking may, 
 doubtless, be hastened both by a choice of the 
 period of the year in which drains are made, and 
 by such a management of the surface as to expose 
 it to the full force of atmospheric evaporation." 
 
 With reference to drains, we have known a case 
 in the Queen's County in which inch pipes had to 
 be taken up, and pipes of 2J-inch bore substituted. 
 
358 THE DISCHARGE OF WATEK FROM 
 
 The drains were 40 feet apart, and 4 feet deep, and 
 the pipes had collars. The minor drains should 
 discharge into submains at convenient distances, 
 say 100 yards, on flat grounds. Small pipes will 
 choke unless the velocity in them be sufficient to 
 carry off deposits, and the diameters should vary 
 according to the inclinations of the ground, and 
 distance apart of the drains. 
 
 Mr. Mechi, in 1844, lays down the following 
 rules : 
 
 " 1st. That it is not the size or form of the drains 
 that regulate perfect drainage ; but the depth at 
 which they are placed. The depth aiso governs the 
 distances at which the drains should be cut accord- 
 ing to the quality of the soil. 
 
 " 2nd. The pipes of 1-inch bore, without stones, 
 are amply sufficient, placed at 4 feet deep and 30 
 feet wide in dense soils, and the same depth and 50 
 feet wide in mixed soils. 
 
 " 3rd. The deep drains receive more water than 
 shallow ones, and consequently lay dry a greater 
 extent of ground. 
 
 "4th. The deep drains begin and end running 
 sooner than shallow ones, and carry off more water 
 in a given time. 
 
 "5th. That where shallow drains are made and 
 deep ones cut below them, the shallow ones no longer 
 act, all the water passing to the deeper drains. 
 
 " 6th. That when round stones are used as well as 
 pipes, the latter should always be placed at the 
 bottom, as I find, practically, water flows more 
 quickly through pipes than amongst stones. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 359 
 
 " Before persons begin draining. I would recommend 
 their perusing attentively the facts developed by 
 Mr. Parkes, at pages 39 and 40, and my remarks at 
 page 36 of Letters on Agricultural Improvements. 
 
 u Pipes made to socket into each other (by Ford's 
 
 TABLE showing a Return of the number of Acres thorough-drained 
 in the years 1843 and 1844, by the different Competitors for 
 Sir Richard O'DonnelVs Gold Medal, together with the Average 
 Prices per Perch, and Cost per Acre respectively. (Given in 
 Reports to the Royal Agricultural Improvement Society of Ireland.) 
 
 Competitors. 
 
 Number 
 of statute 
 acres. 
 
 Number 
 of 
 statute 
 perches. 
 
 Average 
 price per 
 statute 
 perch. 
 
 Average 
 price per 
 statute 
 acre. 
 
 Cost to the 
 tenants. 
 
 Kate of wages 
 per day and 
 by task work. 
 
 Marquess of Waterford 
 
 A. R. P. 
 
 501 2 15 
 
 66*900 
 
 9d. in 
 1842, re- 
 duced to 
 5 id. in 
 1845. 
 
 5 in 
 1842, re- 
 duced to 
 3 2s. 8d. 
 in 1845. 
 
 5 per cent, 
 charged to 
 the rent. 
 
 
 
 Viscount Templetown 
 
 564 25 
 
 54,351 
 
 4R 
 
 1 14 4 
 
 500 
 
 .. 
 
 Sir R. O'Donnell, Bart. 
 
 551 
 
 53,478 
 
 U. 
 
 1 12 4 
 
 408 6 
 
 Sd. a day. 
 
 The Earl of Caledon . 
 
 321 19 
 
 47,183 
 
 d. 
 
 295 
 
 396 9 9 
 
 Wd. to 1. a 
 day. 
 
 J. L. W. Naper, Esq. . 
 
 254 1 29 
 
 34,433 
 
 9fd. 
 
 583 
 
 700 7 4 
 
 Id. a day 
 in Winter, 
 andlOd.in 
 Summer. 
 
 Lord Blayney . . . 
 
 201 2 30 
 
 34,634 
 
 5d. 
 
 3 12 
 
 427 18 
 
 lOd. to Is. & 
 day. 
 
 TABLE showing a Return of the number of Acres thorough -drained 
 by Proprietors, for the Society's Gold Medal, and the Average 
 Prices per Perch and per Acre respectively. 
 
 The Earl of Erne . . 
 
 A. R. P. 
 
 110 37 
 
 p. 
 
 .. 
 
 8. d. 
 
 3 12 3 
 
 .. 
 
 .. 
 
 Lord Dufferin and 
 Clanboye . . . . 
 
 203 1 
 
 29,478 
 
 lOfd. 
 
 6 11 9 
 
 , . 
 
 Is. per day. 
 
 Messrs. Andrews . . 
 
 117 1 4 
 
 16,614 
 
 9d. 
 
 5 15 
 
 . 
 
 13d. per day. 
 
 Dr. O'Neill . . . . 
 
 115 12 
 
 
 
 
 
 2 16 3 
 
 
 
 7d. to 8d. a 
 
 day. 
 
360 THE DISCHAEGE OF WATER FROM 
 
 Patent Socketing Machine) are best adapted to loose 
 or mixed soils.'* 
 
 Pipes laid, however, too near the surface, are 
 frequently choked with the roots of plants. The 
 principal advantage of submains alongside open 
 mains is, that the mouths of the minor drains 
 should not be choked from vegetation, and that the 
 water from them, flowing into and taken up by this 
 submain, may be discharged by a few apertures 
 only, and thereby keep themselves open, or as much 
 so as the nature of the case will admit. The fore- 
 going tables show the cost per statute acre, in 
 Ireland, of thorough-drainage, which must vary with 
 circumstances, locality, and the value of labour. 
 
 The average cost per statute acre for Sir Eichard 
 O'Donnell's Gold Medal, was 3 5*. Id., and 
 4 13s. IQd. for the Society's Gold Medal ; average 
 of both, 4 per statute acre nearly. 
 
 The average number of acres now annually im- 
 proved in Ireland, is about 5530, at an average cost 
 per acre of 4 17 s. 
 
 In Ireland, thorough-drainage is almost generally 
 carried out by loan, under the Commissioners of 
 Public Works, and there is no branch of the public 
 service has given more satisfaction to owners of pro- 
 perty. The works are, we believe, always executed 
 within the estimates, and the owner having the ex- 
 penditure in his own hands, can satisfy himself of 
 its proper application. No loans are made unless 
 where immediately, or prospectively, a return of 
 6i per cent, is estimated on the expenditure, a rent 
 charge for this amount being made for 22 years. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 361 
 
 TABLE showing Estimates of the Quantities and General Cost for 
 the Thorough- Drain age of a Statute Acre of Land, with Broken 
 Stones or Tiles, with the distances apart for different class soils. 
 
 Description of land, section 
 of parallel drain, and cost. 
 
 Distance between! 
 the 
 parallel drains. 1 
 
 Lineal 
 perches 
 of 
 drains 
 per 
 statute 
 acre. 
 
 Cubic yards per 
 acre. 
 
 Expense 
 per statute 
 acre. 
 
 Number 
 of feet 
 of tiles 
 per 
 statute 
 acre. 
 
 Of Clay 
 excav. 
 
 Of 
 
 broken 
 stones. 
 
 / 
 
 Feet. 
 
 Perches. 
 
 Cu. yds. 
 
 Cu. yds. 
 
 8. d. 
 
 Number 
 
 
 12 
 
 220 
 
 277 
 
 38-5 
 
 768 
 
 3,630 
 
 
 13 
 
 203 
 
 255| 
 
 35-5 
 
 6 15 4 
 
 3,351 
 
 
 14 
 
 188 
 
 237 
 
 33-0 
 
 658 
 
 3,111 
 
 Hard subsoil stiff and 
 
 15 
 
 176 
 
 221 
 
 30-8 
 
 5 17 4 
 
 2,904 
 
 sandy clay drains, from 
 
 16 
 
 165 
 
 207 
 
 28-9 
 
 5 10 
 
 2,722 
 
 12 to 20 feet apart, at 
 
 17 
 
 155J 
 
 iftj 
 
 27-2 
 
 5 3 6 
 
 2,562 
 
 Sd. per lineal perch. 
 
 18 
 
 146 
 
 184 
 
 25-9 
 
 4 17 6 
 
 2,420 
 
 
 19 
 
 139 
 
 175 
 
 24-3 
 
 4 12 
 
 2,293 
 
 
 20 
 
 132 
 
 166J 
 
 23-1 
 
 480 
 
 2,178 
 
 
 
 
 
 
 
 
 / 
 
 21 
 
 125J 
 
 182 
 
 29-3 
 
 4 14 6 
 
 2,074 
 
 
 22 
 
 120 
 
 174f 
 
 28-1 
 
 4 10 
 
 1,971 
 
 
 23 
 
 115 
 
 167 
 
 23-9 
 
 463 
 
 1,894 
 
 Freestone bottom drains, 
 
 24 
 
 110 
 
 159i 
 
 25-7 
 
 426 
 
 1,815 
 
 from 20 to 30 feet 
 
 25 
 
 105 
 
 153J 
 
 24-7 
 
 3 19 1J 
 
 1,742 
 
 apart, at 9d. per 
 
 26 
 
 101 
 
 147* 
 
 23-7 
 
 3 16 1J 
 
 1,675 
 
 
 27 
 
 97f 
 
 142 
 
 22-9 
 
 3 13 4 
 
 1,613 
 
 
 28 
 
 94 
 
 136J 
 
 22-0 
 
 3 10 6 
 
 1,556 
 
 
 29 
 
 91 
 
 132 
 
 21-3 
 
 383 
 
 1,502 
 
 
 30 
 
 88 
 
 127} 
 
 20-6 
 
 360 
 
 1,452 
 
 
 
 
 
 
 
 
 j 
 
 31 
 
 85 
 
 151J 
 
 24-7 
 
 3 10 10 
 
 1,405 
 
 
 32 
 
 82 
 
 147 
 
 24-0 
 
 389 
 
 1,361 
 
 
 33 
 
 80 
 
 142| 
 
 23-2 
 
 368 
 
 1,320 
 
 Beds of gravel, sand, and 
 
 34 
 
 77f 
 
 188} 
 
 22-6 
 
 3 4 8 
 
 1,280 
 
 rocky stratification, 
 
 35 
 
 75| 
 
 134 
 
 21-9 
 
 3 2 9J 
 
 1,245 
 
 from 30 to 100 feet < 
 
 36 
 
 73* 
 
 1301 
 
 21-3 
 
 3 1 1J 
 
 1,210 
 
 apart, at IQd. per 
 
 37 
 
 71 
 
 126 
 
 20-6 
 
 2 19 2 
 
 1,177 
 
 lineal perch . . 
 
 38 
 
 69 
 
 123 
 
 20-0 
 
 2 17 6 
 
 1,146 
 
 
 
 39 
 
 67* 
 
 119| 
 
 19-6 
 
 2 16 1J 
 
 1,117 
 
 
 40 
 
 65| 
 
 H6 
 
 19-1 
 
 2 14 8 
 
 1,089 
 
 
 100 
 
 27 
 
 47 
 
 8-0 
 
 126 
 
 436 
 
362 THE DISCHARGE OF WATER FROM 
 
 ARTERIAL DRAINAGE. 
 
 The effect of thorough-drainage on the arterial 
 channels of a district, is to discharge the rain-fall 
 into the main channels in a shorter time than before, 
 particularly during wet seasons. This frequently 
 causes floods to rise higher as well as more rapidly. 
 During dry seasons the supply is less, and so far, 
 when it is limited, an injury is done to the adjacent 
 districts requiring it for use. The effect of ob- 
 structions in the main channel is to impound the 
 upland water, sometimes made available for water 
 power or navigation purposes, but in general, to the 
 injury of the drainage of adjacent lands, and the 
 regimen of the river, particularly in flat districts. 
 The arterial drainage in Ireland has effected a vast 
 amount of good, but up to 1853 the estimates appear 
 to have been usually doubled ; the estimates for 
 eleven of these works being 124,647, and the ex- 
 penditure, 293,532. The average cost per acre, on 
 the land improved by these projects, varied from 
 1 19s. Sd. to 5 17s. Id., the average of the 
 eleven districts being 4 3s., which is about the 
 average for thorough-drainage.* 
 
 The following table affords valuable information of 
 the cost of arterial drainage works in Ireland : it 
 is extracted from the Report of the Commissioners 
 of Inquiry, presented to the House of Commons 
 June 16th, 1853. 
 
 * See Parliamentary Eeport, by Sir Richard Griffith, Sir 
 W. Cubitt, and Jas. M. Eendel, June 16th, 1853. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 363 
 
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364 
 
 THE DISCHARGE OF WATEK FROM 
 
 At 
 
 The abstract of 84 arterial drainage awards, made 
 by the Commissioners of Public Works in Ireland, 
 in 1854, gives for different years, 1849 to 1854 
 
 Number of 
 districts 
 drained. 
 
 Total 
 combined 
 catchment 
 acreage of 
 districts. 
 
 Area of 
 flooded lands. 
 
 Average cost of 
 arterial drainage 
 per acre improved 
 by drainage. 
 s. d. 
 
 12 districts 
 
 90,332 
 
 9,453 
 
 3 14 2 
 
 27 
 
 95,582 
 
 11,579 
 
 3 16 7 
 
 19 
 
 237,466 
 
 13,707 
 
 4 17 3 
 
 16 
 
 374,427 
 
 29,452 
 
 3 13 4 
 
 2 
 
 49,840 
 
 3,275 
 
 500 
 
 8 
 
 266,420 
 
 21,033 
 
 394 
 
 84 
 
 1,114,067 
 
 88,501 
 
 3 17 7 
 
 The last line gives the general average, and shows 
 that in these 84 districts, about 1 acre in 13 is the 
 average of flooded lands to^ the catchment area, or 
 8 per cent, nearly. 
 
 l*. 
 
 SE 
 
 f 
 
 CTION 
 
 XIV. 
 
 WATER AND HORSE POWER. FRICTION BRAKE, OR DYNAMO- 
 METER, CALCULATION OF THE EFFECTIVE POWER OF 
 WATER WHEELS. OVERSHOT, UNDERSHOT, AND BREAST 
 VERTICAL WHEELS. HORIZONTAL WHEELS AND TURBINES. 
 HYDRAULIC RAM. WATER ENGINE. 
 
 Taking the representative of a horse's power at 
 33,000 foot-pounds* or 33,000 Ibs. raised one foot 
 
 * 16,500 foot-pounds, or one half of the above, is much nearer 
 the average power of a horse, working for 10 hours only, as the 
 work is ordinarily done through the country ; 33,000 Ibs. raised 
 one foot per minute, is equivalent to 884 tons, nearly, raised one 
 foot in an hour. Therefore, a river discharging 884 tons, over a 
 fall one foot high in an hour, or 884 tons, over a fall 24 feet high 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 365 
 
 high in one minute, the theoretical horse-power of an 
 overfall is expressed by the fall in feet, multiplied by 
 the discharge in cubic feet per minute, the product 
 multiplied by 62 (the weight in Ibs., nearly, of a 
 cubic foot of water), and divided by 33,000. The 
 following table gives the weight in air of a cubic foot 
 of pure water at different temperatures, Fahrenheit's 
 thermometer. 
 
 WEIGHT OF A CUBIC FOOT OF WATER. 
 
 The weight of 36 cubic feet of water is one ton, nearly. 
 
 Tempe- 
 rature, in 
 degrees. 
 
 Weight of a 
 cubic foot of 
 water. Pounds 
 Avoirdupois. 
 
 Tempe- 
 rature, in 
 degrees. 
 
 Weight of a 
 cubic foot of 
 water. Pounds 
 Avoirdupois. 
 
 Tempe- 
 rature, in 
 degrees. 
 
 Weight of a 
 cubic foot of 
 water. Pounds 
 Avoirdupois. 
 
 32 
 
 62 ' 375 
 
 51 
 
 62-365 
 
 69 
 
 62-278 
 
 33 
 
 62-377 
 
 52 
 
 62-363 
 
 70 
 
 62-272 
 
 34 
 
 62-378 
 
 53 
 
 62-359 
 
 71 
 
 62 264 
 
 35 
 
 62-379 
 
 54 
 
 C2-356 
 
 72 
 
 62 257 
 
 36 
 
 62-380 
 
 55 
 
 62-352 
 
 73 
 
 62 249 
 
 37 
 
 62 381 
 
 56 
 
 62 349 
 
 74 
 
 62 242 
 
 38 
 
 62-381 
 
 57 
 
 62 345 
 
 75 
 
 62-234 
 
 39 
 
 62 382 
 
 58 
 
 62-340 
 
 76 
 
 62-225 
 
 40 
 
 62-382 
 
 59 
 
 62 ' 336 
 
 77 
 
 62-217 
 
 41 
 
 62-381 
 
 60 
 
 62-331 
 
 78 
 
 62-208 
 
 42 
 
 62-381 
 
 61 
 
 62-326 
 
 79 
 
 62-199 
 
 43 
 
 62-380 
 
 62 
 
 62-321 
 
 80 
 
 62-190 
 
 44 
 
 62*379 
 
 63 
 
 62-316 
 
 81 
 
 62-181 
 
 45 
 
 62-378 
 
 64 
 
 62-310 
 
 82 
 
 62 172 
 
 46 
 
 62-376 
 
 65 
 
 62-304 
 
 83 
 
 62-162 
 
 47 
 
 62 '375 
 
 66 
 
 62'298 
 
 84 
 
 62-152 
 
 48 
 
 62 '373 
 
 67 
 
 62-292 
 
 85 
 
 62-142 
 
 49 
 
 62-371 
 
 68 
 
 62-285 
 
 86 
 
 62-132 
 
 50 
 
 62-368 
 
 
 
 87 
 
 62-122 
 
 The effective power of a fall depends on the nature, 
 
 in 24 hours, has also a horse power. The drainage of 10 square 
 miles, with an average collection of 12 inches annually of rain in 
 depth, will give an annual unceasing one-horse power for each 
 foot of fall in a receiving channel ; or five square miles will give 
 the same result, if the collection amounts to 24 inches in depth. 
 The collection of 10 square miles, one foot deep, yearly, is nearly 
 equal to the delivery of 530 cubic feet per minute, for the same 
 period. 
 
366 THE DISCHARGE OF WATER FROM 
 
 proportions, and construction of the wheel or 
 machine, and also upon the manner in which the 
 theoretical power is applied. When the velocity of 
 a stream acting on a wheel only is known, the theo- 
 retical head, A, due to it is found in feet from the 
 formula h -0155 v 2 , v being the volocity in feet per 
 second. 
 
 In order to gauge the quantity of water applied to 
 a wheel, and thereby determine with accuracy its 
 effective power, the water used must be passed 
 through a notch, orifice, or over a weir, the coefficients 
 for which had been previously ascertained from expe- 
 riment. Greater accuracy can be obtained from 
 gaugings through thin plates, or planks having the 
 down-stream arrises chamfered, than with any other 
 form of orifice or notch ; and when it can be effected, 
 the channel above should be sufficiently enlarged to 
 prevent the effects of an approaching current. We 
 have already in the body of this work dwelt in detail 
 on the various formulae required for gauging under 
 different circumstances. The accuracy of the results 
 showing the effective powers of wheels depends in 
 the first place, on the accuracy of the gaugings or 
 estimates of the quantity of water used, and next on 
 the fall employed. 
 
 FRICTION BRAKE, OE DYNAMOMETER. 
 
 The power applied to a revolving shaft through a 
 water wheel of any construction, is the weight of 
 water multiplied by the fall. It is evident that the 
 portion of this power available to turn a shaft and 
 machinery, or the effective power, must depend on 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 367 
 
 the construction of the wheel, as a portion of the 
 theoretical power is lost mechanically, in applying it; 
 in changes of direction, friction, eddies, and discharg- 
 ing currents. The greater the effective power con- 
 veyed to a shaft, the greater becomes the power of 
 the wheel, or medium through which the original 
 power is transmitted. The mechanical effect produced 
 by a revolving shaft is best measured by a friction 
 brake, the principle of which is as follows. In 
 Fig. 44, let the friction pulley AA be 
 
 diagram 1, 
 
 FIG. 44. 
 
 FRICTION BRAKES, OR DYNAMOMETERS. 
 
 firmly fixed to the revolving shaft or axis of the wheel ; 
 E and E, two wooden clamps grasping the friction 
 pully by means of the screw bolts, delineated, which 
 can be tightened on the axis, and also to the arm F, 
 by means of suitable nuts. The more tightly the 
 bolts are screwed, the greater will be the friction 
 
368 THE DISCHARGE OF WATEE FROM 
 
 between the friction pulley AA, and the clamping 
 pieces EE. If, while the axis and friction pulley A A, 
 are revolving in the direction indicated by the arrow, 
 a weight be applied in the scale at i, so that the arm 
 F shall not be carried round, but remain fixed ; it is 
 clear that the work done by the revolving shaft in 
 one revolution, will be measured by the circumference 
 of the friction pulley, multiplied by the friction due 
 to the pressure on it, or by its equivalent, the weight 
 in the scale i, multiplied by the circumference of a 
 circle whose radius is L, or by 2 L x 3*14159 x w, in 
 which expression w is the weight in Ibs. in the 
 scale i. If n be the number of revolutions in a 
 given time, say one minute, we shall therefore have 
 the useful effect of the wheel on the shaft in foot- 
 pounds per minute, equal to 
 
 2LX 314159 Xwxn. 
 
 We have also the power of the water acting on the 
 wheel, equal to 
 
 hx DX 62-37, 
 
 in which h is the head and D the discharge in cubic 
 feet per minute ; therefore, we shall have for the ratio 
 of the effect to the power the expression 
 
 2LX 3-14159 x w xn 
 
 hXVX 62-37 
 
 If the revolving shaft be horizontal, the weight of 
 the arm F, acting at its centre of gravity and reduced 
 to the length L, where the weight w is suspended, will 
 have to be included in the weight w. If the weight 
 w be suspended at the end of a connexion of levers, 
 or other mechanical powers, the length L will have to 
 be determined accordingly. Diagram 2, Fig. 44, 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 369 
 
 shows the Armstrong-brake ; Diagram 3, the common 
 form ; and Diagram 4, Egen's brake. 
 
 Fig. 45, is a general representation of the brake 
 
 FIG. 45. 
 
 DYNAMOMETER FOE DETERMINING THE USEFUL EFFECT OF THE 
 TREMONT TURBINE. 
 
 used by Francis, in the Lowell experiments. The 
 length of the arm of the brake L, was 9-745 feet; 
 the length of the vertical arm I of the bell crank 4-5 
 
 BB 
 
370 THE DISCHARGE OF WATER FROM 
 
 feet ; and the length of the horizontal arm /' 5 feet. 
 The following detailed description is by Francis : 
 
 " The Friction Pulley A is of cast-iron, 5 -5 feet in 
 diameter, two feet wide on the face, and three inches 
 thick. It is attached to the vertical shaft by the 
 spider B, the hub of which occupies the place on the 
 shaft intended for the bevel gear. 
 
 " The friction pulley has, cast on its interior circum- 
 ference, six lugs, c c, corresponding to the six arms 
 of the spider. The bolt holes in the ends of the arms 
 are slightly elongated in the direction of the radius, 
 for the purpose of allowing the friction pulley to 
 expand a little as it becomes heated, without throwing 
 much strain upon the spider. When the spider and 
 friction pulley are at the same temperature, the ends 
 of the arms are in contact with the friction pulley. 
 The friction pulley was made of great thickness for 
 two reasons. When the pulley is heated, the arms 
 cease to be in contact with the interior circumference 
 of the pulley, consequently they would not prevent 
 the pressure of the brake from altering the form of 
 the pulley. This renders great stiffness necessary in 
 the pulley itself. Again, it is found that a heavy 
 friction pulley insures more regularity in the motion, 
 operating, in fact, as a fly-wheel, in equalizing small 
 irregularities. 
 
 " The brakes E and F are of maple wood ; the two 
 parts are drawn together by the wrought-iron bolts 
 c c, which are two inches square. 
 
 " The bell crank F', carries at one end the scale i, and 
 at the other the piston of the hydraulic regulator K ; 
 this end carries also the pointer L, which indicates 
 
ORIFICES, WEIRS, PIPES, AND RIVERS, 371 
 
 the level of the horizontal arm. The vertical arm is 
 connected with the brake F, by the link M. 
 
 " The hydraulic regulator K, shown in the figures, is a 
 very important addition to the Prony dynamometer, 
 first suggested to the author by Mr. Boy den, in 1844. 
 Its office is to control and modify the violent shocks 
 and irregularities, which usually occur in the action 
 of this valuable instrument, and are the cause of some 
 uncertainty in its indications. 
 
 "The hydraulic regulator used in these experiments, 
 consisted of the cast-iron cylinder K, about 1*5 feet 
 in diameter, with a bottom of plank, which was 
 strongly bolted to the capping stone of the wheel pit, 
 as represented in figure 1. In this cylinder, moves 
 the piston N, formed of plate-iron 0-5 inches thick, 
 which is connected with the horizontal arm of the bell 
 crank by the piston rod o. The circumference of the 
 piston is rounded off, and its diameter is about TV inch 
 less than the diameter of the interior of the cylinder. 
 The action of the hydraulic regulator is as follows. 
 The cylinder should be nearly filled with water, or 
 other heavy inelastic fluid. In case of any irregularity 
 in the force of the wheel, or in the friction of the 
 brake, the tendency will be, either to raise or lower 
 the weight, in either case the weight cannot move, 
 except with a corresponding movement of the piston. 
 In consequence of the inelasticity of the fluid, the 
 piston can move only by the displacement of a portion 
 of the fluid, which must evidently pass between the 
 edge of the piston and the cylinder ; and the area of 
 this space being very small, compared to the area of 
 the piston, the motion of the latter must be slow, 
 
 BBS 
 
372 THE DISCHARGE OF WATER FROM 
 
 giving time to alter the tension of the brake screws 
 before the piston has moved far. It is plain that 
 this arrangement must arrest all violent shocks, but, 
 however violent and irregular they may be, it is 
 evident, that, if the mean force of them is greater in 
 one direction than in the other, the piston must move 
 in the direction of the preponderating force, the 
 resistance to a slow movement being very slight. A 
 small portion of the useful effect of the turbine must 
 be expended in this instrument, probably less, how- 
 ever, than in the rude shocks the brake would be 
 subject to without its use. 
 
 " For the purpose of ascertaining the velocity of 
 the wheel, a counter was attached to the top of the 
 vertical shaft, so arranged, that a bell was struck at 
 the end of every fifty revolutions of the wheel. 
 
 " To lubricate the friction pulley, and at the same 
 time to keep it cool, water was let on to its surface 
 in four jets, two of which are shown. These jets were 
 supplied from a large cistern, in the attic of the 
 neighbouring cotton mill, kept full, during the work- 
 ing hours of the mill, by force pumps. The quantity 
 of water discharged by the four jets was, by a mean 
 of two trials, 0-0288 cubic feet per second. 
 
 " In many of the experiments with heavy weights, 
 and consequently slow velocities, oil was used to 
 lubricate the brake, the water, during the experiment, 
 being shut off. It is found, that, with a small 
 quantity of oil, the friction between the brake and 
 the pulley is much greater than when the usual 
 quantity of water is applied; consequently, the 
 requisite tension of the brake screws is much less 
 
OKIFICES, WEIES, PIPES, AND RIVERS. 373 
 
 with the oil, as a lubricator, than with water. This 
 may not be the whole cause of the phenomenon, but, 
 whatever it may be, the ease of regulating in slow 
 velocities is incomparably greater with oil as a 
 lubricator, than with water applied in a quantity 
 sufficient to keep the pulley cool. The oil was 
 allowed to flow on in two fine continuous streams ; 
 it did not, however, prevent the pulley from becoming 
 heated sufficiently to decompose the oil, after running 
 some time, which was distinctly indicated by the 
 smoke and peculiar odour. When these indications 
 became very apparent, the experiment was stopped, 
 and water let on by the jets, until the pulley was 
 cooled. As the pulley became heated, the brake 
 screws required to be gradually slackened. Water, 
 linseed oil, and resin oil, were each used for lubrica- 
 tion." 
 
 Fig. 46 is a representation of a brake used by 
 Professor Thomson, at Crawford and Lindsay's mill, 
 to determine the power of a turbine put up there, 
 by Mr. Gardner, of Armagh. One of the common 
 causes of the swinging or vibratory jumps of the arms 
 F, in Figs. 44, 45, and 46 is, that very often the 
 friction pulley, or drum A, A, must be made in two 
 parts, so as to be fixed to its place on the shaft. 
 This fixing is liable to give an oval shape, and causes 
 an irregular action with the clamps E, E. 0S gives 
 greater regularity of motion than water, but without 
 the use of the latter abundantly, the friction pulley 
 would usually get too much heated. The following 
 calculations from practical operations will point out 
 pretty clearly the use of the brake, and the manner 
 
374 
 
 THE DISCHARGE OF WATER FROM 
 
 of determining the useful effect in the tables of expe- 
 riments, by Francis and Thomson, pp. 383 and 392 : 
 
 FIG. 46. 
 
 FRICTION BRAKE, USED AT CRAWFORD AND LINDSAY'S MILL, BY PROFESSOR 
 JAMES THOMSON, TO DETERMINE THE POWER OF THE TURBINE. 
 
 Length of the brake, L, Fig. 45, adjusted . . . 9-745 feet. 
 
 Effective length of vertical arm I 4-5 
 
 Effective length of horizontal arm I 1 .... 5'0 
 
 The effective length of the brake was therefore 
 9-745 x 5 
 
 4-5 
 
 = 2 x 10 -827778 feet; and the circumference 
 
 of a circle of this radius = 10-827778 x 3-14159 = 
 68-0329 feet. 
 In the first experiment on the Tremont turbine, 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 375 
 
 page 393, the number of revolutions of the wheel per 
 second was '89374, and the weight in the scale 
 1443-34 Ibs. The useful effect of the brake was 
 therefore in foot-pounds per second 68-0329 x 
 89374 x 1443-34 = 87680-3 Ibs. raised one foot per 
 second. The quantity of water which passed the 
 gauge-weir in cubic feet per second was 139-4206, 
 and the total fall acting on the wheel 12-864 feet; 
 therefore, the total power of the water acting on the 
 wheel was 12-864x1394206x62-375=111870 feet- 
 pounds per second, 62-375 being taken as the weight 
 in Ibs. of a cubic foot of water at 32 Fahrenheit. 
 The ratio of the useful effect, at the given velocity of 
 the wheel (viz. 450 revolutions in 503*5 seconds), to 
 
 Qrr/^or\ .0 
 
 the power expended, is therefore i 11070 = '784, or 
 
 about 78j per cent. The effect in the experiments 
 generally appears to have been a maximum, when 
 the velocity of the interior circumference of the wheel 
 was about 66 per cent, of the velocity due to the fall ; 
 and this was about half of the maximum velocity, 
 which was 1-333 times that due to the fall alone, 
 when the turbine was doing no work. 
 
 In Thomson's brake for determining the useful 
 effect of the vortex turbine, erected from his designs 
 at Bally sillan, Ireland, L = 4 ft. 2 in., and the cir- 
 cumference of a circle that would be described by the 
 arm 3-14159 x 8 feet 4 inches. = 26-18 feet. In the 
 first experiment, taken from the tabulated results, 
 page 383, we get 26-18, the circumference multiplied 
 by 46-31, the weight in Ibs., and the product by 
 323*3, the number of revolutions per minute, equal 
 
376 THE DISCHARGE OF WATER FROM 
 
 to 391,967 foot-pounds, for the effect transmitted 
 from the turbine or work done. We have also 3544, 
 the number of cubic feet of water passed to the 
 wheel per minute, multiplied by 62*37, the weight of 
 a cubic foot of water in Ibs., multiplied by 23*73 feet, 
 
 the available fall, equal to 524,526 foot-pounds : 
 
 391 967 
 therefore Swa '747 is the useful effect, that in 
 
 the table being -7481, which probably arose from 
 taking a different weight per cubic foot for the water. 
 Of course the difference is immaterial. The drum 
 attached to the vortex wheel shaft for fixing the 
 brake to, was in two parts, bolted together, and 
 firmly enclosing the shaft. It was of cast-iron, 20 
 inches diameter, and 8 inches wide ; the shaft to 
 which it was attached was 2f inches diameter. The 
 arm of the brake was 5''4" x 6" x 4", of timber, and 
 extending 1 foot 2 inches beyond the centre of the 
 shaft and drum. The clamping pieces were about 
 2 feet 5 inches long externally. 
 
 FOR OVERSHOT WHEELS the ratio of the power to 
 the effect may be taken as 3 to 2, and therefore the 
 effective horse-power, taking 33,000 foot-pounds 
 per minute as a standard, will be 49,500 Ibs. of 
 water falling one foot in one minute. The maximum 
 effect varies with the construction of the wheel. 
 Smeaton found it '76 times the theoretical power ; 
 Weisbach -78 times for the wheel of a Stamp Mill 
 at Frieberg, which was 23 feet high, 3 feet wide, 
 carrying 48 buckets.* To find the effective horse- 
 
 * Some valuable experiments on the power of water wheels are 
 given by Rennie, in Weale's Quarterly Papers of Engineering, 
 vol. vi. They however require reduction. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 377 
 
 power, the theoretical horse-power must be here di- 
 vided by the coefficient of effect -76 or -78, which will 
 give 43,600 foot-pounds, or 43,300 foot-pounds per 
 minute. The following experimental results from a 
 model wheel are by Smeaton. 
 
 TABLE containing the Result of Sixteen Experiments, on a Model 
 Overshot Wheel, by Smeaton. 
 
 
 .9 
 
 S . 
 
 S 
 
 -1 
 
 4J 
 
 o 
 
 
 
 iS ^ 
 
 If . 
 
 
 
 4 
 
 'SJ! 
 
 'p_g 
 
 |.J 
 
 1 . 
 
 J 
 
 
 "f o 
 
 
 
 
 
 <g 
 
 "a .9 
 
 S a 
 
 a 1 
 
 jl 
 
 "o" 
 
 -s 
 
 I.S 
 
 oS^ 
 
 1 
 
 No 
 
 
 I.r 
 
 S 
 
 
 
 *< r 
 
 
 
 ^ *"* "S 
 
 
 
 o.9 
 
 
 *" eS 
 
 t'B 
 
 
 
 |p 
 
 N 
 
 oo 
 
 Ifl 
 
 i 
 
 
 o 
 
 |1 
 
 !ji 
 
 g 
 
 
 ft 
 
 
 
 I? 1 
 
 S 
 
 
 
 P 
 
 s 
 
 ^ 
 
 ft 
 
 
 
 fl 
 
 B 
 
 
 1 
 
 27 
 
 30 
 
 19 
 
 9| 
 
 810 
 
 720 
 
 556 
 
 69 
 
 77 
 
 
 2 
 
 27 
 
 56f 
 
 161 
 
 14J 
 
 1530 
 
 1360 
 
 1060 
 
 69 
 
 78 
 
 
 3 
 
 27 
 
 56| 
 
 20| 
 
 12^ 
 
 1530 
 
 1360 
 
 1167 
 
 76 
 
 84 
 
 rH 
 
 00 
 
 4 
 
 27 
 
 63| 
 
 20* 
 
 13f 
 
 1710 
 
 1524 
 
 1245 
 
 73 
 
 82 
 
 
 5 
 
 27 
 
 78| 
 
 21* 
 
 15i 
 
 2070 
 
 1840 
 
 1500 
 
 73 
 
 82 
 
 
 6 
 
 28* 
 
 73* 
 
 18f 
 
 17J 
 
 2090 
 
 1764 
 
 1476 
 
 70 
 
 84 
 
 S3 
 
 7 
 
 28* 
 
 96| 
 
 20J 
 
 
 2755 
 
 2320 
 
 1868 
 
 68 
 
 80 
 
 
 8 
 
 30 
 
 90 
 
 20 
 
 1H 
 
 2700 
 
 2160 
 
 1755 
 
 65 
 
 81 
 
 
 9 
 
 30 
 
 96| 
 
 aof 
 
 20^ 
 
 2900 
 
 2320 
 
 1914 
 
 66 
 
 82 
 
 00 
 
 10 
 
 30 
 
 113* 
 
 21 
 
 23* 
 
 3400 
 
 2720 
 
 2221 
 
 65 
 
 82 
 
 
 11 
 
 33 
 
 56| 
 
 20* 
 
 13* 
 
 1870 
 
 1360 
 
 1230 
 
 66 
 
 90 
 
 
 12 
 13 
 
 33 
 33 
 
 106f 
 146f 
 
 23 
 
 dil 
 
 27* 
 
 3520 
 4840 
 
 2560 
 3520 
 
 2153 
 2846 
 
 61 
 
 59 
 
 84 
 81 
 
 * 
 
 14 
 
 35 
 
 65 
 
 19f 
 
 16* 
 
 2275 
 
 1560 
 
 1466 
 
 65 
 
 95 
 
 
 15 
 
 35 
 
 120 
 
 9if 
 
 25* 
 
 4200 
 
 2880 
 
 2467 
 
 59 
 
 86 
 
 00 
 
 16 
 
 35 
 
 163* 
 
 25 
 
 
 5728 
 
 3924 
 
 2981 
 
 52 
 
 76 
 
 
 In this table the effective power of the water 
 must be reckoned upon the whole descent, because it 
 must be raised that height, in order to be in a 
 condition of producing the same effect a second time. 
 
378 THE DISCHARGE OF WATER FROM 
 
 The ratios between the powers so estimated, and 
 the effects at the maximum, deduced from the several 
 sets of experiments, are exhibited at one view, in 
 column 9, of Table II. ; and from hence it appears, 
 that those ratios differ from that of 10 to 7-6 to that 
 of 10 to 5-2, that is, nearly from 4 to 3 to 4 to 2. In 
 those experiments where the heads of water and 
 quantities expended are least, the proportion is nearly 
 as 4 to 3, but where the heads and quantities are 
 greatest, it approaches nearer to that of 4 to 2 ; and 
 by a medium of the whole, the ratio is that of 3 to 2, 
 nearly. We have seen before, in our observations 
 upon the effects of undershot wheels, that the general 
 ratio of the power to the effect, when greatest, was 3 
 to 1 ; the effect, therefore, of overshot wheels, under the 
 same circumstances of quantity and fall, is at a 
 medium double to that of the undershot ; and, as a 
 consequence thereof y that non-elastic bodies, when acting 
 by their impulse or collision, communicate only a part 
 of their original power ; the other part being spent 
 in changing their figure, in consequence of the 
 stroke. 
 
 The powers of water, computed from the height of 
 the wheel only, compared with the effects as in 
 column 10, appear to observe a more constant ratio : 
 for, if we take the medium of each class, which is set 
 down in column 11, we shall find the extremes to 
 differ no more than from the ratio of 10 to 8*1 to that 
 of 10 to 8'5 ; and as the second term of the ratio 
 gradually increases from 8-1 to 8-5, by an increase of 
 head from 3 inches to 11, the excess of 8-5 above 8-1 
 is to be imputed to the superior impulse of the water 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 379 
 
 at the head of 11 inches, above that of 3 inches : so 
 that if we reduce 8-1 to 8, on account of the impulse 
 of the 3-inch head, we shall have the ratio of the power, 
 computed upon the height of the wheel only, to the 
 effect at a maximum, as 10 to 8 or as 5 to 4, nearly ; 
 and from the equality of the ratio between power and 
 effect subsisting, where the constructions are similar, 
 we must infer, that the effects, as well as the powers, 
 are as the quantities of water and perpendicular 
 heights, multiplied together respectively. 
 
 FOR BREAST WHEELS, the ratio of the theoretical 
 power to the effective power must vary considerably, 
 the mean value being about 1 to *5 and, therefore, the 
 effective horse-power would be 66,000 foot-pounds 
 in one minute. Morin gives an efficiency of from *52 
 to -7. Egen, with a wheel 23 feet in diameter, 4i feet 
 wide, having 69 ventilated buckets, very well con- 
 structed, found at best an efficiency of only -52, 
 under ordinary circumstances '48, the mean amount 
 being -5. Very wide wheels give a larger effect, 
 sometimes as high as '7 ; but a great deal depends on 
 the manner of bringing on the water and the con- 
 struction of the wheel and buckets. 
 
 FOR UNDERSHOT WHEELS the mean effect may be 
 taken at one-third or -33, or 100,000 foot-pounds in 
 one minute for an effective horse's power; a maximum 
 effect of -5 is sometimes approached, and a minimum 
 of -26 or less. The following results, obtained from 
 a model, are given by Smeaton. The virtual or 
 effective head is here termed the theoretical head 
 due to the velocity of the wheel, at the circum- 
 ference, which was 75 inches girth. 
 
380 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE containing the Eesult of Twenty-seven Experiments, on a 
 Model Undershot Wheel, by Smeaton. 
 
 
 1 
 
 
 'S rf 
 
 
 
 
 c! 
 
 
 
 1 
 
 
 
 
 
 
 SI 
 
 1 
 
 g S 
 
 Srfl 
 
 TJj 
 
 . 
 
 
 & 
 
 
 
 "5 ,2 
 
 '"SI! 
 
 ill 
 
 
 fi 
 
 ti 
 
 j| 
 
 || 
 
 l& 
 
 *l 8 
 
 1| 
 
 | 
 
 | 
 
 . < 
 
 m 
 
 J2 
 
 <B 
 
 
 
 "o g 
 
 11 
 
 11 
 
 " a 
 
 02 
 
 || 
 
 ^21 
 
 l| 
 
 i 
 
 H 
 
 jl o 
 
 || 
 
 ?|S 
 
 
 i) 
 
 p 
 
 p3 
 
 Q 
 
 fl 
 
 +* -3 
 
 4-> 
 
 
 
 
 o f* 
 
 o'S * 
 
 
 .SP.fH 
 
 H 
 
 P* 
 
 H 
 
 
 
 * 
 
 
 
 2 I 
 
 H 
 
 ri 
 
 1 
 
 33 
 
 88 
 
 15-85 
 
 so- 
 
 13 10 
 
 10 9 
 
 275- 
 
 4358 
 
 1411 
 
 324 
 
 340 
 
 775 
 
 2 
 
 30 
 
 86 
 
 15-0 
 
 so- 
 
 12 10 
 
 9 6 
 
 264-7 
 
 3970 
 
 1266 
 
 320 
 
 350 
 
 740 
 
 3 
 
 27 
 
 82 
 
 13-7 
 
 28- 
 
 11 2 
 
 8 6 
 
 243- 
 
 3329 
 
 1044 
 
 315 
 
 340 
 
 750 
 
 4 
 
 24 
 
 78 
 
 12-3 
 
 27-7 
 
 9 10 
 
 7 5 
 
 235- 
 
 2890 
 
 901-4 
 
 312 
 
 355 
 
 753 
 
 5 
 
 21 
 
 76 
 
 11-4 
 
 25-9 
 
 8 10 
 
 6 5 
 
 214- 
 
 2439 
 
 735-7 
 
 302 
 
 345 
 
 732 
 
 6 
 
 18 
 
 70 
 
 9-95 
 
 23-5 
 
 6 10 
 
 5 5 
 
 199- 
 
 1970 
 
 561-8 
 
 285 
 
 336 
 
 802 
 
 7 
 
 15 
 
 65 
 
 8-54 
 
 23-4 
 
 5 2 
 
 4 4 
 
 178-5 
 
 1524 
 
 442-5 
 
 290 
 
 360 
 
 830 
 
 8 
 
 12 
 
 60 
 
 7-29 
 
 22- 
 
 3 10 
 
 3 5 
 
 161- 
 
 1173 
 
 328 
 
 280 
 
 377 
 
 910 
 
 9 
 
 9 
 
 52 
 
 5-47 
 
 19- 
 
 2 12 
 
 2 8 
 
 134- 
 
 733 
 
 213-7 
 
 290 
 
 365 
 
 910 
 
 10 
 
 6 
 
 42 
 
 3-55 
 
 16- 
 
 1 12 
 
 1 10 
 
 114- 
 
 404-7 
 
 117-0 
 
 282 
 
 380 
 
 930 
 
 11 
 
 24 
 
 84 
 
 14-2 
 
 30-75 
 
 13 10 
 
 10 14 
 
 342- 
 
 4890 
 
 1505 
 
 307 
 
 366 
 
 790 
 
 12 
 
 21 
 
 81 
 
 13-5 
 
 29- 
 
 11 10 
 
 9 6 
 
 297- 
 
 4009 
 
 1223 
 
 301 
 
 362 
 
 805 
 
 13 
 
 18 
 
 72 
 
 10-5 
 
 26- 
 
 9 10 
 
 8 7 
 
 285- 
 
 2993 
 
 975 
 
 325 
 
 360 
 
 875 
 
 14 
 
 15 
 
 69 
 
 9-6 
 
 25- 
 
 7 10 
 
 6 14 
 
 277- 
 
 2659 
 
 774 
 
 292 
 
 362 
 
 900 
 
 15 
 
 12 
 
 63 
 
 8-0 
 
 25- 
 
 5 10 
 
 4 14 
 
 234- 
 
 1872 
 
 549 
 
 294 
 
 397 
 
 870 
 
 16 
 
 9 
 
 56 
 
 6-37 
 
 23- 
 
 4 
 
 3 13 
 
 201- 
 
 1280 
 
 390 
 
 305 
 
 410 
 
 950 
 
 17 
 
 6 
 
 46 
 
 4-25 
 
 21- 
 
 2 8 
 
 2 4 
 
 167-5 
 
 712 
 
 212 
 
 298 
 
 455 
 
 900 
 
 18 
 
 15 
 
 72 
 
 10-5 
 
 29- 
 
 11 10 
 
 9 6 
 
 357- 
 
 3748 
 
 1210 
 
 323 
 
 402 
 
 805 1 
 
 19 
 
 12 
 
 66 
 
 8-75 
 
 26-75 
 
 8 10 
 
 7 6 
 
 330- 
 
 2887 
 
 878 
 
 305 
 
 405 
 
 810 
 
 20 
 
 9 
 
 58 
 
 6-8 
 
 24-5 
 
 5 8 
 
 5 
 
 255- 
 
 1734 
 
 541 
 
 301 
 
 422 
 
 910 
 
 21 
 
 6 
 
 48 
 
 4-7 
 
 23-5 
 
 3 2 
 
 3 
 
 228- 
 
 1064 
 
 317 
 
 299 
 
 490 
 
 960 
 
 22 
 
 12 
 
 68 
 
 9-3 
 
 27- 
 
 9 2 
 
 8 6 
 
 359- 
 
 3338 
 
 1006 
 
 302 
 
 397 
 
 917. 1 
 
 23 
 
 9 
 
 58 
 
 6-8 
 
 26-25 
 
 6 2 
 
 5 13 
 
 332- 
 
 2257 
 
 686 
 
 904 
 
 452 
 
 950 
 
 24 
 
 6 
 
 48 
 
 4-7 
 
 24-5 
 
 3 12 
 
 3 8 
 
 262- 
 
 1231 
 
 385 
 
 313 
 
 510 
 
 935 
 
 25 
 
 9 
 
 60 
 
 7-29 
 
 27-3 
 
 6 12 
 
 6 6 
 
 355- 
 
 2588 
 
 785 
 
 303 
 
 455 
 
 945 1 
 
 26 
 
 6 
 
 50 
 
 5-03 
 
 24-6 
 
 4 6 
 
 4 1 
 
 307- 
 
 1544 
 
 450 
 
 292 
 
 490 
 
 930 I 
 
 27 
 
 6 
 
 50 
 
 5-03 
 
 26- 
 
 4 15 
 
 4 9 
 
 360- 
 
 1811 
 
 534 
 
 295 
 
 520 
 
 925 1 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 381 
 
 Smeaton derived the following " maxims " from 
 the foregoing experiments. Their truth, independent 
 of any experiment, will be apparent : 
 
 /. That the virtual or effective head being the same, the effect will 
 be nearly as the quantity of water expended. 
 
 II. That the eoopense of water being the same, the effect will be 
 nearly as the height of the virtual or effective head. 
 
 III. That the quantity of water expended being the same, the effect 
 is nearly as the square of the velocity. 
 
 IV. The aperture being the same, the effect will be nearly as the cube 
 of the velocity of the water. 
 
 FOR TURBINES OR HORIZONTAL WHEELS, a USeM 
 
 effect of two-thirds or '67 may be assumed, or 
 49,500 foot-pounds in a minute for a horse-power, 
 and the efficiency varies from -5 to -8, or less.* Pon- 
 celet's turbine gives an efficiency of *5 to *6. Float- 
 ing wheels -38, impact wheels from -16 to *4, and 
 Barker's mill from -16 to -35. We believe that the 
 efficiency of the turbine has been too often over- 
 estimated, and that the great advantage of this 
 wheel, as a medium of power, is derived from its 
 capability of employment for all falls, whether large 
 or small, without any considerable loss of effect. 
 In Ireland, Mr. Gardner, of Armagh, was amongst 
 the first, if not the first, to apply this wheel to prac- 
 tical purposes ; and Professor Thomson has, in his 
 
 * In our first edition we gave an efficiency of -821, on the 
 authority of a paper by Dr. Robinson, Armagh, in the Proceed- 
 ings of the Royal Irish Academy, vol. iv., p. 914. On again 
 glancing over this paper, we believe there are mistakes, which 
 vitiate the results there given; first, in the formula for calcu- 
 lating the discharge over the weir, and next, in the formula for 
 finding the effect of the brake. Francis gives an efficiency of -88, 
 p. 3, his book. 
 
382 THE DISCHARGE OF WATER FROM 
 
 vortex wheels, produced, we believe, the highest effi- 
 ciencies which have yet been obtained in practice. 
 In the experiments on the Ballysillan wheel, higher 
 efficiencies would probably have been attained with a 
 supply pipe of larger diameter. It will be seen from 
 the remarks, at pp. 171 and 172, and the tables, at 
 pp. 152 and 191, that quite apart from bends, &c., 
 a loss of mechanical power always results from the 
 passage through orifices and pipes ; and that it 
 is necessary to take this loss into account, before 
 the head acting on the wheel can be accurately used 
 to determine its effective power. The table, next page, 
 contains the experiments on the Ballysillan turbine. 
 
 The following remarks on the vortex turbine, read 
 at the meeting of the British Association at Belfast, 
 in 1852, are also by Professor Thomson : 
 
 "Numberless are the varieties, both of principle 
 and of construction, in the mechanisms by which 
 motive power may be obtained from falls of water. 
 The chief modes of action of the water are, however, 
 reducible to three, as follows : First, the water may 
 act directly by its weight on a part of the mechanism 
 which descends while loaded with water, and ascends 
 while free from load. The most prominent example 
 of the application of this mode is afforded by the 
 ordinary bucket water wheel. Secondly, the water 
 may act by fluid pressure, and drive before it some 
 yielding part of a vessel by which it is confined. 
 This is the mode in which the water acts in the water 
 pressure engine, analogous to the ordinary high-pres- 
 sure steam-engine. Thirdly, the water, having been 
 brought to its place of action subject to the pressure 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 383 
 
 the 
 ppo 
 nts 
 
 time 
 o the 
 xperi 
 
 at 
 atin 
 f th 
 
 Remarks 
 ments, 
 
 VOM a 
 
 Aq :mo U<JA]# H-'OM. ao 
 
 aaujft. am jo Xouapuja 
 
 a^nujtu jad 
 spunod-}ooj uj Ja 
 ire* am 0} anp H 
 
 ^3 
 
 rH C5 O I I O5 iH CO O 00 II H< i I CO 
 00 C2 ir ** ^ ' ?ri ""^ f^t c^ f*d co 
 4i< CO 
 t- t~ 
 
 oocooo 000000 
 
 OOt-OO'f O O O C CO 
 
 co co co co co O? c: ac X ac oo co 
 
 OOOOOO 
 
 t-COOC?CO t CO O5 O CO b- O 
 i O C 
 TH iH 
 
 agnujtu jad spunod 
 ooj ' 
 
 ye :mo uaAj;H H-IO.M. 
 
 a*v 0} papnpap autaq 
 .( 'paiiM am oj pa^d 
 ns 'ajuujni aad ja^^V 
 
 o ^aaj oiquo jo aaqmn.M 
 
 a^nujm aad .lai^AV. 
 o jaaj ojqno jo aaqiunij 
 
 0000 
 
 oooo 
 
 ooo 
 
 ooo 
 
 ooo 
 
 ooo 
 oocooooosoo* 
 
 CO O CO CO CO O O 
 CO CO CO CO CO CO CO 
 
 qp op op op op op 
 
 o o o o o o 
 o o o o o o 
 
 CO CO CC CO CO CO 
 
 OD X X 00 3D O CO 
 CO CO CO CO CO CO CO 
 
 xaoooooaooo 
 
 7 1 rH rl i ii 1 i 1 
 
 oococo 
 
 COCOCOCO 
 
 CO i I C? ^^ i ( r-l b-OOOO>OO 
 
 cococbcocbcb cocbcbcbcbcoco 
 
 luacujjadxg Sujjnp 
 uoijniOAa^ jo aaqoim 
 
 spunod ui y 
 pioo uo }n3j 
 
 CO O t-- CO C* 
 
 CO O rH ^H O rH 
 
 l>OCOCOrtHr-IOO 
 
 in 
 
 
 COCO7OCO CO'CO CO 
 
 TtfrJtCOCCCOOO 
 
 T-IC?COTJ<OCO t- 00 C: O r- 1 C CO 
 
 *3|fS 
 
 1 filial 
 s *:i^s| 
 
 a &'" & . 
 rs 3* fi ^ "g 5 fe 
 
 3 5 I" 3 5 
 
 - Mitt 
 .? vffJlS 
 
 S -g M .3 
 
 a 
 
 N^l^iap 
 
 tl 
 
 
384 THE DISCHAEGE OF WATEE FEOM 
 
 due to the height of fall, may be allowed -to issue 
 through small orifices with a high velocity, its inertia 
 being one of the forces essentially involved in the 
 communication of the power to the moving part of 
 the mechanism. Throughout the general class of 
 water wheels called turbines, which is of wide extent, 
 the water acts according to some of the variations of 
 which this third mode is susceptible. In our own 
 country, and more especially on the Continent, turbines 
 have attracted much attention, and many forms of 
 them have been made known by published descrip- 
 tions. The subject of the present communication is a 
 new water wheel, which belongs to the same general 
 class, and which has recently been invented and 
 brought successfully into use by the author. 
 
 "In this machine the moving wheel is placed within 
 a chamber of a nearly circular form. The water is 
 injected into the chamber tangentially at the circum- 
 ference, and thus it receives a rapid motion of rotation. 
 Retaining this motion it passes onwards towards the 
 centre, where alone it is free to make its exit. The 
 wheel, which is placed within the chamber, and which 
 almost entirely fills it, is divided by thin partitions 
 into a great number of radiating passages. Through 
 these passages the water must flow on its course 
 towards the centre ; and in doing so it imparts its own 
 rotatory motion to the wheel. The whirlpool of water 
 acting within the wheel chamber, being one principal 
 feature of this turbine, leads to the name Vortex as 
 a suitable designation for the machine as a whole. 
 
 " The vortex admits of several modes of construc- 
 tion, but the two principal forms are the one adapted 
 
OKITICES, WEIKS, PIPES, AND RIVERS. 385 
 
 for high falls and the one for low falls. The former 
 may be called the High-pressure Vortex, and the 
 latter the Low-pressure Vortex. Examples of these 
 two kinds are in operation at two mills near Belfast. 
 
 " The height of the fall for the first vortex is about 
 37 feet, and the standard or medium quantity of water, 
 for which the dimensions of the various parts of the 
 wheel and case are calculated, is 540 cubic feet per 
 minute. With this fall and water-supply the esti- 
 mated power is 28 horse-power, the efficiency being 
 taken at 75 per cent. The proper speed of the wheel, 
 calculated in accordance with its diameter and the 
 velocity of the water entering its chamber, is 355 
 revolutions per minute. The diameter of the wheel 
 is 22* inches, and the extreme diameter of the case is 
 4 feet 8 inches. 
 
 "In the second vortex, the fall being taken at 7 
 feet, the calculated quantity of water admitted, at the 
 standard opening of the [guide-blades, is 2,460 cubic 
 feet per minute. Then, the efficiency of the wheel 
 being taken at 75 per cent., its power will be 24 
 horse-power. Also, the speed at which the wheel is 
 calculated to revolve is 48 revolutions per minute. 
 
 "The two examples which have now been described 
 of vortex water wheels, adapted for very distinct cir- 
 cumstances, will serve to indicate the principal features 
 in the structural arrangements of these new machines 
 in general. Respecting their principles of action some 
 further explanations will next be given. In these 
 machines the velocity of the circumference is made 
 the same as the velocity of the entering water, and 
 thus there is no impact between the water and the 
 
 c c 
 
386 THE DISCHARGE OF WATER FROM 
 
 wheel ; but, on the contrary, the water enters the 
 radiating conduits of the wheel gently, that is to say, 
 with scarcely any motion in relation to their mouths. 
 In order to attain the equalization of these velocities, 
 it is necessary that the circumference of the wheel should 
 move with the velocity which a heavy body would attain, 
 in falling through a vertical space equal to half the 
 vertical fall of the water, or in other words, with the 
 velocity due to half the fall ; and that the orifices 
 through which the water is injected into the wheel- 
 chamber should be conjointly of such area that when 
 all the water required is flowing through them, it also 
 may have a velocity due to half the fall. Thus one-half 
 only of the fall is employed in producing velocity in 
 the water ; and, therefore, the other half still remains 
 acting on the water within the wheel-chamber at the 
 circumference of the wheel, in the condition of fluid 
 pressure. Now, with the velocity already assigned 
 to the wheel, it is found that this fluid pressure is 
 exactly that which is requisite to overcome the cen- 
 trifugal force of the water in the wheel, and to bring 
 the water to a state of rest at its exit ; the mechanical 
 work due to both halves of the fall being transferred 
 to the wheel during the combined action of the moving 
 water and the moving wheel. In the foregoing state- 
 ments, the effects of fluid friction, and of some other 
 modifying influences, are, for simplicity, left out of 
 consideration ; but in the practical application of the 
 principle, the skill and judgment of the designer 
 must be exercised in taking all such elements, as far 
 as possible, into account. To aid in this, some prac- 
 tical rules, to which the author as yet closely adheres, 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 387 
 
 were made out by him previously to the date of his 
 patent. These are to be found in the specification of 
 the patent, published in the Mechanics' Magazine for 
 January 18 and January 25, 1851 (London). 
 
 " In respect to the numerous modifications of con- 
 struction and arrangement which are admissible in 
 the vortex, while the leading principles of action are 
 retained, it may be sufficient here merely to advert, 
 first, to the use of straight instead of curved radiating 
 passages in the wheel ; secondly, to the employment, 
 for simplicity, of invariable entrance orifices, or of 
 fixed instead of moveable guide-blades ; and lastly, to 
 the placing of the wheel at any height, less than about 
 thirty feet, above the water in the tail-race, combined 
 with the employment of suction pipes descending 
 from the central discharge orifices, and terminating 
 in the water of the tail-race, so as to render available 
 the part of the fall below the wheel. 
 
 " In relation to the action of turbines in general, 
 the chief and most commonly recognized conditions, 
 of which the accomplishment is to be aimed at, are 
 that the water should flow through the whole machine 
 with the least possible resistance, and that it should 
 enter the moving wheel without shock, and be dis- 
 charged from it with only a very inconsiderable 
 velocity. The vortex is in a remarkable degree 
 adapted for the fulfilment of these conditions. The 
 water moving centripetally (instead of centrifugally, 
 which is more usual in turbines), enters at the period 
 of its greatest velocity (that is, just after passing the 
 injection orifices) into the most rapidly moving part 
 of the wheel, the circumference ; and, at the period 
 
 cc3 
 
388 THE DISCHARGE OF WATER FROM 
 
 when it ought to be as far as possible deprived of 
 velocity, it passes away by the central part of the 
 wheel, the part which has the least motion. Thus, in 
 each case, that of the entrance and that of the dis- 
 charge, there is an accordance between the velocities 
 of the moving mechanism and the proper velocities of 
 the water. 
 
 " The principle of injection from without inwards, 
 adopted in the vortex, affords another important 
 advantage in comparison with turbines having the 
 contrary motion of the water; as it allows ample 
 room, in the space outside of the wheel, for large and 
 well-formed injection channels, in which the water 
 can be made very gradually and regularly to converge 
 to the most contracted parts, where it is to have its 
 greatest velocity. It is as a concomitant also of the 
 same principle, that the very simple and advantageous 
 mode of regulating the power of the wheel, by the 
 jnoveable guide-blades already described, can be intro- 
 duced. This mode, it is to be observed; while giving 
 great variation to the areas of the entrance orifices^ 
 retains at all times very suitable forms for the con- 
 verging water channels. 
 
 u Another adaptation in the vortex is to be remarked 
 as being highly beneficial, that, namely, according to 
 which, by the balancing of the contrary fluid pressures 
 due to half the head of water and to the centrifugal 
 force of the water in the wheel, combined with the 
 pressure due to the ejection of the water backwards 
 from the inner ends of the vanes of the wheel when 
 .they are curved, only one-half of the work due to the 
 fall is spent in communicating vis viva to the watery 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 389 
 
 to be afterwards taken from it during its passage 
 through the wheel ; the remainder of the work being 
 communicated through the fluid pressure to the wheel, 
 without any intermediate generation of vis viva. 
 Thus the velocity of the water, where it moves fastest 
 in the machine, is kept comparatively low ; not ex- 
 ceeding that due to half the height of the fall, while 
 in other turbines the water usually requires to act at 
 much higher velocities. In many of them it attains 
 at two successive times the velocity due to the whole 
 fall. The much smaller amount of action, or agita- 
 tion, with which the water in the vortex performs 
 its work, causes a material saving of power by 
 diminishing the loss necessarily occasioned by fluid 
 friction. 
 
 " In the vortex, further, a very favourable influence 
 on the regularity of the motion proceeds from the 
 centrifugal force of the water, which, on any increase 
 of the velocity of the wheel, increases, and so checks 
 the water supply ; and on any diminution of the 
 velocity of the wheel, diminishes, and so admits the 
 water more freely; thus counteracting, in a great 
 degree, the irregularities of speed arising from varia- 
 tions in the work to be performed. When the work 
 is subject to great variations, as for instance in saw- 
 mills, in bleaching works, or in forges, great incon- 
 venience often arises with the ordinary bucket water- 
 wheels and with turbines which discharge at the 
 circumference, from their running too quickly when 
 any considerable diminution occurs in the resistance 
 to their motion. 
 "The first vortex which was constructed on the 
 
390 THE DISCHARGE OF WATER FROM 
 
 large scale was made in Glasgow, to drive a new 
 beetling-mill of Messrs. C. Hunter and Co., of 
 Dunadry, in County Antrim. It was the only one in 
 action at the time of the meeting of the British 
 Association in Belfast ; but the two which have been 
 particularly described in the present article, and one for 
 an unusually high fall, 100 feet, have since been com- 
 pleted, and brought into operation. There are also 
 several others in progress ; of which it may be suffi- 
 cient to particularize one of great dimensions and 
 power, for a new flax-mill at Ballyshannon in the 
 West of Ireland. It is calculated for working at 150 
 horse-power, on a faU of 14 feet, and it is to be 
 impelled by the water of the Eiver Erne. This great 
 river has an ample reservoir in the Lough of the 
 same name ; so that the water of wet weather is long 
 retained, and continues to supply the river abun- 
 dantly even in the dryest weather. The lake has 
 also the effect of causing the floods to be of long 
 duration, and the vortex will consequently be, 
 through a considerable part of the year, and for long 
 periods at a time, deeply submerged under back- 
 water. The water of the tail-race will frequently be 
 seven feet above its ordinary summer level ; but as 
 the water of the head-race will also rise to such a 
 height as to maintain a sufficient difference of levels, 
 the action of the wheel will not be deranged or 
 impeded by the floods. These circumstances have 
 had a material influence in leading to the adoption in 
 the present case of this new wheel in preference to 
 the old breast or undershot wheels." 
 The next tables have been arranged by us from 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 391 
 
 Mr. Francis' valuable experiments. They show the 
 ratio of the effect to the power in two wheels, the 
 first a centre-vent wheel, erected at the Boott Cotton 
 Mills, and the second a turbine, erected at the 
 Tremont Mills, Lowell, Massachusetts. 
 
 The maximum effect -794 was obtained from the 
 Tremont turbine experiments, when the velocity of 
 the interior circumference of the wheel was to that 
 due to the whole fall as -63 to 1 ; and an effect of 78 
 per cent, was obtained when these velocities were as 
 51 to 1. In the Boott centre-vent wheel the max- 
 imum effect -797 was obtained when the velocity of 
 the exterior circumference was to that due to the fall 
 as *64 to 1 ; and a like effect was produced when 
 this ratio was -708 to 1. Indeed, between these 
 ratios the useful effect was nearly the same; an 
 effect of *78 to '79 was obtained for all such ratios 
 between limits of *59 and -71 to 1, averaging a ratio 
 of -65 to 1. If a turbine have a variable fall, say 
 from 2 to 1, and be of sufficient capacity to give the 
 required power always, the dimensions should be 
 determined from the lesser fall, and if correctly so 
 determined, it will not have sufficient velocity for 
 the greater fall. When the fall is greatest the 
 quantity in the same place is generally least, giving 
 thereby a lessened effect when most is required. For 
 such cases two turbines may be used with advantage. 
 
392 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE showing the Results of Experiments upon a Model of a Centre-vent Water Wheel, and 
 also upon a Centre-vent Water Wheel at the Boott Cotton Mills, Lowell, Massachusetts, 
 arranged from Mr. Francis 1 valuable Experiments. Diameter of Wheel to the outside of 
 the Buckets, about 9'3 feet. Depths of External Guide Curves about -75 foot. External 
 height of Wheel about 1-5 feet. Number of Buckets, 40. Mean height of the Orifices be- 
 tween the Guides, I foot. Diameter of Supply Pipe, 8 feet. The first Seven Experiments 
 were made on a Model, the Exterior Diameter of the Wheel being 22| inches, Interior 
 Diameter 19 inches, height between the Crowns 2^| inches, and the number of Buckets 36. 
 The Construction of the Wheel is shown in Mr. Francis' Book, and all necessary Details, 
 lent Wheel of Francis' and Thomson's Vortex Wheel 
 Blades being fewer in the latter, and capable of 
 
 The general principle of the Centre-vent Wheel of Francis' and Thomson's Vortex Wheel 
 appears to be the same; the Guide Bla 
 
 adjustment. 
 
 (lumbers of the 1 
 Experiments, j 
 
 Falls acting on 1 
 the Wheels. 
 
 i Depth on the 1 
 Weirs in feet. 1 
 
 Cubic feet of 
 Water acting on 1 
 the Wheel per 1 
 second. 
 
 Number of 
 pounds avoir- 1 
 dupois, if raised 3 
 one foot per 
 second. 
 
 Revolutions of 1 
 the Wheel per 1 
 second. 
 
 Weight in the I 
 scale in pounds I 
 avoirdupois. 1 
 
 Ratios of the ef- 
 fect to the power, 
 calculated by 
 means of Prony's 
 Brake. 
 
 Velocity due to 
 the fall acting on 
 the Wheel, in 
 feet per second. 
 
 Velocity of the 
 outside circum- 
 ference of the 
 "Wheel, in feet 
 per second. 
 
 1 
 
 2-52 
 
 365 
 
 2-15 
 
 337-7 
 
 1-14 
 
 16- 
 
 679 
 
 12-73 
 
 6-83 
 
 2 
 
 2-46 
 
 366 
 
 2-16 
 
 331-5 
 
 1-34 
 
 14- 
 
 711 
 
 12-58 
 
 8-03 
 
 3 
 
 2-50 
 
 367 
 
 2-17 
 
 338-0 
 
 1-54 
 
 12-5 
 
 716 
 
 12-68 
 
 9-23 
 
 4 
 
 2-60 
 
 372 
 
 2-21 
 
 358-2 
 
 1-70 
 
 12- 
 
 705 
 
 12-93 
 
 10-03 
 
 5 
 
 2-60 
 
 373 
 
 2-22 
 
 361-3 
 
 1-73 
 
 11-5 
 
 692 
 
 12-95 
 
 10-36 
 
 6 
 
 2-60 
 
 373 
 
 2-22 
 
 559-6 
 
 1-71 
 
 11-5 
 
 689 
 
 12-93 
 
 10-26 
 
 7 
 
 2-60 
 
 374 
 
 2-23 
 
 360-8 
 
 1-90 
 
 10-0 
 
 648 
 
 12-93 
 
 11-15 
 
 8 
 
 14-60 
 
 1-296 
 
 67-53 
 
 61493-4 
 
 1-48 
 
 390-9 
 
 377 
 
 30-65 
 
 17-39 
 
 9 
 
 14-67 
 
 1-262 
 
 64-89 
 
 59364-4 
 
 1-26 
 
 775-6 
 
 203 
 
 30-72 
 
 25-77 
 
 10 
 
 14-57 
 
 1-282 
 
 66-43 
 
 60347-0 
 
 1-14 
 
 963-3 
 
 332 
 
 30-61 
 
 21-21 
 
 11 
 
 14-16 
 
 1-284 
 
 66-61 
 
 58821-6 
 
 1-08 
 
 1069- 
 
 382 
 
 30-18 
 
 15-97 
 
 12 
 
 14-20 
 
 1-290 
 
 67-03 
 
 59351-7 
 
 1-02 
 
 1150-8 
 
 381 
 
 30-22 
 
 14-65 
 
 13 
 
 14-14 
 
 1-288 
 
 66-89 
 
 59002-2 
 
 96 
 
 1243-0 
 
 373 
 
 30-16 
 
 13-18 
 
 14 
 
 14-24 
 
 1-294 
 
 67-37 
 
 59858-1 
 
 93 
 
 1293-6 
 
 269 
 
 30-27 
 
 7-46 
 
 15 
 
 14-gO 
 
 1-211 
 
 61-08 
 
 54486-4 
 
 90 
 
 1345-5 
 
 . . 
 
 30-33 
 
 29-75 
 
 16 
 
 14-29 
 
 1-514 
 
 85- 
 
 75732-8 
 
 86 
 
 1396-1 
 
 303 
 
 30-32 
 
 29-63 
 
 17 
 
 14-23 
 
 1-531 
 
 86-35 
 
 76608-0 
 
 83 
 
 1444-0 
 
 376 
 
 30-25 
 
 28-16 
 
 18 
 
 14-20 
 
 1-539 
 
 87-08 
 
 77093-2 
 
 80 
 
 1494-7 
 
 413 
 
 30-22 
 
 27-35 
 
 19 
 
 14-19 
 
 1-547 
 
 87'68 
 
 77607-2 
 
 77 
 
 1548-7 
 
 444 
 
 30-21 
 
 26-43 
 
 20 
 
 14-19 
 
 1-554 
 
 88'28 
 
 78143 8 
 
 71 
 
 1657-0 
 
 489 
 
 30-22 
 
 25-12 
 
 21 
 
 13-78 
 
 1-576 
 
 90'17 
 
 77480-4 
 
 1-55 
 
 
 582 
 
 29-77 
 
 20-37 
 
 22 
 
 13-61 
 
 1-594 
 
 9170 
 
 77812-1 
 
 1-35 
 
 316-0 
 
 596 
 
 29-58 
 
 17-54 
 
 23 
 
 13-94 
 
 1-418 
 
 77-11 
 
 67076-7 
 
 1-22 
 
 5197 
 
 
 29-9 
 
 34-51 
 
 24 
 
 13-52 
 
 1-642 
 
 95-76 
 
 80736-3 
 
 1-10 
 
 720-2 
 
 245 
 
 29-49 
 
 32-44 
 
 25 
 
 13-37 
 
 1*673 
 
 98-49 
 
 82145-2 
 
 1-00 
 
 832'3 
 
 437 
 
 29-33 
 
 29-14 
 
 26 
 
 13-37 
 
 1-695 
 
 100-42 
 
 83728-2 
 
 92 
 
 934-7 
 
 576 
 
 2932 
 
 26-43 
 
 27 
 
 13-40 
 
 1-718 
 
 102-42 
 
 85571-4 
 
 84 
 
 1033-3 
 
 657 
 
 29-35 
 
 24-20 
 
 28 
 
 13-38 
 
 1-723 
 
 102-82 
 
 85800-0 
 
 77 
 
 1115-0 
 
 690 
 
 29-34 
 
 23-07 
 
 29 
 
 13-34 
 
 1-731 
 
 103-52 
 
 86138'0 
 
 77 
 
 1115-0 
 
 710 
 
 29 -30 
 
 21-89 
 
 30 
 
 13-32 
 
 1-734 
 
 103-77 
 
 86218'S 
 
 69 
 
 1204-8 
 
 720 
 
 29-27 
 
 21-26 
 
 31 
 
 13-33 
 
 1-733 
 
 103-70 
 
 86229-3 
 
 62 
 
 1278-0 
 
 728 
 
 29-29 
 
 20-86 
 
 32 
 
 13-30 
 
 1-739 
 
 104-23 
 
 86483*7 
 
 46 
 
 1482-6 
 
 731 
 
 29-25 
 
 20-50 
 
 33 
 
 13-70 
 
 1-598 
 
 92*02 
 
 78648-0 
 
 85 
 
 1482-6 
 
 
 29-70 
 
 3678 
 
 34 
 
 13-40 
 
 1-832 
 
 112-52 
 
 94057-5 
 
 82 
 
 1544-9 
 
 797 
 
 29-36 
 
 20-81 
 
 35 
 
 13-43 
 
 1-837 
 
 112-99 
 
 94662-2 
 
 79 
 
 1604-8 
 
 796 
 
 29-39 
 
 20-51 
 
 36 
 
 13-33 
 
 1-832 
 
 112-56 
 
 93603-9 
 
 1-25 
 
 .. 
 
 797 
 
 29-28 
 
 19-93 
 
 37 
 
 13-38 
 
 1-837 
 
 113-00 
 
 94296-4 
 
 1-15 
 
 118-6 
 
 797 
 
 29-33 
 
 19-71 
 
 88 
 
 13-39 
 
 1-838 
 
 113-07 
 
 94415-2 
 
 96 
 
 325-4 
 
 796 
 
 29-34 
 
 19-36 
 
 39 
 
 13-38 
 
 1-839 
 
 113-16 
 
 94471-1 
 
 77 
 
 519-9 
 
 796 
 
 29-34 
 
 19-03 
 
 40 
 
 13-36 
 
 1-838 
 
 113-09 
 
 94219-0 
 
 67 
 
 612-2 
 
 797 
 
 29-31 
 
 18-67 
 
 41 
 
 13-38 
 
 L-844 
 
 113-67 
 
 94881-9 
 
 56 
 
 704-4 
 
 791 
 
 29-34 
 
 18-32 
 
 42 
 
 13-40 
 
 1-851 
 
 11429 
 
 95571-2 
 
 46 
 
 777-6 
 
 787 
 
 29-36 
 
 18-00 
 
 43 
 
 1332 
 
 1-848 
 
 113-97 
 
 94703-1 
 
 30 
 
 882-1 
 
 781 
 
 29-27 
 
 17-38 
 
 44 
 
 13-54 
 
 1-809 
 
 110-45 
 
 93270-1 
 
 62 
 
 118-6 
 
 
 29-51 
 
 
 45 
 
 13-57 
 
 1-807 
 
 110-32 
 
 93422-4 
 
 69 
 
 73-1 
 
 
 29-55 
 
 
 46 
 
 13-60 
 
 1-088 
 
 99-79 
 
 84635-0 
 
 39 
 
 296-4 
 
 
 
 29-57 
 
 37-70 
 
ORIFICES, WEIRS, PIPES, AND EIVERS. 
 
 393 
 
 TABLE showing the Results of Experiments upon the Turbine at Tremont Mills, Lowell, 
 Massachusetts, arranged from Mr. Francis' valuable Experiments. Diameter, measured 
 to the Exterior Circumference of Crowns of the Wheel, 8'333 feet. Height of Buckets 
 from top of the Disc to the bottom of the Garniture, '97 feet. Number of Buckets, 44. 
 Width, of the Buckets '8 foot, nearly. Width of Guide Curves, fr'dfect, nearly. Number of 
 Ditto, 33. A Double Weir with 4 end constructions and 16'98 feet long, used for gauging 
 the Water the Crest being &5feet above the floor of the Wheel Pit. The Falls show the 
 difference of heads in the Forebay and Wheel Pit. For further details, see Francis' 
 Lowell Hydraulic Experiments, pp. 1 to 43. Tlie supply pipe is fully a, quadrant, and 
 varies from 6 to 9 feet in diameter. 
 
 1 Numbers of the 1 
 Experiments. I 
 
 Falls actinp: on I 
 the Wheel. 
 
 Depths on the I 
 Weir in feet. 1 
 
 Cubic feet of 
 Water acting on 
 the Wheel per 
 second. 
 
 Number of pounds! 
 avoirdupois, if 1 
 raised one foot perl 
 second. 
 
 Number of revolu-S 
 tions of the Wheelj 
 per second. 
 
 1 1 
 
 ill 
 
 C o "p 
 
 pl 
 * 
 
 Ratio of the effect! 
 to the power, cal- 1 
 dilated by meansl 
 ofProny's brake! 
 or dynamometer. 1 
 
 Velocity due to 1 
 the fall acting on 1 
 the Wheel, in feet I 
 per second. 
 
 Velocity of the in- 
 terior circumfer- 
 ence of the Wheel, 
 in feet per second. 
 
 1 
 
 12-864 
 
 1-88 
 
 139-42 
 
 111870-0 
 
 894 
 
 1443-34 
 
 784 
 
 28-76 
 
 18-95 
 
 2 
 
 12-869 
 
 1-88 
 
 139-47 
 
 111951-2 
 
 892 
 
 1443-34 
 
 784 
 
 28-77 
 
 18-96 
 
 3 
 
 12-611 
 
 2-02 
 
 154-40 
 
 121444-2 
 
 893 
 
 1443-34 
 
 353 
 
 28-48 
 
 32-49 
 
 4 
 
 12-696 
 
 1-97 
 
 149-46 
 
 118363-5 
 
 1-596 
 
 307-03 
 
 507 
 
 28-58 
 
 29-32 
 
 5 
 
 12-777 
 
 1-94 
 
 146-02 
 
 116373-2 
 
 1-532 
 
 411-48 
 
 622 
 
 28-67 
 
 26-40 
 
 6 
 
 12-819 
 
 1-92 
 
 143-91 
 
 115067-3 
 
 1-461 
 
 519-77 
 
 703 
 
 28-71 
 
 23-86 
 
 7 
 
 12-856 
 
 1-91 
 
 142-52 
 
 114284-2 
 
 1-382 
 
 638-36 
 
 735 
 
 28-76 
 
 22-63 
 
 8 
 
 12-888 
 
 1-90 
 
 142-04 
 
 114187-1 
 
 1-309 
 
 750-42 
 
 750 
 
 28-79 
 
 21-71 
 
 9 
 
 12-896 
 
 1-90 
 
 141-28 
 
 113640-9 
 
 1-245 
 
 854-87 
 
 766 
 
 28-80 
 
 20-57 
 
 10 
 
 12-883 
 
 1-89 
 
 140-08 
 
 112568-7 
 
 1-184 
 
 957-35 
 
 779 
 
 28-79 
 
 19-57 
 
 11 
 
 12-899 
 
 1-88 
 
 139-90 
 
 112563-3 
 
 1-125 
 
 1057-49 
 
 781 
 
 28-80 
 
 19-13 
 
 12 
 
 12-905 
 
 1-88 
 
 139-01 
 
 111893-5 
 
 1-784 
 
 
 789 
 
 28-81 
 
 18-92 
 
 13 
 
 12-899 
 
 1-88 
 
 139-03 
 
 111859-4 
 
 1-784 
 
 
 788 
 
 28-80 
 
 18-77 
 
 14 
 
 12-902 
 
 1-87 
 
 138-85 
 
 111740-3 
 
 1-067 
 
 1156*27 
 
 790 
 
 28-81 
 
 18-66 
 
 15 
 
 12-906 
 
 1-87 
 
 138-51 
 
 111504-9 
 
 1-02J: 
 
 1229-41 
 
 792 
 
 28-81 
 
 18-37 
 
 16 
 
 12-915 
 
 1-87 
 
 138-27 
 
 111384-0 
 
 999 
 
 1269-42 
 
 794 
 
 28-82 
 
 17-73 
 
 17 
 
 12-934 
 
 1-87 
 
 138-23 
 
 111521-1 
 
 970 
 
 1319-22 
 
 792 
 
 28-84 
 
 17-25 
 
 18 
 
 12-939 
 
 1-86 
 
 137-71 
 
 111139-7 
 
 945 
 
 1359-23 
 
 783 
 
 28-85 
 
 15-74 
 
 19 
 
 12-940 
 
 1-84 
 
 135-14 
 
 109077-1 
 
 923 
 
 1397-12 
 
 770 
 
 28-85 
 
 13-69 
 
 20 
 
 12-963 
 
 1-84 
 
 135-34 
 
 109433-4 
 
 911 
 
 1416-70 
 
 769 
 
 28-88 
 
 13-72 
 
 21 
 
 12-977 
 
 1-83 
 
 133-75 
 
 108265-8 
 
 902 
 
 1433-43 
 
 725 
 
 28-89 
 
 11-29 
 
 22 
 
 12-948 
 
 1-S2 
 
 133-43 
 
 107764-7 
 
 897 
 
 1443-06 
 
 679 
 
 28-86 
 
 9-63 
 
 23 
 
 12-954 
 
 1-87 
 
 138-62 
 
 112009-3 
 
 1-781 
 
 
 791 
 
 28-86 
 
 17-64 
 
 24 
 
 12-932 
 
 1-87 
 
 138-50 
 
 111720-6 
 
 892 
 
 1454-24 
 
 791 
 
 28-84 
 
 17-32 
 
 25 
 
 12-951 
 
 1-87 
 
 138-37 
 
 111777-2 
 
 885 
 
 1464-80 
 
 788 
 
 28-86 
 
 16-74 
 
 26 
 
 12-758 
 
 1-92 
 
 143-33 
 
 114060-7 
 
 875 
 
 1474-37 
 
 346 
 
 28-65 
 
 31-45 
 
 27 
 
 12-909 
 
 1-86 
 
 137-75 
 
 110917-6 
 
 880 
 
 1474-37 
 
 675 
 
 28-82 
 
 24-21 
 
 28 
 
 12-950 
 
 1-86 
 
 137-00 
 
 110664-7 
 
 873 
 
 1485-63 
 
 725 
 
 28-86 
 
 21-73 
 
 29 
 
 12-965 
 
 1-84 
 
 135-10 
 
 109252-2 
 
 866 
 
 1498-66 
 
 750 
 
 28-88 
 
 19-74 
 
 30 
 
 12-999 
 
 1-82 
 
 133-30 
 
 108082-5 
 
 851 
 
 1524-67 
 
 760 
 
 28-92 
 
 18-35 
 
 31 
 
 13-026 
 
 1-81 
 
 131-99 
 
 107246-3 
 
 1-790 
 
 
 763 
 
 38-95 
 
 17-05 
 
 32 
 
 13-028 
 
 1-80 
 
 130-89 
 
 106366-6 
 
 836 
 
 1552'-44 
 
 750 
 
 28-95 
 
 15-01 
 
 33 
 
 13-077 
 
 1-68 
 
 118-55 
 
 96699-5 
 
 813 
 
 1597-08 
 
 300 
 
 29-00 
 
 28-66 
 
 34 
 
 13-134 
 
 1-66 
 
 116-10 
 
 95112-0 
 
 784 
 
 1648-87 
 
 453 
 
 29-06 
 
 25-84 
 
 35 
 
 13-215 
 
 1-63 
 
 113-24 
 
 93346-0 
 
 742 
 
 1724-49 
 
 609 
 
 29-15 
 
 21-28 
 
 36 
 
 13-282 
 
 1-59 
 
 109-71 
 
 90893-3 
 
 695 
 
 1816-71 
 
 652 
 
 29-23 
 
 17-88 
 
 37 
 
 13-310 
 
 1-58 
 
 107-95 
 
 89620-7 
 
 646 
 
 1911-45 
 
 656 
 
 29-26 
 
 16-44 
 
 38 
 
 13-362 
 
 1-54 
 
 103-85 
 
 86555-6 
 
 1-786 
 
 ;; 
 
 619 
 
 29-32 
 
 13-06 
 
 39 
 
 12-883 
 
 1-86 
 
 137-36 
 
 110380-8 
 
 647 
 
 1911-45 
 
 .-781 
 
 28-79 
 
 18-12 
 
 40 
 
 12-896 
 
 1-86 
 
 136-97 
 
 110176-6 
 
 600 
 
 2011-52 
 
 784 
 
 28-80 
 
 17-42 
 
 41 
 
 12-912 
 
 1-85 
 
 136-55 
 
 109973-0 
 
 532 
 
 2167-38 
 
 783 
 
 28-82 
 
 16-73 
 
 42 
 
 13-369 
 
 1-27 
 
 78-84 
 
 65746-8 
 
 454 
 
 2367-88 
 
 141 
 
 29-32 
 
 24-30 
 
 43 
 
 13-395 
 
 1-25 
 
 76-62 
 
 64018-1 
 
 
 4213-38 
 
 ,332 
 
 29-35 
 
 20-36 
 
 44 
 
 13-435 
 
 1-22 
 
 74-05 
 
 62062-0 
 
 
 3946-38 
 
 440 
 
 29-40 
 
 16-37 
 
 45 
 
 13-478 
 
 1-19 
 
 71-87 
 
 60424-6 
 
 1-783 
 
 . 
 
 463 
 
 29-44 
 
 14-25 
 
 46 
 
 13-513 
 
 1-17 
 
 70-01 
 
 59006-4 
 
 832 
 
 156V-21 
 
 455 
 
 29-48 
 
 11-87 
 
 47 
 
 13-559 
 
 1-11 
 
 64-50 
 
 54554-9 
 
 817 
 
 1590-50 
 
 330 
 
 29-53 
 
 6-36 
 
 48 
 
 13-985 
 
 ' -78 
 
 38-22 
 
 33340-0 
 
 803 
 
 1614-79 
 
 -150 
 
 29-99 
 
 13-14 
 
 49 
 
 14-001 
 
 78' 
 
 38-57 
 
 33683v 
 
 789 
 
 1641-34 
 
 102 
 
 30-01 
 
 14-61 
 
 50 
 
 14-020 
 
 76 : 
 
 37-17 
 
 32508-Q 
 
 769 
 
 1679-62 
 
 240 
 
 30-03 
 
 8-32 
 
394 
 
 THE DISCHAKGE OF WATER FROM 
 
 TABLE for Turbines of different Diameters, modified from Francis, 
 operating with different Falls; assuming the useful effect is 
 seventy-five per cent, of the power expended, that the Velocity of the 
 Interior Circumference is fifty-six per cent, of the Velocity due 
 to the Fall ; and that also, the Height between the Crowns is one- 
 tenth of the Outside Diameter. 
 
 
 Outside diameter 2 ft. 
 
 Outside diameter 3 ft. 
 
 Outside diameter 4 ft. 
 
 3utside diameter 5 ft. 
 
 
 Inside 1-56 
 Number of buckets 36. 
 
 Inside ,. 2-38 
 Number of buckets 39. 
 
 tnslde 3-24 ,. 
 Number of buckets 42. 
 
 Inside 4-11 
 Number of buckets 45. 
 
 1 
 
 11 
 
 jj 
 
 1 
 
 || 
 
 i 
 
 1 
 
 ft 
 
 j. 
 
 ft 
 
 11 
 
 ^ 
 
 fe 
 P. 
 
 i 
 
 Hi 
 
 i 
 
 1! 
 
 |1| 
 
 i 
 
 oj 
 
 ll 
 
 ||,3 
 
 1 
 
 i! 
 
 hi 
 
 1 
 
 li 
 
 
 *j g 
 
 -23* 
 
 I 
 
 
 
 I! 
 
 fll 
 
 |g n 
 
 1 
 
 33 
 o a 
 
 m 
 
 III 
 
 S 
 
 | 
 
 P 
 
 W 
 
 1 
 
 li 
 
 o n 
 
 
 1* 
 
 B 
 
 ) 
 
 
 
 a 
 
 II 
 
 B 
 
 s 
 
 K 
 
 B 
 
 
 
 PH 
 
 5 
 
 4-5 
 
 1-9 
 
 123 
 
 10-1 
 
 4-3 
 
 80 
 
 17-88 
 
 7-6 
 
 59 
 
 27-9 
 
 11-9 
 
 47 
 
 6 
 
 4-9 
 
 2-5 
 
 135 
 
 11-0 
 
 5-6 
 
 88 
 
 19-6 
 
 10-0 
 
 65 
 
 30-6 
 
 15-6 
 
 51 
 
 7 
 
 5-3 
 
 3-1 
 
 146 
 
 11-9 
 
 7-1 
 
 95 
 
 21-17 
 
 12-6 
 
 70 
 
 33-1 
 
 19-7 
 
 55 
 
 8 
 
 5-7 
 
 3-8 
 
 156 
 
 12-7 
 
 8-7 
 
 102 
 
 22-63 
 
 15-4 
 
 75 
 
 35-3 
 
 24-0 
 
 59 
 
 9 
 
 6-0 
 
 4-6 
 
 165 
 
 13-5 
 
 10-3 
 
 108 
 
 24-00 
 
 18-4 
 
 79 
 
 37-5 
 
 28-7 
 
 63 
 
 10 
 
 6-3 
 
 5-4 
 
 174 
 
 14-2 
 
 12-1 
 
 114 
 
 25-30 
 
 21-5 
 
 84 
 
 39-5 
 
 33-6 
 
 66 
 
 11 
 
 6-6 
 
 6-2 
 
 183 
 
 14-9 
 
 13-9 
 
 119 
 
 26-53 
 
 24-8 
 
 88 
 
 41-5 
 
 38-8 
 
 69 
 
 12 
 
 6-9 
 
 7-1 
 
 191 
 
 15-6 
 
 15-9 
 
 125 
 
 27-71 
 
 28-3 
 
 92 
 
 43-3 
 
 44-2 
 
 72 
 
 13 
 
 7-2 
 
 8-0 
 
 199 
 
 16-2 
 
 17-9 
 
 130 
 
 28-84 
 
 31-9 
 
 95 
 
 45-1 
 
 49-8 
 
 75 
 
 14 
 
 7-5 
 
 8-9 
 
 206 
 
 168 
 
 20-0 
 
 135 
 
 29-93 
 
 35-6 
 
 99 
 
 46-8 
 
 55-7 
 
 78 
 
 15 
 
 7-7 
 
 9-9 
 
 213 
 
 17-4 
 
 22-2 
 
 139 
 
 30-98 
 
 39-5 
 
 103 
 
 48-4 
 
 61-7 
 
 81 
 
 16 
 
 8-0 
 
 10-9 
 
 220 
 
 18-0 
 
 24-5 
 
 144 
 
 32-00 
 
 43-5 
 
 106 
 
 50-0 
 
 68-0 
 
 83 
 
 17 
 
 8-2 
 
 11-9 
 
 227 
 
 18-5 
 
 26-8 
 
 148 
 
 32-99 
 
 47-7 
 
 109 
 
 51-5 
 
 74-5 
 
 86 
 
 18 
 
 8-5 
 
 13-0 
 
 234 
 
 19-1 
 
 29-2 
 
 153 
 
 33-94 
 
 51-9 
 
 112 
 
 53-0 
 
 81-1 
 
 88 
 
 19 
 
 8-7 
 
 14-1 
 
 240 
 
 19-6 
 
 31-7 
 
 157 
 
 34-87 
 
 56-3 
 
 115 
 
 54-5 
 
 88-0 
 
 91 
 
 20 
 
 8-9 
 
 15-2 
 
 247 
 
 20-1 
 
 34-2 
 
 161 
 
 35-78 
 
 60-8 
 
 118 
 
 55-9 
 
 95-0 
 
 93 
 
 21 
 
 9-2 
 
 16-4 
 
 253 
 
 20-6 
 
 36-8 
 
 165 
 
 36-66 
 
 65-4 
 
 121 
 
 57-3 
 
 102-2 
 
 96 
 
 22 
 
 9-4 
 
 17-5 
 
 259 
 
 21-1 
 
 39-5 
 
 169 
 
 37-52 
 
 70-2 
 
 124 
 
 58-6 
 
 109-6 
 
 98 
 
 23 
 
 9-6 
 
 18-7 
 
 264 
 
 21-6 
 
 42-2 
 
 172 
 
 38-37 
 
 75-0 
 
 127 
 
 59-9 
 
 117-2 
 
 100 
 
 24 
 
 9-8 
 
 20-0 
 
 270 
 
 220 
 
 45-0 
 
 176 
 
 39-19 
 
 79-9 
 
 130 
 
 61-2 
 
 124-9 
 
 102 
 
 25 
 
 10-0 
 
 21-2 
 
 276 
 
 22-5 
 
 47-8 
 
 180 
 
 40-00 
 
 85-0 
 
 132 
 
 62-5 
 
 132-8 
 
 104 
 
 26 
 
 10-2 
 
 22-5 
 
 281 
 
 22-9 
 
 50-7 
 
 183 
 
 40-79 
 
 90-1 
 
 135 
 
 63-7 
 
 140-9 
 
 106 
 
 27 
 
 10-4 
 
 23-8 
 
 286 
 
 23-4 
 
 53-7 
 
 187 
 
 41-57 
 
 95-4 
 
 138 
 
 65-0 
 
 149-1 
 
 108 
 
 28 
 
 10-6 
 
 25-2 
 
 292 
 
 23-8 
 
 56-7 
 
 190 
 
 42-33 
 
 100-7 
 
 140 
 
 66-1 
 
 157-4 
 
 110 
 
 29 
 
 10-8 
 
 26-5 
 
 297 
 
 24-2 
 
 59-7 
 
 194 
 
 43-08 
 
 106-2 
 
 143 
 
 67-3 
 
 165-9 
 
 112 
 
 30 
 
 10-9 
 
 27-9 
 
 302 
 
 24-6 
 
 62-8 
 
 197 
 
 43-82 
 
 111-7 
 
 145 
 
 68-5 
 
 174-6 
 
 114 
 
 31 
 
 11-1 
 
 29-3 
 
 307 
 
 25-0 
 
 66-0 
 
 200 
 
 44-54 
 
 117-4 
 
 147 
 
 89-6 
 
 183-4 
 
 116 
 
 32 
 
 11-3 
 
 30-8 
 
 312 
 
 25-5 
 
 69-2 
 
 203 
 
 45-25 
 
 123-1 
 
 150 
 
 70-7 
 
 192-3 
 
 118 
 
 33 
 
 11-5 
 
 32-2 
 
 317 
 
 25-8 
 
 72-5 
 
 207 
 
 45-96 
 
 128-9 
 
 152 
 
 71-8 
 
 201-4 
 
 120 
 
 34 
 
 11-7 
 
 33-7 
 
 321 
 
 26-2 
 
 75-8 
 
 210 
 
 46-65 
 
 134-8 
 
 154 
 
 72-9 
 
 210-6 
 
 122 
 
 35 
 
 11-8 
 
 35-2 
 
 326 
 
 26-6 
 
 79-2 
 
 213 
 
 47-33 
 
 140-8 
 
 157 
 
 73-9 
 
 220-0 
 
 123 
 
 36 
 
 12-0 
 
 36-7 
 
 331 
 
 27-0 
 
 82-6 
 
 216 
 
 48-00 
 
 146-9 
 
 159 
 
 75-0 
 
 229-5 
 
 125 
 
 37 
 
 12-2 
 
 38-3 
 
 335 
 
 27-4 
 
 86-1 
 
 219 
 
 48-66 
 
 153-0 
 
 161 
 
 76-0 
 
 239-1 
 
 127 
 
 38 
 
 12-3 
 
 39-8 
 
 340 
 
 27-7 
 
 89-6 
 
 222 
 
 49-32 
 
 159-3 
 
 163 
 
 77-0 
 
 248-9 
 
 129 
 
 39 
 
 12-5 
 
 41-4 
 
 344 
 
 28-1 
 
 93-2 
 
 225 
 
 49-96 
 
 165-6 
 
 165 
 
 78-1 
 
 258-8 
 
 130 
 
 40 
 
 12-6 
 
 43-0 
 
 349 
 
 28-5 
 
 96-8 
 
 227 
 
 50-60 
 
 172-0 
 
 167 
 
 79-1 
 
 268-8 
 
 132 
 
ORIFICES, WEIRS, PIPES, AND EIVEES. 
 
 395 
 
 TABLE of Turbines of different Diameters, modified from Francis, 
 operating with different Falls; assuming the useful effect is 
 seventy-Jive per cent, of the power expended, that the Velocity of 
 the Interior Circumference is fifty-six per cent, of the Velocity due 
 to the Fall, and that also, the Height between the Crowns is one- 
 tenth of the Outside Diameter. 
 
 
 Outside diameter 6 ft. 
 
 Outside diameter 7 ft. 
 
 Outside diameter 8 ft. 
 
 Outside diameter 10 ft. 
 
 
 Inside 5 
 Number of buckets 48. 
 
 Inside 5-90 ,, 
 Number of buckets 51. 
 
 Inside 6-81 
 Number of buckets 54. 
 
 Inside 8-67 ,. 
 Number of buckets 60. 
 
 I 
 
 
 
 || 
 
 O 
 
 1 
 
 IK 
 
 i 
 
 1 
 
 fj 
 
 *3 
 
 1 
 
 Is. 
 
 j 
 
 1 
 
 1 
 
 ||l 
 
 1 
 
 T 
 
 Jf 
 
 llH 
 lal 
 
 o 
 
 r 
 
 V 
 
 K V 
 53 
 
 l|l 
 
 p 
 
 II 
 
 ^3 
 
 | 
 
 || 
 
 
 |f 
 
 w 
 
 fs 
 
 If 
 
 1 
 
 I 
 
 "3 a 
 
 Water 
 in cub) 
 se 
 
 1 
 
 11 
 
 1 
 
 m 
 
 I 
 
 1 
 
 5 
 
 40-2 
 
 17-1 
 
 38 
 
 54-8 
 
 23-3 
 
 32-5 
 
 71-5 
 
 30-4 
 
 28-1 
 
 111-8 
 
 47-5 
 
 22-1 
 
 6 
 
 44-1 
 
 22-5 
 
 42 
 
 60-0 
 
 30-6 
 
 35-6 
 
 78-4 
 
 40-0 
 
 30-8 
 
 122-5 
 
 62-5 
 
 24-2 
 
 7 
 
 47-6 
 
 28-3 
 
 45 
 
 64-8 
 
 38-6 
 
 38-4 
 
 84-7 
 
 50-4 
 
 33-3 
 
 132-3 
 
 78-7 
 
 26-2 
 
 8 
 
 50-9 
 
 34-6 
 
 48 
 
 69-3 
 
 47-1 
 
 41-1 
 
 90-5 
 
 61-5 
 
 35-6 
 
 141-4 
 
 96-2 
 
 28-0 
 
 9 
 
 54-0 
 
 41-3 
 
 51 
 
 73-5 
 
 56-2 
 
 43-6 
 
 96-0 
 
 73-4 
 
 37-8 
 
 150-0 
 
 114-7 
 
 29-7 
 
 10 
 
 56-9 
 
 48-4 
 
 54 
 
 77-5 
 
 65-9 
 
 46-0 
 
 101-2 
 
 86-0 
 
 39-8 
 
 158-1 
 
 134-4 
 
 31-3 
 
 11 
 
 59-7 
 
 55-8 
 
 57 
 
 81-3 
 
 76-0 
 
 48-2 
 
 106-1 
 
 99-2 
 
 41-7 
 
 165-8 
 
 1550 
 
 32-8 
 
 12 
 
 62-4 
 
 63-6 
 
 59 
 
 84-9 
 
 86-6 
 
 50-3 
 
 110-8 
 
 113-1 
 
 43-6 
 
 173-2 
 
 176-7 
 
 34-3 
 
 13 
 
 64-9 
 
 71-7 
 
 62 
 
 88-3 
 
 97-6 
 
 52-4 
 
 115-4 
 
 127-5 
 
 45-4 
 
 180-3 
 
 199-2 
 
 35-7 
 
 14 
 
 67-3 
 
 80-1 
 
 64 
 
 91-7 
 
 109-1 
 
 54-4 
 
 119-7J142-5 
 
 47-1 
 
 187-1 
 
 222-6 
 
 37-0 
 
 15 
 
 69-7 
 
 88-9 
 
 66 
 
 94-9 
 
 121-0 
 
 56-3 
 
 123-9 158-0 
 
 48-7 
 
 193-6 
 
 246-9 
 
 38-3 
 
 16 
 
 72-0 
 
 97-9 
 
 69 
 
 98-0 
 
 133-3 
 
 58-1 
 
 128-0 174-1 
 
 50-3 
 
 200-0 
 
 272-0 
 
 39-6 
 
 17 
 
 74*2 
 
 107-2 
 
 71 
 
 101-0 
 
 146-0 
 
 59-9 
 
 131-9190-6 
 
 51-9 
 
 206-2 
 
 297-9 
 
 40-8 
 
 18 
 
 76-4 
 
 116-8 
 
 73 
 
 103-9 
 
 159-0 
 
 61-7 
 
 135-8207-7 
 
 53-4 
 
 212-1 
 
 324-6 
 
 42-0 
 
 19 
 
 78-5 
 
 126-7 
 
 75 
 
 106-8 
 
 172-5 
 
 63-3 
 
 139-5225-3 
 
 54-9 
 
 217-9 
 
 352-0 
 
 43-1 
 
 20 
 
 80-5 
 
 136-8 
 
 77 
 
 109-6 
 
 186-3 
 
 65-D 
 
 143-1 243-3 
 
 56-3 
 
 223-6 
 
 380-1 
 
 44-3 
 
 21 
 
 82-5 
 
 147-2 
 
 79 
 
 112-3 
 
 200-4 
 
 66-6 
 
 146-6261-7 57-7 
 
 229-1 
 
 4090 
 
 45-4 
 
 22 
 
 84-4 
 
 157-9 
 
 80 
 
 114-9 
 
 214-9 
 
 68*2 
 
 150-1:280-7 
 
 59-0 
 
 234-5 
 
 438-5 
 
 46-4 
 
 23 
 
 86-3 
 
 168*8 
 
 82 
 
 117-5 
 
 229-7 
 
 69-7 
 
 153-5 300*0 
 
 60-4 
 
 239-8 
 
 468-8 
 
 47-5 
 
 24 
 
 88-2 
 
 179*9 
 
 84 
 
 120-0 
 
 244-8 
 
 71-2 
 
 156-8 319*8 
 
 61-7 
 
 244-9 
 
 499-7 
 
 48-5 
 
 25 
 
 90-0 
 
 191*2 
 
 86 
 
 122'5 260-3 
 
 72-7 
 
 160'0 340'0 
 
 62-9 
 
 250-0 
 
 531 '2 
 
 49-5 
 
 26 
 
 91-8 
 
 202-8 
 
 87 
 
 124-9 
 
 276-1 
 
 74-1 
 
 163'2 360'6 
 
 64-2 
 
 254-9 
 
 563*4 
 
 50-5 
 
 27 
 
 93-5 
 
 214-6 
 
 89 
 
 127-3 
 
 292-2 
 
 75*5 
 
 166*3 
 
 381-6 
 
 65-4 
 
 259-8 
 
 596*3 
 
 51-4 
 
 28 
 
 95-2 
 
 226-7 
 
 91 
 
 129-6 
 
 308-5 
 
 76'9 
 
 169*3 
 
 403*0 
 
 66-6 
 
 264-6 
 
 629*7 
 
 52-4 
 
 29 
 
 96-9 
 
 238-9 
 
 92 
 
 131-9 
 
 325-2 
 
 78*3 
 
 172-3 
 
 424-8 
 
 67-8 
 
 269-3 
 
 663*7 
 
 53-3 
 
 30 
 
 98-6 
 
 251-4 
 
 94 
 
 134-2 
 
 342-2 
 
 79-6 
 
 175-3 
 
 446-9 
 
 68-9 
 
 273-9 
 
 698*3 
 
 54'2 
 
 31 
 
 100-2 
 
 264-1 
 
 95 
 
 136-4 
 
 359-4 
 
 80-9 
 
 178-2 
 
 469-5 
 
 70-1 
 
 278-4 
 
 733'5 
 
 55-1 
 
 32 
 
 101-8 
 
 277-0 
 
 97 
 
 138-6 
 
 377-0 
 
 82-2 
 
 181-0 
 
 492-4 
 
 71-2 
 
 282-8 
 
 769*3 
 
 56-0 
 
 33 
 
 103-4 
 
 290-0 
 
 98 
 
 140-7 
 
 394-8 
 
 83-5 
 
 183-8 
 
 515-6 
 
 72-3 
 
 287-2 
 
 805*7 
 
 56-9 
 
 34 
 
 105-0 
 
 303-3 
 
 100 
 
 142-9 
 
 412-9 
 
 84-7 
 
 186-6 
 
 539-2 
 
 73-4 
 
 291-5 
 
 842'6 
 
 57-7 
 
 35 
 
 106-5 
 
 316-8 
 
 101 
 
 144-9 431-2 
 
 86-0 
 
 189*3 
 
 563-2 
 
 74-5 
 
 295-8 
 
 880*0 
 
 58-5 
 
 36 
 
 108-0 
 
 330-5 
 
 103 
 
 147-0 
 
 449-8 
 
 87-2 
 
 192-0 
 
 587-5 
 
 75-5 
 
 300-0 
 
 918'0 
 
 59-4 
 
 37 
 
 109-5 
 
 344-3 
 
 104 
 
 149-0 
 
 468-7 
 
 88-4 
 
 194-6 
 
 612-2 
 
 766 
 
 304-1 
 
 956'5 
 
 60-2 
 
 38 
 39 
 
 111-0 
 112-4 
 
 358-4 
 372-6 
 
 106 
 107 
 
 151-0487-8 
 153-OJ507-2 
 
 89-6 
 90-8 
 
 197-3 
 199-8 
 
 637-1 
 662-5 
 
 77-6 
 78-6 
 
 308-2 
 312-2 
 
 995-5 
 1035-1 
 
 61-0 
 61.8 
 
 40 
 
 113-8 
 
 387-1 
 
 108 
 
 154-9 
 
 526-8 
 
 91-9 
 
 202-4 
 
 688-1 
 
 79-6 
 
 316-2 
 
 1075-2 
 
 62-6 
 
396 ,THE DISCHARGE OF WATER FROM 
 
 THE HYDRAULIC RAM has been applied with advan- 
 tage in raising water to a considerable height by the 
 momentum of a larger quantity at a lower level. 
 The shock of the valves, and vibration of the ma- 
 chine, require heavy and strong setting, and con- 
 siderable strength in all the parts. This limits its 
 application, and prevents its use for raising large 
 quantities of water. The work done by the ram, in 
 over one thousand experiments by Eytelwein, did 
 not exceed in any of them 1480 Ibs. raised one 
 foot in one minute ; and in France, the ram put 
 up by the younger Montgolfier, said to be the 
 largest constructed, raised only 7400 Ibs. one foot 
 high per minute, and had a useful effect, it is re- 
 ported, of -65. This ram was put up at Mello, near 
 Clermont-sur-Oise. Its dimensions were 
 
 Length of the body pipe or injection pipe 108 feet. 
 Diameter . ; ... . 4-3 inches. 
 Weight of body pipe . . v . 3190 Ibs. 
 Weight of head . . |'^- '^'V ^ ; 440 Ibs. 
 Contents of air-chamber . ->v 1 J gallons. 
 
 This ram worked under a head of 37 feet, discharg- 
 ing in use 31i gallons each minute, and raising 
 3-85 gallons a height of 195 feet. 
 
 The largest ram employed by Eytelwein in his ex- 
 periments had the following dimensions 
 
 Length of the body pipe or injection pipe 43 feet 9 inches. 
 Diameter of ditto . . . .0 feet 2 inches. 
 Contents of air-chamber . . ; 1-94 gallons. 
 Area of tail or escape valve . -^ ,' T ' . 3-74 square inches ; 
 
 and his experiments led to the following practical 
 formula by D'Aubuisson 
 
OKIFICES, WEIKS, PIPES, AND EIVEKS. 
 
 397 
 
 in which D is the water used per minute in gallons, 
 d the quantity raised in gallons, h the head used, 
 and h' the lift of the quantity d. By a slight re- 
 duction we get 
 
 dh' = 1-42 D (h - -28 v/AAO 
 
 for the effect produced, which is reduced nearly one- 
 sixth for practical application, giving the formula 
 
 dA'=l-2 D (A - -2 v/A"A') 
 for the work done. 
 
 EXPERIMENTAL RESULTS. HYDEAULIC RAM. 
 
 umber 
 of 
 
 strokes 
 
 minute. 
 
 Height in feet 
 of 
 
 Ratio of 
 
 leights. 
 
 Gallons of water per 
 minute. 
 
 ** 
 
 Ratio 
 D 
 d~ 
 
 Fall 
 h 
 
 Elevation 
 h' 
 
 h 1 
 h 
 
 Expended 
 D 
 
 Raised 
 d 
 
 Experi- 
 ments. 
 
 Formula. 
 
 
 Ft. Iu. 
 
 Ft. In. 
 
 
 
 
 
 
 
 66 
 
 icy o" 
 
 26' 4" 
 
 2-63 
 
 10-65 
 
 3-39 
 
 
 
 97 
 
 2-92 
 
 04 
 
 10 2 
 
 32 4 
 
 3-18 
 
 13-97 
 
 383 
 
 873 
 
 92 
 
 3-67 
 
 00 
 
 9 11 
 
 38 8 
 
 3-9 
 
 12-01 
 
 2-622 
 
 85 
 
 87 
 
 4-58 
 
 52 
 
 8 
 
 32 4 
 
 4- 
 
 8-16 
 
 1-687 
 
 847 
 
 85 
 
 4-72 
 
 45 
 
 8 9 
 
 38 8 
 
 4-4 
 
 10-85 
 
 2-09 
 
 845 
 
 84 
 
 5-2 
 
 42 
 
 7 5 
 
 38 8 
 
 5-21 
 
 9-92 
 
 1-5 
 
 787 
 
 78 
 
 6-62 
 
 36 
 
 6 
 
 38 8 
 
 6-5 
 
 8-89 
 
 1-05 
 
 754 
 
 71 
 
 8-62 
 
 26 
 
 4 6 
 
 32 4 
 
 7-2 
 
 5-23 
 
 495 
 
 672 
 
 67 
 
 10-7 
 
 31 
 
 5 
 
 38 7 
 
 7-7 
 
 8-05 
 
 704 
 
 667 
 
 65 
 
 11-54 
 
 23 
 
 4 1 
 
 38 8 
 
 9-47 
 
 11-11 
 
 649 
 
 548 
 
 56 
 
 17-2 
 
 17 
 
 3 
 
 32 2 
 
 10-7 
 
 10-8 
 
 479 
 
 473 
 
 51 
 
 22-6 
 
 15 
 
 3 3 
 
 38 8 
 
 11-9 
 
 12-34 
 
 363 
 
 352 
 
 45 
 
 33-8 
 
 14 
 
 2 6 
 
 38 8 
 
 15-5 
 
 11-95 
 
 22 
 
 284 
 
 32 
 
 54-6 
 
 10 
 
 \, 
 
 MIJ 
 
 38 8 
 
 19-3 
 
 9-81 
 
 088 
 
 181 
 
 18 
 
 106-6 
 
39 
 
 THE DISCHARGE OF WATER FROM 
 
 Eytelwein recommends, that the length of the 
 body pipe should not be less than three-fourths of 
 the height to which the water is to be raised ; its 
 diameter in inches equal -58 V/D"; the diameter of 
 the rising pipe -3 \/:D ; and the contents of the air- 
 chamber equal to that of the rising pipe. 
 
 The following table gives the result of experiments 
 made by Montgolfier and his son : 
 
 TABLE OF EXPEEIMENTAL KESULTS HYDEAULIC HAM. 
 
 Height. 
 
 Water per Minute. 
 
 dh 
 Dh 
 
 Mean Batio 
 dlif 
 Dh 
 
 Fall 
 h 
 
 Elevation 
 h 
 
 Expended 
 D 
 
 Delivered 
 
 d 
 
 Ft. In. 
 
 Ft. In. 
 
 Gallons. 
 
 Gallons. 
 
 
 
 8' 6" 
 
 52' 8" 
 
 15 
 
 1-37 
 
 57 
 
 , . 
 
 37 2 
 
 195 
 
 31 
 
 3-85 
 
 653 
 
 
 34 9 
 
 111 11 
 
 18-5 
 
 3-74 
 
 651 
 
 65 
 
 3 3 
 
 14 11 
 
 437 
 
 59-18 
 
 629 
 
 .. 
 
 22 10 
 
 196 10 
 
 2-86 
 
 0-22 
 
 671 
 
 '" 
 
 Latterly, the Messrs. Easton and Amos have 
 patented improvements in this machine, and have 
 raised water to a height of 330 feet. The injection 
 pipe is laid by them at an inclination of about one 
 in four for high falls, and varies down to one in 
 eighteen for smaller falls. The quantities raised in 
 their practice vary up to six gallons per minute. 
 
 WATER PRESSURE ENGINES give a useful effect 
 varying up to 70 per cent, for the best constructed. 
 An immense amount of mechanical skill and inven- 
 tion has been brought to bear on their construction, 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 399 
 
 and in Weisbach's book* a useful effect of 83 per 
 cent, has been calculated ; this is, however, a result 
 seldom obtained in practice, where two-thirds, or 66 
 per cent., is nearer to the general efficiency. Jordan 
 got a maximum efficiency of '66 from one of the 
 Clausthal engines, making four strokes per minute, 
 and -71 making three strokes per minute. These re- 
 sults were for the combined engine and pumps, from 
 which it was calculated that the efficiency of the 
 engine alone, was in the first case -83, and in the 
 second -85. It would be a great mistake to calculate 
 on such high efficiencies. 
 
 CORN MILLS will grind about a bushel of corn per 
 horse-power per hour, but much depends on the state 
 of the stones and of the grain. The value of the work 
 done in an hour being once known, the value of the 
 standard horse-power can be determined accordingly. 
 
 * Vol. ii., p. 342. 
 
400 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE I. Coefficients of Discharge from Square and differently 
 proportioned Rectangular Lateral Orifices in thin Vertical Plates, 
 arranged from Poncelet and Lesbros. 
 
 , 
 
 
 Square orifice 
 
 Rectangular 
 
 Rectangular 
 
 
 
 
 8" X 8". 
 
 orifice 8" X 4". 
 
 orifice 8" X 2", 
 
 J" p " 
 
 *i ( 
 
 Ratio of the sides 
 
 Ratio of the sides 
 
 Ratio of the sides 
 
 . _-- 
 
 S 
 
 Itol. 
 
 2tol. 
 
 4 to 1. 
 
 |!|i 
 
 iff 
 
 |J 
 
 Jfj 
 
 Li 
 
 Sjj 
 
 1 | 
 
 |l ^ 
 
 1 | 
 
 ill 
 
 1 1 
 
 IM! 
 
 
 
 i 
 
 ill 
 
 'rt 
 
 i 
 
 ~ 
 
 J "o 
 
 fi 
 
 If 
 
 w a 
 
 11 
 
 w s 
 
 o-ooo 
 
 
 
 619 
 
 
 667 
 
 
 713 
 
 0-197 
 
 025 
 
 
 597 
 
 
 630 
 
 
 668 
 
 0-394 
 
 050 
 
 
 595 
 
 
 618 
 
 607 
 
 642 
 
 0-591 
 
 075 
 
 
 594 
 
 593 
 
 615 
 
 612 
 
 639 
 
 0-787 
 
 100 
 
 572 
 
 594 
 
 596 
 
 614 
 
 615 
 
 638 
 
 1-181 
 
 150 
 
 578 
 
 593 
 
 600 
 
 613 
 
 -620 
 
 637 
 
 1-575 
 
 200 
 
 582 
 
 593* 
 
 603 
 
 612 
 
 623 
 
 636 
 
 1969 
 
 250 
 
 585 
 
 593 
 
 605 
 
 612* 
 
 625 
 
 636 
 
 2-362 
 
 300 
 
 587 
 
 594 
 
 607 
 
 613 
 
 627 
 
 635 
 
 2-756 
 
 350 
 
 588 
 
 594 
 
 609 
 
 613 
 
 628 
 
 635 
 
 3-150 
 
 400 
 
 589 
 
 594 
 
 610 
 
 613 
 
 629 
 
 635 
 
 3-545 
 
 450 
 
 591 
 
 595 
 
 610 
 
 614 
 
 629 
 
 634 
 
 3-937 
 
 500 
 
 592 
 
 595 
 
 611 
 
 614 
 
 630 
 
 634 
 
 4-724 
 
 600 
 
 593 
 
 596 
 
 612 
 
 614 
 
 630 
 
 633 
 
 5-512 
 
 700 
 
 595 
 
 597 
 
 613 
 
 614 
 
 630 
 
 632 
 
 6-299 
 
 800 
 
 596 
 
 597 
 
 614 
 
 615 
 
 631* 
 
 631 
 
 7-087 
 
 900 
 
 597 
 
 598 
 
 615 
 
 615 
 
 630 
 
 631 
 
 7-874 
 
 1-000 
 
 598 
 
 599 
 
 615 
 
 615 
 
 630 
 
 630 
 
 9-843 
 
 1-250 
 
 599 
 
 600 
 
 616 
 
 616 
 
 630 
 
 630 
 
 11-811 
 
 1-500 
 
 600 
 
 601 
 
 616 - 
 
 616 
 
 629 
 
 629 
 
 15-748 
 
 2-000 
 
 602 
 
 602 
 
 617 
 
 617 
 
 628 
 
 629 
 
 19-685 
 
 2-500 
 
 603 
 
 603 
 
 617* 
 
 617* 
 
 628 
 
 628 
 
 23-622 
 
 3-000 
 
 604 
 
 604 
 
 617 
 
 617 
 
 627 
 
 627 
 
 27-560 
 
 3-500 
 
 604 
 
 604 
 
 616 
 
 616 
 
 627 
 
 627 
 
 31-497 
 
 4-000 
 
 605 
 
 605 
 
 616 
 
 616 
 
 627 
 
 627 
 
 35-434 
 
 4-500 
 
 605* 
 
 605* 
 
 615 
 
 615 
 
 526 
 
 626 
 
 39-371 
 
 5-000 
 
 605 
 
 605 
 
 615 
 
 615 
 
 626 
 
 626 
 
 43-307 
 
 5-500 
 
 604 
 
 604 
 
 614 
 
 614 
 
 625 
 
 625 
 
 47-245 
 
 6-000 
 
 604 
 
 604 
 
 614 
 
 614 
 
 624 
 
 624 
 
 51-182 
 
 6-500 
 
 603 
 
 603 
 
 613 
 
 613 
 
 622 
 
 622 
 
 55-119 
 
 7-000 
 
 603 
 
 603 
 
 612 
 
 612 
 
 621 
 
 621 
 
 59-056 
 
 7-500 
 
 602 
 
 602 
 
 611 
 
 611 
 
 620 
 
 620 
 
 62-993 
 
 8-000 
 
 602 
 
 602 
 
 611 
 
 611 
 
 618 
 
 618 
 
 66-930 
 
 8-500 
 
 602 
 
 602 
 
 610 
 
 610 
 
 617 
 
 617 
 
 70-867 
 
 9-000 
 
 601 
 
 601 
 
 609 
 
 609 
 
 615 
 
 615 
 
 74-805 
 
 9-500 
 
 601 
 
 601 
 
 608 
 
 608 
 
 614 
 
 614 
 
 78-742 
 
 10-000 
 
 601 
 
 601 
 
 607 
 
 607 
 
 613 
 
 614 
 
 118-112 
 
 15-000 
 
 601 
 
 601 
 
 603 
 
 603 
 
 606 
 
 606 
 
 See pages 71, 72, and 73. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 401 
 
 TABLE J. -Coefficients of Discharge from Square and differently 
 proportioned Rectangular Lateral Orifices in thin Vertical Plates t 
 arranged from Poncelet and Lesbros. 
 
 Rectangular 
 
 Rectangular 
 
 Rectangular 
 
 
 o ,0 
 
 orifice 8" Xl-18". 
 
 orifice 8" X 0-8". 
 
 orifice 8"X 0-4". 
 
 3 
 
 j *o 
 
 Ratio of the sides 
 
 Ratio of the sides 
 
 Ratio of the sides 
 
 r-j 
 
 03 60 
 
 7 to 1 nearly. 
 
 10 to 1. 
 
 20 to 1. 
 
 J^ J 
 
 S-I W . 
 
 si 
 
 If 
 
 |l 
 
 3s 
 
 1 
 
 sl 
 
 Has 
 
 a 
 
 1 1 
 
 Jll 
 
 ll'!l 
 
 li 1 
 
 %'% 
 1 1 
 
 Sell 
 
 IF 
 
 -I] 
 
 
 
 W 
 
 Heads 1 
 back fro 
 orifk 
 
 ~m~a o 
 
 -H ^ 
 
 *o <o *fl 
 
 3$ 
 
 CS <0 
 
 1*3 
 
 
 766 
 
 
 783 
 
 
 795 
 
 
 
 
 725 
 
 
 750 
 
 705 
 
 778 
 
 025 
 
 0-197 
 
 630 
 
 687 
 
 660 
 
 720 
 
 701 
 
 762 
 
 050 
 
 0-394 
 
 632 
 
 674 
 
 660 
 
 707 
 
 697 
 
 745 
 
 075 
 
 0-591 
 
 634 
 
 668 
 
 659 
 
 697 
 
 694 
 
 729 
 
 100 
 
 0-787 
 
 638 
 
 659 
 
 659 
 
 685 
 
 688 
 
 708 
 
 150 
 
 1-181 
 
 640 
 
 654 
 
 658 
 
 678 
 
 683 
 
 695 
 
 200 
 
 1-575 
 
 640* 
 
 651 
 
 658 
 
 672 
 
 679 
 
 686 
 
 250 
 
 1-969 
 
 640 
 
 647 
 
 657 
 
 668 
 
 676 
 
 681 
 
 300 
 
 2-362 
 
 639 
 
 645 
 
 656 
 
 665 
 
 673 
 
 677 
 
 350 
 
 2-756 
 
 638 
 
 643 
 
 656 
 
 662 
 
 670 
 
 675 
 
 400 
 
 3-150 
 
 637 
 
 641 
 
 655 
 
 659 
 
 668 
 
 672 
 
 450 
 
 3-543 
 
 637 
 
 640 
 
 654 
 
 657 
 
 666 
 
 669 
 
 500 
 
 3-937 
 
 636 
 
 637 
 
 653 
 
 655 
 
 663 
 
 665 
 
 600 
 
 4-724 
 
 635 
 
 636 
 
 651 
 
 653 
 
 660 
 
 661 
 
 700 
 
 5-512 
 
 634 
 
 635 
 
 650 
 
 651 
 
 658 
 
 659 
 
 800 
 
 6-299 
 
 634 
 
 634 
 
 649 
 
 650 
 
 657 
 
 657 
 
 900 
 
 7-087 
 
 633 
 
 633 
 
 648 
 
 649 
 
 655 
 
 656 
 
 1-000 
 
 7-874 
 
 632 
 
 632 
 
 646 
 
 646 
 
 653 
 
 653 
 
 1-250 
 
 9'843 
 
 632 
 
 632 
 
 644 
 
 644 
 
 650 
 
 651 
 
 1-500 
 
 11-811 
 
 631 
 
 631 
 
 642 
 
 642 
 
 647 
 
 647 
 
 2-000 
 
 15-748 
 
 630 
 
 630 
 
 640 
 
 640 
 
 644 
 
 645 
 
 2-500 
 
 19-685 
 
 630 
 
 630 
 
 638 
 
 638 
 
 642 
 
 643 
 
 3-000 
 
 23-622 
 
 629 
 
 629 
 
 637 
 
 637 
 
 640 
 
 640 
 
 3-500 
 
 27-560 
 
 629 
 
 629 
 
 636 
 
 636 
 
 637 
 
 637 
 
 4-000 
 
 31-497 
 
 628 
 
 628 
 
 634 
 
 634 
 
 635 
 
 635 
 
 4-500 
 
 35-434 
 
 628 
 
 628 
 
 633 
 
 633 
 
 632 
 
 632 
 
 5-000 
 
 39-371 
 
 627 
 
 627 
 
 631 
 
 631 
 
 629 
 
 629 
 
 5-500 
 
 43-307 
 
 626 
 
 626 
 
 628 
 
 628 
 
 626 
 
 626 
 
 6-000 
 
 47-245 
 
 624 
 
 624 
 
 625 
 
 625 
 
 622 
 
 622 
 
 6-500 
 
 51-182 
 
 622 
 
 622 
 
 622 
 
 622 
 
 618 
 
 618 
 
 7-000 
 
 55-119 
 
 620 
 
 620 
 
 619 
 
 619 
 
 615 
 
 615 
 
 7-500 
 
 59-056 
 
 618 
 
 618 
 
 617 
 
 617 
 
 613 
 
 613 
 
 8-000 
 
 62-993 
 
 616 
 
 616 
 
 615 
 
 615 
 
 612 
 
 612 
 
 8-500 
 
 66-930 
 
 615 
 
 615 
 
 614 
 
 614 
 
 612 
 
 612 
 
 9-000 
 
 70-867 
 
 613 
 
 613 
 
 612 
 
 612 
 
 611 
 
 611 
 
 9-500 
 
 74-805 
 
 612 
 
 612 
 
 612 
 
 612 
 
 611 
 
 611 
 
 10-000 
 
 78-742 
 
 608 
 
 608 
 
 610 
 
 610 
 
 609 
 
 609 
 
 15-000 
 
 118-112 
 
 See pages 71, 72, and 73. 
 
402 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE II. For finding the Velocities from the Altitudes, and the 
 Altitudes from the Velocities. 
 
 Altitudes feet 
 
 inch to feet 3|- inches. 
 
 1 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 
 
 discharge in inches per second. 
 
 r 
 
 
 o , 
 
 
 
 
 
 
 .S 2 
 
 ^ "a jf 
 
 "s^ si 
 
 ^5*5. 
 
 "o"*^ 
 
 ^"*3 in 
 
 "o 
 
 *J 
 
 l?|l 
 
 jii 
 
 ill 
 
 111 
 
 ill 
 
 ill 
 
 :j3 
 
 k S| 1 
 
 ^1 
 
 ^| 
 
 **gi 
 
 If 1 
 
 * B! 
 
 5 
 
 T-i H | "3 
 
 <* *j 
 
 rf ! 
 
 Tj5 5> g 
 
 *>J 
 
 5 *! 
 
 6 '6^ 
 
 2-78 
 
 2-71 
 
 2-66 
 
 2-39 
 
 2-27 
 
 2-22 
 
 0^ 
 
 3-48 
 
 3-38 
 
 3-32 
 
 2-99 
 
 2-83 
 
 278 
 
 0^-g. 
 
 6-95 
 
 6-77 
 
 6-64 
 
 5-98 
 
 5-66 
 
 5-56 
 
 Oi 
 
 9-829 
 
 957 
 
 9-40 
 
 8-45 
 
 8-01 
 
 7-86 
 
 -JL 
 
 12038 
 
 11-72 
 
 11-51 
 
 1035 
 
 9-81 
 
 9-63 
 
 Oi 
 
 13-900 
 
 13-54 
 
 1329 
 
 11 95 
 
 11-33 
 
 
 o o-j? 
 
 15541 
 
 15-14 
 
 14-86 
 
 13-36 
 
 12-67 
 
 12-43 
 
 Of 
 
 17024 
 
 16-58 
 
 16-27 
 
 1464 
 
 1387 
 
 13-62 
 
 S of 
 
 18-388 
 19-658 
 
 17-91 
 19 15 
 
 17-58 
 1879 
 
 15-81 
 16-91 
 
 14-99 
 16-02 
 
 14-71 
 1573 
 
 T 9 
 
 20850 
 
 20-31 
 
 19-93 
 
 1793 
 
 1699 
 
 16-68 
 
 Of 
 
 21-978 
 
 21-41 
 
 21-01 
 
 18-90 
 
 17-91 
 
 17-58 
 
 o ott 
 
 23-051 
 
 22-45 
 
 22-04 
 
 19-82 
 
 18-79 
 
 1844 
 
 o of 
 
 24-076 
 
 23-45 
 
 23-02 
 
 20-70 
 
 19-62 
 
 1926 
 
 o olf 
 
 25-059 
 
 24-41 
 
 24-00 
 
 21-55 
 
 20-4-2 
 
 20-05 
 
 Oi 
 
 26-005 
 
 25-33 
 
 24-86 
 
 22-36 
 
 21-19 
 
 2080 
 
 0^-f 
 
 26-917 
 
 26-22 
 
 25-73 
 
 23-15 
 
 21-94 
 
 21-53 
 
 1 
 
 27-800 
 
 27-08 
 
 26-58 
 
 23-91 
 
 22-66 
 
 22-24 
 
 1 
 
 29-486 
 
 28'72 
 
 28-19 
 
 25-36 
 
 24-03 
 
 23-59 
 
 o u 
 
 31-081 
 
 30-27 
 
 29-71 
 
 26-73 
 
 2533 
 
 24-87 
 
 l| 
 
 32-598 
 
 3175 
 
 31-16 
 
 28-03 
 
 26-57 
 
 26-08 
 
 1| 
 
 34-048 
 
 33-19 
 
 32-58 
 
 29-30 
 
 27-75 
 
 2726 
 
 l| 
 
 35-438 
 
 3452 
 
 33-88 
 
 30-48 
 
 28-88 
 
 28-35 
 
 1| 
 
 36-776 
 
 3582 
 
 35 16 
 
 31-63 
 
 29-97 
 
 29-4-2 
 
 13 
 
 38-067 
 
 3708 
 
 3639 
 
 32-74 
 
 31 02 
 
 30-45 
 
 2 8 
 
 39-315 
 
 38-29 
 
 37-59 
 
 33-81 
 
 32-04 
 
 31-45 
 
 2i 
 
 40525 
 
 39-47 
 
 38-74 
 
 34-85 
 
 33-03 
 
 3242 
 
 2| 
 
 41-700 
 
 40*62 
 
 3987 
 
 35-86 
 
 33-99 
 
 33-36 
 
 2s 
 
 42-843 
 
 4173 
 
 40-96 
 
 36-84 
 
 34-92 
 
 3427 
 
 2| 
 
 43-956 
 
 42-81 
 
 42-02 
 
 37-80 
 
 35-82 
 
 3516 
 
 2| 
 
 45-041 
 
 43-87 
 
 43-06 
 
 38-74 
 
 36-71 
 
 3603 
 
 2f 
 
 46-101 
 
 44-90 
 
 44-07 
 
 39-65 
 
 3757 
 
 36-88 
 
 2 
 
 47-137 
 
 45-90 
 
 45-06 
 
 40-54 
 
 38-42 
 
 37-71 
 
 3 
 
 48-151 
 
 46-90 
 
 46-03 
 
 41-41 
 
 39-24 
 
 38-52 
 
 3$. 
 
 49-144 
 
 47-87 
 
 4698 
 
 42-26 
 
 40-05 
 
 39-32 
 
 3| 
 
 50-117 
 
 4881 
 
 47-91 
 
 43-10 
 
 40-85 
 
 40-09 
 
 3f 
 
 51072 
 
 49-74 
 
 4882 
 
 4392 
 
 4162 
 
 4086 
 
 3| 
 
 52-009 
 
 50-66 
 
 49-72 
 
 44-73 
 
 4239 
 
 41 61 
 
 3| 
 
 52-930 
 
 5155 
 
 5060 
 
 45-52 
 
 43 14 
 
 42-34 
 
 3f 
 
 53-834 
 
 52-43 
 
 51-47 
 
 46-30 
 
 43-88 
 
 43-07 
 
 3| 
 
 54725 
 
 53-30 
 
 52-3-2 
 
 47-06 
 
 44-60 
 
 4378 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 403 
 
 TABLE II. For finding the Velocities from the Altitudes, and the 
 
 Altitudes from the Velocities. 
 Altitudes feet T g- inch to feet 3| inches. 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 1 
 
 discharge in inches per second. 
 
 
 
 1 
 
 Hi 
 
 t*g 
 
 *sS 
 
 "o^ 
 
 *ag 
 
 **.$ 
 
 ,M 
 
 ^ ^* 
 
 ffi r * fc ^P 
 
 re " 
 
 f/3 -* 
 
 so r* ^ 
 
 o> 10 
 
 ** 2 
 
 ill 
 
 S3 
 a QO .2 
 
 fS 
 
 li?l 
 
 J oo "a 
 g <b .2 
 
 1 f 1 
 
 1 
 
 "1 
 
 ii 
 
 *71 
 
 k 7 e 
 
 1 1 
 
 7 1 
 
 3 
 
 
 
 ^ *< 
 
 ' & O 
 
 * & 3 
 
 s *! 
 
 s * J 
 
 9 *J 
 
 3 
 
 195 
 
 1-85 
 
 1-75 
 
 1-72 
 
 1-68 
 
 1-62 
 
 0/ <Uo 
 
 2-43 
 
 2-31 
 
 2-l 
 
 2-15 
 
 2-11 
 
 2-03 
 
 0^ 
 
 487 
 
 4-63 
 
 4-36 
 
 4-29 
 
 4-21 
 
 4-06 
 
 i 
 
 6-88 
 
 655 
 
 6-17 
 
 6-06 
 
 5-96 
 
 5-74 
 
 o o{ 
 
 8-43 
 
 8-02 
 
 7-56 
 
 7-43 
 
 7-29 
 
 7-03 
 
 03 
 
 9-73 
 
 9-26 
 
 8-73 
 
 8-58 
 
 8-42 
 
 8 12 
 
 0| 
 
 10-88 
 
 10-35 
 
 9-76 
 
 9-59 
 
 9'42 
 
 908 
 
 3 
 
 1192 
 
 11-24 
 
 10-69 
 
 10-50 
 
 10-32 
 
 994 
 
 o ojr 
 
 1-2-87 
 
 1225 
 
 11-55 
 
 11-35 
 
 11-14 
 
 10-74 
 
 o oX 
 
 13-76 
 
 12-97 
 
 1234 
 
 12 13 
 
 11-91 
 
 11-48 
 
 o o| 6 
 
 1460 
 
 13-89 
 
 1309 
 
 12'86 
 
 12-64 
 
 1218 
 
 0_9 
 
 15-38 
 
 1464 
 
 13-80 
 
 1356 
 
 13*32 
 
 12-84 
 
 (15^ 
 
 16-14 
 
 15-35 
 
 1448 
 
 1422 
 
 13-97 
 
 13-46 
 
 oli 
 
 16-85 
 
 1603 
 
 15-12 
 
 1485 
 
 14-59 
 
 14-06 
 
 0| 
 
 17-54 
 
 16-69 
 
 15-74 
 
 1546 
 
 15-19 
 
 14-63 
 
 o 0*1 
 
 18-20 
 
 17-32 
 
 16-33 
 
 16-04 
 
 1576 
 
 15-09 
 
 Oi 
 
 18-84 
 
 17-93 
 
 16-90 
 
 16-61 
 
 16-31 
 
 15-72 
 
 o olf 
 
 19-46 
 
 1851 
 
 1746 
 
 17-15 
 
 16-85 
 
 1624 
 
 1 
 
 20-64 
 
 19-64 
 
 18-52 
 
 18'i9 
 
 17-87 
 
 17-22 
 
 li 
 
 21-76 
 
 20-70 
 
 19-52 
 
 19 18 
 
 18-84 
 
 1815 
 
 o 4 
 
 22-82 
 
 21-71 
 
 20-47 
 
 20*11 
 
 19-75 
 
 19-04 
 
 If 
 
 23-85 
 
 22-69 
 
 21-38 
 
 21-01 
 
 20-63 
 
 19-88 
 
 ] 
 
 24-81 
 
 23-60 
 
 22-26 
 
 21-87 
 
 21-48 
 
 20-70 
 
 If 
 
 25-74 
 
 24-49 
 
 23-10 
 
 22-69 
 
 22-29 
 
 21-48 
 
 If 
 
 26-65 
 
 25-35 
 
 23-91 
 
 23-49 
 
 23-07 
 
 22-23 
 
 If 
 
 2752 
 
 26-18 
 
 24-69 
 
 24-26 
 
 23-82 
 
 22-96 
 
 2 
 
 28-37 
 
 2699 
 
 2545 
 
 25-00 
 
 24-50 
 
 23-67 
 
 2| 
 
 29-19 
 
 27-77 
 
 26-19 
 
 25-73 
 
 25-27 
 
 24-35 
 
 2| 
 
 29-99 
 
 28-53 
 
 26-91 
 
 2643 
 
 25-96 
 
 25-02 
 
 2i 
 
 30-77 
 
 29-27 
 
 27-60 
 
 27-12 
 
 26-64 
 
 25-67 
 
 2 
 
 3153 
 
 3000 
 
 28-29 
 
 2779 
 
 27-29 
 
 26-30 
 
 2| 
 
 32-27 
 
 30-70 
 
 28-95 
 
 28-44 
 
 27-94 
 
 26-92 
 
 2| 
 
 3300 
 
 31-39 
 
 29-60 
 
 29-08 
 
 28-57 
 
 27-53 
 
 2| 
 
 3371 
 
 32-07 
 
 30-24 
 
 29-71 
 
 29 18 
 
 28-12 
 
 3 
 
 34-40 
 
 32-73 
 
 3086 
 
 30-32 
 
 29-78 
 
 28-70 
 
 3| 
 
 3508 
 
 33-38 
 
 31-47 
 
 30-92 
 
 30-37 
 
 29-27 
 
 3} 
 
 35-75 
 
 34-01 
 
 32-07 
 
 31-51 
 
 30-95 
 
 29-83 
 
 3| 
 
 3641 
 
 34-64 
 
 32-66 
 
 32-09 
 
 31-52 
 
 30-37 
 
 3j 
 
 37-05 
 
 35-25 
 
 33-24 
 
 32-66 
 
 32-08 
 
 30-91 
 
 85 
 
 3768 
 
 35-85 
 
 3381 
 
 3322 
 
 32-62 
 
 31-44 
 
 3f 
 
 38-31 
 
 36-45 
 
 34-37 
 
 33-77 
 
 33-16 
 
 31-96 
 
 3| 
 
 DD3 
 
404 
 
 THE DISCHARGE OF WATEE FROM 
 
 TABLE II. For finding the Velocities from the Altitudes, and the 
 
 Altitudes from the Velocities. 
 Altitudes feet 4 inches to 1 foot. 
 
 T3 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 
 
 "8 
 
 discharge in inches per second. 
 
 1 i 
 
 sill 
 
 r^ 
 
 r .| 
 
 TO 00 * 
 
 i*i 
 
 -o-s | 
 
 -^ 
 
 s^^ "3 
 
 o "S 
 
 ,2 > "S 
 
 5 S 9 
 
 to "S 
 
 c5 "3 
 
 J 
 
 ?||'.a 
 
 ^ 85 
 
 S -1 
 
 a 05 .2 
 
 f> cq jS 
 
 gal 
 
 1 
 
 e ^ Q 
 
 -'ij 
 
 -i| 
 
 1 1 
 
 ii| 
 
 -i| 
 
 6// 
 4 
 
 55-600 
 
 54-15 
 
 53-15 
 
 47-82 
 
 45-31 
 
 44-48 
 
 4ft 
 
 56-462 
 
 54-99 
 
 53-98 
 
 48-56 
 
 46-02 
 
 45-17 
 
 4j 
 
 57-311 
 
 55-82 
 
 54-79 
 
 49-29 
 
 46-71 
 
 45-85 
 
 4f 
 
 58-148 
 
 56-64 
 
 55-59 
 
 50-01 
 
 47-39 
 
 46-52 
 
 4J 
 
 58-973 
 
 57-44 
 
 56-38 
 
 50-72 
 
 48-06 
 
 47-18 
 
 4f 
 
 59-786 
 
 58-23 
 
 57-16 
 
 51-42 
 
 48-73 
 
 47-83 
 
 4f 
 
 60-589 
 
 59-01 
 
 57-92 
 
 52-11 
 
 49-38 
 
 48-47 
 
 4|- 
 
 61-368 
 
 59-77 
 
 58-67 
 
 52-78 
 
 50-02 
 
 49-09 
 
 5 
 
 62-163 
 
 60-55 
 
 69*43 
 
 53-46 
 
 50-66 
 
 49-73 
 
 5J 
 
 62-935 
 
 61-30 
 
 60-17 
 
 54-12 
 
 51-29 
 
 50-35 
 
 51 
 
 63-698 
 
 62-04 
 
 60-90 
 
 54-78 
 
 51-91 
 
 50-96 
 
 5| 
 
 64-452 
 
 62-78 
 
 61-62 
 
 55-43 
 
 52-53 
 
 51-56 
 
 5 
 
 65-197 
 
 63-50 
 
 62-33 
 
 56-07 
 
 53-14 
 
 52-16 
 
 5f 
 
 65-933 
 
 64-22 
 
 63-03 
 
 56-70 
 
 53-74 
 
 52-75 
 
 5f 
 
 66-662 
 
 64-93 
 
 63-73 
 
 57-33 
 
 54-33 
 
 53-33 
 
 5 
 
 67-383 
 
 65-63 
 
 64-42 
 
 57-95 
 
 54-92 
 
 53-91 
 
 6 
 
 68-096 
 
 66-33 
 
 65-10 
 
 58-56 
 
 55-50 
 
 54-48 
 
 61 
 
 69-500 
 
 67-69 
 
 66-44 
 
 59-77 
 
 56-64 
 
 55-60 
 
 6| 
 
 70-876 
 
 69-03 
 
 67-76 
 
 60-95 
 
 57-24 
 
 56-70 
 
 6f 
 
 72-227 
 
 70-35 
 
 69-05 
 
 62-11 
 
 58-86 
 
 57-78 
 
 7 
 
 73-552 
 
 71-64 
 
 70-32 
 
 63-25 
 
 59-95 
 
 58-84 
 
 7i 
 
 74-854 
 
 72-91 
 
 71-56 
 
 64-37 
 
 61-01 
 
 59-88 
 
 7$ 
 
 76-133 
 
 74-15 
 
 72-78 
 
 65-47 
 
 62-05 
 
 60-91 
 
 7| 
 
 77-392 
 
 75-38 
 
 73-99 
 
 66-56 
 
 63-07 
 
 61-91 
 
 0> 8 
 
 78-630 
 
 76-59 
 
 75-17 
 
 67-62 
 
 64-08 
 
 62-90 
 
 8J 
 
 79-849 
 
 77-77 
 
 76-34 
 
 68-67 
 
 65-08 
 
 63-88 
 
 8 
 
 81-050 
 
 78-94 
 
 77-48 
 
 69-70 
 
 66-06 
 
 64-84 
 
 8| 
 
 82-234 
 
 80-10 
 
 78-62 
 
 70-72 
 
 67-02 
 
 65-79 
 
 9 
 
 83-40 
 
 81-23 
 
 79-73 
 
 71-72 
 
 67-97 
 
 66-72 
 
 9 
 
 84-550 
 
 82-35 
 
 80-83 
 
 72-71 
 
 68-91 
 
 67-64 
 
 9J 
 
 85*685 
 
 83-46 
 
 81-92 
 
 73-69 
 
 69-83 
 
 68-55 
 
 Of 
 
 86-805 
 
 84-55 
 
 82-99 
 
 74-65 
 
 70-75 
 
 69-44 
 
 10 
 
 87-911 
 
 85-63 
 
 84-04 
 
 75-60 
 
 71-65 
 
 70-33 
 
 10i 
 
 89-004 
 
 86-69 
 
 85-09 
 
 76-54 
 
 72-54 
 
 71-20 
 
 10 
 
 90-082 
 
 87-74 
 
 86-12 
 
 77-47 
 
 73-42 
 
 72-07 
 
 lOf 
 
 91-148 
 
 88-79 
 
 87-14 
 
 78-39 
 
 74-29 
 
 72-92 
 
 11 
 
 92-202 
 
 89-80 
 
 88-15 
 
 79-29 
 
 75-14 
 
 73-76 
 
 111 
 
 93-244 
 
 90-82 
 
 89-14 
 
 80-19 
 
 75-99 
 
 74-59 
 
 Hi 
 
 94-274 
 
 91-82 
 
 90-13 
 
 81-08 
 
 76-83 
 
 75-42 
 
 11| 
 
 95-294 
 
 92-82 
 
 91.10 
 
 81-95 
 
 77-66 
 
 76-23 
 
 1 
 
 96-302 
 
 93-80 
 
 92-06 
 
 82-82 
 
 78-49 
 
 77-04 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 405 
 
 TABLE II. For finding the Velocities from the Altitudes, and the 
 
 Altitudes from the Velocities. 
 Altitudes feet 4 inches to 1 foot. 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 ti 
 
 discharge in inches per second. 
 
 ' | 
 
 S~ 
 
 w .g 
 
 **| 
 
 H* 
 
 ** 
 
 *ftg 
 
 J J 
 
 1 S 13 
 
 Hi 
 
 |I| 
 
 Hj 
 
 ||j 
 
 I1J 
 
 jl 
 
 -1*1 
 
 ij 
 
 OS J 
 
 SfjJ 
 
 rfi| 
 
 **j 
 
 1 
 
 38-92 
 
 37-03 
 
 34-92 
 
 34-31 
 
 33-69 
 
 32-47 
 
 6 I 
 
 39-52 
 
 37-60 
 
 35-46 
 
 34-84 
 
 34-22 
 
 32-97 
 
 4* 
 
 40-12 
 
 38-17 
 
 35-99 
 
 35-36 
 
 34-73 
 
 33-47 
 
 4J- 
 
 40-70 
 
 38-73 
 
 36-52 
 
 35-88 
 
 35-24 
 
 33-96 
 
 4f 
 
 41-28 
 
 39-28 
 
 37-03 
 
 36-39 
 
 35-74 
 
 34-44 
 
 4 
 
 41-85 
 
 39-82 
 
 37-55 
 
 36-89 
 
 36-23 
 
 34-92 
 
 4f 
 
 42-41 
 
 40-35 
 
 38-05 
 
 37-38 
 
 36-72 
 
 35-38 
 
 4| 
 
 42-96 
 
 40-87 
 
 38-54 
 
 37-86 
 
 37-19 
 
 35-84 
 
 4| 
 
 43-51 
 
 41-40 
 
 39-04 
 
 38-35 
 
 37-67 
 
 36-30 
 
 5 
 
 44-05 
 
 41-91 
 
 39-52 
 
 38-83 
 
 38-14 
 
 36-75 
 
 6k 
 
 44-59 
 
 42-42 
 
 40-00 
 
 39-30 
 
 38-60 
 
 37-20 
 
 5i 
 
 45-12 
 
 42-92 
 
 40-48 
 
 39-77 
 
 39-06 
 
 37-64 
 
 51 
 
 45-64 
 
 43-42 
 
 40-94 
 
 40-23 
 
 39-51 
 
 38-07 
 
 5| 
 
 46-15 
 
 43-91 
 
 41-41 
 
 40-68 
 
 39-96 
 
 38-51 
 
 5f 
 
 46-66 
 
 44-40 
 
 41-86 
 
 41-13 
 
 40-40 
 
 38-93 
 
 5| 
 
 47-17 
 
 44-88 
 
 42-32 
 
 41-58 
 
 40-83 
 
 39-35 
 
 5 
 
 47-67 
 
 45-35 
 
 42-76 
 
 42-02 
 
 41-27 
 
 39-77 
 
 6 
 
 48-65 
 
 46-29 
 
 43-65 
 
 42-88 
 
 42-12 
 
 40-59 
 
 6J 
 
 49-61 
 
 47-20 
 
 44-51 
 
 43-73 
 
 42-95 
 
 41-39 
 
 6 
 
 50-56 
 
 48-10 
 
 45-36 
 
 44-56 
 
 43-77 
 
 42-18 
 
 6| 
 
 51-49 
 
 48-99 
 
 46-19 
 
 45-38 
 
 44-57 
 
 42-95 
 
 7 
 
 52-40 
 
 49-85 
 
 47-01 
 
 46-18 
 
 45-36 
 
 43-71 
 
 7J 
 
 ^3-29 
 
 50-70 
 
 47-81 
 
 46-97 
 
 46-14 
 
 44-46 
 
 7f 
 
 54-17 
 
 51-54 
 
 48-60 
 
 47-75 
 
 46-90 
 
 45-20 
 
 7f 
 
 55-04 
 
 52-37 
 
 49-38 
 
 48-51 
 
 47.65 
 
 45-92 
 
 8 
 
 55-89 
 
 53-18 
 
 50-15 
 
 49-27 
 
 48-39 
 
 46-63 
 
 8J 
 
 56-74 
 
 53-98 
 
 50-90 
 
 50-01 
 
 49-12 
 
 47-33 
 
 8| 
 
 57-56 
 
 54-77 
 
 51-64 
 
 50-74 
 
 49-83 
 
 48-02 
 
 8f 
 
 58-38 
 
 55-54 
 
 52-38 
 
 51-46 
 
 50-54 
 
 48-71 
 
 9 
 
 59-19 
 
 56-31 
 
 53-10 
 
 52-17 
 
 51-24 
 
 49-38 
 
 9| 
 
 59-98 
 
 57-07 
 
 53-81 
 
 52-87 
 
 51-93 
 
 50-04 
 
 9| 
 
 60-76 
 
 57-81 
 
 54-51 
 
 53-56 
 
 52-60 
 
 50-69 
 
 9| 
 
 61-54 
 
 58-55 
 
 55-22 
 
 54-24 
 
 53-27 
 
 51-34 
 
 10 
 
 62-30 
 
 59-28 
 
 55-89 
 
 54-92 
 
 53-94 
 
 51-98 
 
 10 
 
 63-06 
 
 60-00 
 
 56-57 
 
 55-58 
 
 54-59 
 
 52-61 
 
 10J 
 
 63-80 
 
 60-70 
 
 57-24 
 
 56-24 
 
 55-24 
 
 53-23 
 
 10f 
 
 64-54 
 
 61-41 
 
 57-90 
 
 56-89 
 
 55-87 
 
 53-85 
 
 11 
 
 65-27 
 
 62-10 
 
 58-56 
 
 57-53 
 
 56-51 
 
 54-45 
 
 Hi 
 
 65-99 
 
 62-79 
 
 59-70 
 
 58-17 
 
 57-13 
 
 55-06 
 
 HJ 
 
 66-71 
 
 63-47 
 
 59-84 
 
 58-80 
 
 57-75 
 
 55-65 
 
 llf 
 
 67-41 
 
 64-14 
 
 60-48 
 
 59-42 
 
 58-36 
 
 56-24 
 
 1 
 
406 
 
 THE DISCHABGE OF WATEK FEOM 
 
 TABLE II. For finding the Velocities from the Altitudes, and the 
 
 Altitudes from the Velocities. 
 Altitudes 1 foot O inch to 5 feet 3 inches. 
 
 *2 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 1 
 
 discharge in inches per second. 
 
 i| 
 
 *8 *" "i 8 
 
 r^ 
 
 Is 
 
 * -U 
 
 |.s? 
 
 !~;i 
 
 
 
 ,5 ^ .3 
 
 "3 - H 
 
 a 
 
 " 5 
 
 * *? g 
 
 i H 9 
 
 a 
 
 k | -3 
 
 ^ ^i 
 
 ^ i 
 
 > S3 - 
 
 8 -3 
 
 > * 1 
 
 52 
 
 * 
 
 ei & J 
 
 rfij 
 
 4>| 
 
 >! 
 
 rfV| 
 
 0} 
 
 98-288 
 
 95-73 
 
 93-96 
 
 84-53 
 
 80-10 
 
 78-63 
 
 1 
 
 100-234 
 
 97-63 
 
 95-82 
 
 86-20 
 
 81-69 
 
 80-19 
 
 1 
 
 102-144 
 
 99-49 
 
 97-65 
 
 87-84 
 
 83-25 
 
 81-71 
 
 2 
 
 104-018 
 
 101-31 
 
 99-44 
 
 89-46 
 
 8477 
 
 83-21 
 
 2} 
 
 105-859 
 
 103-11 
 
 101-20 
 
 91-04 
 
 86-28 
 
 8469 
 
 3 
 
 107-669 
 
 10487 
 
 102-93 
 
 9-260 
 
 87-75 
 
 86-14 
 
 3} 
 
 109 449 
 
 10660 
 
 104-63 
 
 94-13 
 
 89-20 
 
 87-56 
 
 4 
 
 111-200 
 
 108-31 
 
 106-31 
 
 95-63 
 
 90-63 
 
 88-96 
 
 4} 
 
 112-924 
 
 10999 
 
 107 96 
 
 97 11 
 
 9203 
 
 90-34 
 
 5 
 
 114-6-22 
 
 111-42 
 
 109-58 
 
 9858 
 
 93-42 
 
 91-70 
 
 5} 
 
 116-296 
 
 113-27 
 
 111-18 
 
 100-01 
 
 94-78 
 
 93-04 
 
 6 
 
 117-945 
 
 114-78 
 
 11276 
 
 10143 
 
 96-13 
 
 94-36 
 
 7 
 
 1-21177 
 
 118-03 
 
 115-85 
 
 10421 
 
 98-76 
 
 96-94 
 
 8 
 
 124 325 
 
 121 -09 
 
 11886 
 
 106-92 
 
 101-33 
 
 99-46 
 
 9 
 
 127-896 
 
 124-08 
 
 121-79 
 
 109-56 
 
 103-83 
 
 101-92 
 
 10 
 
 130-394 
 
 127-00 
 
 12466 
 
 112 14 
 
 106-27 
 
 10431 
 
 11 
 
 183-324 
 
 129-86 
 
 .127-46 
 
 114-66 
 
 108-66 
 
 106-66 
 
 2 
 
 136-192 
 
 132-65 
 
 130-20 
 
 117-12 
 
 111-00 
 
 108-95 
 
 2 H 
 
 140-383 
 
 13673 
 
 13421 
 
 120-73 
 
 11441 
 
 112-31 
 
 2 3 
 
 144-453 
 
 140-70 
 
 138-10 
 
 1-24-23 
 
 117-73 
 
 115-56 
 
 2 4} 
 
 148-411 
 
 144-55 
 
 141-88 
 
 127-64 
 
 120-96 
 
 118-73 
 
 2 6 
 
 152-267 
 
 148-31 
 
 14557 
 
 130-95 
 
 124-10 
 
 12181 
 
 2 7} 
 
 156027 
 
 151-97 
 
 149-16 
 
 134-18 
 
 127-16 
 
 124-82 
 
 2 9 
 
 159-699 
 
 155-55 
 
 152-67 
 
 137-34 
 
 130-15 
 
 127-76 
 
 2 10} 
 
 163-288 
 
 159-04 
 
 156-10 
 
 140-43 
 
 133-80 
 
 130-63 
 
 3 
 
 166-800 
 
 162-46 
 
 159-46 
 
 143-45 
 
 13594 
 
 133-44 
 
 3 1} 
 
 170-240 
 
 165-81 
 
 162-75 
 
 14641 
 
 138-75 
 
 136-19 
 
 3 3 
 
 173-611 
 
 169-10 
 
 165-97 
 
 149*31 
 
 141-49 
 
 13889 
 
 3 4} 
 
 176-918 
 
 172-32 
 
 169-13 
 
 152-15 
 
 144-19 
 
 141-53 
 
 3 6 
 
 180-165 
 
 175-48 
 
 172-24 
 
 154-94 
 
 146-83 
 
 144-13 
 
 3 7} 
 
 183-354 
 
 178-59 
 
 175-29 
 
 157-68 
 
 149-43 
 
 146-68 
 
 3 9 
 
 186-488 
 
 181-64 
 
 178-28 
 
 1' 0'38 
 
 151-99 
 
 149-19 
 
 3 10} 
 
 189-571 
 
 184-64 
 
 181-23 
 
 163-03 
 
 154-50 
 
 151-66 
 
 4 
 
 192604 
 
 187-60 
 
 184-13 
 
 165-64 
 
 156-97 
 
 154-08 
 
 4 2 
 
 196576 
 
 191-46 
 
 187-93 
 
 169-06 
 
 16021 
 
 157-26 
 
 4 4 
 
 200-469 
 
 195-26 
 
 191-65 
 
 172-40 
 
 16338 
 
 160-37 
 
 4 6 
 
 204287 
 
 198-98 
 
 195-30 
 
 17569 
 
 16I5-49 
 
 163-43 
 
 4 8 
 
 208-036 
 
 202-63 
 
 198-88 
 
 178-91 
 
 169-55 
 
 166-43 
 
 4 10 
 
 211-718 
 
 206-21 
 
 202-40 
 
 182-08 
 
 172-55 
 
 16937 
 
 5 
 
 215338 
 
 209-74 
 
 205-86 
 
 185-19 
 
 17550 
 
 172-27 
 
 5 3 
 
 220-656 
 
 214-92 
 
 210-95 
 
 189-76 
 
 179-83 
 
 176-52 
 
OEIFICES, WEIES, PIPES, AND EIVEES. 
 
 407 
 
 TABLE II. For finding the Velocities from the Altitudes, and tlie 
 
 Altitudes from the Velocities. 
 Altitudes 1 foot OJ inch to 5 feet 3 inches. 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 
 discharge in inches per second. 
 
 9 
 
 s 
 
 *j 
 
 o^i 
 
 **g 
 
 oa ** S 
 
 M 
 
 1*1 
 
 s 
 
 o to .^ 
 
 1 is 
 
 ! 2 ^ 
 
 QJ CO " 
 
 |S1 
 
 H ? s 
 
 fj 
 
 > 7 i 
 
 3 * 
 
 
 
 > 5 -3 
 
 s| 
 
 s| 
 
 
 II fi 
 
 *J1 
 
 <!! 
 
 r-l & O 
 
 ri'll 
 
 *ij 
 
 9 
 
 
 
 
 
 
 
 / // 
 
 68-80 
 
 65-46 
 
 61-72 
 
 60-64 
 
 59-56 
 
 67-40 
 
 1 
 
 70-16 
 
 66-76 
 
 62-95 
 
 61-84 
 
 60-74 
 
 58-54 
 
 1 1 
 
 71-50 
 
 68-03 
 
 64-15 
 
 63-02 
 
 61-90 
 
 59-65 
 
 1 1J 
 
 72-81 
 
 6928 
 
 65 32 
 
 64-18 
 
 6303 
 
 60-75 
 
 1 2 
 
 74-10 
 
 70-50 
 
 6648 
 
 65-32 
 
 64-15 
 
 61-82 
 
 1 2 
 
 75-37 
 
 71-71 
 
 67-62 
 
 66-43 
 
 6525 
 
 C2-88 
 
 1 3 
 
 76-61 
 
 72-89 
 
 68-73 
 
 67-53 
 
 66-33 
 
 63-92 
 
 3 
 
 77-84 
 
 74-06 
 
 69-83 
 
 68-61 
 
 67-34 
 
 64-94 
 
 4 
 
 79-05 
 
 75-21 
 
 70-92 
 
 6967 
 
 68-43 
 
 65-95 
 
 41 
 
 80-24 
 
 76-34 
 
 71-98 
 
 70-72 
 
 69-46 
 
 66-94 
 
 5 
 
 81-41 
 
 77-45 
 
 73-03 
 
 71-75 
 
 70-48 
 
 67-92 
 
 5J 
 
 82-56 
 
 78*55 
 
 74-07 
 
 72-77 
 
 71-47 
 
 68-88 
 
 6 
 
 84-82 
 
 80-70 
 
 76-10 
 
 7477 
 
 73-43 
 
 70-77 
 
 1 7 
 
 87-03 
 
 82-80 
 
 78-08 
 
 76-71 
 
 7534 
 
 72-61 
 
 1 8 
 
 8918 
 
 84-85 
 
 80-00 
 
 78-60 
 
 77-20 
 
 74-40 
 
 1 9 
 
 9128 
 
 86-84 
 
 81 89 
 
 80-45 
 
 79-02 
 
 76-15 
 
 1 10 
 
 9333 
 
 88-79 
 
 83-73 
 
 82-26 
 
 80 79 
 
 77-86 
 
 1 11 
 
 9533 
 
 90-70 
 
 8553 
 
 84-03 
 
 82-53 
 
 7954 
 
 2 
 
 98-27 
 
 93-50 
 
 88-16 
 
 86-62 
 
 85-07 
 
 81-98 
 
 2 1 
 
 101 12 
 
 96-21 
 
 90 72 
 
 89-13 
 
 87-54 
 
 84-36 
 
 2 3 
 
 103-89 
 
 98-84 
 
 93-20 
 
 91-57 
 
 8994 
 
 8667 
 
 2 4 
 
 106-59 
 
 101-41 
 
 95-62 
 
 9395 
 
 92-27 
 
 88-92 
 
 2 6 
 
 10922 
 
 103-91 
 
 97-99 
 
 96-27 
 
 94-55 
 
 91-12 
 
 2 7 
 
 111-79 
 
 10636 
 
 100-29 
 
 98-53 
 
 96-78 
 
 9326 
 
 2 9 
 
 114-30 
 
 108-75 
 
 102-54 
 
 100-75 
 
 98-95 
 
 9V36 
 
 2 10 
 
 11676 
 
 111-09 
 
 104-75 
 
 102-92 
 
 101-03 
 
 97-41 
 
 3 
 
 119-17 
 
 11338 
 
 106-91 
 
 105-04 
 
 103-17 
 
 99-42 
 
 3 1 
 
 121-53 
 
 115-62 
 
 109-03 
 
 107-12 
 
 105-21 
 
 101-39 
 
 3 3 
 
 123-84 
 
 117-83 
 
 111-10 
 
 109-16 
 
 10721 
 
 103-32 
 
 3 4 
 
 126-12 
 
 119-99 
 
 113-14 
 
 111-16 
 
 109-18 
 
 105-22 
 
 3 6 
 
 128-35 
 
 122*11 
 
 115-15 
 
 11313 
 
 111-11 
 
 107-08 
 
 3 1\ 
 
 130-54 
 
 124-20 
 
 117-11 
 
 115-06 
 
 11301 
 
 108-91 
 
 3 9 
 
 13270 
 
 126-25 
 
 119*5 
 
 11697 
 
 114-88 
 
 110-71 
 
 3 10 
 
 134-82 
 
 12827 
 
 120-96 
 
 118-84 
 
 11672 
 
 112-48 
 
 4 
 
 137-60 
 
 130-92 
 
 123-45 
 
 12129 
 
 119-12 
 
 114-80 
 
 4 2 
 
 14033 
 
 133-51 
 
 125-89 
 
 12369 
 
 121-48 
 
 117-07 
 
 4 4 
 
 H300 
 
 13606 
 
 128-29 
 
 12605 
 
 123 80 
 
 11930 
 
 4 6 
 
 145-63 
 
 138-55 
 
 130-65 
 
 128-36 
 
 126-07 
 
 121-49 
 
 4 8 
 
 148-20 
 
 141-00 
 
 132-96 
 
 130-63 
 
 128 30 
 
 123-64 
 
 4 10 
 
 150-74 
 
 143 42 
 
 135-23 
 
 132-86 
 
 130-49 
 
 12576 
 
 6 
 
 15446 
 
 146-96 
 
 138-57 
 
 136-14 
 
 133-72 
 
 128-86 
 
 5 3 
 
408 
 
 THE DISCHAEGE OF WATEE FKOM 
 
 TABLE IL For finding the Velocities from the Altitudes, and the 
 
 Altitudes from the Velocities. 
 Altitudes 5 feet 6 inches to 17 feet. 
 
 . 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 ! 
 
 discharge in inches per second. 
 
 lj 
 
 *jAj 
 
 "s^S 
 
 <3^S 
 
 **! 
 
 ^rf 
 
 <3~g 
 
 TO 
 
 SsT *" *o 
 
 3 t^ " 
 
 5 r- ^ 
 
 S oo ^ 
 
 3 >~- ^ 
 
 3 ^< ^ 
 
 I 
 
 *!* 
 
 |j 
 
 5iS 
 
 Jij 
 
 P S | 
 
 10 ?> J 
 
 <o 8 
 
 o 
 
 5 6 
 
 225-848 
 
 219-98 
 
 215-91 
 
 194-23 
 
 184-07 
 
 180-68 
 
 5 9 
 
 230-924 
 
 224-92 
 
 220-76 
 
 198-59 
 
 188-20 
 
 184-74 
 
 6 
 
 235-891 
 
 229-76 
 
 225*51 
 
 202-87 
 
 192-25 
 
 188-71 
 
 6 3 
 
 240-755 
 
 234-50 
 
 230-16 
 
 207-05 
 
 196-22 
 
 192-60 
 
 6 6 
 
 245-524 
 
 239-14 
 
 234-72 
 
 211-15 
 
 200-10 
 
 196-42 
 
 6 9 
 
 250-200 
 
 243-69 
 
 239-19 
 
 215-17 
 
 203-91 
 
 200-16 
 
 7 
 
 254-791 
 
 248-17 
 
 243-58 
 
 219-12 
 
 207-65 
 
 203-83 
 
 7 3 
 
 259-301 
 
 252-56 
 
 247-89 
 
 222-99 
 
 211-33 
 
 207-44 
 
 7 6 
 
 263-734 
 
 256-88 
 
 252-13 
 
 226-81 
 
 214-94 
 
 210-99 
 
 7 9 
 
 268-093 
 
 261-12 
 
 256-30 
 
 230-56 
 
 218-50 
 
 214-47 
 
 8 
 
 272-383 
 
 265-30 
 
 260-40 
 
 234-25 
 
 221-99 
 
 217-91 
 
 8 3 
 
 276-607 
 
 269-41 
 
 264-44 
 
 237-88 
 
 225-43 
 
 221-29 
 
 8 6 
 
 280-766 
 
 273-47 
 
 268-41 
 
 241-46 
 
 228-82 
 
 224-61 
 
 8 9 
 
 284- 65 
 
 277-46 
 
 272-33 
 
 244-98 
 
 232-17 
 
 227-89 
 
 9 
 
 288-906 
 
 281-39 
 
 276-19 
 
 248-46 
 
 235-46 
 
 231-12 
 
 9 3 
 
 292-891 
 
 285-28 
 
 280-00 
 
 251-89 
 
 238-71 
 
 234-31 
 
 9 6 
 
 296-823 
 
 289-11 
 
 283-76 
 
 255-27 
 
 241-91 
 
 237-46 
 
 9 9 
 
 300-703 
 
 292-88 
 
 287-47 
 
 258-60 
 
 245-07 
 
 240-56 
 
 10 
 
 304-534 
 
 296-62 
 
 291-13 
 
 261-90 
 
 248-19 
 
 243 '63 
 
 10 3 
 
 308-317 
 
 300-30 
 
 294-75 
 
 265-15 
 
 251-28 
 
 245-65 
 
 10 6 
 
 312-054 
 
 303-94 
 
 297-32 
 
 268-37 
 
 254-32 
 
 249-64 
 
 10 9 
 
 315-747 
 
 307-54 
 
 301-85 
 
 271-54 
 
 257-33 
 
 252-60 
 
 11 
 
 319-398 
 
 311-09 
 
 305-34 
 
 274-68 
 
 260-31 
 
 255-52 
 
 11 3 
 
 323-007 
 
 314-61 
 
 308-79 
 
 277-79 
 
 262-25 
 
 258-41 
 
 11 6 
 
 326-576 
 
 318-09 
 
 312-21 
 
 280-86 
 
 266-16 
 
 261-26 
 
 11 9 
 
 330-107 
 
 321-52 
 
 315-58 
 
 283-89 
 
 269-04 
 
 264-09 
 
 12 
 
 333-600 
 
 324-93 
 
 318-92 
 
 286-90 
 
 271-88 
 
 266-88 
 
 12 3 
 
 337-057 
 
 328-29 
 
 322-23 
 
 289-87 
 
 274-70 
 
 269-65 
 
 12 6 
 
 340-479 
 
 331-63 
 
 325-50 
 
 292-81 
 
 277-49 
 
 272-38 
 
 12 9 
 
 343-867 
 
 334-93 
 
 328*74 
 
 295-73 
 
 280-25 
 
 275-09 
 
 13 
 
 347-222 
 
 338-19 
 
 331-94 
 
 298-61 
 
 282-99 
 
 277-78 
 
 13 3 
 
 350-545 
 
 341-43 
 
 335-12 
 
 301-47 
 
 285-69 
 
 280-44 
 
 13 6 
 
 353-836 
 
 344-64 
 
 338-27 
 
 304-30 
 
 288-38 
 
 283-07 
 
 13 9 
 
 357-097 
 
 347-81 
 
 341-39 
 
 307-10 
 
 291-03 
 
 285-68 
 
 14 
 
 360-329 
 
 350-96 
 
 344-47 
 
 309-88 
 
 293-67 
 
 288-26 
 
 14 6 
 
 366-707 
 
 357-17 
 
 350-57 
 
 315-37 
 
 298-87 
 
 293-37 
 
 15 
 
 372-976 
 
 363-28 
 
 356-57 
 
 320-76 
 
 303-98 
 
 298-38 
 
 15 6 
 
 379-141 
 
 369-28 
 
 362-46 
 
 326-06 
 
 309-00 
 
 303-31 
 
 16 
 
 385-208 
 
 375-19 
 
 368-26 
 
 331-28 
 
 313-94 
 
 308-17 
 
 16 6 
 
 391-181 
 
 381-01 
 
 373-97 
 
 336-42 
 
 318-81 
 
 312-94 
 
 17 
 
 397-063 
 
 386-74 
 
 379-50 
 
 341-47 
 
 323-61 
 
 317-65 
 
ORIFICES, WEIRS, PIPES, AND EIVEES. 
 
 409 
 
 TABLE II. For finding the Velocities from the Altitudes, and the 
 
 Altitudes from the Velocities. 
 Altitudes 5 feet 6 inches to 17 feet. 
 
 Coefficients of velocity, and the corresponding velocities of 
 
 
 discharge in inches per second. 
 
 a 
 
 a 
 
 ** 
 
 **! 
 
 ^i 
 
 *^g 
 
 **i 
 
 *** 
 
 f| 
 
 Is 2 
 
 I'g j 
 
 ill 
 
 S co ^ 
 
 a s 
 
 igi 
 
 JgS 
 
 jl 
 
 3| 
 
 l& I; 
 
 ?;1 
 
 ii SB 
 
 ^fi 
 
 Is I 
 
 1 
 
 *;| 
 
 00 J 
 
 . ii % 
 
 it! 
 
 d&i 
 
 CQ " 1 
 
 1 
 
 158-09 
 
 150-41 
 
 141-83 
 
 139-35 
 
 136-86 
 
 131-90 
 
 t 8 
 
 161-65 
 
 153-80 
 
 145-02 
 
 142-48 
 
 139-94 
 
 134-86 
 
 5 9 
 
 165-12 
 
 157-10 
 
 148-14 
 
 145-55 
 
 142-95 
 
 137-76 
 
 6 
 
 168-53 
 
 160-34 
 
 151-19 
 
 14855 
 
 145-90 
 
 140-60 
 
 6 3 
 
 171-87 
 
 163-52 
 
 154-19 
 
 151-49 
 
 148-79 
 
 143-39 
 
 6 6 
 
 175-14 
 
 166-63 
 
 157-13 
 
 154-37 
 
 151-62 
 
 146-12 
 
 6 9 
 
 178-35 
 
 169-69 
 
 160-01 
 
 157-21 
 
 154-40 
 
 148-80 
 
 7 
 
 181-51 
 
 172-69 
 
 162-84 
 
 159-99 
 
 157-14 
 
 151-43 
 
 7 3 
 
 184-61 
 
 175-65 
 
 165-62 
 
 162-72 
 
 159-82 
 
 154-02 
 
 7 6 
 
 187-67 
 
 178-55 
 
 168-36 
 
 165-41 
 
 162-46 
 
 156-57 
 
 7 9 
 
 190-67 
 
 181-41 
 
 171-06 
 
 168-06 
 
 165-06 
 
 159-07 
 
 8 
 
 193-62 
 
 184-22 
 
 17371 
 
 170-67 
 
 167-62 
 
 161-54 
 
 8 3 
 
 196-54 
 
 186-99 
 
 176-32 
 
 173-23 
 
 170-14 
 
 163-97 
 
 8 6 
 
 199-41 
 
 189-72 
 
 178-90 
 
 175-76 
 
 172-63 
 
 166-36 
 
 8 9 
 
 202-23 
 
 192-41 
 
 181-43 
 
 178-26 
 
 175-08 
 
 168-72 
 
 9 
 
 205-02 
 
 195-07 
 
 183-94 
 
 180-71 
 
 177-49 
 
 171-05 
 
 9 3 
 
 207-78 
 
 197-68 
 
 186-40 
 
 183-14 
 
 179-87 
 
 173-34 
 
 9 6 
 
 210-49 
 
 200-27 
 
 188-84 
 
 185-53 
 
 182-23 
 
 175-61 
 
 9 9 
 
 213-17 
 
 202-82 
 
 191-25 
 
 187-90 
 
 184-55 
 
 177*85 
 
 10 
 
 215-82 
 
 205-34 
 
 193-62 
 
 190-23 
 
 186-84 
 
 180-06 
 
 10 3 
 
 218-44 
 
 207-83 
 
 195-97 
 
 192-54 
 
 189-10 
 
 182-24 
 
 10 6 
 
 221-02 
 
 210-29 
 
 198-29 
 
 194-82 
 
 191-34 
 
 184-40 
 
 10 9 
 
 223-58 
 
 212-72 
 
 200-58 
 
 197-07 
 
 193-55 
 
 186-53 
 
 11 
 
 226-10 
 
 215-12 
 
 202*85 
 
 199-30 
 
 195-74 
 
 188-64 
 
 11 3 
 
 228-60 
 
 217-50 
 
 205-09 
 
 201-50 
 
 197-91 
 
 190-72 
 
 11 6 
 
 231-07 
 
 219-85 
 
 207-31 
 
 203-68 
 
 200-04 
 
 192-78 
 
 11 9 
 
 233-52 
 
 222-18 
 
 209-50 
 
 205-83 
 
 202-16 
 
 194-82 
 
 12 
 
 235-94 
 
 224-48 
 
 211-67 
 
 . 207-96 
 
 204-26 
 
 196-84 
 
 12 3 
 
 238-34 
 
 226-76 
 
 213-82 
 
 210-08 
 
 206-33 
 
 198-84 
 
 12 6 
 
 240-71 
 
 229-02 
 
 215-95 
 
 212-17 
 
 208-38 
 
 200-82 
 
 12 9 
 
 243-06 
 
 231-25 
 
 218-06 
 
 214-24 
 
 210-42 
 
 202-78 
 
 13 
 
 245-38 
 
 233-46 
 
 220*14 
 
 216-29 
 
 212-43 
 
 204-72 
 
 13 3 
 
 247-69 
 
 235-65 
 
 222-21 
 
 218-32 
 
 214-42 
 
 206*64 
 
 13 6 
 
 249-97 
 
 237-83 
 
 224-26 
 
 220-33 
 
 216-40 
 
 208-54 
 
 13 9 
 
 252-23 
 
 239-98 
 
 226-29 
 
 22-2-32 
 
 218'36 
 
 210-43 
 
 14 
 
 256-70 
 
 244-23 
 
 230-29 
 
 226-26 
 
 222-22 
 
 214-16 
 
 14 6 
 
 261-08 
 
 248-40 
 
 234-23 
 
 230-13 
 
 226-02 
 
 217-82 
 
 15 
 
 265-40 
 
 252-51 
 
 238-10 
 
 233-93 
 
 229-76 
 
 221-42 
 
 15 6 
 
 269-65 
 
 256-55 
 
 241-91 
 
 237'67 
 
 233-44 
 
 224-96 
 
 16 
 
 273-83 
 
 260-53 
 
 245-66 
 
 241-36 
 
 237-06 
 
 228-45 
 
 16 6 
 
 277-94 
 
 264-44 
 
 249-36 
 
 244-99 
 
 240-62 
 
 231-89 
 
 17 
 
410 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE II. For finding the Velocities from the Altitudes, and the 
 
 Altitudes from the Velocities. 
 Altitudes 17 feet 6 inches to 40 feet. 
 
 >CJ 
 
 Coefficient of velocity, and the corresponding velocities of 
 
 1 
 
 S*" 
 
 discharge in inches per second. 
 
 2| 
 
 **ij 
 
 **f 
 
 **S 
 
 "* CO 
 
 "3^5; 00 
 
 **| 
 
 JI 
 
 l^'rt "I 
 
 Is! 
 
 1 -*> 
 
 
 3s 
 
 J9 j 
 
 I 
 
 > * 1 '* 
 
 Ill 
 
 u I 
 
 III 
 
 i 1 . W 
 
 
 jj 
 
 Jijt 
 
 ci s> 8 
 
 *M 
 
 II 1 
 
 . <u 
 
 IO c* 
 
 <IJ 
 
 17 6 
 
 402-860 
 
 392-39 
 
 385-13 
 
 346-46 
 
 328-33 
 
 322-29 
 
 18 
 
 408-575 
 
 397-95 
 
 390-60 
 
 351-37 
 
 332-99 
 
 326-86 
 
 18 6 
 
 414-211 
 
 403-44 
 
 395-99 
 
 356-22 
 
 337-58 
 
 331-37 
 
 19 
 
 419-772 
 
 408-86 
 
 401-30 
 
 361-00 
 
 34-2 11 
 
 33582 
 
 19 6 
 
 425-258 
 
 414-20 
 
 406-55 
 
 365-72 
 
 346-59 
 
 340-21 
 
 20 
 
 430-676 
 
 419-48 
 
 411-73 
 
 370-38 
 
 351-00 
 
 344-54 
 
 20 6 
 
 436-026 
 
 424-69 
 
 416-84 
 
 374-98 
 
 35536 
 
 348-82 
 
 21 
 
 441-311 
 
 429-84 
 
 421-89 
 
 379-53 
 
 359-59 
 
 353-05 
 
 21 6 
 
 446-534 
 
 43492 
 
 426-89 
 
 384-02 
 
 363-93 
 
 357-23 
 
 22 
 
 451-697 
 
 439-95 
 
 431-82 
 
 388-46 
 
 368-13 
 
 361-36 
 
 22 6 
 
 456-801 
 
 444-92 
 
 436-70 
 
 392-85 
 
 372-29 
 
 365-44 
 
 23 
 
 461-848 
 
 449-84 
 
 441-53 
 
 397-19 
 
 376-41 
 
 369-48 
 
 23 6 
 
 466-841 
 
 450-70 
 
 446-30 
 
 401-48 
 
 380-48 
 
 373-47 
 
 24 
 
 471-782 
 
 459-52 
 
 451-02 
 
 405-73 
 
 384-50 
 
 377-43 
 
 24 6 
 
 476671 
 
 464-28 
 
 455-70 
 
 409-94 
 
 388-49 
 
 38134 
 
 25 
 
 481-510 
 
 468-99 
 
 460-32 
 
 414-10 
 
 392-43 
 
 385-21 
 
 25 6 
 
 486-301 
 
 473-66 
 
 464-90 
 
 418-22 
 
 396-34 
 
 389-04 
 
 26 
 
 491-046 
 
 478-28 
 
 469-44 
 
 422*30 
 
 40020 
 
 392-84 
 
 26 6 
 
 495745 
 
 482-86 
 
 473-93 
 
 426*34 
 
 40403 
 
 396-60 
 
 27 
 
 500-400 
 
 487-39 
 
 478-38 
 
 430-34 
 
 40783 
 
 400-32 
 
 27 6 
 
 505-012 
 
 491-88 
 
 482-79 
 
 434-31 
 
 411-58 
 
 404-01 
 
 28 
 
 509 682 
 
 49H'33 
 
 487-16 
 
 438-24 
 
 41531 
 
 40767 
 
 28 6 
 
 514-112 
 
 500-75 
 
 491 49 
 
 442-14 
 
 419-00 
 
 411-29 
 
 29 
 
 518-602 
 
 505-12 
 
 49578 
 
 446-00 
 
 4-22-66 
 
 414-88 
 
 29 6 
 
 5-23-054 
 
 509-45 
 
 500-04 
 
 449-83 
 
 42629 
 
 418-44 
 
 30 
 
 527-468 
 
 513-75 
 
 504-26 
 
 453-62 
 
 429-89 
 
 421-97 
 
 30 6 
 
 531-845 
 
 518-02 
 
 508-44 
 
 457-31 
 
 433-45 
 
 425-48 
 
 31 
 
 536-187 
 
 522-25 
 
 512-59 
 
 461-12 
 
 436-99 
 
 428-95 
 
 31 6 
 
 540-494 
 
 526-44 
 
 516-71 
 
 464-8-2 
 
 440-50 
 
 432-40 
 
 32 
 
 544-767 
 
 530-00 
 
 520-80 
 
 468-50 
 
 443-98 
 
 435-81 
 
 3-2 6 
 
 549006 
 
 534-73 
 
 5-24-85 
 
 472-15 
 
 447-44 
 
 43920 
 
 33 
 
 553-213 
 
 53883 
 
 5-28-87 
 
 475*76 
 
 450-87 
 
 442-57 
 
 33 6 
 
 557-388 
 
 54-2-90 
 
 532-86 
 
 479 35 
 
 454-27 
 
 445-91 
 
 34 
 
 561-532 
 
 54'r93 
 
 536-83 
 
 48-2-92 
 
 407-65 
 
 449-23 
 
 34 6 
 
 565-646 
 
 550-94 
 
 540-76 
 
 186-46 
 
 461-00 
 
 45-2-5-2 
 
 35 
 
 569730 
 
 554-92 
 
 544-66 
 
 489-97 
 
 464-33 
 
 455-78 
 
 36 
 
 577-812 
 
 56279 
 
 552-39 
 
 496-92 
 
 4709-2 
 
 46-2-25 
 
 37 
 
 585 782 
 
 570-55 
 
 560-01 
 
 503-77 
 
 477-41 
 
 468-63 
 
 38 
 
 593-646 
 
 578-21 
 
 567-53 
 
 510-54 
 
 483 82 
 
 4749-2 
 
 39 
 
 601 406 
 
 585-77 
 
 574-94 
 
 517-21 
 
 49015 
 
 481 12 
 
 40 
 
 609-067 
 
 593-23 
 
 58227 
 
 523-80 
 
 496-39 
 
 487-25 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 411 
 
 TABLE II. For finding the Velocities from the Altitudes, and the 
 
 Altitudes from the Velocities. 
 Altitudes 17 feet 6 inches to 40 feet. 
 
 Coefficients of Telocity, and the corresponding velocities of 
 
 ri 
 
 discharge in inches per second. 
 
 1 
 
 Hg 
 
 ^| 
 
 **i 
 
 ^g 
 
 *%i 
 
 * % : 4 
 
 .a | 
 
 iij 
 
 ^| 
 
 Hi 
 
 III 
 
 ycd 
 
 II S 
 
 ill 
 
 01 C 
 
 *! 
 
 00 
 
 e I o 
 
 V 
 
 r4 H 
 
 N o 
 
 1 
 
 sJ 
 
 u 
 
 a 
 
 
 
 ^ o 
 
 
 
 
 
 
 
 
 
 
 / // 
 
 282-00 
 
 268-30 
 
 253-00 
 
 248-56 
 
 244-13 
 
 235-27 
 
 17 6 
 
 286-00 
 
 272-11 
 
 256-59 
 
 252-09 
 
 247-60 
 
 238-61 
 
 18 
 
 28995 
 
 275-86 
 
 260 12 
 
 255-57 
 
 251-01 
 
 241-90 
 
 18 6 
 
 29384 
 
 279-57 
 
 263-32 
 
 259-00 
 
 254-38 
 
 245-14 
 
 19 
 
 297-68 
 
 283-22 
 
 267-06 
 
 26238 
 
 257*71 
 
 248-35 
 
 19 6 
 
 301-47 
 
 28683 
 
 270-46 
 
 265-73 
 
 260-99 
 
 251-51 
 
 20 
 
 305-22 
 
 290-39 
 
 273-82 
 
 269-03 
 
 264-23 
 
 254-64 
 
 20 6 
 
 30892 
 
 293-91 
 
 277-08 
 
 272-23 
 
 26737 
 
 257-67 
 
 21 
 
 312-57 
 
 297-39 
 
 2SO-42 
 
 275-51 
 
 27060 
 
 260-78 
 
 21 6 
 
 316-19 
 
 300-83 
 
 283-67 
 
 27870 
 
 27373 
 
 263-79 
 
 22 
 
 319-76 
 
 304-23 
 
 286-87 
 
 281-85 
 
 276-82 
 
 266-77 
 
 22 6 
 
 323-29 
 
 307-59 
 
 290-04 
 
 284-96 
 
 279-88 
 
 269-72 
 
 23 
 
 326-79 
 
 310-92 
 
 293-18 
 
 288-04 
 
 28291 
 
 272-64 
 
 23 6 
 
 330-25 
 
 314-21 
 
 296-28 
 
 291-09 
 
 285-90 
 
 275-52 
 
 24 
 
 333 67 
 
 317-46 
 
 299-35 
 
 294-11 
 
 288-86 
 
 278-38 
 
 24 6 
 
 33706 
 
 320-69 
 
 302-39 
 
 297-09 
 
 291-80 
 
 281-20 
 
 25 
 
 340-41 
 
 323-88 
 
 305-40 
 
 300-05 
 
 294-70 
 
 284-00 
 
 25 6 
 
 34373 
 
 32704 
 
 308-38 
 
 302-98 
 
 297-57 
 
 286-77 
 
 26 
 
 347-02 
 
 330-17 
 
 311-33 
 
 305-87 
 
 30042 
 
 289-52 
 
 26 6 
 
 350-28 
 
 333-13 
 
 314-25 
 
 308-75 
 
 303-94 
 
 292-23 
 
 27 
 
 353-51 
 
 336-34 
 
 317-15 
 
 311-59 
 
 30604 
 
 29493 
 
 27 6 
 
 35671 
 
 339 38 
 
 32002 
 
 314-41 
 
 308-81 
 
 297-60 
 
 28 
 
 359-88 
 
 342-40 
 
 322-86 
 
 317-20 
 
 311-55 
 
 300-24 
 
 28 6 
 
 363-02 
 
 34539 
 
 325-68 
 
 319-98 
 
 314-27 
 
 302 86 
 
 29 
 
 366-14 
 
 34835 
 
 328-48 
 
 322-72 
 
 316-97 
 
 305-46 
 
 29 6 
 
 369-23 
 
 351-29 
 
 331-25 
 
 325-45 
 
 319-65 
 
 3<'8-04 
 
 30 
 
 372-29 
 
 354-21 
 
 334-00 
 
 3-28-15 
 
 32230 
 
 310-60 
 
 30 6 
 
 375-33 
 
 357 10 
 
 386-73 
 
 330-83 
 
 324-93 
 
 313-13 
 
 31 
 
 37835 
 
 359-97 
 
 339-43 
 
 333-48 
 
 327-54 
 
 315-60 
 
 31 6 
 
 381-34 
 
 362-81 
 
 342-11 
 
 336-12 
 
 330-13 
 
 31814 
 
 32 
 
 384-30 
 
 36564 
 
 344-78 
 
 338-74 
 
 332-70 
 
 320-62 
 
 32 6 
 
 387 25 
 
 368-44 
 
 347-42 
 
 341-33 
 
 335-25 
 
 323-08 
 
 33 
 
 390-17 
 
 371-22 
 
 350-04 
 
 34301 
 
 337-78 
 
 325-51 
 
 33 6 
 
 393-07 
 
 37398 
 
 352-64 
 
 346-47 
 
 340-29 
 
 327-93 
 
 34 
 
 395-95 
 
 376-72 
 
 355-23 
 
 349-00 
 
 34-2- / 8 
 
 330-34 
 
 34 6 
 
 39881 
 
 379-44 
 
 35779 
 
 351-52 
 
 345-26 
 
 332-72 
 
 35 
 
 404-47 
 
 384-82 
 
 36287 
 
 356-51 
 
 350-15 
 
 337-44 
 
 36 
 
 410-05 
 
 390-13 
 
 367-87 
 
 361-43 
 
 35498 
 
 342 10 
 
 37 
 
 41555 
 
 395-37 
 
 37-2-81 
 
 366-28 
 
 35975 
 
 346-69 
 
 38 
 
 420-98 
 
 40054 
 
 377-68 
 
 371 11 
 
 36445 
 
 35122 
 
 39 
 
 42635 
 
 405-64 
 
 382-49 
 
 375-79 
 
 369-09 
 
 355-70 
 
 40 
 
412 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE III. Square Roots for finding the effects of the Velocity of 
 Approach ivhen the Orifice is small in proportion to the Head. 
 Also for finding the Increase in the Discharge from an Increase 
 of Head. (See p. 101.) 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 1-000 
 
 1-0000 
 
 1-115 
 
 1-0559 
 
 1-475 
 
 1-2141 
 
 1-975 
 
 4053 
 
 1-001 
 
 1-0005 
 
 1-120 
 
 1-0583 
 
 1-49 
 
 1-2207 
 
 1-99 
 
 4107 
 
 1-002 
 
 1-0010 
 
 1-125 
 
 1-0607 
 
 1-5 
 
 1-2247 
 
 2-00 
 
 4142 
 
 1-004 
 
 1-0020 
 
 1-13 
 
 1-0630 
 
 1-51 
 
 1-2288 
 
 2-01 
 
 4177 
 
 1-005 
 
 1-0025 
 
 1-135 
 
 1-0654 
 
 1-525 
 
 1-2349 
 
 2-025 
 
 4230 
 
 1-006 
 
 1-0030 
 
 1-14 
 
 1-0677 
 
 1-54 
 
 1-2410 
 
 2-04 
 
 4283 
 
 1-008 
 
 1-0040 
 
 1-145 
 
 1-0700 
 
 1-55 
 
 1-2450 
 
 2-05 
 
 4318 
 
 1-009 
 
 1-0044 
 
 1-15 
 
 1-0723 
 
 1-56 
 
 1-2490 
 
 2-06 
 
 4353 
 
 1-010 
 
 1-0050 
 
 1-155 
 
 1-0747 
 
 1-575 
 
 1-2550 
 
 2-075 
 
 1-4405 
 
 1-011 
 
 1-0055 
 
 1-16 
 
 1-0770 
 
 1-58 
 
 1-2570 
 
 2-09 
 
 1-4457 
 
 1-012 
 
 1-0060 
 
 165 
 
 1-0794 
 
 1-59 
 
 1-2610 
 
 2-10 
 
 1-4491 
 
 1-014 
 
 1-0070 
 
 17 
 
 1-0817 
 
 1-6 
 
 1-2649 
 
 2-11 
 
 1-4526 
 
 1-015 
 
 1-0075 
 
 175 
 
 1-0840 
 
 1-61 
 
 1-2689 
 
 2-125 
 
 1-4577 
 
 1-016 
 
 1-0080 
 
 18 
 
 1-0863 
 
 1-625 
 
 1-2748 
 
 2-14 
 
 1-4629 
 
 1-018 
 
 1-0090 
 
 185 
 
 0886 
 
 1-64 
 
 1-2806 
 
 2-15 
 
 1-4663 
 
 1-019 
 
 1-0095 
 
 19 
 
 0909 
 
 1-65 
 
 1-2845 
 
 2-16 
 
 1-4697 
 
 1-020 
 
 1-0100 
 
 195 
 
 0932 
 
 1-66 
 
 1-2884 
 
 2-175 
 
 1-4748 
 
 1-0225 
 
 1-0112 
 
 2 
 
 0954 
 
 675 
 
 1-2942 
 
 2-19 
 
 1-4799 
 
 1-025 
 
 1-0124 
 
 21 
 
 1000 
 
 69 
 
 1-3000 
 
 2-2 
 
 1-4832 
 
 1-0275 
 
 1-0137 
 
 22 
 
 1045 
 
 7 
 
 1-3038 
 
 2-21 
 
 1-4866 
 
 1-03 
 
 1-0149 
 
 23 
 
 1091 
 
 71 
 
 1-3077 
 
 2-225 
 
 1-4916 
 
 1-0325 
 
 1-0161 
 
 24 
 
 1136 
 
 725 
 
 1-3134 
 
 2-24 
 
 1-4967 
 
 1-035 
 
 1-0174 
 
 25 
 
 1-1180 
 
 74 
 
 1-3191 
 
 2-25 
 
 1-5000 
 
 1-0375 
 
 1-0186 
 
 26 
 
 1-1225 
 
 75 
 
 1-3229 
 
 2-26 
 
 1-5033 
 
 1-04 
 
 1-0198 
 
 27 
 
 1-1269 
 
 76 
 
 1-3267 
 
 2275 
 
 1-5083 
 
 1-0425 
 
 1-0210 
 
 28 
 
 1-1314 
 
 775 
 
 1-3323 
 
 2-29 
 
 1-5133 
 
 1-045 
 
 1-0223 
 
 29 
 
 1-1358 
 
 79 
 
 1-3379 
 
 2-3 
 
 1-5166 
 
 1-0475 
 
 1-0235 
 
 30 
 
 1402 
 
 80 
 
 1-3416 
 
 2-31 
 
 1-5199 
 
 1-05 
 
 1-0247 
 
 31 
 
 1446 
 
 81 
 
 1-3454 
 
 2-325 
 
 1-5248 
 
 1-055 
 
 1-0271 
 
 325 
 
 1511 
 
 825 
 
 1-3509 
 
 2-34 
 
 1-5297 
 
 1-06 
 
 1-0296 
 
 34 
 
 1576 
 
 84 
 
 1-3565 
 
 2-35 
 
 1-5330 
 
 1-065 
 
 1-0320 
 
 35 
 
 1619 
 
 85 
 
 1-3601 
 
 2-36 
 
 1-5362 
 
 1-07 
 
 1-0344 
 
 36 
 
 1662 
 
 86 
 
 1-3638 
 
 2-375 
 
 1-5411 
 
 1-075 
 
 1-0368 
 
 375 
 
 1726 
 
 875 
 
 1-3693 
 
 2-39 
 
 1-5460 
 
 1-08 
 
 1-0392 
 
 39 
 
 1-1790 
 
 89 
 
 1-3748 
 
 2-4 
 
 1-5492 
 
 1-085 
 
 1-0416 
 
 40 
 
 1-1832 
 
 9 
 
 1-3784 
 
 2'41 
 
 1-5524 
 
 1-09 
 
 1-0440 
 
 41 
 
 1-1874 
 
 91 
 
 1-3820 
 
 2'425 
 
 1-5572 
 
 1-095 
 
 1-0464 
 
 1-425 
 
 1-1937 
 
 925 
 
 1-3875 
 
 2-44 
 
 1-5621 
 
 1-1 
 
 1-0488 
 
 1-44 
 
 1-2000 
 
 94 
 
 1-3928 
 
 2-45 
 
 1-5652 
 
 1-105 
 
 1-0512 
 
 1-45 
 
 1-2042 
 
 1-95 
 
 1-3964 
 
 2-46 
 
 1-5684 
 
 1-110 
 
 1-0536 
 
 1-46 
 
 1-2083 
 
 1-96 
 
 1-4000 
 
 2-475 
 
 1-5732 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 413 
 
 TABLE III. Square Eoots for finding the effects of the Velocity of 
 Approach when the Orifice is small in proportion to the Head. 
 Also for finding the Increase in the Discharge from an Increase 
 of Head. (See p. 101.) 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 No. 
 
 Square 
 root. 
 
 2-49 
 
 1-5780 
 
 3-0000 
 
 1-7321 
 
 4-5 
 
 2-1213 
 
 26 
 
 5-0990 
 
 2-5 
 
 1-5811 
 
 3-025 
 
 1-7393 
 
 5'0 
 
 2-2361 
 
 27 
 
 5-1962 
 
 2-51 
 
 1-5843 
 
 3-05 
 
 1-7464 
 
 5-5 
 
 2-3452 
 
 28 
 
 5-2915 
 
 2-525 
 
 1-5890 
 
 3-075 
 
 1-7536 
 
 6-0 
 
 2-4495 
 
 29 
 
 5-3852 
 
 254 
 
 1-5937 
 
 3-1 
 
 1-7607 
 
 6-5 
 
 2-5495 
 
 30 
 
 5-4772 
 
 2-55 
 
 1-5969 
 
 3-125 
 
 1-7678 
 
 7-0 
 
 2-6458 
 
 31 
 
 5-5678 
 
 2-56 
 
 1-6000 
 
 3-15 
 
 1-7748 
 
 7-5 
 
 2-7386 
 
 32 
 
 5-6569 
 
 2-575 
 
 1-6047 
 
 3-175 
 
 7819 
 
 8'0 
 
 2-8284 
 
 33 
 
 5-7446 
 
 2-59 
 
 1-6093 
 
 3-2 
 
 7889 
 
 8'5 
 
 2-9155 
 
 34 
 
 5-8310 
 
 2-6 
 
 1-6125 
 
 3*225 
 
 7958 
 
 9-0 
 
 3-0000 
 
 35 
 
 5-9161 
 
 2-61 
 
 1-6155 
 
 3-25 
 
 8028 
 
 9-5 
 
 3-0822 
 
 36 
 
 6-0000 
 
 2-625 
 
 1-6202 
 
 3-275 
 
 8097 
 
 10-0 
 
 3-1623 
 
 37 
 
 6-0828 
 
 2-64 
 
 1-6248 
 
 3-3 
 
 8166 
 
 10-5 
 
 2-2404 
 
 38 
 
 6-1644 
 
 2-65 
 
 1-6279 
 
 3-325 
 
 8235 
 
 11-0 
 
 3-3166 
 
 39 
 
 6-2450 
 
 2'66 
 
 1-6310 
 
 3-35 
 
 8303 
 
 11-5 
 
 3-3912 
 
 40 
 
 6-3246 
 
 2-675 
 
 1-6355 
 
 3-375 
 
 8371 
 
 12'0 
 
 3-4641 
 
 41 
 
 6-4031 
 
 2-69 
 
 1-6401 
 
 3-4 
 
 8439 
 
 12-5 
 
 3'5355 
 
 42 
 
 6-4807 
 
 2-7 
 
 1-6432 
 
 3-425 
 
 8507 
 
 13'0 
 
 3-6056 
 
 43 
 
 6-5574 
 
 2-71 
 
 1-6462 
 
 3-45 
 
 8574 
 
 13-5 
 
 3-6742 
 
 44 
 
 6-6332 
 
 2-725 
 
 1-6508 
 
 3-475 
 
 8641 
 
 14-0 
 
 3-7417 
 
 45 
 
 6-7082 
 
 2-74 
 
 1-6553 
 
 3-5 
 
 8708 
 
 14-5 
 
 3-8079 
 
 46 
 
 6-7823 
 
 2-75 
 
 1-6583 
 
 3-525 
 
 8775 
 
 15-0 
 
 3-8730 
 
 47 
 
 6-8557 
 
 2-76 
 
 1-6613 
 
 3-55 
 
 8841 
 
 15-5 
 
 3-9370 
 
 48 
 
 6-9282 
 
 2-775 
 
 1-6658 
 
 3-575 
 
 8908 
 
 16-0 
 
 4-0000 
 
 49 
 
 7-0000 
 
 2-79 
 
 1-6703 
 
 3-6 
 
 8974 
 
 16-5 
 
 4-0620 
 
 50 
 
 7-0711 
 
 2-8 
 
 1-6733 
 
 3-625 
 
 9039 
 
 17-0 
 
 4-1231 
 
 51 
 
 7-1414 
 
 2-81 
 
 1-6763 
 
 3-65 
 
 9105 
 
 17-5 
 
 4-1833 
 
 52 
 
 7-2111 
 
 2-825 
 
 1-6808 
 
 3-675 
 
 9170 
 
 18-0 
 
 4-2426 
 
 53 
 
 7-2810 
 
 2-84 
 
 1-6852 
 
 3-7 
 
 1-9235 
 
 18-5 
 
 4-3012 
 
 54 
 
 7-3485 
 
 2-85 
 
 6882 
 
 3-725 
 
 1-9300 
 
 19-0 
 
 4-3589 
 
 55 
 
 7-4162 
 
 2-86 
 
 6912 
 
 3-75 
 
 1-9365 
 
 19'5 
 
 4-4159 
 
 56 
 
 7-4833 
 
 2-875 
 
 6956 
 
 3-775 
 
 1-9429 
 
 20-0 
 
 4-4721 
 
 57 
 
 7-5498 
 
 2-89 
 
 7000 
 
 3-8 
 
 1-9494 
 
 20-5 
 
 4-5277 
 
 58 
 
 7-6158 
 
 2-9 
 
 7029 
 
 3-825 
 
 1-9558 
 
 21-0 
 
 4-5826 
 
 59 
 
 7-6811 
 
 2-91 
 
 7059 
 
 3-85 
 
 1-9621 
 
 21-5 
 
 4-6368 
 
 60 
 
 7-7460 
 
 2-925 
 
 7103 
 
 3-875 
 
 1-9685 
 
 22-0 
 
 4-6904 
 
 61 
 
 7-8102 
 
 2-94 
 
 7146 
 
 3-9 
 
 1-9748 
 
 22-5 
 
 4-7434 
 
 62 
 
 7-8740 
 
 2-95 
 
 7176 
 
 3-925 
 
 1-9812 
 
 230 
 
 4-7958 
 
 63 
 
 7-9373 
 
 2-96 
 
 7205 
 
 3-95 
 
 1-9875 
 
 23-5 
 
 4-8477 
 
 64 
 
 8-0000 
 
 2-975 
 
 7248 
 
 3-975 
 
 1-9938 
 
 24-0 
 
 4-8990 
 
 65 
 
 8-0623 
 
 2-99 
 
 7292 
 
 4-0 
 
 2-0000 
 
 25-0 
 
 5-0000 
 
 66 
 
 8-1240 
 
414 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE IV. For finding the Discharge through Eectangular Ori- 
 fices ; in which n = -. Also for finding the effects of the 
 ct 
 
 Velocity of Approach to Weirs, and the Depression on the Crest. 
 (See p. 101.) 
 
 1+ n 
 
 . 
 
 (1 + n)* 
 
 (H) f -n f 
 
 l + 
 
 .* 
 
 (1 4 n)* 
 
 (1+71)2 n f 
 
 1-000 
 
 0000 
 
 1-0000 
 
 i-oooo 
 
 115 
 
 0390 
 
 1-1774 
 
 1-1384 
 
 1-<01 
 
 0000 
 
 1-0015 
 
 1-0015 
 
 120 
 
 0416 
 
 1-1853 
 
 1-1437 
 
 1 -002 
 
 0001 
 
 1-0030 
 
 1-0029 
 
 125 
 
 0442 
 
 1-1932 
 
 1-1491 
 
 1004 
 
 0003 
 
 1-0060 
 
 1-0058 
 
 13 
 
 0469 
 
 1-2012 
 
 1-1543 
 
 1-005 
 
 0004 
 
 1-0075 
 
 1-0072 
 
 135 
 
 0496 
 
 1-2092 
 
 1-1596 
 
 1-006 
 
 0005 
 
 1-0090 
 
 1-0086 
 
 14 
 
 0524 
 
 1 2172 
 
 1 1648 
 
 1 008 
 
 0007 
 
 1-0120 
 
 1-0113 
 
 145 
 
 0552 
 
 1-2251 
 
 1-1700 
 
 1-009 
 
 0009 
 
 1-0135 
 
 1 0127 
 
 15 
 
 0581 
 
 1-2332 
 
 1-1751 
 
 1-010 
 
 0010 
 
 1-0150 
 
 1-0140 
 
 155 
 
 0610 
 
 1-2413 
 
 1-1803 
 
 1-011 
 
 0012 
 
 T0165 
 
 1-0154 
 
 16 
 
 0^40 
 
 1-2494 
 
 1-1854 
 
 1-012 
 
 0013 
 
 1-0181 
 
 1-0167 
 
 165 
 
 0670 
 
 1-2574 
 
 1-1904 
 
 1-014 
 
 0017 
 
 1-0211 
 
 1-0194 
 
 17 
 
 0701 
 
 1-2655 
 
 1-1955 
 
 1-015 
 
 0018 
 
 T0226 
 
 1-0207 
 
 175 
 
 0732 
 
 1-2737 
 
 1-2005 
 
 1-016 
 
 0020 
 
 1 0241 
 
 1-0221 
 
 18 
 
 0764 
 
 1-2818 
 
 1 -2054 
 
 1-018 
 
 0024 
 
 1-0271 
 
 1-0247 
 
 185 
 
 0796 
 
 1-2900 
 
 1-2104 
 
 1-019 
 
 0026 
 
 1-0286 
 
 1-0260 
 
 19 
 
 0828 
 
 1 2981 
 
 1-2153 
 
 1-020 
 
 0028 
 
 1-0301 
 
 1-0273 
 
 1-195 
 
 0861 
 
 1 -3063 
 
 1-2^02 
 
 1-0225 
 
 0034 
 
 1-0339 
 
 1 0306 
 
 1-2 
 
 0894 
 
 1-3145 
 
 1-2251 
 
 1-025 
 
 0040 
 
 1 0377 
 
 1-0338 
 
 1-21 
 
 0962 
 
 1-3310 
 
 1-2348 
 
 1-0275 
 
 0046 
 
 1-0415 
 
 1-0370 
 
 1-22 
 
 1032 
 
 1-3475 
 
 1-2443 
 
 1-03 
 
 0052 
 
 1-0453 
 
 1-0401 
 
 123 
 
 1103 
 
 1-3641 
 
 1-2538 
 
 1-0325 
 
 0059 
 
 1-0491 
 
 1-0433 
 
 1-24 
 
 1176 
 
 1 3808 
 
 1-2632 
 
 1-035 
 
 0065 
 
 1-0530 
 
 1-0464 
 
 1 25 
 
 1250 
 
 1-3975 
 
 2725 
 
 1-0375 
 
 0073 
 
 1-0568 
 
 1-0495 
 
 1-26 
 
 1326 
 
 1 4143 
 
 2818 
 
 1-04 
 
 0080 
 
 1-0606 
 
 T0526 
 
 1-27 
 
 1403 
 
 1 4312 
 
 2909 
 
 1-0425 
 
 0088 
 
 1-0644 
 
 1-0557 
 
 1-28 
 
 1482 
 
 1-4482 
 
 3000 
 
 1-045 
 
 '0095 
 
 1-0683 
 
 1-0587 
 
 -29 
 
 1562 
 
 T4652 
 
 3090 
 
 1-0475 
 
 0104 
 
 1-0721 
 
 1-0617 
 
 30 
 
 1643 
 
 1-4822 
 
 3179 
 
 1-05 
 
 0112 
 
 1-0759 
 
 1-0648 
 
 31 
 
 1726 
 
 1 4994 
 
 3268 
 
 1-055 
 
 0129 
 
 1-0836 
 
 1-0707 
 
 325 
 
 18 3 
 
 1 5252 
 
 3399 
 
 1-06 
 
 0147 
 
 T0913 
 
 1-0766 
 
 34 
 
 1983 
 
 1-5512 
 
 3529 
 
 1-065 
 
 0166 
 
 1-0991 
 
 1-0825 
 
 35 
 
 2071 
 
 1-5686 
 
 3615 
 
 P07 
 
 0185 
 
 1 1068 
 
 1-0883 
 
 36 
 
 2160 
 
 1-5860 
 
 3700 
 
 1-075 
 
 0205 
 
 1 1146 
 
 1-0940 
 
 375 
 
 -2296 
 
 1-6123 
 
 3827 
 
 1-08 
 
 0226 
 
 1 1-224 
 
 1-0997 
 
 1-39 
 
 2436 
 
 1-6388 
 
 3952 
 
 1 085 
 
 0248 
 
 1 1302 
 
 1-1054 
 
 1 40 
 
 2530 
 
 1 -6565 
 
 4035 
 
 l-i <9 
 
 0270 
 
 1-13NO 
 
 rino 
 
 T41 
 
 2625 
 
 1-6743 
 
 4118 
 
 1 -095 
 
 0-293 
 
 1-1458 
 
 1-1166 
 
 1-425 
 
 2771 
 
 1 7011 
 
 4240 
 
 1-1 
 
 03 1 6 
 
 1-1537 
 
 1 1221 
 
 1-44 
 
 2919 
 
 1-7-J80 
 
 4361 
 
 1-105 
 
 0340 
 
 1-1616 
 
 1 1275 
 
 1-45 
 
 3019 
 
 1-7460 
 
 4442 
 
 1-110 
 
 0365 
 
 1-1695 
 
 1-1330 
 
 1-4G 
 
 3120 
 
 1-7641 
 
 4521 
 
 
 
 
 
 
 
 i 
 
 
 Values Of n from to '46. ^Continued on next page. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 415 
 
 TABLE IV. For finding the Discharge through 'Rectangular Ori- 
 fices ; in which n = - . Also for finding the effects of the 
 Velocity of Approach to Weirs, <&c. f See p. 101.) 
 
 l+tt 
 
 .< 
 
 (1 + n)^ 
 
 (1+nf-J 
 
 1 +n 
 
 .* 
 
 (l + ) f . 
 
 (l-fn)*-n* 
 
 1*475 
 
 3274 
 
 17914 
 
 4640 
 
 1-975 
 
 9627 
 
 2-7756 
 
 1-8128 
 
 1-49 
 
 3430 
 
 8188 
 
 4758 
 
 1-99 
 
 9850 
 
 2-8072 
 
 18222 
 
 1-5 
 
 3536 
 
 8371 
 
 4836 
 
 2- 
 
 1-0000 
 
 2-8284 
 
 1-8284 
 
 1-51 
 
 3642 
 
 8555 
 
 4913 
 
 2-01 
 
 1-0150 
 
 2-8497 
 
 18346 
 
 1525 
 
 3804 
 
 8832 
 
 5028 
 
 2-025 
 
 1-0377 
 
 2-8816 
 
 8439 
 
 1-54 
 
 3968 
 
 9111 
 
 5143 
 
 2-04 
 
 1-0606 
 
 2-9137 
 
 8531 
 
 155 
 
 4079 
 
 9297 
 
 5218 
 
 2-05 
 
 1-0759 
 
 2-9352 
 
 8592 
 
 1-56 
 
 4191 
 
 9484 
 
 5294 
 
 2-06 
 
 1-0913 
 
 2-9567 
 
 8653 
 
 1-575 
 
 4360 
 
 9766 
 
 5406 
 
 2'075 
 
 1-1146 
 
 2-9890 
 
 8744 
 
 1-58 
 
 4417 
 
 9860 
 
 5443 
 
 2-09 
 
 1-1380 
 
 30215 
 
 8835 
 
 1-59 
 
 4532 
 
 2-0049 
 
 5517 
 
 2-10 
 
 1 -1537 
 
 3-0432 
 
 1-8895 
 
 16 
 
 4648 
 
 2-0239 
 
 1-5591 
 
 211 
 
 1-1695 
 
 30650 
 
 1-8955 
 
 1-61 
 
 4764 
 
 2-0429 
 
 1-5664 
 
 2-125 
 
 1-1932 
 
 3-0977 
 
 1-9045 
 
 1 -625 
 
 4941 
 
 2-0715 
 
 1-5774 
 
 2-14 
 
 1-2172 
 
 3-1306 
 
 19134 
 
 164 
 
 5120 
 
 2-1002 
 
 1-5882 
 
 2-15 
 
 1-2332 
 
 3-1525 
 
 1-9193 
 
 1 65 
 
 5240 
 
 2-1195 
 
 1-5954 
 
 2-16 
 
 1-2494 
 
 3-1745 
 
 1-9252 
 
 1-66 
 
 5362 
 
 2-1388 
 
 1-6026 
 
 2-175 
 
 1-2737 
 
 3-2077 
 
 1-9340 
 
 1-675 
 
 5546 
 
 2-1678 
 
 1-6132 
 
 2-19 
 
 1'2981 
 
 3-2409 
 
 1-9428 
 
 1-69 
 
 5732 
 
 2-1970 
 
 1-6238 
 
 2-2 
 
 1-3145 
 
 3-2631 
 
 1-9-186 
 
 1-7 
 
 5857 
 
 22165 
 
 1-6309 
 
 2-21 
 
 1-3310 
 
 3-2854 
 
 1-9544 
 
 1-71 
 
 5983 
 
 2-2361 
 
 1-6379 
 
 2225 
 
 1-3558 
 
 3-3189 
 
 T9631 
 
 1-725 
 
 6173 
 
 2-2656 
 
 1-6483 
 
 0-04 
 
 1-3808 
 
 33525 
 
 T9717 
 
 1-74 
 
 6366 
 
 2-2952 
 
 1-6586 
 
 2-25 
 
 13975 
 
 3-3750 
 
 1-9775 
 
 1-75 
 
 6495 
 
 2-3150 
 
 1-6655 
 
 2-26 
 
 1-4143 
 
 3-3975 
 
 1-9832 
 
 1-76 
 
 6626 
 
 2-3349 
 
 1-6724 
 
 2"275 
 
 1-4397 
 
 3-4314 
 
 1-9917 
 
 1-775 
 
 6823 
 
 2-3648 
 
 1-6826 
 
 2'29 
 
 1-4652 
 
 3-4654 
 
 2-0002 
 
 1-79 
 
 7322 
 
 2-3949 
 
 1 6927 
 
 2-3 
 
 1-4822 
 
 3-4881 
 
 2-0059 
 
 1-80 
 
 7155 
 
 2-4150 
 
 6994 
 
 2-31 
 
 1-4994 
 
 3-5109 
 
 2-0115 
 
 1-81 
 
 7290 
 
 2-4351 
 
 7061 
 
 2-325 
 
 T5252 
 
 3-5451 
 
 2-0200 
 
 1-825 
 
 7493 
 
 2-4654 
 
 7161 
 
 2-34 
 
 1-5512 
 
 3-5795 
 
 2-0284 
 
 1-84 
 
 7699 
 
 2-4959 
 
 7260 
 
 2-35 
 
 1-5686 
 
 36025 
 
 2-0339 
 
 1-85 
 
 7837 
 
 2-5163 
 
 7326 
 
 2-36 
 
 1-5860 
 
 36255 
 
 2-0395 
 
 1-80 
 
 7975 
 
 2-5367 
 
 7392 
 
 2-375 
 
 1-6123 
 
 3-6601 
 
 2-0478 
 
 1-875 
 
 8185 
 
 2-5674 
 
 7490 
 
 2-39 
 
 T6388 
 
 3-6948 
 
 2-0561 
 
 1-89 
 
 8396 
 
 2-5983 
 
 7587 
 
 2'4 
 
 V6565 
 
 37181 
 
 2-0616 
 
 1-9 
 
 8538 
 
 2-6190 
 
 7652 
 
 2-41 
 
 1-6743 
 
 3-7U3 
 
 2-0670 
 
 1-91 
 
 8681 
 
 2-6397 
 
 7716 
 
 2425 
 
 17011 
 
 3-7763 
 
 2-0752 
 
 1-925 
 
 8896 
 
 2-6709 
 
 7813 
 
 244 
 
 1-7280 
 
 38114 
 
 2-0834 
 
 194 
 
 9114 
 
 2-7021 
 
 7907 
 
 245 
 
 1-7460 
 
 38349 
 
 2-0888 
 
 1-95 
 
 9259 
 
 2-7230 
 
 7971 
 
 2-46 
 
 1-7641 
 
 3-8584 
 
 2-0942 
 
 1-96 
 
 9406 
 
 2-7440 
 
 8034 
 
 2475 
 
 1-7914 
 
 3-8937 
 
 2-1023 
 
 Values Of n from -475 to 1'475. [Continued on next page. 
 
416 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE IV. For finding the Discharge through Eectangular Ori- 
 fices ; in which n = -,. Also for finding the effects of the 
 Velocity of Approach to Weirs, dc. (See p. 101.) 
 
 14-* 
 
 **. 
 
 (l + n) f . 
 
 (!+)*_*. 
 
 l + n. 
 
 nl 
 
 d + nf. 
 
 (l+n) f -n f . 
 
 2-49 
 
 1-8188 
 
 3-9292 
 
 2-1104 
 
 3- 
 
 2-8284 
 
 5-1962 
 
 2-3677 
 
 2-5 
 
 1-8371 
 
 3-9528 
 
 2-1157 
 
 3-025 
 
 2-8816 
 
 5-2612 
 
 2-3796 
 
 2-51 
 
 1-8555 
 
 3-9766 
 
 2-1211 
 
 3-05 
 
 2-9352 
 
 5-3626 
 
 2-3914 
 
 2-525 
 
 1-8832 
 
 4-0123 
 
 2-1291 
 
 3-075 
 
 2-9890 
 
 5-3922 
 
 2-4032 
 
 2-54 
 
 1-9111 
 
 4-0481 
 
 2-1370 
 
 3-1 
 
 3-0432 
 
 5-4581 
 
 2-4149 
 
 2-55 
 
 1-9297 
 
 4-0720 
 
 2-1423 
 
 3-125 
 
 3-0977 
 
 5-5243 
 
 2-4266 
 
 2-56 
 
 1-9484 
 
 4-0960 
 
 2-1476 
 
 3-15 
 
 3-1525 
 
 5-5907 
 
 2-4382 
 
 2-575 
 
 1-9766 
 
 4-1321 
 
 2-1554 
 
 3-175 
 
 3-2077 
 
 5-6574 
 
 2-4497 
 
 2-59 
 
 2-0049 
 
 4-1682 
 
 2-1633 
 
 3-2 
 
 3-2631 
 
 5-7243 
 
 2-4612 
 
 2'6 
 
 2-0239 
 
 4-1924 
 
 2-1685 
 
 3-225 
 
 3-3189 
 
 5-7915 
 
 2-4726 
 
 2-61 
 
 2-0429 
 
 4-2166 
 
 2-1737 
 
 3-25 
 
 3-3750 
 
 5-8590 
 
 2-4840 
 
 2-625 
 
 2-0715 
 
 4-2530 
 
 2-1815 
 
 3-275 
 
 3-4314 
 
 ,5-9268 
 
 2-4953 
 
 2-64 
 
 2-1002 
 
 4-2895 
 
 2-1893 
 
 33 
 
 3-4881 
 
 5-9947 
 
 2-5066 
 
 2-65 
 
 2-1195 
 
 4-3139 
 
 2-1944 
 
 3-325 
 
 35451 
 
 6-0630 
 
 2-5179 
 
 2'66 
 
 2-1388 
 
 4-3383 
 
 2-1996 
 
 3-35 
 
 3-6025 
 
 6-1315 
 
 2-5290 
 
 2-675 
 
 2-1678 
 
 4-3751 
 
 2-2073 
 
 3-375 
 
 3-6601 
 
 6-2003 
 
 2-5401 
 
 2-69 
 
 2-1970 
 
 4-4119 
 
 2-2149 
 
 3-4 
 
 3-7181 
 
 6-2693 
 
 2-5512 
 
 2-7 
 
 2-2165 
 
 4-4366 
 
 2-2200 
 
 3-425 
 
 3-7763 
 
 6-3386 
 
 2-5623 
 
 2-71 
 
 2-2361 
 
 4-4612 
 
 2-2251 
 
 3-45 
 
 3-8349 
 
 6-4081 
 
 2-5732 
 
 2-725 
 
 2-2656 
 
 4-4983 
 
 22327 
 
 3-475 
 
 3-8937 
 
 6-4779 
 
 2-5842 
 
 2-74 
 
 2-2952 
 
 4-5355 
 
 2-2403 
 
 3-5 
 
 3-9528 
 
 6-5479 
 
 2-5951 
 
 2-75 
 
 2-3150 
 
 4-5604 
 
 2-2453 
 
 3-525 
 
 4-0123 
 
 6-6182 
 
 2-6059 
 
 2-76 
 
 2-3349 
 
 4-5853 
 
 2-2504 
 
 3-55 
 
 4-0720 
 
 6-6887 
 
 2-6167 
 
 2-775 
 
 2-3648 
 
 4-6227 
 
 2-2579 
 
 3-575 
 
 4-1321 
 
 6-7595 
 
 2-6274 
 
 2-79 
 
 2-3949 
 
 4-6602 
 
 2-2654 
 
 36 
 
 4-1924 
 
 6-8305 
 
 2-6381 
 
 2'8 
 
 2-4150 
 
 4-6853 
 
 2-2703 
 
 3-625 
 
 4-2530 
 
 6-9018 
 
 2-6488 
 
 2-81 
 
 2-4351 
 
 4-7104 
 
 2-2753 
 
 3-65 
 
 4-3139 
 
 69733 
 
 2-6594 
 
 2'825 
 
 2-4654 
 
 4-7482 
 
 2-2827 
 
 3-675 
 
 4-3751 
 
 7-0451 
 
 2-6700 
 
 2-84 
 
 2-4959 
 
 4-7861 
 
 2-2902 
 
 3-7 
 
 4-4366 
 
 7-1171 
 
 2-6805 
 
 2-85 
 
 2-5163 
 
 4-8114 
 
 2-2951 
 
 3725 
 
 4-4983 
 
 7-1893 
 
 2-6910 
 
 2-86 
 
 2-5367 
 
 4-8367 
 
 2-3000 
 
 3-75 
 
 4-5604 
 
 7-2618 
 
 2-7015 
 
 2-875 
 
 2-5674 
 
 4-8748 
 
 2-3074 
 
 3-775 
 
 4-6227 
 
 7-3346 
 
 2-7119 
 
 2-89 
 
 2-5983 
 
 4-9130 
 
 2-3147 
 
 3-8 
 
 4-6853 
 
 7-4076 
 
 2-7223 
 
 2'9 
 
 2-6190 
 
 4-9385 
 
 2-3196 
 
 3-825 
 
 4-7482 
 
 7-4808 
 
 2-7326 
 
 2-91 
 
 2-6397 
 
 4-9641 
 
 2-3244 
 
 3-85 
 
 4-8114 
 
 7-5542 
 
 2-7429 
 
 2-925 
 
 2-6708 
 
 5-0025 
 
 2-3317 
 
 3875 
 
 4-8748 
 
 7-6279 
 
 2-7531 
 
 2'94 
 
 2-7021 
 
 5-0411 
 
 2-3389 
 
 3-9 
 
 4-9385 
 
 7-7019 
 
 27634 
 
 2-95 
 
 2-7230 
 
 5-0668 
 
 2-3438 
 
 3925 
 
 5-0025 
 
 77761 
 
 2-7735 
 
 2-96 
 
 2-7440 
 
 5-0926 
 
 2-3486 
 
 3-95 
 
 50668 
 
 7-8505 
 
 2-7837 
 
 2-975 
 
 2-7756 
 
 5-1313 
 
 2-3558 
 
 3-975 
 
 5-1313 
 
 7-9251 
 
 2-7838 
 
 2-99 
 
 2-8072 
 
 5' 1702 
 
 2-3630 
 
 4- 
 
 5-1962 
 
 8- 
 
 2-8038 
 
 Values of n from *475 to 1-475. 
 
 [ Continued on next page. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 417 
 
 TABLE IV. For finding the Discharge through Eectangular Ori- 
 fices ; in which n = T. Also for finding the effects of the 
 Velocity of Approach to Weirs, &c. (See p. 101.) 
 
 l-t-n. 
 
 4 
 
 <l + n)*. 
 
 i*M 
 
 1 + n. 
 
 f. 
 
 d + ) f . 
 
 (l+n) 2 -7i^ 
 
 4-5 
 
 6-5479 
 
 9-5459 
 
 2-9980 
 
 26- 
 
 125-0000 
 
 132-5745 
 
 7-5745 
 
 5-0 
 
 8-0000 
 
 11-1803 
 
 3-1803 
 
 27- 
 
 132-5745 
 
 140-2961 
 
 77216 
 
 5-5 
 
 9-5459 
 
 128986 
 
 3-3527 
 
 28- 
 
 140-2961 
 
 148-1621 
 
 7-8660 
 
 60 
 
 11-1803 
 
 14-6969 
 
 3-5166 
 
 29- 
 
 148-1621 
 
 156-1698 
 
 8-0077 
 
 6-5 
 
 12-8986 
 
 16-5718 
 
 3-6732 
 
 30- 
 
 156-1698 
 
 164-3168 
 
 8-1470 
 
 7-0 
 
 146969 
 
 18-5203 
 
 3-8234 
 
 31- 
 
 164-3168 
 
 172-6007 
 
 8-2839 
 
 7-5 
 
 16-5718 
 
 20-5396 
 
 3-9678 
 
 32- 
 
 172-6007 
 
 181-0193 
 
 8-4186 
 
 8-0 
 
 18-5203 
 
 226274 
 
 4-1071 
 
 33- 
 
 181-0193 
 
 189-5706 
 
 8-5513 
 
 8-5 
 
 20-5396 
 
 24-7815 
 
 42419 
 
 34- 
 
 189-5706 
 
 198-2524 
 
 8-6818 
 
 9-0 22-6-274 
 
 27-0000 
 
 4-3726 
 
 35- 
 
 198-2524 
 
 207-0628 
 
 8-8104 
 
 95 
 
 24-7815 
 
 29-2810 
 
 4-4995 
 
 36- 
 
 207-0628 
 
 216-0000 
 
 8-9372 
 
 10-0 
 
 27-0000 
 
 31-6-2-28 
 
 4-6-228 
 
 37- 
 
 216-0000 
 
 225-0622 
 
 9-0622 
 
 10 5 
 
 29-2810 
 
 34-0239 
 
 4-7429 
 
 38- 
 
 225-0622 
 
 234'2477 
 
 9-1855 
 
 11-0 
 
 31-62-28 
 
 36-4829 
 
 4-8601 
 
 39- 
 
 234-2477 
 
 243-5549 
 
 9-3072 
 
 11-5 
 
 34-0239 
 
 38-9984 
 
 4-9745 
 
 40- 
 
 2435549 
 
 252-9822 
 
 9-4273 
 
 12-0 
 
 36-4829 
 
 41-5692 
 
 5-0863 
 
 41- 
 
 252-9822 
 
 262-5281 
 
 9-5459 
 
 12-5 
 
 38-9984 
 
 44-1942 
 
 5-1958 
 
 42- 
 
 262-5281 
 
 272-1911 
 
 9-6630 
 
 13-0 
 
 41-5692 
 
 46-8722 
 
 5-3030 
 
 43- 
 
 272-1911 
 
 281-9699 
 
 9-7788 
 
 13-5 
 
 44-1942 I 49-6022 
 
 5-4080 
 
 44- 
 
 281-9699 
 
 291-8630 
 
 9 8931 
 
 14-0 
 
 46-8722 | 52-3832 
 
 5-5110 
 
 45- 
 
 291-8630 
 
 301-8692 
 
 10-0062 
 
 14-5 
 
 49-6022 
 
 55-2144 
 
 5-6122 
 
 46- 
 
 301-8692 
 
 311-9872 
 
 10-1180 
 
 15-0 
 
 52-3832 
 
 58-0947 
 
 5-7115 
 
 47- 
 
 311-9872 
 
 322-2158 
 
 10-2286 
 
 15-5 
 
 55-2144 
 
 61-0236 
 
 5-8092 
 
 48- 
 
 322-2158 
 
 332-5538 
 
 10-3380 
 
 16-0 
 
 58-0947 
 
 64- 
 
 5-9053 
 
 49- 
 
 332-5538 
 
 343-0000 
 
 10-4462 
 
 16-5 
 
 61-0236 
 
 67-0247 
 
 6-0011 
 
 50- 
 
 343-0000 
 
 353-5534 
 
 10-5534 
 
 17-0 
 
 64- 
 
 70-0928 
 
 6-0928 
 
 51- 
 
 353-5534 
 
 364-2128 
 
 10-6594 
 
 17-5 
 
 67-0247 
 
 73-2078 
 
 6-1831 
 
 52- 
 
 364-2128 
 
 374-9773 
 
 10-7645 
 
 18-0 
 
 70-0928 
 
 76-3675 
 
 6-2747 
 
 53- 
 
 374-9773 
 
 385 8458 
 
 10-8685 
 
 18-5 
 
 73*2078 
 
 795715 
 
 6-3637 
 
 54- 
 
 385-8458 
 
 396-8173 
 
 10-9715 
 
 19-0 
 
 76-3675 
 
 82.8191 
 
 6-4516 
 
 55- 
 
 396-8173 
 
 407-8909 
 
 11-0736 
 
 19-5 
 
 79-5715 
 
 86-1097 
 
 6-5382 
 
 56- 
 
 407-8909 
 
 419-0656 
 
 11-1747 
 
 20-0 
 
 82-8191 
 
 89-4427 
 
 6-6236 
 
 57- 
 
 419-0656 
 
 430-3406 
 
 11-2750 
 
 205 
 
 86-1097 
 
 92-8)77 
 
 6-7080 
 
 58- 
 
 430-3406 
 
 441-7148 
 
 11-3742 
 
 21-0 
 
 894427 
 
 96-2341 
 
 6-7914 
 
 59- 
 
 441-7148 
 
 453-1876 
 
 1 1-4728 
 
 21-5 
 
 92-8177 
 
 99-6914 
 
 6-8737 
 
 60- 
 
 453-1876 
 
 464-7580 
 
 11-5704 
 
 2-2-0 
 
 96-2341 
 
 103-1892 
 
 6-9551 
 
 61- 
 
 464-7580 
 
 4764252 
 
 11-6672 
 
 22-5 
 
 99-6914 
 
 1067269 
 
 7 0355 
 
 62- 
 
 476-4252 
 
 488-1885 
 
 11-7633 
 
 23- 
 
 103-1892 
 
 110-3041 
 
 7-1149 
 
 63- 
 
 488-1885 
 
 500-0470 
 
 11-8585 
 
 23-5 
 
 106-7269 
 
 1139205 
 
 7-1936 
 
 64- 
 
 500-0470 
 
 512-0000 
 
 11-9530 
 
 24- 
 
 110-3041 
 
 117-5755 
 
 7-2714 
 
 65- 
 
 512-0000 
 
 524-0468 
 
 12-0468 
 
 25- 
 
 117-5755 
 
 125- 
 
 7-4245 
 
 66- 
 
 524-0468 
 
 5361865 
 
 12-1397 
 
 Values of n from 3-5 to 
 
 E E 
 
418 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE V. Coefficients of Discharge for Different Eatios of the 
 
 Channel to the Orifice. 
 Coefficients for Heads in still water '550 and -573. 
 
 
 Coefficient -550 for heads in still 
 
 Coefficient -573 for heads in still 
 
 
 water. 
 
 water. 
 
 Ratio 
 of the 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 channel 
 
 the height 
 
 for orifices : 
 
 for weirs : 
 
 the height 
 
 for orifices : 
 
 for weirs : 
 
 to the 
 
 due to the 
 
 the heads 
 
 the heads 
 
 due to the 
 
 : the heads 
 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 so- 
 
 000 
 
 550 
 
 550 
 
 000 
 
 573 
 
 573 
 
 so- 
 
 001 
 
 550 
 
 551 
 
 001 
 
 573 
 
 574 
 
 15- 
 
 001 
 
 550 
 
 551 
 
 001 
 
 573 
 
 574 
 
 lo- 
 
 003 
 
 551 
 
 552 
 
 003 
 
 574 
 
 576 
 
 o- 
 
 004 
 
 551 
 
 553 
 
 004 
 
 574 
 
 576 
 
 s' 
 
 005 
 
 551 
 
 554 
 
 005 
 
 574 
 
 577 
 
 7- 
 
 006 
 
 552 
 
 5,55 
 
 007 
 
 575 
 
 578 
 
 6- 
 
 008 
 
 552 
 
 557 
 
 009 
 
 576 
 
 580 
 
 5-5 
 
 010 
 
 553 
 
 558 
 
 Oil 
 
 576 
 
 582 
 
 5-0 
 
 012 
 
 553 
 
 559 
 
 013 
 
 577 
 
 584 
 
 4-5 
 
 015 
 
 554 
 
 562 
 
 016 
 
 578 
 
 586 
 
 4-0 
 
 019 
 
 555 
 
 565 
 
 021 
 
 579 
 
 589 
 
 3-75 
 
 022 
 
 556 
 
 566 
 
 024 
 
 580 
 
 592 
 
 3-50 
 
 025 
 
 557 
 
 569 
 
 028 
 
 581 
 
 594 
 
 3-25 
 
 029 
 
 558 
 
 572 
 
 032 
 
 582 
 
 598 
 
 3-0 
 
 035 
 
 559 
 
 575 
 
 038 
 
 584 
 
 602 
 
 2-75 
 
 042 
 
 561 
 
 580 
 
 045 
 
 586 
 
 607 
 
 2-50 
 
 051 
 
 564 
 
 586 
 
 055 
 
 589 
 
 614 
 
 2'25 
 
 064 
 
 567 
 
 594 
 
 069 
 
 593 
 
 623 
 
 2'00 
 
 082 
 
 572 
 
 606 
 
 089 
 
 598 
 
 636 
 
 1'95 
 
 086 
 
 573 
 
 609 
 
 094 
 
 599 
 
 639 
 
 1-90 
 
 091 
 
 575 
 
 612 
 
 100 
 
 601 
 
 643 
 
 1-85 
 
 097 
 
 576 
 
 615 
 
 106 
 
 603 
 
 647 
 
 1-80 
 
 103 
 
 578 
 
 619 
 
 113 
 
 604 
 
 651 
 
 1-75 
 
 110 
 
 579 
 
 623 
 
 120 
 
 606 
 
 655 
 
 1-70 
 
 117 
 
 581 
 
 627 
 
 128 
 
 609 
 
 660 
 
 1-65 
 
 125 
 
 583 
 
 632 
 
 137 
 
 611 
 
 666 
 
 1-60 
 
 134 
 
 586 
 
 637 
 
 147 
 
 614 
 
 671 
 
 1-55 
 
 144 
 
 588 
 
 643 
 
 158 
 
 617 
 
 678 
 
 1-50 
 
 155 
 
 591 
 
 649 
 
 171 
 
 620 
 
 6H5 
 
 1-45 
 
 168 
 
 594 
 
 656 
 
 185 
 
 624 
 
 694 
 
 1-40 
 
 183 
 
 598 
 
 664 
 
 201 
 
 628 
 
 703 
 
 1-35 
 
 199 
 
 602 
 
 673 
 
 220 
 
 633 
 
 713 
 
 1-30 
 
 218 
 
 607 
 
 683 
 
 241 
 
 638 
 
 724 
 
 1-25 
 
 240 
 
 612 
 
 695 
 
 266 
 
 645 
 
 737 
 
 1-20 
 
 265 
 
 619 
 
 707 
 
 295 
 
 652 
 
 753 
 
 1-15 
 
 297 
 
 626 
 
 723 
 
 330 
 
 661 
 
 770 
 
 MO 
 
 333 
 
 635 
 
 741 
 
 372 
 
 671 
 
 791 
 
 1-05 
 
 378 
 
 646 
 
 762 
 
 424 
 
 684 
 
 816 
 
 1-00 
 
 434 
 
 659 
 
 787 
 
 489 
 
 699 
 
 845 
 
 See the auxiliary table, p. 136. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 419 
 
 TABLE V. Coefficients of Discharge for different Ratios of the 
 
 Channel to the Ori/ice. 
 Coefficients for heads in still water *584 and *595. 
 
 
 Coefficient -584 for heads in still 
 
 Coefficient -595 for heads in still 
 
 
 water. 
 
 water. 
 
 Ratio of 
 the 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 channel 
 
 the h eight 
 
 for orifices: 
 
 for weirs : 
 
 the height 
 
 for orifices : 
 
 for weirs : 
 
 to the 
 
 due to the 
 
 the heads 
 
 the heads 
 
 due to the 
 
 the heads 
 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 30- 
 
 000 
 
 584 
 
 584 
 
 000 
 
 595 
 
 595 
 
 20- 
 
 001 
 
 584 
 
 585 
 
 001 
 
 595 
 
 596 
 
 15- 
 
 002 
 
 584 
 
 585 
 
 002 
 
 595 
 
 596 
 
 10- 
 
 003 
 
 585 
 
 587 
 
 004 
 
 596 
 
 598 
 
 9-0 
 
 004 
 
 585 
 
 588 
 
 004 
 
 596 
 
 599 
 
 1-0 
 
 005 
 
 586 
 
 588 
 
 006 
 
 597 
 
 600 
 
 7-0 
 
 007 
 
 586 
 
 590 
 
 007 
 
 597 
 
 601 
 
 6-0 
 
 010 
 
 587 
 
 592 
 
 010 
 
 598 
 
 603 
 
 5-5 
 
 Oil 
 
 587 
 
 593 
 
 012 
 
 599 
 
 605 
 
 5-0 
 
 014 
 
 588 
 
 595 
 
 014 
 
 599 
 
 607 
 
 4-5 
 
 017 
 
 589 
 
 598 
 
 018 
 
 600 
 
 610 
 
 4-0 
 
 022 
 
 590 
 
 601 
 
 023 
 
 602 
 
 613 
 
 3-75 
 
 025 
 
 591 
 
 604 
 
 026 
 
 603 
 
 616 
 
 3-50 
 
 029 
 
 592 
 
 606 
 
 030 
 
 604 
 
 619 
 
 3-25 
 
 033 
 
 594 
 
 610 
 
 035 
 
 605 
 
 622 
 
 3-0 
 
 039 
 
 595 
 
 614 
 
 041 
 
 607 
 
 627 
 
 275 
 
 047 
 
 598 
 
 620 
 
 049 
 
 609 
 
 633 
 
 2-50 
 
 058 
 
 601 
 
 627 
 
 060 
 
 613 
 
 641 
 
 2-25 
 
 072 
 
 605 
 
 637 
 
 075 
 
 617 
 
 651 
 
 20 
 
 093 
 
 611 
 
 651 
 
 097 
 
 623 
 
 666 
 
 1-95 
 
 099 
 
 612 
 
 654 
 
 103 
 
 625 
 
 669 
 
 1-90 
 
 104 
 
 614 
 
 660 
 
 109 
 
 627 
 
 673 
 
 1-85 
 
 111 
 
 615 
 
 662 
 
 115 
 
 628 
 
 678 
 
 1-80 
 
 118 
 
 617 
 
 666 
 
 123 
 
 630 
 
 682 
 
 1-75 
 
 125 
 
 620 
 
 671 
 
 131 
 
 633 
 
 687 
 
 T70 
 
 134 
 
 '622 
 
 676 
 
 140 
 
 635 
 
 693 
 
 1-65 
 
 143 
 
 624 
 
 682 
 
 149 
 
 638 
 
 699 
 
 1-60 
 
 154 
 
 627 
 
 689 
 
 160 
 
 641 
 
 706 
 
 155 
 
 166 
 
 631 
 
 696 
 
 173 
 
 644 
 
 713 
 
 1-50 
 
 179 
 
 634 
 
 703 
 
 187 
 
 648 
 
 721 
 
 1-45 
 
 194 
 
 638 
 
 712 
 
 202 
 
 652 
 
 730 
 
 1-40 
 
 211 
 
 643 
 
 722 
 
 220 
 
 657 
 
 741 
 
 1-35 
 
 230 
 
 648 
 
 732 
 
 241 
 
 663 
 
 752 
 
 T30 
 
 253 
 
 654 
 
 745 
 
 265 
 
 669 
 
 765 
 
 1-25 
 
 279 
 
 661 
 
 759 
 
 293 
 
 677 
 
 780 
 
 1-20 
 
 310 
 
 669 
 
 775 
 
 325 
 
 685 
 
 797 
 
 1-15 
 
 .348 
 
 678 
 
 794 
 
 366 
 
 695 
 
 818 
 
 4-10 
 
 393 
 
 689 
 
 816 
 
 414 
 
 707 
 
 842 
 
 1-05 
 
 448 
 
 703 
 
 842 
 
 473 
 
 722 
 
 870 
 
 1-00 
 
 518 
 
 719 
 
 874 
 
 548 
 
 740 
 
 905 
 
 See the auxiliary table, p. 136. 
 
 EE 3 
 
420 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE V. Coefficients of Discharge for different Eatios of the 
 
 Channel to the Orifice. 
 Coefficients for heads in still water -606 and -617. 
 
 
 Coefficient -606 for heads in 
 
 Coefficient -617 for heads in 
 
 
 still water. 
 
 still water. 
 
 Ratio 
 of the 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 channel 
 
 the height 
 
 for orifices : 
 
 for weirs : 
 
 the height 
 
 for orifices : 
 
 for weirs : 
 
 to the 
 
 due to the 
 
 the heads 
 
 the heads 
 
 due to the 
 
 the heads 
 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 30- 
 
 000 
 
 606 
 
 606 
 
 000 
 
 617 
 
 617 
 
 20- 
 
 001 
 
 606 
 
 607 
 
 001 
 
 617 
 
 618 
 
 15- 
 
 002 
 
 607 
 
 607 
 
 002 
 
 618 
 
 619 
 
 1O 
 
 004 
 
 607 
 
 609 
 
 004 
 
 618 
 
 620 
 
 9-0 
 
 005 
 
 607 
 
 610 
 
 005 
 
 618 
 
 621 
 
 8-0 
 
 006 
 
 608 
 
 611 
 
 006 
 
 619 
 
 622 
 
 7-0 
 
 008 
 
 608 
 
 612 
 
 008 
 
 619 
 
 624 
 
 6'0 
 
 010 
 
 609 
 
 615 
 
 on 
 
 620 
 
 626 
 
 5-5 
 
 012 
 
 610 
 
 616 
 
 013 
 
 621 
 
 628 
 
 5-0 
 
 015 
 
 611 
 
 619 
 
 015 
 
 622 
 
 630 
 
 4'5 
 
 018 
 
 612 
 
 621 
 
 019 
 
 623 
 
 633 
 
 4-0 
 
 023 
 
 613 
 
 625 
 
 024 
 
 624 
 
 637 
 
 375 
 
 027 
 
 614 
 
 628 
 
 028 
 
 626 
 
 640 
 
 3'50 
 
 031 
 
 615 
 
 631 
 
 032 
 
 627 
 
 643 
 
 3'25 
 
 036 
 
 617 
 
 635 
 
 037 
 
 628 
 
 647 
 
 3'00 
 
 043 
 
 619 
 
 640 
 
 044 
 
 630 
 
 653 
 
 275 
 
 051 
 
 621 
 
 646 
 
 053 
 
 633 
 
 660 
 
 2'50 
 
 062 
 
 625 
 
 654 
 
 065 
 
 637 
 
 668 
 
 2'25 
 
 078 
 
 629 
 
 665 
 
 081 
 
 642 
 
 679 
 
 2-00 
 
 101 
 
 636 
 
 681 
 
 105 
 
 649 
 
 696 
 
 1-95 
 
 107 
 
 638 
 
 685 
 
 111 
 
 650 
 
 700 
 
 1'90 
 
 113 
 
 639 
 
 689 
 
 118 
 
 652 
 
 704 
 
 1-85 
 
 119 
 
 641 
 
 693 
 
 125 
 
 654 
 
 709 
 
 T80 
 
 128 
 
 644 
 
 698 
 
 133 
 
 657 
 
 714 
 
 1-75 
 
 136 
 
 646 
 
 703 
 
 142 
 
 659 
 
 720 
 
 1-70 
 
 146 
 
 649 
 
 709 
 
 152 
 
 662 
 
 726 
 
 1-65 
 
 156 
 
 652 
 
 716 
 
 163 
 
 665 
 
 733 
 
 1-60 
 
 167 
 
 655 
 
 723 
 
 175 
 
 669 
 
 741 
 
 T55 
 
 180 
 
 658 
 
 731 
 
 188 
 
 673 
 
 749 
 
 1-50 
 
 195 
 
 662 
 
 739 
 
 204 
 
 677 
 
 759 
 
 1-45 
 
 212 
 
 667 
 
 749 
 
 221 
 
 681 
 
 768 
 
 1'40 
 
 231 
 
 672 
 
 760 
 
 241 
 
 687 
 
 780 
 
 1-35 
 
 252 
 
 678 
 
 772 
 
 264 
 
 694 
 
 793 
 
 1-30 
 
 278 
 
 685 
 
 -786 
 
 291 
 
 701 
 
 808 
 
 125 
 
 307 
 
 693 
 
 803 
 
 322 
 
 709 
 
 825 
 
 1-20 
 
 342 
 
 702 
 
 821 
 
 359 
 
 719 
 
 845 
 
 1-15 
 
 384 
 
 713 
 
 843 
 
 404 
 
 731 
 
 868 
 
 1-10 
 
 436 
 
 726 
 
 868 
 
 459 
 
 745 
 
 895 
 
 1-05 
 
 499 
 
 742 
 
 898 
 
 527 
 
 763 
 
 928 
 
 1-00 
 
 580 
 
 762 
 
 936 
 
 615 
 
 784 
 
 969 
 
 See the auxiliary table, p. 136. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 421 
 
 TABLE V. Coefficients of Discharge for different Ratios 
 of the Channel to the Orifice. 
 
 Mean Coefficient -628. 
 Coefficients for heads in still water -628 and -639. 
 
 
 Coefficient -628 for heads in 
 
 Coefficient -639 for heads in 
 
 
 still water. 
 
 still water. 
 
 Ratio 
 of the 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 channel 
 
 the height 
 
 for orifices 
 
 for weirs: 
 
 the height 
 
 for orifices 
 
 for weirs : 
 
 to the 
 
 due to the 
 
 the heads 
 
 the heads 
 
 due to the 
 
 the heads 
 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 Telocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 to the head 
 
 centres. 
 
 depth. 
 
 to the head 
 
 centres. 
 
 depth. 
 
 30- 
 
 000 
 
 628 
 
 628 
 
 000 
 
 639 
 
 639 
 
 20- 
 
 001 
 
 628 
 
 629 
 
 001 
 
 639 
 
 640 
 
 15- 
 
 002 
 
 629 
 
 630 
 
 002 
 
 640 
 
 641 
 
 10- 
 
 004 
 
 629 
 
 632 
 
 004 
 
 640 
 
 643 
 
 9-0 
 
 005 
 
 630 
 
 632 
 
 005 
 
 641 
 
 644 
 
 8-0 
 
 006 
 
 630 
 
 634 
 
 006 
 
 641 
 
 645 
 
 7-0 
 
 008 
 
 631 
 
 635 
 
 008 
 
 642 
 
 647 
 
 6-0 
 
 on 
 
 631 
 
 638 
 
 Oil 
 
 643 
 
 649 
 
 5-5 
 
 013 
 
 632 
 
 640 
 
 014 
 
 643 
 
 651 
 
 5-0 
 
 016 
 
 633 
 
 642 
 
 017 
 
 644 
 
 654 
 
 4-5 
 
 020 
 
 634 
 
 645 
 
 021 
 
 646 
 
 657 
 
 4-0 
 
 025 
 
 636 
 
 649 
 
 026 
 
 647 
 
 662 
 
 375 
 
 029 
 
 637 
 
 652 
 
 030 
 
 648 
 
 665 
 
 3-50 
 
 033 
 
 638 
 
 656 
 
 034 
 
 650 
 
 668 
 
 3-25 
 
 039 
 
 639 
 
 659 
 
 040 
 
 652 
 
 673 
 
 3-0 
 
 046 
 
 642 
 
 666 
 
 048 
 
 654 
 
 678 
 
 2-75 
 
 055 
 
 645 
 
 672 
 
 057 
 
 657 
 
 686 
 
 2-50 
 
 067 
 
 649 
 
 682 
 
 070 
 
 661 
 
 695 
 
 2-25 
 
 084 
 
 654 
 
 694 
 
 088 
 
 666 
 
 708 
 
 2-0 
 
 109 
 
 661 
 
 711 
 
 114 
 
 674 
 
 727 
 
 1-95 
 
 116 
 
 663 
 
 715 
 
 120 
 
 676 
 
 731 
 
 1-90 
 
 123 
 
 665 
 
 720 
 
 128 
 
 679 
 
 736 
 
 1-85 
 
 130 
 
 668 
 
 725 
 
 135 
 
 681 
 
 741 
 
 1-80 
 
 139 
 
 670 
 
 731 
 
 144 
 
 684 
 
 747 
 
 75 
 
 148 
 
 673 
 
 737 
 
 154 
 
 686 
 
 753 
 
 70 
 
 158 
 
 676 
 
 743 
 
 165 
 
 690 
 
 760 
 
 65 
 
 169 
 
 679 
 
 750 
 
 176 
 
 693 
 
 768 
 
 60 
 
 182 
 
 683 
 
 758 
 
 190 
 
 797 
 
 776 
 
 55 
 
 196 
 
 687 
 
 767 
 
 205 
 
 701 
 
 786 
 
 50 
 
 213 
 
 692 
 
 777 
 
 222 
 
 706 
 
 796 
 
 45 
 
 231 
 
 697 
 
 788 
 
 241 
 
 712 
 
 808 
 
 40 
 
 252 
 
 703 
 
 800 
 
 262 
 
 718 
 
 820 
 
 35 
 
 276 
 
 709 
 
 814 
 
 289 
 
 725 
 
 836 
 
 30 
 
 304 
 
 717 
 
 830 
 
 319 
 
 734 
 
 853 
 
 25 
 
 338 
 
 726 
 
 846 
 
 354 
 
 743 
 
 872 
 
 20 
 
 377 
 
 734 
 
 866 
 
 396 
 
 755 
 
 895 
 
 15 
 
 425 
 
 750 
 
 894 
 
 447 
 
 769 
 
 921 
 
 10 
 
 484 
 
 765 
 
 924 
 
 509 
 
 785 
 
 953 
 
 05 
 
 557 
 
 784 
 
 "959 
 
 588 
 
 805 
 
 991 
 
 00 
 
 651 
 
 807 
 
 1-OC2 
 
 690 
 
 831 
 
 1-038 
 
 See the auxiliary table, p. 136. 
 
422 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE V. Coefficients of Discharge for Different Eatios of the 
 Channel to the Orifice. 
 
 Coefficients for heads in still water *650 and '667. 
 
 
 Coefficient -650 for heads in 
 
 Coefficient -667 for heads in 
 
 
 still water. 
 
 still water. 
 
 Ratio 
 of the 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 channel 
 to the 
 
 the height 
 due to the 
 
 for orifices : 
 the heads 
 
 for weirs : 
 the heads 
 
 the height 
 due to the 
 
 for orifices : 
 the heads 
 
 for weirs : 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 to the head. 
 
 centres. 
 
 depth. 
 
 so- 
 
 000 
 
 650 
 
 650 
 
 000 
 
 667 
 
 667 
 
 so- 
 
 001 
 
 650 
 
 651 
 
 001 
 
 667 
 
 668 
 
 15- 
 
 002 
 
 651 
 
 652 
 
 002 
 
 667 
 
 669 
 
 lo- 
 
 004 
 
 651 
 
 654 
 
 004 
 
 668 
 
 671 
 
 g- 
 
 005 
 
 652 
 
 655 
 
 006 
 
 669 
 
 67-2 
 
 s' 
 
 007 
 
 652 
 
 656 
 
 007 
 
 669 
 
 673 
 
 7-0 
 
 009 
 
 653 
 
 658 
 
 009 
 
 670 
 
 675 
 
 6-0 
 
 012 
 
 654 
 
 661 
 
 012 
 
 671 
 
 678 
 
 5-5 
 
 014 
 
 655 
 
 663 
 
 015 
 
 672 
 
 680 
 
 5-0 
 
 017 
 
 656 
 
 665 
 
 018 
 
 673 
 
 682 
 
 4-5 
 
 021 
 
 657 
 
 669 
 
 022 
 
 674 
 
 687 
 
 4-0 
 
 027 
 
 659 
 
 674 
 
 029 
 
 676 
 
 692 
 
 3-75 
 
 031 
 
 660 
 
 677 
 
 033 
 
 678 
 
 6'c>6 
 
 3-50 
 
 036 
 
 662 
 
 681 
 
 038 
 
 679 
 
 700 
 
 3-25 
 
 042 
 
 663 
 
 686 
 
 044 
 
 681 
 
 705 
 
 3-0 
 
 049 
 
 666 
 
 692 
 
 052 
 
 684 
 
 711 
 
 275 
 
 059 
 
 669 
 
 699 
 
 062 
 
 687 
 
 720 
 
 2-50 
 
 073 
 
 673 
 
 709 
 
 077 
 
 692 
 
 731 
 
 2-25 
 
 091 
 
 679 
 
 723 
 
 096 
 
 698 
 
 745 
 
 2-0 
 
 118 
 
 687 
 
 742 
 
 125 
 
 707 
 
 766 
 
 1-95 
 
 125 
 
 689 
 
 747 
 
 132 
 
 709 
 
 771 
 
 1-90 
 
 133 
 
 692 
 
 752 
 
 140 
 
 712 
 
 -777 
 
 1-85 
 
 141 
 
 694 
 
 758 
 
 149 
 
 715 
 
 783 
 
 1-80 
 
 150 
 
 697 
 
 764 
 
 159 
 
 718 
 
 790 
 
 1-75 
 
 160 
 
 700 
 
 771 
 
 170 
 
 721 
 
 797 
 
 1-70 
 
 172 
 
 704 
 
 779 
 
 182 
 
 725 
 
 805 
 
 1-65 
 
 184 
 
 707 
 
 786 
 
 195 
 
 729 
 
 814 
 
 1-60 
 
 198 
 
 711 
 
 795 
 
 210 
 
 733 
 
 823 
 
 1-55 
 
 213 
 
 716 
 
 805 
 
 227 
 
 738 
 
 833 
 
 50 
 
 231 
 
 721 
 
 816 
 
 '246 
 
 744 
 
 846 
 
 45 
 
 251 
 
 727 
 
 828 
 
 268 
 
 751 
 
 859 
 
 40 
 
 275 
 
 734 
 
 842 
 
 293 
 
 758 
 
 874 
 
 35 
 
 302 
 
 742 
 
 858 
 
 322 
 
 764 
 
 888 
 
 30 
 
 333 
 
 751 
 
 876 
 
 356 
 
 776 
 
 911 
 
 1-25 
 
 371 
 
 761 
 
 896 
 
 398 
 
 788 
 
 934 
 
 1-20 
 
 415 
 
 773 
 
 920 
 
 446 
 
 802 
 
 961 
 
 1-15 
 
 469 
 
 788 
 
 949 
 
 506 
 
 818 
 
 992 
 
 1-10 
 
 537 
 
 806 
 
 983 
 
 580 
 
 838 
 
 1-030 
 
 1-05 
 
 621 
 
 828 
 
 T024 
 
 675 
 
 863 
 
 1-076 
 
 i-oo 
 
 732 
 
 855 
 
 1-074 
 
 800 
 
 894 
 
 1-133 
 
 See the auxiliary table, p. 136. 
 
ORIFICES, WEIES, PIPES, AND RIVERS. 
 
 423 
 
 TABLE V. Coefficients of Discharge for different Ratios of the 
 Channel to the Orifice. 
 
 Coefficients for heads in still water N /~ 7 5~ = -7071 and 1. 
 
 
 Coefficient '7071 for heads in still 
 
 Coefficient 1-000 for heads in still 
 
 
 water. 
 
 water. 
 
 Ratio of 
 the 
 
 Ratio of 
 
 Coefficients 
 
 Coefficient 
 
 Ratio of 
 
 Coefficients 
 
 Coefficients 
 
 channel 
 
 the height 
 
 for orifices 
 
 for weirs : 
 
 the height 
 
 for orifices 
 
 for weirs : 
 
 to the 
 
 due to the 
 
 the heads 
 
 the heads 
 
 due to the 
 
 the heads 
 
 the heads 
 
 orifice. 
 
 velocity of 
 
 measured 
 
 measured 
 
 velocity of 
 
 measured 
 
 measured 
 
 
 approach 
 
 to the 
 
 the full 
 
 approach 
 
 to the 
 
 the full 
 
 
 to the head 
 
 centres. 
 
 depth. 
 
 to the, head 
 
 centres. 
 
 depth. 
 
 30- 
 
 001 
 
 707 
 
 708 
 
 001 
 
 1-001 
 
 1-002 
 
 20- 
 
 001 
 
 708 
 
 708 
 
 003 
 
 1-001 
 
 1-004 
 
 15- 
 
 001 
 
 708 
 
 709 
 
 005 
 
 1-002 
 
 1-006 
 
 lo- 
 
 005 
 
 709 
 
 712 
 
 010 
 
 1-005 
 
 1-014 
 
 o- 
 
 006 
 
 709 
 
 713 
 
 013 
 
 1-006 
 
 1-017 
 
 s' 
 
 008 
 
 710 
 
 714 
 
 016 
 
 1-008 
 
 1-021 
 
 7- 
 
 010 
 
 711 
 
 717 
 
 021 
 
 1-010 
 
 1-028 
 
 6- 
 
 014 
 
 712 
 
 721 
 
 029 
 
 1-014 
 
 1-038 
 
 5-5 
 
 017 
 
 713 
 
 723 
 
 034 
 
 1017 
 
 1-045 
 
 5-0 
 
 020 
 
 714 
 
 727 
 
 041 
 
 1-021 
 
 1-055 
 
 4-5 
 
 025 
 
 716 
 
 731 
 
 052 
 
 1-026 
 
 1-067 
 
 4-0 
 
 032 
 
 718 
 
 737 
 
 067 
 
 1-033 
 
 1-084 
 
 3-75 
 
 037 
 
 720 
 
 742 
 
 077 
 
 1-038 
 
 1-096 
 
 3-50 
 
 043 
 
 722 
 
 747 
 
 089 
 
 1-044 
 
 rno 
 
 3-25 
 
 050 
 
 724 
 
 753 
 
 105 
 
 1-051 
 
 1-127 
 
 300 
 
 059 
 
 728 
 
 760 
 
 125 
 
 1-061 
 
 1-149 
 
 2-75 
 
 071 
 
 732 
 
 770 
 
 152 
 
 1-073 
 
 1-178 
 
 2-50 
 
 087 
 
 737 
 
 783 
 
 190 
 
 1-091 
 
 1-216 
 
 2-25 
 
 110 
 
 745 
 
 801 
 
 246 
 
 1-116 
 
 1-269 
 
 200 
 
 143 
 
 756 
 
 826 
 
 333 
 
 1-155 
 
 1-347 
 
 T95 
 
 151 
 
 759 
 
 832 
 
 356 
 
 1-165 
 
 1-367 
 
 1-90 
 
 161 
 
 762 
 
 839 
 
 383 
 
 1-176 
 
 T389 
 
 T85 
 
 171 
 
 765 
 
 846 
 
 412 
 
 1-188 
 
 1-413 
 
 1-80 
 
 182 
 
 769 
 
 854 
 
 446 
 
 1-203 
 
 1-441 
 
 1-75 
 
 195 
 
 773 
 
 863 
 
 484 
 
 1-218 
 
 1-471 
 
 1-70 
 
 209 
 
 778 
 
 873 
 
 529 
 
 1-237 
 
 1-505 
 
 1-65 
 
 225 
 
 783 
 
 883 
 
 579 
 
 1-257 
 
 1-543 
 
 1-60 
 
 243 
 
 788 
 
 895 
 
 641 
 
 1-281 
 
 1-589 
 
 1-55 
 
 263 
 
 795 
 
 908 
 
 711 
 
 1-308 
 
 1-638 
 
 1-50 
 
 286 
 
 802 
 
 923 
 
 800 
 
 1-342 
 
 1-699 
 
 1-45 
 
 312 
 
 810 
 
 939 
 
 903 
 
 1-379 
 
 1-767 
 
 1-40 
 
 342 
 
 819 
 
 958 
 
 1-042 
 
 1-429 
 
 1-854 
 
 1-35 
 
 378 
 
 830 
 
 980 
 
 1 216 
 
 1-489 
 
 1-958 
 
 1-30 
 
 421 
 
 842 
 
 1'003 
 
 1-449 
 
 1-565 
 
 2-088 
 
 T25 
 
 471 
 
 857 
 
 1-033 
 
 1-778 
 
 1-667 
 
 2-259 
 
 1-20 
 
 532 
 
 875 
 
 1-066 
 
 2-273 
 
 T810 
 
 2-499 
 
 1-15 
 
 608 
 
 897 
 
 1-107 
 
 3-100 
 
 2-025 
 
 2-844 
 
 1-10 
 
 704 
 
 923 
 
 1-155 
 
 4-762 
 
 2-400 
 
 3-440 
 
 1-05 
 
 830 
 
 957 
 
 1-216 
 
 9756 
 
 3-280 
 
 4-803 
 
 1-00 
 
 '1-000 
 
 1-000 
 
 1-293 
 
 infinite. 
 
 infinite. 
 
 infinite. 
 
 See the auxiliary table, p. 136, also p. 138, 
 
424 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VI. The Discharge over Weirs or Notches of one foot in length, 
 in Cubic feet per minute. 
 
 Depths I inch to 10 inches. Coefficients -667 to -617. 
 
 GREATER COEFFICIENTS. 
 
 The Formulce at the heads of the Columns give the Value of the Discharge, 
 D, in Cubic feet per minute, when I, the length of the Weir, is taken 
 feet, and the head, h, in inches. For I \fh* we may substitute I h 
 retaining tJie same standards.. 
 
 Heads 
 in 
 inches. 
 
 Theoretical 
 discharge, 
 D = 
 7-72 l\/h3. 
 
 Coefficient 
 667, 
 D = 
 5-15 l\fh*. 
 
 Coefficient 
 650. 
 D = 
 5-02 J\A3. 
 
 Coefficient 
 639. 
 D = 
 4-93 l\/hf. 
 
 Coefficient 
 628. 
 D = 
 4-85 Z\A 3 - 
 
 Coefficient 
 617. 
 D = 
 4-76 l\/h*. 
 
 25 
 
 965 
 
 644 
 
 627 
 
 617 
 
 606 
 
 596 
 
 5 
 
 2730 
 
 1-821 
 
 1-775 
 
 1-744 
 
 1-714 
 
 1-684 
 
 75 
 
 5-016 
 
 3-345 
 
 3-260 
 
 3-205 
 
 3-150 
 
 3-095 
 
 1- 
 
 7-722 
 
 5-151 
 
 5-019 
 
 4-934 
 
 4-849 
 
 4-764 
 
 1-25 
 
 10-792 
 
 7-198 
 
 7-015 
 
 6-896 
 
 6*777 
 
 6-659 
 
 1-5 
 
 14186 
 
 9-462 
 
 9221 
 
 9-065 
 
 8-909 
 
 8-753 
 
 1-75 
 
 17-877 
 
 11-924 
 
 11-620 
 
 11-423 
 
 11-227 
 
 11-030 
 
 2- 
 
 21-842 
 
 14-569 
 
 14-197 
 
 13957 
 
 13-717 
 
 13-477 
 
 2-25 
 
 26-062 
 
 17-383 
 
 16-940 
 
 16-654 
 
 16-367 
 
 16-080 
 
 2-5 
 
 30-524 
 
 20-360 
 
 19-841 
 
 19-505 
 
 19-169 
 
 18-833 
 
 2-75 
 
 35-215 
 
 23-489 
 
 22-890 
 
 22-503 
 
 22'115 
 
 21-728 
 
 3- 
 
 40-125 
 
 26-763 
 
 26-081 
 
 25-640 
 
 25-199 
 
 24-757 
 
 3-25 
 
 45-244 
 
 30-178 
 
 29-408 
 
 28-911 
 
 28-413 
 
 27-915 
 
 3-5 
 
 50563 
 
 33-726 
 
 32-866 
 
 32310 
 
 31-754 
 
 31-197 
 
 3-75 
 
 56-077 
 
 37-403 
 
 36-450 
 
 35-833 
 
 35-216 
 
 34-599 
 
 4' 
 
 61-777 
 
 41-205 
 
 40-155 
 
 39-476 
 
 38-796 
 
 38-116 
 
 4-25 
 
 67-658 
 
 45-128 
 
 43978 
 
 43-233 
 
 42-489 
 
 41-745 
 
 4-5 
 
 73-714 
 
 49-167 
 
 47-914 
 
 47-103 
 
 46*292 
 
 45-482 
 
 4-75 
 
 79-942 
 
 53-321 
 
 51-962 
 
 51-083 
 
 50-203 
 
 49324 
 
 5' 
 
 86-335 
 
 57-585 
 
 56-118 
 
 55-168 
 
 54-218 
 
 63269 
 
 5-25 
 
 92-891 
 
 61-958 
 
 60-379 
 
 59-357 
 
 58-335 
 
 57-314 
 
 5-5 
 
 99-604 
 
 66-436 
 
 64-743 
 
 63-647 
 
 62-551 
 
 61-456 
 
 5-75 
 
 106-472 
 
 71-017 
 
 69-207 
 
 68-036 
 
 66-864 
 
 65-693 
 
 6- 
 
 113-491 
 
 75698 
 
 73-769 
 
 72521 
 
 71-272 
 
 70024 
 
 625 
 
 120 657 
 
 80-478 
 
 78-427 
 
 77-100 
 
 75-772 
 
 74-445 
 
 6-5 
 
 127-969 
 
 85355 
 
 83-180 
 
 81-772 
 
 80-365 
 
 78-957 
 
 6-75 
 
 135-422 
 
 90-326 
 
 88-024 
 
 86-535 
 
 85-045 
 
 83-555 
 
 7* 
 
 143-015 
 
 95-391 
 
 9-2-960 
 
 91-387 
 
 89-813 
 
 88-240 
 
 7-25 
 
 150-744 
 
 100-546 
 
 97'983 
 
 96-325 
 
 94-667 
 
 93-009 
 
 7-5 
 
 158-608 
 
 105-792 
 
 103-095 
 
 101-350 
 
 99 606 
 
 97-861 
 
 7-75 
 
 166-604 
 
 111-125 
 
 108-292 
 
 106-460 
 
 1046-^7 
 
 102-795 
 
 8' 
 
 174-731 
 
 116-546 
 
 113-575 
 
 111-653 
 
 109731 
 
 107-809 
 
 8-25 
 
 182-984 
 
 122-05] 
 
 118-940 
 
 116927 
 
 114-914 
 
 112-901 
 
 8'5 
 
 191-365 
 
 127-640 
 
 124-387 
 
 122-282 
 
 120-177 
 
 118-072 
 
 875 
 
 199-869 
 
 133-313 
 
 129915 
 
 127716 
 
 1-25-518 
 
 123-319 
 
 9- 
 
 208-496 
 
 139-067 
 
 135-52-2 
 
 133-229 
 
 130935 
 
 128-642 
 
 9-25 
 
 217243 
 
 144-901 
 
 141-207 
 
 138-818 
 
 136-428 
 
 134-039 
 
 9'5 
 
 226-111 
 
 150-816 
 
 146972 
 
 144-485 
 
 141-997 
 
 139-510 
 
 9-75 
 
 235-093 
 
 156-807 
 
 152-810 
 
 150225 
 
 147-639 
 
 145-053 
 
 10- 
 
 244-193 
 
 162-877 
 
 158-725 
 
 156-039 
 
 153-353 
 
 150-666 
 
 See pp. Ill to 127. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 425 
 
 TABLE VI. The Discharge over Weirs or Notches of one foot in length, 
 
 in Cubic feet per minute. 
 Depths 10-25 inches to 32 inches. Coefficients -667 to -617. 
 
 GEEATER COEFFICIENTS. 
 
 The Formula at the heads of the Columns give the Value of the Discharge 
 D, in Cubic feet per minute, when I, the length of the Weir, is taken in 
 feet, and the head, h, in inches. For I \fh? we may substitute I h \/Ji, 
 retaining the same standards. 
 
 Heads 
 in 
 inches. 
 
 Theoretical 
 discharge 
 D- 
 7-72 I \A 3 - 
 
 Coefficient 
 667. 
 D = 
 5-15 l\/lfl. 
 
 Coefficient 
 650. 
 D = 
 5-02 I \/W. 
 
 Coefficient 
 639. 
 D = 
 4-93 I </h?. 
 
 Coefficient 
 628. 
 D = 
 4'85 I v/P. 
 
 Coefficient 
 617. 
 D = 
 4-76 I \/h~3. 
 
 10-25 
 
 253-407 
 
 169.023 
 
 164-715 
 
 161-927 
 
 159-140 
 
 156-352 
 
 10-5 
 
 262-734 
 
 175-244 
 
 170-777 
 
 167-887 
 
 164-997 
 
 162-107 
 
 10-75 
 
 272-173 
 
 181-540 
 
 176-913 
 
 173-919 
 
 170-925 
 
 167-931 
 
 IV 
 
 281-723 
 
 187909 
 
 183-120 
 
 180-021 
 
 176-92-2 
 
 173-823 
 
 11-25 
 
 291-382 
 
 194-352 
 
 189-398 
 
 186-193 
 
 182-988 
 
 179 782 
 
 11-5 
 
 301-148 
 
 200-866 
 
 195-746 
 
 192-434 
 
 189' 121 
 
 185-808 
 
 11-75 
 
 311-024 
 
 207-451 
 
 202-164 
 
 198-743 
 
 195-321 
 
 191-900 
 
 12- 
 
 321- 
 
 214-107 
 
 208-650 
 
 205-119 
 
 201-5&8 
 
 198-057 
 
 12-5 
 
 341-275 
 
 227-628 
 
 221-826 
 
 218-072 
 
 214-318 
 
 210-564 
 
 13- 
 
 361-950 
 
 241-421 
 
 235-268 
 
 231-286 
 
 227-305 
 
 223-323 
 
 13-5 
 
 383-031 
 
 255-482 
 
 248-970 
 
 244-757 
 
 240-543 
 
 236-330 
 
 14- 
 
 404-507 
 
 269806 
 
 262-930 
 
 258-480 
 
 254-030 
 
 249-581 
 
 14-5 
 
 426-368 
 
 284-387 
 
 277-139 
 
 272-449 
 
 267-759 
 
 263-069 
 
 15- 
 
 448-611 
 
 299-223 
 
 291-597 
 
 286-662 
 
 281-728 
 
 276-793 
 
 15-5 
 
 471-228 
 
 314-309 
 
 306-298 
 
 301-115 
 
 295-931 
 
 290-748 
 
 16- 
 
 494-212 
 
 329-639 
 
 321-238 
 
 315801 
 
 310-365 
 
 304-929 
 
 165 i 517-558 
 
 345-211 
 
 336-413 
 
 330-720 
 
 325-026 
 
 319-333 
 
 17- 
 
 541-261 
 
 361 021 
 
 351-820 
 
 345-866 
 
 339-912 
 
 333-958 
 
 17-5 
 
 565-315 
 
 377 065 
 
 367-455 
 
 361-236 
 
 355-018 
 
 348-799 
 
 18- 
 
 589715 
 
 393340 
 
 383'315 
 
 376-828 
 
 370-341 
 
 363-854 
 
 18-5 
 
 614-443 
 
 409-833 
 
 399'388 
 
 392-629 
 
 385-870 
 
 379-111 
 
 19- 
 
 639*533 
 
 426-569 
 
 415*696 
 
 408-662 
 
 401 627 
 
 394-592 
 
 19'5 
 
 664-944 
 
 443-518 
 
 432'214 
 
 424-899 
 
 417-585 
 
 410-270 
 
 20- 
 
 690-682 
 
 460-685 
 
 448*943 
 
 441-346 
 
 433-748 
 
 426-151 
 
 20-5 
 
 716737 
 
 478-064 
 
 465-879 
 
 457*995 
 
 450-111 
 
 442-227 
 
 21- 
 
 743 1-25 
 
 495 664 
 
 483-031 
 
 474-857 
 
 466-683 
 
 458-508 
 
 21-5 
 
 769-823 
 
 513 472 
 
 500-385 
 
 491-917 
 
 483-449 
 
 474 981 
 
 22- 
 
 796 832 
 
 531 487 
 
 517*941 
 
 509-176 
 
 500-410 
 
 491-645 
 
 22-5 
 
 824-151 
 
 549-709 
 
 535*698 
 
 526-632 
 
 517-567 
 
 508-501 
 
 23- 
 
 851 775 
 
 568-134 
 
 553-654 
 
 544284 
 
 534-915 
 
 525-545 
 
 235 
 
 879 700 
 
 586.760 
 
 571-805 
 
 562-128 
 
 552-452 
 
 542-775 
 
 24- 
 
 907-925 
 
 605-586 
 
 590-151 
 
 580-164 
 
 570-177 
 
 560-190 
 
 25' 
 
 965-253 
 
 643 824 
 
 627-414 
 
 616797 
 
 606-179 
 
 595-561 
 
 26- 
 
 1023-748 
 
 682-840 
 
 665*436 
 
 654-175 
 
 642-914 
 
 631-653 
 
 27- 
 
 1083-375 
 
 722611 
 
 704-194 
 
 692-277 
 
 680-360 
 
 668-442 
 
 28- 
 
 1144-116 
 
 763 125 
 
 743*675 
 
 731-090 
 
 718-505 
 
 705-920 
 
 29' 
 
 1205 950 
 
 804-369 
 
 783*868 
 
 770-602 
 
 757-337 
 
 744-071 
 
 30 
 
 1268-864 
 
 846-332 
 
 824-762 
 
 810-804 
 
 796-847 
 
 782-889 
 
 31- 
 
 1332-833 
 
 889 000 
 
 866-341 
 
 851-680 
 
 837019 
 
 822-358 
 
 32- 
 
 1397-842 
 
 982-361 
 
 908-597 
 
 893-221 
 
 877-845 
 
 862-469 
 
 See pp. Ill to 127. 
 
426 
 
 THE DISCHARGE OF "WATER FROM 
 
 TABLE VI. The Discharge over Weirs or Notches of one foot in length, 
 
 in Cubic feet per minute. 
 Depths 33 inches to 72 inches. Coefficients '607 to -617. 
 
 GEEATEE COEFFICIENTS. 
 
 The Formula at the heads of the Columns give the Value of the Discharge, 
 D, in Cubic feet per minute, when I, the length of the Weir, is taken in 
 feet, and the head, h f in inches. For I \/h 3 we may substitute 
 retaining the same standards. 
 
 Heads 
 in 
 inches. 
 
 Theoretical 
 discharge 
 D = 
 7'72 l\/M. 
 
 Coefficient 
 667. 
 D = 
 5-15 I -V//A 
 
 Coefficient 
 650. 
 D = 
 5-02 I \A 3 . 
 
 Coefficient 
 639. 
 D = 
 4-93 I \/~ti. 
 
 Coefficient 
 628. 
 D = 
 4-85 I \/W. 
 
 Coefficient 
 617. 
 D = 
 4-76 l\/M. 
 
 33- 
 
 1463-875 
 
 976-405 
 
 951-519 
 
 935-416 
 
 919314 
 
 903-211 
 
 34- 
 
 1530-917 
 
 1021-122 
 
 995-096 
 
 978-256 
 
 961-416 
 
 944-576 
 
 35- 
 
 1598-951 
 
 1066-500 
 
 1039-318 
 
 1021-730 
 
 1004-141 
 
 986-553 
 
 36- 
 
 1667-964 
 
 1112-532 
 
 1084-177 
 
 1065-829 
 
 1047-481 
 
 1029-134 
 
 37- 
 
 1737-943 
 
 1159208 
 
 1129-663 
 
 1110-546 
 
 1091-428 
 
 1072-311 
 
 38- 
 
 1808-875 
 
 1206-520 
 
 1175-769 
 
 1155-871 
 
 1135-974 
 
 1116-076 
 
 39- 
 
 1880-746 
 
 1254-458 
 
 1222-485 
 
 1201-797 
 
 1181-108 
 
 1160-420 
 
 40- 
 
 1953-544 
 
 1303-014 
 
 1269-804 
 
 1248-315 
 
 1226-826 
 
 1205337 
 
 41- 
 
 ^027-258 
 
 1352181 
 
 1317-718 
 
 1295-418 
 
 1273-118 
 
 1250*818 
 
 42- 
 
 2101-876 
 
 1401-951 
 
 1366-219 
 
 1343-099 
 
 ,1319-978 
 
 1296-857 
 
 43- 
 
 2177-387 
 
 1452'317 
 
 1415-302 
 
 1391-350 
 
 1367-399 
 
 1343-448 
 
 44- 
 
 2253-783 
 
 1503-273 
 
 1464-959 
 
 1440-167 
 
 1415376 
 
 1390-584 
 
 45- 
 
 2331-052 
 
 1554'812 
 
 1515-184 
 
 1489-542 
 
 1463-901 
 
 1438259 
 
 46- 
 
 2409-183 
 
 1606-925 
 
 1565-969 
 
 1539*468 
 
 1512-967 
 
 1486466 
 
 47- 
 
 2488-170 
 
 1659-609 
 
 1617-311 
 
 1589-941 
 
 1562-571 
 
 1535-201 
 
 48- 
 
 2568- 
 
 1712856 
 
 1669-200 
 
 1640-952 
 
 1612-704 
 
 1584-456 
 
 49- 
 
 2648*666 
 
 1766660 
 
 1721-633 
 
 1692-498 
 
 1663-362 
 
 1634-227 
 
 50- 
 
 2730-160 
 
 1821-021 
 
 1774-604 
 
 1744-572 
 
 1714-540 
 
 1684-509 
 
 51- 
 
 2812-474 
 
 1875'920 
 
 1828-108 
 
 1797-171 
 
 1766-234 
 
 1735-296 
 
 52- 
 
 2895-597 
 
 1931-363 
 
 1882-138 
 
 1850-286 
 
 1818-435 
 
 1786-583 
 
 53- 
 
 2979'525 
 
 1987-313 
 
 1936-691 
 
 1903*916 
 
 1871-142 
 
 1838-367 
 
 54- 
 
 3064-253 
 
 2043-857 
 
 1991-764 
 
 1958-058 
 
 1924-351 
 
 1890-644 
 
 55- 
 
 3149755 
 
 2100-887 
 
 2047-341 
 
 2012-693 
 
 1978-046 
 
 1943-399 
 
 56- 
 
 3236050 
 
 2158-445 
 
 2103433 
 
 2067-836 
 
 2032-239 
 
 1996-643 
 
 57- 
 
 3323-117 
 
 2216-519 
 
 2160-026 
 
 2123-472 
 
 2086-917 
 
 2050-363 
 
 58- 
 
 3410-946 
 
 2275-101 
 
 2217-115 
 
 2179-594 
 
 2142-074 
 
 2104-554 
 
 59- 
 
 3499-542 
 
 2334-195 
 
 2274-702 
 
 2236-207 
 
 2197712 
 
 2159-217 
 
 60- 
 
 3588'889 
 
 2393-789 
 
 2332-778 
 
 2293-300 
 
 22o3-822 
 
 2214-344 
 
 61- 
 
 3678 984 
 
 2453-882 
 
 2391-340 
 
 2350-871 
 
 2310-402 
 
 2269-933 
 
 6*2- 
 
 3769-825 
 
 2514-473 
 
 2450-386 
 
 2408-918 
 
 2367-450 
 
 2325-982 
 
 63- 
 
 3861-393 
 
 2575-549 
 
 2509-905 
 
 2467-430 
 
 2424-955 
 
 2382-479 
 
 64- 
 
 3953-694 
 
 2637-114 
 
 2569-901 
 
 2526-410 
 
 2482-920 
 
 2439*429 
 
 65- 
 
 4046720 
 
 2699-162 
 
 2630-368 
 
 2585-854 
 
 2541340 
 
 2496-826 
 
 06- 
 
 4140-465 
 
 2761-690 
 
 2691-302 
 
 2645-757 
 
 2600-212 
 
 2554-667 
 
 (37- 
 
 4234-922 
 
 2824-693 
 
 2752-699 
 
 2706-115 
 
 2659 531 
 
 2612-947 
 
 68- 
 
 4330-086 
 
 2888-167 
 
 2814-556 
 
 2766-925 
 
 2719-294 
 
 2671-663 
 
 69- 
 
 4425-954 
 
 2952-111 
 
 2876-870 
 
 2828-185 
 
 2779-499 
 
 2730-814 
 
 70- 
 
 4522-516 
 
 3016-518 
 
 2939-635 
 
 2889-888 
 
 2840-140 
 
 2790-392 
 
 71- 
 
 4619-774 
 
 3081-389 
 
 3002-853 
 
 2952-036 
 
 2901-218 
 
 2850-401 
 
 7-2- 
 
 4717-718 
 
 3146-718 
 
 3066 518 
 
 3014622 
 
 3962-727 
 
 2910-832 
 
 See pp, 111 to 
 
ORIFICES, WEIRS, PIPES, AND RIVEES. 
 
 427 
 
 TABLE VI. The Discharge over Weirs or Notches of one foot in length, 
 in Cubic feet per minute. 
 
 Depths J inch to 10 inches. Coefficients -606 to '518. 
 
 LESSER COEFFICIENTS. 
 
 The Formula at the heads of the Columns give the Value of the Discharge, 
 D, in Cubic feet per minute, when I, the length of the Weir, is taken in 
 feet, and the head, h, in inches. For l^/ h* we may substitute 
 retaining the same standards. 
 
 Heads 
 in 
 inches. 
 
 Coefficient 
 606 
 D = 
 
 44B {*/*? 
 
 Coefficient 
 595 
 D = 
 4-59 I \A 3 - 
 
 Coefficent 
 584 
 D = 
 4-51 1</M. 
 
 Coefficient 
 562 
 D = 
 4-34 I \/W. 
 
 Coefficient 
 540 
 D = 
 4-17 l*/h*. 
 
 Coefficient 
 518 
 D = 
 l\/ti*. 
 
 25 
 
 585 
 
 574 
 
 564 
 
 542 
 
 521 
 
 500 
 
 5 
 
 1-654 
 
 1-624 
 
 1-504 
 
 1-534 
 
 1-474 
 
 1-414 
 
 75 
 
 3-039 
 
 2-985 
 
 2-929 
 
 2-819 
 
 2-708 
 
 2-598 
 
 1- 
 
 4-680 
 
 4-595 
 
 4-510 
 
 4-340 
 
 4-170 
 
 4-000 
 
 i-25 
 
 6-540 
 
 6-421 
 
 6-303 
 
 6-065 
 
 5-828 
 
 5-590 
 
 To 
 
 8-597 
 
 8-441 
 
 8-284 
 
 7-973 
 
 7-660 
 
 7-348 
 
 1-75 
 
 10-833 
 
 10-637 
 
 10-440 
 
 10-047 
 
 9-653 
 
 9-260 
 
 2- 
 
 13-236 
 
 12-996 
 
 12-756 
 
 12-275 
 
 11-795 
 
 11-314 
 
 225 
 
 15-794 
 
 15-507 
 
 15-220 
 
 14-647 
 
 14-073 
 
 13-500 
 
 2-5 
 
 18-498 
 
 18-162 
 
 17-826 
 
 17-155 
 
 16-483 
 
 15-811 
 
 2-75 
 
 21-340 
 
 20-953 
 
 20-566 
 
 19-791 
 
 19-016 
 
 18-241 
 
 3- 
 
 24-316 
 
 23-874 
 
 23-433 
 
 22-550 
 
 21-668 
 
 20-785 
 
 3-25 
 
 27-418 
 
 26-920 
 
 26-422 
 
 25-427 
 
 24M32 
 
 23-436 
 
 3'5 
 
 30-641 
 
 30-085 
 
 29-529 
 
 28-416 
 
 27-304 
 
 26-192 
 
 3'75 
 
 33-982 
 
 33-366 
 
 32-749 
 
 31-515 
 
 30-281 
 
 29-048 
 
 4- 
 
 37-437 
 
 36-757 
 
 36-078 
 
 34-719 
 
 33-i60 
 
 32-000 
 
 4-25 
 
 41-001 
 
 40-256 
 
 39-512 
 
 38-024 
 
 36-535 
 
 35-047 
 
 4-5 
 
 44-671 
 
 43-860 
 
 43-049 
 
 41-427 
 
 39-806 
 
 38-184 
 
 4-75 
 
 48-445 
 
 47-565 
 
 46-686 
 
 44-927 
 
 43-169 
 
 41-410 
 
 5- 
 
 52-319 
 
 51-369 
 
 50-420 
 
 48-520 
 
 46-621 
 
 44-722 
 
 5'25 
 
 56-292 
 
 55-270 
 
 54-248 
 
 52-205 
 
 50-161 
 
 48-117 
 
 5'5 
 
 60-360 
 
 59-264 
 
 58-169 
 
 55-977 
 
 53-786 
 
 51-595 
 
 575 
 
 64-52-2 
 
 63-351 
 
 02-180 
 
 59-837 
 
 57-495 
 
 55-153 
 
 6- 
 
 68-776 
 
 67-527 
 
 66-279 
 
 63-782 
 
 61-285 
 
 58-788 
 
 6-25 
 
 73-118 
 
 71-791 
 
 70-464 
 
 67-a09 
 
 65-155 
 
 62-500 
 
 6'5 
 
 77-549 
 
 76-142 
 
 74-734 
 
 71-919 
 
 69-103 
 
 66-288 
 
 6'75 
 
 82-066 
 
 80-576 
 
 79-086 
 
 76-107 
 
 73-128 
 
 70-149 
 
 7- 
 
 86-667 
 
 85-094 
 
 83-521 
 
 80-374 
 
 77-228 
 
 74-082 
 
 7-25 
 
 91-351 
 
 89*693 
 
 88-034 
 
 84-718 
 
 81-402 
 
 78-085 
 
 75 
 
 96-116 
 
 94-372 
 
 92*627 
 
 89-138 
 
 85-648 
 
 82-159 
 
 7-75 
 
 100-962 
 
 99-129 
 
 97-297 
 
 93-631 
 
 89-966 
 
 86-301 
 
 8- 
 
 105-887 
 
 103-965 
 
 102-043 
 
 98-199 
 
 94-355 
 
 90511 
 
 8-25 
 
 110-889 
 
 108-876 
 
 106-863 
 
 102-837 
 
 98-812 
 
 94-786 
 
 8-5 
 
 115-967 
 
 113-862 
 
 111-757 
 
 107-547 
 
 103-337 
 
 99-127 
 
 8-75 
 
 121-121 
 
 118-922 
 
 116-723 
 
 112-326 
 
 107-929 
 
 103-532 
 
 9- 
 
 126-349 
 
 124-055 
 
 121-762 
 
 117-175 
 
 112-588 
 
 108-001 
 
 9-25 
 
 131-649 
 
 129-259 
 
 126-870 
 
 122-090 
 
 117-311 
 
 112-532 
 
 95 
 
 137-023 
 
 134-535 
 
 132-048 
 
 127-074 
 
 122-100 
 
 117-125 
 
 975 
 
 142-467 
 
 139-881 
 
 137-294 
 
 132122 
 
 126-950 
 
 121-778 
 
 10- 
 
 147-991 
 
 145-295 
 
 142-609 
 
 137-237 
 
 131-864 
 
 126-492 
 
 See pp. Ill to 
 
428 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VI. The Discharge over Weirs or Notches of one foot in length, 
 in Cubic feet per minute. 
 
 Depths 10-25 inches to 32 inches. Coefficients -606 to -518. 
 
 LESSEE COEFFICIENTS. 
 
 The Formula at the heads of the Columns give the Value of the Discharge, 
 D, in Cubic Jeet per minute, when I, the length of the Weir, is taken in 
 feet, and the head, h, in inches. For l*J h 3 we may substitute I h <J h> 
 retaining the same standards. 
 
 Heads 
 in 
 inches. 
 
 Coefficient 
 606. 
 D = 
 
 ^eSZ-v/k 3 - 
 
 Coefficient 
 595. 
 D = 
 4-59 l\/W. 
 
 Coefficient 
 584. 
 D = 
 4-51 1 \/l#. 
 
 Coefficient 
 562. 
 D = 
 4-34 1 -vA. 
 
 Coefficient 
 540. 
 D = 
 4-17 l\/h*. 
 
 Coefficient 
 518. 
 D = 
 4 I */h\ 
 
 10-25 
 
 153-565 
 
 150-777 
 
 147-990 
 
 142-415 
 
 136-840 
 
 131-265 
 
 10-5 
 
 159-217 
 
 156-327 
 
 153-437 
 
 147-657 
 
 141-876 
 
 136-096 
 
 1075 
 
 164-937 
 
 161-943 
 
 158-949 
 
 152961 
 
 146-974 
 
 140-986 
 
 11- 
 
 170-724 
 
 167625 
 
 164-526 
 
 158-328 
 
 152-130 
 
 145-933 
 
 11-25 
 
 176-577 
 
 173372 
 
 170 167 
 
 163-756 
 
 157-346 
 
 150-936 
 
 11-5 
 
 182-496 
 
 179-183 
 
 175-870 
 
 169-245 
 
 162-620 
 
 155-995 
 
 11-75 
 
 188-479 
 
 185-059 
 
 181-636 
 
 174-794 
 
 167-9,52 
 
 161-109 
 
 12- 
 
 194-526 
 
 190-995 
 
 187-464 
 
 180-402 
 
 173-340 
 
 166-278 
 
 12-5 
 
 206-810 
 
 203-056 
 
 199 302 
 
 191-794 
 
 184-286 
 
 176-778 
 
 13- 
 
 219-342 
 
 215-360 
 
 211-379 
 
 203-415 
 
 195-453 
 
 187-490 
 
 13-5 
 
 232-117 
 
 227903 
 
 223690 
 
 215263 
 
 206-837 
 
 198-410 
 
 14- 
 
 245-131 
 
 240-682 
 
 236-232 
 
 227333 
 
 218-434 
 
 209-535 
 
 145 
 
 258-379 
 
 253-689 
 
 248-999 
 
 239619 
 
 230-239 
 
 220-859 
 
 15- 
 
 271-858 
 
 266-924 
 
 261 989 
 
 252-119 
 
 242250 
 
 232-380 
 
 15-5 
 
 285-564 
 
 280-381 
 
 275-197 
 
 264.830 
 
 254-463 
 
 244-096 
 
 16- 
 
 299-492 
 
 294*056 
 
 288-6-20 
 
 277-747 
 
 266875 
 
 256001 
 
 16-5 
 
 313-640 
 
 307*947 
 
 302-253 
 
 290-868 
 
 279-481 
 
 268-095 
 
 17- 
 
 328-004 
 
 322-050 
 
 316-096 
 
 304-189 
 
 29-2-281 
 
 280-373 
 
 17-5 
 
 342-581 
 
 336-362 
 
 330-144 
 
 317-707 
 
 305-270 
 
 292-833 
 
 18- 
 
 357-367 
 
 350-880 
 
 344-394 
 
 381-4-0 
 
 318-446 
 
 305-472 
 
 18-5 
 
 372-352 
 
 365 594 
 
 358-835 
 
 345-317 
 
 331-799 
 
 318-241 
 
 19 
 
 387-557 
 
 380'o22 
 
 373-487 
 
 359-418 
 
 345-348 
 
 331-278 
 
 19-5 
 
 402-956 
 
 395-642 
 
 388-3-27 
 
 373-699 
 
 359-070 
 
 344-441 
 
 20- 
 
 418*553 
 
 410-959 
 
 403-358 
 
 388-163 
 
 372-968 
 
 357-773 
 
 20'5 
 
 434-343 
 
 426-458 
 
 418-574 
 
 402-806 
 
 387-038 
 
 371-270 
 
 21- 
 
 450-334 
 
 442-159 
 
 433-985 
 
 417-636 
 
 401-288 
 
 384-939 
 
 21'5 
 
 466-513 
 
 458-045 
 
 449-577 
 
 432-641 
 
 415-704 
 
 398-768 
 
 22- 
 
 482-880 
 
 474-115 
 
 465-350 
 
 447-819 
 
 430-289 
 
 41-2759 
 
 22-5 
 
 499-436 
 
 490-370 
 
 481 -304 
 
 463-173 
 
 445-042 
 
 426-910 
 
 23- 
 
 517-176 
 
 506-806 
 
 497-437 
 
 478-698 
 
 459*959 
 
 441-219 
 
 23'5 
 
 533-098 
 
 523-421 
 
 513-745 
 
 494-391 
 
 475-038 
 
 455 685 
 
 24- 
 
 550-203 
 
 540-215 
 
 530-228 
 
 510254 
 
 490280 
 
 470-305 
 
 25- 
 
 584943 
 
 574-326 
 
 563*708 
 
 542-472 
 
 521-237 
 
 500-001 
 
 26- 
 
 620-391 
 
 609-130 
 
 597 869 
 
 575-346 
 
 552'824 
 
 530-301 
 
 27- 
 
 656-525 
 
 644-608 
 
 632-691 
 
 608-857 
 
 585-023 
 
 561-188 
 
 28- 
 
 693334 
 
 680-749 
 
 668-164 
 
 642-993 
 
 61 7-828 
 
 592-652 
 
 29' 
 
 730-806 
 
 717-540 
 
 734-275 
 
 677-744 
 
 651-213 
 
 624-682 
 
 30- 
 
 768-932 
 
 754-974 
 
 711-017 
 
 713-102 
 
 6^5-187 
 
 657-272 
 
 31- 
 
 807-697 
 
 793-036 
 
 778-374 
 
 749-052 
 
 739730 
 
 690-407 
 
 32- 
 
 847-092 
 
 831-716 
 
 8A6-340 
 
 785-587 
 
 7b4-835 
 
 724-082 
 
 See pp. Ill to 127. 
 
OKIFICES, WEIES, PIPES, AND EIVEES. 
 
 429 
 
 TABLE VI. The. Discharge over Weirs or Notches of one foot in length, 
 in Cubic feet per minute. 
 
 Depths 33 inches to 72 inches. Coefficients -606 to '518. 
 
 LESSER COEFFICIENTS. 
 
 The Formula at the heads of the Columns give the Value of the Discharge, 
 D, in Cubic feet per minute, whence I, the length of the Weir, is taken in 
 feet, and the head, h, in inches. For l^/ h 3 we may substitute I h^/ h, 
 retaining the same standards. 
 
 TT__ J_ 
 
 Coefficient 
 
 Coefficient 
 
 Coefficient 
 
 Coefficient 
 
 Coefficient 
 
 Coefficient 
 
 Heads 
 
 606. 
 
 595. 
 
 584. 
 
 562. 
 
 540. 
 
 518. 
 
 in 
 
 D= 
 
 D = 
 
 D = 
 
 D = 
 
 D = 
 
 D = 
 
 inches. 
 
 4-88 I\A*' 
 
 4-59 I \/W. 
 
 4-51 1 \/h*. 
 
 4-34 14/h*. 
 
 4-17/ <sA a . 
 
 4^/13. 
 
 33- 
 
 887-108 
 
 871-006 
 
 854-903 
 
 822-698 
 
 790-493 
 
 758-287 
 
 34- 
 
 927736 
 
 910-896 
 
 894-056 
 
 860-375 
 
 8^6-695 
 
 793-015 
 
 35- 
 
 968-964 
 
 951-376 
 
 933-787 
 
 898-610 
 
 863-434 
 
 828-257 
 
 36- 
 
 1010-786 
 
 992-439 
 
 974-091 
 
 937-396 
 
 900-701 
 
 864-005 
 
 37- 
 
 1053-193 
 
 1034-076 
 
 1014-959 
 
 976-724 
 
 938 489 
 
 900-254 
 
 38- 
 
 1096-178 
 
 1076-281 
 
 1056-383 
 
 1016-588 
 
 976793 
 
 936-997 
 
 39- 
 
 1139-732 
 
 1119-044 
 
 1098-356 
 
 1056-979 
 
 1015-603 
 
 974-226 
 
 40- 
 
 1183-848 
 
 116-2-359 
 
 1140-870 
 
 1097-892 
 
 1054-914 
 
 1011-936 
 
 41- 
 
 1228-518 
 
 1206-219 
 
 1183919 
 
 1139-319 
 
 1094-719 
 
 1050-120 
 
 42- 
 
 1273-737 
 
 1250-616 
 
 1227-496 
 
 1181-254 
 
 1135-013 
 
 1088-772 
 
 43- 
 
 1319-497 1295-545 
 
 1271-594 
 
 1223-691 
 
 1175-789 
 
 1127-886 
 
 44- 
 
 1365792 
 
 1341-001 
 
 1316-209 
 
 1266-626 
 
 1217-043 
 
 1167-460 
 
 45- 
 
 1412-618 
 
 1386-976 
 
 1361334 
 
 1310-051 
 
 1258-768 
 
 1207-485 
 
 46- 
 
 1459-965 
 
 1433-464 
 
 1406-963 
 
 1353-961 
 
 1300-959 
 
 1247-957 
 
 47- 
 
 1507831 
 
 1480-461 
 
 1453-091 
 
 1398-352 
 
 1343-612 
 
 1288-872 
 
 48' 
 
 1556-208 
 
 1527-960 
 
 1499-712 
 
 1443216 
 
 1386720 
 
 1330-224 
 
 49' 
 
 1605-092 
 
 1575-956 
 
 1546-821 
 
 1488-550 
 
 1430-280 
 
 1372-009 
 
 50- 
 
 1654-477 
 
 1624-445 
 
 1594-413 
 
 1534-350 
 
 1474-286 
 
 1414-223 
 
 51- 
 
 1704-359 
 
 1 673-422 
 
 1642-485 
 
 1580-610 
 
 1518-736 
 
 1456-862 
 
 52- 
 
 1754732 
 
 1722-880 
 
 1691-029 
 
 1627-326 
 
 1563-622 
 
 1499-919 
 
 53- 
 
 1805-592 
 
 1772-817 
 
 1740-043 
 
 1674-493 
 
 1608-944 
 
 1543-394 
 
 54- 
 
 1856-937 
 
 1823-231 
 
 1789-524 
 
 1722-110 
 
 1654-697 
 
 1587-283 
 
 55- 
 
 1908-751 
 
 1874-104 
 
 1839-457 
 
 1770-162 
 
 1700-868 
 
 1631-573 
 
 56- 
 
 1961-046 1925-450 
 
 1889-853 
 
 1818-660 
 
 1747-467 
 
 1676-274 
 
 57- 
 
 2013-809 1977-255 
 
 1940-700 
 
 1867592 
 
 1794-483 
 
 1721-375 
 
 58- 
 
 2067-033 
 
 2029-513 
 
 1991992 
 
 1916-952 
 
 1841-911 
 
 1766870 
 
 59- 
 
 2120-722 
 
 2082-227 
 
 2043-733 
 
 1966-743 
 
 1889-753 
 
 1812-763 
 
 60- 
 
 2174-867 
 
 2135-389 
 
 2095-911 
 
 2016-956 
 
 1938-000 
 
 1859045 
 
 61- 
 
 2229-464 
 
 2188-995 
 
 2148-527 
 
 2067-589 
 
 1986-651 
 
 1905-714 
 
 62- 
 
 2284-514 
 
 2243-046 
 
 2201-578 
 
 2118-642 
 
 2035-706 
 
 1952-769 
 
 63- 
 
 2340-004 
 
 2297-529 
 
 2255-054 
 
 2170-103 
 
 2085-152 
 
 2000-202 
 
 64- 
 
 2395-939 
 
 2352-448 
 
 2308-957 
 
 2221-976 
 
 2134995 
 
 2048-013 
 
 65- 
 
 2452 312 
 
 2407-798 
 
 2363-284 
 
 2274 257 
 
 2185-229 
 
 2096-201 
 
 66- 
 
 2509-122 
 
 2463-577 
 
 2418-032 
 
 2326-941 
 
 2235-851 
 
 2144-761 
 
 67- 
 
 2566-363 
 
 2519-779 
 
 2473194 
 
 2380-026 
 
 2286-858 
 
 2193-690 
 
 68- 
 
 2624-032 
 
 2576-401 
 
 2528-770 
 
 2433-508 
 
 2338-246 
 
 2242-985 
 
 69- 
 
 2682-128 
 
 2633-443 
 
 2584-757 
 
 2487-386 
 
 2390-015 
 
 2292-644 
 
 70- 
 
 2740-645 
 
 2690-897 
 
 2641-149 
 
 2541 654 
 
 2442-159 
 
 2342-663 
 
 71- 
 
 2799-583 
 
 2748-766 
 
 2697-948 
 
 2596-313 
 
 2494-678 
 
 2393-043 
 
 72- 
 
 2858-937 
 
 2807-042 
 
 2755-147 
 
 2651-358 
 
 2547*568 
 
 2443-778 
 
 See pp. Ill to 127. 
 
430 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VII. For finding the Mean Velocity from the Maximum 
 Velocity at the Surface, in Mill Races, Streams, and Rivers with 
 uniform Channels ; and the Maximum Velocity from the Mean 
 Velocity. (See p. 184.) 
 
 For the Velocity in feet per minute, multiply by 5. 
 
 frfl^ 
 
 
 .S 
 
 > * 
 
 
 S 
 
 * -d 
 
 M " 
 
 ri . 
 
 "S 
 
 "3, 
 
 "3 1 
 
 *O >F * ^ 
 
 .M "-" fl 
 
 '111 
 
 'i 
 
 ','> 
 
 '^"3 
 
 Ml 
 
 B g 
 
 3S 
 > a 
 
 >" p, 
 
 ill 
 
 '3 C 2o 
 
 m 
 
 d QQ 
 
 11$ 
 
 P. 
 
 ?ll 
 
 Sis, 
 
 1 
 
 "is & 
 
 G S 
 
 lls 
 
 w a 
 
 ss 
 
 '5 * J 
 
 gal 
 
 ||I 
 
 Pi 
 
 00 
 
 Sf &>J3 
 
 III 
 
 Pi- 
 
 |&| 
 
 III 
 
 3*1 
 
 m 
 
 ".9 
 
 l-i 
 
 333 
 
 g P 
 
 3*3 
 
 s ^l 
 
 S P 
 
 i 
 
 84 
 
 75 
 
 41 
 
 34-24 
 
 33-37 
 
 81 
 
 67-64 
 
 68-86 
 
 2 
 
 1-67 
 
 1-51 
 
 42 
 
 35-07 
 
 34-23 
 
 82 
 
 68-47 
 
 69-77 
 
 3 
 
 2-51 
 
 2-27 
 
 43 
 
 35-91 
 
 35-09 
 
 83 
 
 69-31 
 
 7068 
 
 4 
 
 3-34 
 
 3-04 
 
 44 
 
 36-74 
 
 35-95 
 
 84 
 
 70-14 
 
 71-59 
 
 5 
 
 4-18 
 
 3-81 
 
 45 
 
 37-58 
 
 36-82 
 
 85 
 
 70-98 
 
 72-50 
 
 6 
 
 5-01 
 
 4-58 
 
 46 
 
 38-41 
 
 37-69 
 
 86 
 
 71-81 
 
 73-42 
 
 7 
 
 5-85 
 
 5-36 
 
 47 
 
 3925 
 
 38-56 
 
 87 
 
 72-65 
 
 74-33 
 
 8 
 
 6-68 
 
 6-14 
 
 48 
 
 40-08 
 
 39-43 
 
 88 
 
 73-48 
 
 75-24 
 
 9 
 
 7-52 
 
 6-92 
 
 49 
 
 40-92 
 
 40-30 
 
 89 
 
 74-32 
 
 76-16 
 
 10 
 
 8-35 
 
 7-71 
 
 50 
 
 41-75 
 
 41-17 
 
 90 
 
 75-15 
 
 77-08 
 
 11 
 
 9-19 
 
 8-50 
 
 51 
 
 42-59 
 
 42-05 
 
 91 
 
 75-99 
 
 77-99 
 
 12 
 
 10-02 
 
 9-29 
 
 52 
 
 43-42 
 
 42-92 
 
 92 
 
 76-82 
 
 78-91 
 
 13 
 
 10-86 
 
 10-09 
 
 53 
 
 44-26 
 
 43-80 
 
 93 
 
 77-66 
 
 79-83 
 
 14 
 
 11-69 
 
 10-88 
 
 54 
 
 45-09 
 
 44-68 
 
 94 
 
 78-49 
 
 80-75 
 
 15 
 
 12-53 
 
 11-69 
 
 55 
 
 45-93 
 
 45-56 
 
 95 
 
 79-33 
 
 81-67 
 
 16 
 
 13-36 
 
 12-49 
 
 56 
 
 46-76 
 
 46-44 
 
 96 
 
 80-16 
 
 82-59 
 
 17 
 
 14-20 
 
 13-30 
 
 57 
 
 47-60 
 
 47*32 
 
 97 
 
 81-00 
 
 83-51 
 
 18 
 
 15-03 
 
 14-11 
 
 58 
 
 48-43 
 
 48'2t 
 
 98 
 
 81-83 
 
 84-43 
 
 19 
 
 15-87 
 
 14-92 
 
 59 
 
 49-27 
 
 49-09 
 
 99 
 
 82-67 
 
 85-36 
 
 20 
 
 16-70 
 
 15-73 
 
 60 
 
 50-10 
 
 49-98 
 
 100 
 
 83-50 
 
 86-28 
 
 21 
 
 17-54 
 
 16-55 
 
 61 
 
 50-94 
 
 50-87 
 
 101 
 
 84-34 
 
 87-20 
 
 22 
 
 18-37 
 
 17-37 
 
 62 
 
 51-77 
 
 51'76 
 
 102 
 
 85-17 
 
 88-13 
 
 23 
 
 19-21 
 
 18-19 
 
 63 
 
 52-61 
 
 52-65 
 
 103 
 
 86-01 
 
 89-06 
 
 24 
 
 20-04 
 
 19-02 
 
 64 
 
 53-44 
 
 53*54 
 
 104 
 
 86-84 
 
 89-98 
 
 25 
 
 20-88 
 
 19-85 
 
 65 
 
 54-28 
 
 54-43 
 
 105 
 
 87-68 
 
 90-91 
 
 26 
 
 21-71 
 
 20-68 
 
 66 
 
 55-11 
 
 55-33 
 
 106 
 
 88-51 
 
 91-84 
 
 27 
 
 22-55 
 
 21-51 
 
 67 
 
 55-95 
 
 56-22 
 
 107 
 
 89-35 
 
 92-77 
 
 28 
 
 23-38 
 
 22-34 
 
 68 
 
 56-78 
 
 57-12 
 
 108 
 
 90-18 
 
 93-69 
 
 29 
 
 24-22 
 
 23-18 
 
 69 
 
 57-62 
 
 58-02 
 
 109 
 
 91-02 
 
 94-62 
 
 30 
 
 25-05 
 
 24-02 
 
 70 
 
 58-45 
 
 58-91 
 
 110 
 
 91-85 
 
 9555 
 
 31 
 
 25-89 
 
 24-86 
 
 71 
 
 59-29 
 
 59-81 
 
 111 
 
 92-69 
 
 96-49 
 
 32 
 
 26-72 
 
 25-70 
 
 72 
 
 60-12 
 
 60-71 
 
 112 
 
 93-52 
 
 97-42 
 
 33 
 
 27-56 
 
 26-54 
 
 73 
 
 60-96 
 
 61-61 
 
 113 
 
 94-36 
 
 98-35 
 
 34 
 
 28-39 
 
 27-39 
 
 74 
 
 61-79 
 
 62-52 
 
 114 
 
 95-19 
 
 99-28 
 
 35 
 
 29-23 
 
 28-24 
 
 75 
 
 62-63 
 
 63-42 
 
 115 
 
 96-03 
 
 100-21 
 
 36 
 
 30-06 
 
 29-09 
 
 76 
 
 63-46 
 
 64-32 
 
 116 
 
 96-86 
 
 101-15 
 
 37 
 
 30-90 
 
 29-94 
 
 77 
 
 64-30 
 
 65-23 
 
 117 
 
 97-70 
 
 102-08 
 
 38 
 
 31-73 
 
 30-79 
 
 78 
 
 65-13 
 
 66-13 
 
 118 
 
 98-53 
 
 103-02 
 
 39 
 
 32-57 
 
 31-65 
 
 79 
 
 65-97 
 
 67-04 
 
 119 
 
 99-37 
 
 103-95 
 
 40 
 
 33-40 
 
 32-51 
 
 80 
 
 66-80 
 
 67-95 
 
 120 
 
 100-20 
 
 104-89 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 431 
 
 TABLE VIII, For finding the Mean Velocities of Water flowing 
 in Pipes, Drains, Streams, and Rivers. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 Diameters of pipes \ inch to 2 inches. Falls per mile 1 inch to 
 
 12 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," and 
 velocities in inches per second. 
 
 Falls. 
 
 Inclinations 
 one in 
 
 ^inch. 
 
 J inch. 
 
 J inch. 
 
 inch. 
 
 J inch. 
 
 F. I. 
 
 1 
 
 63360 
 
 14 
 
 24 
 
 38 
 
 49 
 
 57 
 
 2 
 
 31680 
 
 22 
 
 37 
 
 59 
 
 76 
 
 90 
 
 3 
 
 21120 
 
 28 
 
 48 
 
 75 
 
 97 
 
 1-15 
 
 4 
 
 15840 
 
 34 
 
 57 
 
 89 
 
 1-15 
 
 1-36 
 
 5 
 
 12672 
 
 38 
 
 65 
 
 1-02 
 
 1-30 
 
 1-55 
 
 6 
 
 10560 
 
 42 
 
 72 
 
 1-13 
 
 45 
 
 1-72 
 
 7 
 
 9051 
 
 46 
 
 78 
 
 1-24 
 
 58 
 
 1-88 
 
 8 
 
 7920 
 
 50 
 
 85 
 
 1-33 
 
 71 
 
 202 
 
 9 
 
 7040 
 
 53 
 
 '90 
 
 1-43 
 
 83 
 
 2'16 
 
 10 
 
 6336 
 
 57 
 
 96 
 
 1-51 
 
 94 
 
 2-30 
 
 11 
 
 5760 
 
 60 
 
 I'Ol 
 
 1-60 
 
 96 
 
 2'42 
 
 1 
 
 5280 
 
 63 
 
 1-06 
 
 1-68 
 
 2'15 
 
 2'54 
 
 1 3 
 
 4224 
 
 71 
 
 1'20 
 
 1-90 
 
 2-43 
 
 2'88 
 
 1 6 
 
 3520 
 
 79 
 
 1'33 
 
 2'10 
 
 2'69 
 
 3-19 
 
 1 9 
 
 3017 
 
 87 
 
 1-45 
 
 2-29 2'94 
 
 3'48 
 
 2 
 
 2640 
 
 93 
 
 1'56 
 
 2-47 
 
 3-16 
 
 375 
 
 2 3 
 
 [Interpolated. 
 
 99 
 
 T67 
 
 2'63 
 
 3-37 
 
 399 
 
 2 6 
 
 2112 
 
 1-05 
 
 1-77 
 
 279 
 
 3-58 
 
 4-24 
 
 2 9 
 
 Interpolated. 
 
 1 11 
 
 1-87 
 
 294 
 
 3'77 
 
 4-47 
 
 3 
 
 1760 
 
 1-16 
 
 1-96 
 
 3'09 
 
 3'96 
 
 4-69 
 
 3 3 
 
 [nterpolated. 
 
 1-21 
 
 2-05 
 
 3-23 
 
 4-14 
 
 4-91 
 
 3 6 
 
 1508 
 
 1-26 
 
 2-14 
 
 3'37 
 
 4-32 
 
 5-12 
 
 3 9 
 
 [nterpolated. 
 
 1-31 
 
 2-22 
 
 3'50 
 
 4-48 
 
 5'31 
 
 4 
 
 1320 
 
 1-36 
 
 2-30 
 
 3-63 
 
 4-65 
 
 5-51 
 
 4 6 
 
 Interpolated. 
 
 45 
 
 2-45 
 
 3'87 
 
 4-96 
 
 5'88 
 
 5 
 
 1056 
 
 54 
 
 2'6t 
 
 4-11 
 
 5-27 
 
 6-24 
 
 5 6 
 
 Interpolated. 
 
 62 
 
 2-75 
 
 4-33 
 
 5'55 
 
 6'58 
 
 6 
 
 880 
 
 71 
 
 2-89 
 
 4-55 
 
 5'83 
 
 6-91 
 
 6 6 
 
 Interpolated. 
 
 78 
 
 3-02 
 
 4'76 
 
 6'10 
 
 7-22 
 
 7 
 
 754 
 
 1-86 
 
 3-15 
 
 4-97 
 
 6-36 
 
 7'54 
 
 7 6 
 
 Interpolated. 
 
 193 
 
 3-27 
 
 5-16 
 
 6'61 
 
 7'83 
 
 8 
 
 660 
 
 2-01 
 
 3-39 
 
 5'35 
 
 6-86 
 
 8-12 
 
 8 6 
 
 Interpolated. 
 
 2-07 
 
 3-51 
 
 5-53 
 
 7-09 
 
 8'40 
 
 9 
 
 587 
 
 2-14 
 
 3-62 
 
 5-72 
 
 7-32 
 
 8-68 
 
 9 6 
 
 Interpolated. 
 
 2-20 
 
 3'74 
 
 5-89 
 
 7-55 
 
 8'94 
 
 10 
 
 528 
 
 2-28 
 
 3-85 
 
 6-07 
 
 I'll 
 
 9-21 
 
 10 6 
 
 Interpolated 
 
 2-33 
 
 3-95 
 
 6-24 
 
 7'99 
 
 9-47 
 
 11 
 
 480 
 
 2-40 
 
 4-06 
 
 6-40 
 
 8'20 
 
 972 
 
 11 6 
 
 Interpolated. 
 
 2-46 
 
 4*16 
 
 657 
 
 8-41 
 
 9-97 
 
 12 
 
 440 
 
 2-52 
 
 4-27 
 
 6-73 
 
 8-62 
 
 10-21 
 
 See p. 195. 
 
432 
 
 THE DISCHAEGE OF WATEE FEOM 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing 
 in Pipes, Drains, Streams, and Rivers. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 
 hydraulic mean depth. 
 
 Diameters of pipes J inch to 2 inches. Falls per mile 13 feet to 
 
 5280 feet. 
 
 Falls per mile in feet, and the 
 hydraulic inclination. 
 
 " Hydraulic mean depths," or " mean radii," and 
 velocities in inches per second. 
 
 Falls. 
 
 Inclinations 
 one in 
 
 T V inch. 
 
 J- inch. 
 
 | inch. 
 
 inch. 
 
 \ inch. 
 
 13-2 
 
 400 
 
 2-66 
 
 4-50 
 
 7-10 
 
 9-10 
 
 10-78 
 
 13-6 
 
 Interpolated. 
 
 2-71 
 
 4-59 
 
 7-24 
 
 9-27 
 
 1098 
 
 14-1 
 
 375 
 
 2-76 
 
 4-67 
 
 7-37 
 
 9-44 
 
 11-18 
 
 14-6 
 
 Interpolated. 
 
 2-82 
 
 4-76 
 
 7-52 
 
 9-63 
 
 1141 
 
 15-1 
 
 350 
 
 2-87 
 
 4-85 
 
 7-66 
 
 9-82 
 
 11-63 
 
 15-6 
 
 Interpolated. 
 
 2-94 
 
 4-96 
 
 7'83 
 
 10-03 
 
 11-88 
 
 16'2 
 
 325 
 
 3-00 
 
 5-07 
 
 7-99 
 
 10-24 
 
 12-13 
 
 17'6 
 
 300 
 
 3-14 
 
 5-30 
 
 8'37 
 
 10-72 
 
 12-70 
 
 19'2 
 
 275 
 
 3-30 
 
 5-58 
 
 8-80 
 
 11-27 
 
 13-35 
 
 21-1 
 
 250 
 
 3-48 
 
 5-89 
 
 9-39 
 
 11-90 
 
 14-10 
 
 23-5 
 
 225 
 
 3-70 
 
 6-26 
 
 9-87 
 
 12-65 
 
 14-99 
 
 26-4 
 
 200 
 
 3'96 
 
 670 
 
 10-57 
 
 13*54 
 
 16-04 
 
 30-2 
 
 175 
 
 4-28 
 
 7-24 
 
 11-42 
 
 14-63 
 
 17-33 
 
 35'2 
 
 150 
 
 4-68 
 
 7-92 
 
 12-49 
 
 1600 
 
 18-96 
 
 37-7 
 
 140 
 
 4-88 
 
 8-24 
 
 13-00 
 
 16-66 
 
 19-74 
 
 42-2 
 
 125 
 
 5'21 
 
 8-81 
 
 13-90 
 
 1780 
 
 2r09 
 
 48- 
 
 110 
 
 5-62 
 
 9-50 
 
 14-98 
 
 19-19 
 
 22-74 
 
 52'8 
 
 100 
 
 5-94 
 
 10-05 
 
 15-85 
 
 20-30 
 
 24-06 
 
 587 
 
 90 
 
 6-33 
 
 10-69 
 
 16-87 
 
 21-61 
 
 25-60 
 
 66' 
 
 80 
 
 6-78 
 
 11-47 
 
 18-10 
 
 23-17 
 
 27'46 
 
 75'4 
 
 70 
 
 7'35 
 
 12-42 
 
 19-59 
 
 25-09 
 
 29*73 
 
 88' 
 
 60 
 
 8'05 
 
 13-61 
 
 21'48 
 
 27-51 
 
 32-60 
 
 105'6 
 
 50 
 
 8-99 
 
 15-19 
 
 23-96 
 
 30-69 
 
 36-37 
 
 117-3 
 
 45 
 
 9'57 
 
 16-18 
 
 25-53 
 
 32-70 
 
 38-75 
 
 132-0 
 
 40 
 
 10'28 
 
 17-37 
 
 27'41 
 
 35-11 
 
 41-60 
 
 150-8 
 
 35 
 
 iri4 
 
 18-84 
 
 29-71 
 
 38-06 
 
 45-10 
 
 176- 
 
 30 
 
 12-23 
 
 20-68 
 
 32'62 
 
 41-78 
 
 49-51 
 
 212-2 
 
 25 
 
 13-66 
 
 23-09 
 
 36-43 
 
 4667 
 
 55-30 
 
 264- 
 
 20 
 
 15-64 
 
 26-44 
 
 41-71 
 
 53-43 
 
 63-30 
 
 352' 
 
 15 
 
 18-61 
 
 31-46 
 
 4963 
 
 63-57 
 
 7533 
 
 528- 
 
 10 
 
 23-73 
 
 40-11 
 
 63-28 
 
 81-06 
 
 96-05 
 
 586-7 
 
 9 
 
 25-26 
 
 42-70 
 
 67-37 
 
 86-29 
 
 102-25 
 
 660' 
 
 8 
 
 27-08 
 
 4578 
 
 72-22 
 
 92-51 
 
 109-61 
 
 754-3 
 
 7 
 
 29-29 
 
 49-51 
 
 78-10 
 
 100-04 
 
 118-54 
 
 880-0 
 
 6 
 
 32-05 
 
 54-15 
 
 85-43 
 
 109-43 
 
 129-66 
 
 1056- 
 
 5 
 
 35-08 
 
 60-15 
 
 94-89 
 
 121-54 
 
 144-02 
 
 1320- 
 
 4 
 
 40-40 
 
 68-29 
 
 107-73 
 
 137-99 
 
 163-51 
 
 1760- 
 
 3 
 
 47-48 
 
 80-25 
 
 126-61 
 
 162-17 
 
 192-16 
 
 2640- 
 
 2 
 
 59-47 
 
 100-53 
 
 158-59 
 
 203-14 
 
 240-70 
 
 5280- 
 
 1 
 
 88-13 
 
 148-97 
 
 235-02 
 
 301-04 
 
 356-70 
 
 See p. 195. 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 433 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing in 
 Pipes, Drains, Streams, and Rivers. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 Diameters of pipes 2J inches to 5 inches. Falls per mile 1 inch to 
 
 12 feet.. 
 
 Falls pei 1 mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," and 
 velocities in inches per second. 
 
 Falls. 
 
 Inclinations 
 
 4 inch. 
 
 if inch. 
 
 inch. 
 
 1 inch. 
 
 1 J in. in- 
 
 
 one in 
 
 O 
 
 
 
 
 
 terpolated. 
 
 F. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 65 
 
 73 
 
 79 
 
 85 
 
 96 
 
 2 
 
 31680 
 
 1-02 
 
 1-13 
 
 1-23 
 
 1-33 
 
 1-49 
 
 3 
 
 21120 
 
 1-30 
 
 1-45 
 
 1-58 
 
 1-70 
 
 1-91 
 
 4 
 
 15840 
 
 1-54 
 
 1-71 
 
 1-87 
 
 2-01 
 
 2-26 
 
 5 
 
 12672 
 
 1-76 
 
 1-95 
 
 2-13 
 
 2-29 
 
 2-58 
 
 6 
 
 10560 
 
 1-95 
 
 2-17 
 
 2-36 
 
 2-55 
 
 2-86 
 
 7 
 
 9051 
 
 2-13 
 
 2-37 
 
 2-58 
 
 2-78 
 
 313 
 
 8 
 
 7920 
 
 2-30 
 
 2-55 
 
 2-78 
 
 3-00 
 
 3-37 
 
 9 
 
 7040 
 
 2-46 
 
 2-73 
 
 2-98 
 
 3-21 
 
 361 
 
 10 
 
 6336 
 
 2-61 
 
 2-90 
 
 3-16 
 
 3-40 
 
 3'83 
 
 11 
 
 5760 
 
 2-76 
 
 3-06 
 
 3-33 
 
 3-59 
 
 4-04 
 
 1 
 
 5280 
 
 2-89 
 
 3-21 
 
 3-50 
 
 3-77 
 
 4'24 
 
 1 3 
 
 4224 
 
 3-28 
 
 3-64 
 
 3-97 
 
 4-27 
 
 4-81 
 
 1 6 
 
 3520 
 
 363 
 
 4-03 
 
 4-39 
 
 4-73 
 
 5'32 
 
 1 9 
 
 3017 
 
 3-96 
 
 4'39 
 
 4-79 
 
 5-16 
 
 5'80 
 
 2 
 
 2640 
 
 4-26 
 
 4-73 
 
 5-16 
 
 5-55 
 
 6-25 
 
 2 3 
 
 Interpolated. 
 
 4-55 
 
 5-04 
 
 5-50 
 
 5-92 
 
 6-66 
 
 2 6 
 
 2112 
 
 4-83 
 
 5'35 
 
 5-84 
 
 6-29 
 
 7-07 
 
 2 9 
 
 Interpolated. 
 
 5-09 
 
 5-64 
 
 6-15 
 
 6-12 
 
 7-46 
 
 3 
 
 1760 
 
 5-34 
 
 5'92 
 
 6-46 
 
 6-96 
 
 7-83 
 
 3 3 
 
 Interpolated. 
 
 5-58 
 
 6-19 
 
 6-75 
 
 7-27 
 
 8-18 
 
 3 6 
 
 1508 
 
 5-82 
 
 6-46 
 
 7-04 
 
 7-59 
 
 8-53 
 
 3 9 
 
 Interpolated. 
 
 6-05 
 
 6-71 
 
 7-31 
 
 7-88 
 
 8-86 
 
 4 
 
 1320 
 
 6-27 
 
 6-95 
 
 7-58 
 
 8-17 
 
 9-19 
 
 4 6 
 
 Interpolated. 
 
 6-69 
 
 7-42 
 
 8-09 
 
 8-71 
 
 9-80 
 
 5 
 
 1056 
 
 7-10 
 
 7-88 
 
 8-59 
 
 9-25 
 
 10-41 
 
 5 6 
 
 Interpolated. 
 
 7-48 
 
 8-30 
 
 9-05 
 
 9-76 
 
 10-97 
 
 6 
 
 880 
 
 7-86 
 
 8-72 
 
 9-51 
 
 10-25 
 
 11-53 
 
 6 6 
 
 Interpolated. 
 
 8-22 
 
 9-12 
 
 9-94 
 
 10-71 
 
 12-05 
 
 7 
 
 754 
 
 8-57 
 
 9-51 
 
 10-37 
 
 11-17 
 
 12-57 
 
 7 6 
 
 Interpolated. 
 
 8-92 
 
 9-89 
 
 10-78 
 
 11-62 
 
 13-06 
 
 8 
 
 660 
 
 9-24 
 
 10-25 
 
 11-18 
 
 12-04 
 
 13-54 
 
 8 6 
 
 Interpolated. 
 
 9-55 
 
 10-60 
 
 11-56 
 
 12-45 
 
 14-01 
 
 9 
 
 587 
 
 9-87 
 
 10-95 
 
 11-94 
 
 12-86 
 
 14--17 
 
 9 6 
 
 Interpolated. 
 
 10-18 
 
 11-28 
 
 12-31 
 
 13-26 
 
 14-91 
 
 10 
 
 528 
 
 10-48 
 
 11-62 
 
 1267 
 
 13-65 
 
 15-36 
 
 10 6 
 
 Interpolated. 
 
 10-77 
 
 11-95 
 
 13-03 
 
 14-03 
 
 15-78 
 
 11 
 
 480 
 
 11-00 
 
 12-27 
 
 1338 
 
 14-41 
 
 16-21 
 
 11 6 
 
 Interpolated. 
 
 1134 
 
 12-58 
 
 13-72 
 
 14-82 
 
 16-64 
 
 12 
 
 440 
 
 11-62 
 
 12-89 
 
 14-05 
 
 15-22 
 
 17-07 
 
434 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing in 
 Pipes, Drains, Streams, and Rivers. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 Diameters of pipes 2J inches to 5 inches. Falls per mile 13 feet to 
 
 5280 feet. 
 
 Falls per mile in feet, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second; 
 
 Falls. 
 
 Inclinations 
 one in 
 
 g inch. 
 
 | inch. 
 
 I inch. 
 
 1 inch. 
 
 1 in. in- 
 terpolated. 
 
 F. 
 
 
 
 
 
 
 
 13-2 
 
 400 
 
 12-26 
 
 13-60 
 
 14-83 
 
 15-98 
 
 17-98 
 
 13-6 
 
 Interpolated. 
 
 12-49 
 
 13-86 
 
 15-11 
 
 16-28 
 
 18-31 
 
 14-1 
 
 375 
 
 12-72 
 
 14-11 
 
 15-39 
 
 16-58 
 
 1865 
 
 14-6 
 
 Interpolated. 
 
 12-98 
 
 14-39 
 
 15-70 
 
 16-91 
 
 19-02 
 
 15-1 
 
 350 
 
 13-23 
 
 14-68 
 
 16-00 
 
 17-24 
 
 19-40 
 
 15'6 
 
 Interpolated. 
 
 13-52 
 
 14-99 
 
 16-35 
 
 17-62 
 
 19-81 
 
 16-2 
 
 325 
 
 1380 
 
 1531 
 
 16-79 
 
 17-99 
 
 20-23 
 
 17-6 
 
 300 
 
 14-45 
 
 16-02 
 
 17-48 
 
 18-83 
 
 21-18 
 
 19'2 
 
 275 
 
 15-19 
 
 16-85 
 
 1837 
 
 19-79 
 
 22-26 
 
 21'1 
 
 250 
 
 16-04 
 
 17-80 
 
 1940 
 
 20-91 
 
 23-52 
 
 23-5 
 
 225 
 
 1705 
 
 18-91 
 
 20-62 
 
 22-21 
 
 24-99 
 
 26-4 
 
 200 
 
 18-25 
 
 20-24 
 
 22-07 
 
 23-78 
 
 26-75 
 
 30-2 
 
 175 
 
 19-71 
 
 21-87 
 
 23-85 
 
 25-69 
 
 28-90 
 
 35-2 
 
 150 
 
 21-57 
 
 23-92 
 
 26-09 
 
 28-11 
 
 31-62 
 
 37-7 
 
 140 
 
 22-45 
 
 24-91 
 
 27-16 
 
 29-26 
 
 32-92 
 
 42-2 
 
 125 
 
 23-99 
 
 26-62 
 
 29-03 
 
 31-27 
 
 35-18 
 
 48- 
 
 110 
 
 25-87 
 
 28-69 
 
 31-29 
 
 33-71 
 
 37-92 
 
 62'8 
 
 100 
 
 27-36 
 
 30-35 
 
 33-10 
 
 35-66 
 
 40-11 
 
 587 
 
 90 
 
 29-12 
 
 32-31 
 
 35-23 
 
 37-96 
 
 42-69 
 
 66' 
 
 80 
 
 31-23 
 
 34-64 
 
 37-78 
 
 40-70 
 
 45-79 
 
 75'4 
 
 70 
 
 33-82 
 
 37-51 
 
 40-91 
 
 44-07 
 
 49-58 
 
 88-0 
 
 60 
 
 37-08 
 
 41-13 
 
 44-86 
 
 48-33 
 
 54-36 
 
 105'6 
 
 50 
 
 41-37 
 
 45-78 
 
 50-04 
 
 5391 
 
 60-65 
 
 117-3 
 
 45 
 
 44-08 
 
 4889 
 
 53-32 
 
 57-44 
 
 64-62 
 
 132- 
 
 40 
 
 47-32 
 
 52-49 
 
 57-25 
 
 61-67 
 
 69-37 
 
 150'8 
 
 35 
 
 51-30 
 
 56-90 
 
 62-06 
 
 66-86 
 
 75-20 
 
 176- 
 
 30 
 
 56-32 
 
 62-47 
 
 68-13 
 
 73-40 
 
 82-56 
 
 211-2 
 
 25 
 
 62-90 
 
 69-77 
 
 76-09 
 
 81-97 
 
 92-21 
 
 264- 
 
 20 
 
 72-01 
 
 79-87 
 
 87-11 
 
 93-84 
 
 105-56 
 
 352- 
 
 15 
 
 85-68 
 
 95-05 
 
 103-66 
 
 111-67 
 
 125-61 
 
 528- 
 
 10 
 
 109-26 
 
 121-19 
 
 132-17 
 
 142-39 
 
 160-17 
 
 586-7 
 
 9 
 
 116-31 
 
 129-01 
 
 140-70 
 
 151-58 
 
 170-50 
 
 660- 
 
 8 
 
 124-68 
 
 138-30 
 
 150-83 
 
 162-49 
 
 182-78 
 
 754-3 
 
 7 
 
 134-84 
 
 149-57 
 
 163-12 
 
 17573 
 
 197-67 
 
 880- 
 
 6 
 
 147-69 
 
 163-60 
 
 . 178-42 
 
 192-22 
 
 21622 
 
 1056- 
 
 5 
 
 163-82 
 
 181-71 
 
 198-17 
 
 213-50 
 
 240-15 
 
 1320- 
 
 4 
 
 185-99 
 
 206-31 
 
 225-00 
 
 242-39 
 
 27266 
 
 1760- 
 
 3 
 
 218-58 
 
 242-46 
 
 264-42 
 
 284-86 
 
 320-43 
 
 2640- 
 
 2 
 
 273-79 
 
 303-70 
 
 331 22 
 
 356-82 
 
 401-37 
 
 5280- 
 
 1 
 
 405-74 
 
 450-07 
 
 490-84 
 
 528-79 
 
 594-82 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 435 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing in 
 
 Pipes, Drains, Streams, and Rivers. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 OPEN DEAINS AND PIPES, 
 
 Diameters of pipes 6 indies to 13 inches. Falls per mile 1 inch 
 
 to 12 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," and 
 velocities iu inches per second. 
 
 Falls. 
 
 Inclinations, 
 
 IT? inch. 
 
 If- in. in- 
 
 2 inches. 
 
 2 J inches 
 
 3 inches. 
 
 
 one in 
 
 
 terpolated. 
 
 
 
 
 F. I. 
 
 
 
 
 
 
 
 1 
 
 63360 
 
 MW 
 
 1-15 
 
 1-24 
 
 1-40 
 
 1-55 
 
 2 
 
 31680 
 
 1-66 
 
 1-80 
 
 1-94 
 
 2-19 
 
 2-41 
 
 3 
 
 21120 
 
 2-12 
 
 2-30 
 
 2-48 
 
 2-80 
 
 3-08 
 
 4 
 
 15840 
 
 2-52 
 
 2-73 
 
 2-94 
 
 334 
 
 3-65 
 
 5 
 
 12672 
 
 2-86 
 
 3-11 
 
 3-35 
 
 3-77 
 
 4-16 
 
 6 
 
 10560 
 
 3-18 
 
 3-45 
 
 3-72 
 
 4-19 
 
 4-62 
 
 7 
 
 9051 
 
 3-47 
 
 3-77 
 
 4-06 
 
 4-58 
 
 5-04 
 
 8 
 
 7920 
 
 3-75 
 
 4-06 
 
 4-38 
 
 4-94 
 
 5-44 
 
 9 
 
 7040 
 
 4-01 
 
 4-34 
 
 4-68 
 
 5-28 
 
 5-81 
 
 10 
 
 6336 
 
 4-25 4-61 
 
 4-97 
 
 5-60 
 
 617 
 
 11 
 
 5760 
 
 4-49 
 
 4-86 
 
 5-24 
 
 5-91 
 
 6-51 
 
 1 
 
 5280 
 
 4-71 
 
 5-11 
 
 551 
 
 6-21 
 
 6-84 
 
 1 3 
 
 4224 
 
 5-34 
 
 5-79 
 
 6-24 
 
 7-03 
 
 775 
 
 1 6 
 
 3520 
 
 5-91 
 
 6-41 
 
 6-91 
 
 7-79 
 
 8-58 
 
 1 9 
 
 3017 
 
 6-44 
 
 6-99 
 
 7-53 
 
 8-49 
 
 9-35 
 
 2 
 
 2640 
 
 6-94 
 
 7-53 
 
 8-11 
 
 9-14 
 
 10-07 
 
 2 3 
 
 [nterpolated. 
 
 7-40 
 
 8-03 
 
 8'65 
 
 9-74 
 
 10-74 
 
 2 6 
 
 2112 
 
 7-86 
 
 8-52 
 
 9-18 
 
 10-35 
 
 11-40 
 
 2 9 
 
 Interpolated. 
 
 8-28 
 
 8-98 
 
 9-67 
 
 10-90 
 
 12-01 
 
 3 
 
 1760 
 
 8-70 
 
 9-43 
 
 10-16 
 
 11-45 
 
 12-62 
 
 3 3 
 
 [nterpolated. 
 
 9-09 
 
 9-85 
 
 10-62 
 
 11-97 
 
 13-19 
 
 3 6 
 
 1508 
 
 9-48 
 
 10-28 
 
 11-08 
 
 12'48 
 
 13-76 
 
 3 9 
 
 [nterpolated. 
 
 9-84 
 
 10-67 
 
 11-50 
 
 12-96 
 
 14-29 
 
 4 
 
 1320 
 
 10-21 
 
 11-07 
 
 11-93 
 
 13-44 
 
 14-81 
 
 4 6 
 
 interpolated. 
 
 10-89 
 
 11-80 
 
 12-72 
 
 14-34 
 
 15-80 
 
 5 
 
 1056 
 
 11-56 
 
 12-54 
 
 13-51 
 
 15-23 
 
 16-78 
 
 5 6 
 
 Interpolated. 
 
 12-18 
 
 13-21 
 
 14-24 
 
 1604 
 
 17-68 
 
 6 
 
 880 
 
 1280 
 
 13-88 
 
 14-96 
 
 16-86 
 
 18-58 
 
 6 6 
 
 Interpolated. 
 
 13-38 
 
 14-51 
 
 15-64 
 
 17-62 
 
 19-42 
 
 7 
 
 754 
 
 13-96 
 
 15-14 
 
 16-32 
 
 18-39 
 
 20-26 
 
 7 6 
 
 Interpolated. 
 
 14-51 
 
 15-73 
 
 16-95 
 
 19-10 
 
 21-05 
 
 8 
 
 660 
 
 15-05 
 
 16-32 
 
 17-58 
 
 19-82 
 
 21-84 
 
 8 6 
 
 Interpolated. 
 
 15-56 
 
 16-87 
 
 18-18 
 
 20-49 
 
 2-2-58 
 
 9 
 
 587 
 
 16-07 
 
 17-4*3 
 
 18-78 
 
 21-17 
 
 23-32 
 
 9 6 
 
 Interpolated. 
 
 16-57 
 
 17-97 
 
 1936 
 
 21-82 
 
 24-04 
 
 10 
 
 528 
 
 17-06 
 
 18-50 
 
 1994 
 
 22-47 
 
 24-76 
 
 10 6 
 
 Interpolated, 
 
 17-54 
 
 19-01 
 
 20-49 
 
 23-09 
 
 25-45 
 
 11 
 
 480 
 
 18-01 
 
 1953 
 
 21-04 
 
 23-72 
 
 26-13 
 
 11 6 
 
 Interpolated. 
 
 18-47 
 
 20-02 
 
 21-57 
 
 24-32 
 
 26-79 
 
 12 
 
 440 
 
 18-92 
 
 20-51 
 
 22-11 
 
 24-91 
 
 27-45 
 
 PF2 
 
436 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing in 
 Pipes, Drains, Streams, and Rivers, 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 Diameters of pipes 6 inches to 14 inches. Falls per mile 13 feet 
 
 to 5%8Qfeet. 
 
 Falls per mile in feet, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," and 
 velocities in inches per second. 
 
 Tails. 
 
 Inclinations, 
 one in 
 
 1 J inch. 
 
 2 inches. 
 
 2 J inches. 
 
 3 inches. 
 
 i^- inches. 
 
 13-2 
 
 400 
 
 19-97 
 
 23-34 
 
 26-30 
 
 28-98 
 
 31-44 
 
 13-6 
 
 
 20-34 
 
 23-77 
 
 26-79 
 
 29-52 
 
 32-03 
 
 14-1 
 
 375 
 
 20-72 
 
 24-21 
 
 27-28 
 
 30-06 
 
 32-62 
 
 14-6 
 
 
 21-13 
 
 24-69 
 
 27-83 
 
 30-67 
 
 33-27 
 
 15-1 
 
 350 
 
 21-55 
 
 25-18 
 
 28-38 
 
 31-27 
 
 33-93 
 
 15-6 
 
 
 22-01 
 
 25-72 
 
 28-99 
 
 31-94 
 
 34-66 
 
 16-2 
 
 325 
 
 22-48 
 
 26-27 
 
 29-60 
 
 32-62 
 
 35-39 
 
 17-6 
 
 300 
 
 23-53 
 
 27-50 
 
 30-99 
 
 34-15 
 
 37-05 
 
 19-2 
 
 275 
 
 24-74 
 
 28-90 
 
 32-57 
 
 35-89 
 
 38-94 
 
 21-1 
 
 250 
 
 26-13 
 
 30-53 
 
 34-41 
 
 37-91 
 
 41-14 
 
 23-5 
 
 225 
 
 27-76 
 
 32-44 
 
 36-56 
 
 40-28 
 
 43-71 
 
 26-4 
 
 200 
 
 29-72 
 
 34-72 
 
 39-13 
 
 43-12 
 
 46-79 
 
 30-2 
 
 175 
 
 32-11 
 
 37-52 
 
 42-28 
 
 46-59 
 
 5055 
 
 35-2 
 
 350 
 
 35-13 
 
 41-04 
 
 46-26 
 
 50-97 
 
 55-30 
 
 37-7 
 
 140 
 
 36-57 
 
 42-73 
 
 48-16 
 
 53-07 
 
 57-58 
 
 42-2 
 
 125 
 
 39-08 
 
 45-66 
 
 51-46 
 
 56-71 
 
 61-53 
 
 48* 
 
 110 
 
 42-13 
 
 49-23 
 
 55-48 
 
 61-13 
 
 6633 
 
 52'8 
 
 100 
 
 44-57 
 
 52-07 
 
 58-69 
 
 64-67 
 
 70-17 
 
 587 
 
 90 
 
 47-43 
 
 55-42 
 
 62-46 
 
 68-83 
 
 74-68 
 
 66* 
 
 80 
 
 50-87 
 
 59-44 
 
 66-99 
 
 73-81 
 
 80-09 
 
 75-4 
 
 70 
 
 55-08 
 
 64-36 
 
 72-50 
 
 79-92 
 
 86-72 
 
 88- 
 
 60 
 
 60-39 
 
 70-57 
 
 79-53 
 
 87-63 
 
 95-09 
 
 105*6 
 
 50 
 
 67-38 
 
 78-73 
 
 88-73 
 
 97-77 
 
 106-08 
 
 117-3 
 
 45 
 
 71-79 
 
 83-88 
 
 94-54 
 
 104-17 
 
 113-03 
 
 132' 
 
 40 
 
 77-07 
 
 90-06 
 
 101-50 
 
 118-84 
 
 121-35 
 
 150-8 
 
 35 
 
 83-55 
 
 97-63 
 
 110-03 
 
 121-24 
 
 131-55 
 
 176- 
 
 30 
 
 91-72 
 
 107-18 
 
 120-79 
 
 133 10 
 
 144-41 
 
 211-2 
 
 25 
 
 102-44 
 
 11970 
 
 134-90 
 
 148-65 
 
 161-29 
 
 264- 
 
 20 
 
 117-28 
 
 137-03 
 
 154-44 
 
 170-18 
 
 184-65 
 
 352- 
 
 15 
 
 139-56 
 
 163-06 
 
 183-78 
 
 202-50 
 
 219-72 
 
 528- 
 
 10 
 
 177-95 
 
 207-92 
 
 234-33 
 
 258-21 
 
 280-16 
 
 586-7 
 
 9 
 
 189-43 
 
 221-34 
 
 249-45 
 
 274-87 
 
 298-24 
 
 660- 
 
 8 
 
 203-07 
 
 237-28 
 
 267-42 
 
 294-67 
 
 31972 
 
 754-3 
 
 7 
 
 219-61 
 
 256-61 
 
 289-20 
 
 318-67 
 
 345-77 
 
 880- 
 
 6 
 
 240-22 
 
 281-36 
 
 316-33 
 
 348-57 
 
 378-20 
 
 1056- 
 
 5 
 
 266-81 
 
 311-75 
 
 351-35 
 
 387-15 
 
 420-07 
 
 1320- 
 
 4 
 
 302-92 
 
 353-95 
 
 398-91 
 
 43955 
 
 47693 
 
 1760- 
 
 3 
 
 356-00 
 
 415-96 
 
 468-80 
 
 516-57 
 
 560-49 
 
 2640- 
 
 2 
 
 44593 
 
 521-04 
 
 587-22 
 
 647-06 
 
 702-08 
 
 5280- 
 
 1 
 
 660-84 
 
 772-16 
 
 87023 
 
 958-91 
 
 1040-44 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 437 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing 
 in Pipes, Drains, Streams, and Rivers. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 Diameters of pipes 14 inches to 22 inches. Falls per mile 1 inch 
 
 to 12 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 3 \ inches. 
 
 4 inches. 
 
 4J inches. 
 
 5 inches. 
 
 5 \ inches. 
 
 F. I. 
 
 1 
 
 63360 
 
 1-68 
 
 1-80 
 
 1-91 
 
 2-02 
 
 2-13 
 
 2 
 
 31680 
 
 2-61 
 
 2-81 
 
 2-98 
 
 3-15 
 
 3-32 
 
 3 
 
 21120 
 
 3-34 
 
 3-59 
 
 3-82 
 
 4-03 
 
 4-24 
 
 4 
 
 15840 
 
 3-96 
 
 4-25 
 
 4-52 
 
 4-78 
 
 5-02 
 
 5 
 
 12672 
 
 4-51 
 
 4-84 
 
 5-15 
 
 5-44 
 
 5-72 
 
 6 
 
 10560 
 
 5-01 
 
 5-37 
 
 5-72 
 
 604 
 
 6-35 
 
 7 
 
 9051 
 
 5-47 
 
 5-87 
 
 6-24 
 
 6-60 
 
 6-94 
 
 8 
 
 7920 
 
 5-90 
 
 6-33 
 
 6-74 
 
 7-12 
 
 7-48 
 
 9 
 
 7040 
 
 631 
 
 6-77 
 
 7-20 
 
 7-61 
 
 8-00 
 
 10 
 
 6336 
 
 6-70 
 
 7-18 
 
 7-64 
 
 8-08 
 
 8-49 
 
 11 
 
 5760 
 
 7-06 
 
 7-58 
 
 8-06 
 
 8-52 
 
 8-96 
 
 1 
 
 5280 
 
 7-42 
 
 7-96 
 
 8-47 
 
 895 
 
 9-41 
 
 1 3 
 
 4224 
 
 8-41 
 
 9-02 
 
 9-60 
 
 10-14 
 
 10-66 
 
 1 6 
 
 3520 
 
 9-31 
 
 9-99 
 
 10-63 
 
 11-23 
 
 11-80 
 
 1 9 
 
 3017 
 
 10-15 
 
 10-89 
 
 11-58 
 
 12-24 
 
 12-86 
 
 2 
 
 2640 
 
 1093 
 
 11-73 
 
 12-47 
 
 13-18 
 
 13-86 
 
 2 3 
 
 Interpolated. 
 
 11-65 
 
 12-50 
 
 13-30 
 
 1405 
 
 14-77 
 
 2 6 
 
 2112 
 
 12-37 
 
 13-28 
 
 14-12 
 
 14-93 
 
 15-69 
 
 2 9 
 
 Interpolated. 
 
 13-03 
 
 13-68 
 
 14-88 
 
 15-72 
 
 16-53 
 
 3 
 
 1760 
 
 13-69 
 
 14-69 
 
 15-63 
 
 16-52 
 
 17-36 
 
 3 3 
 
 Interpolated. 
 
 14-31 
 
 15-35 
 
 16-33 
 
 17-26 
 
 18-14 
 
 3 6 
 
 1508 
 
 14-92 
 
 16-01 
 
 17-03 
 
 18-00 
 
 18-92 
 
 3 9 
 
 Interpolated. 
 
 15-50 
 
 16-63 
 
 17-69 
 
 18-70 
 
 19-65 
 
 4 
 
 1320 
 
 16-07 
 
 17-25 
 
 18-35 
 
 1939 
 
 20-38 
 
 4 6 
 
 Interpolated. 
 
 17-14 
 
 18-39 
 
 19-56 
 
 20-68 
 
 21-73 
 
 5 
 
 1056 
 
 18-21 
 
 19-53 
 
 20-78 
 
 2196 
 
 23-08 
 
 5 6 
 
 Interpolated. 
 
 19-18 
 
 20-58 
 
 21-90 
 
 23-14 
 
 24-32 
 
 6 
 
 880 
 
 20-16 
 
 21-63 
 
 23-01 
 
 24-32 
 
 25-56 
 
 6 6 
 
 Interpolated. 
 
 21-07 
 
 22-61 
 
 24-05 
 
 25-42 
 
 26-72 
 
 7 
 
 754 
 
 21-98 
 
 23-59 
 
 25-09 
 
 26-52 
 
 27-87 
 
 7 6 
 
 Interpolated. 
 
 22-84 
 
 24-50 
 
 26-07 
 
 27-55 
 
 28-96 
 
 8 
 
 660 
 
 23-69 
 
 25-42 
 
 27-04 
 
 28-58 
 
 30-04 
 
 8 6 
 
 Interpolated. 
 
 24-50 
 
 26-29 
 
 27-97 
 
 29-55 
 
 31-06 
 
 9 
 
 587 
 
 25-31 
 
 27-54 
 
 28-89 
 
 30-53 
 
 32-09 
 
 9 6 
 
 Interpolated. 
 
 26-09 
 
 27-99 
 
 29-78 
 
 31-47 
 
 3308 
 
 10 
 
 528 
 
 26-87 
 
 28-83 
 
 30-67 
 
 32-41 
 
 34-06 
 
 10 6 
 
 Interpolated. 
 
 27-61 
 
 29-62 
 
 31-52 
 
 3331 
 
 35-01 
 
 11 
 
 480 
 
 28-35 
 
 30-42 
 
 32-37 
 
 34-20 
 
 35-95 
 
 11 6 
 
 Interpolated. 
 
 29-07 
 
 31-19 
 
 33-18 
 
 35-07 
 
 36'86 
 
 12 
 
 440 
 
 29-79 
 
 31-96 
 
 34-00 
 
 35-93 
 
 37-77 
 
438 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing 
 in Pipes, Drains, Streams, and Rivers. 
 
 For a full cylindrical pipe, divide the diameter by 4 to find the 
 hydraulic mean depth. 
 
 Diameters of pipes 16 inches to 2 feet. Falls per mile 13 feet to 
 
 SSSOfeet. 
 
 Falls per mile in feet 
 and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in. 
 
 4: inches. 
 
 4:3% inches. 
 
 5 inches. 
 
 5i-r inches. 
 
 6 inches. 
 
 13-2 
 
 400 
 
 33-74 
 
 35-89 
 
 37-93 
 
 3987 
 
 41-72 
 
 13-6 
 
 Interpolated. 
 
 34-37 
 
 36-56 
 
 38-64 
 
 40-61 
 
 42-50 
 
 14-1 
 
 375 
 
 35-00 
 
 3723 
 
 39-35 
 
 41-36 
 
 43-28 
 
 14-6 
 
 Interpolated. 
 
 35-70 
 
 37-98 
 
 40-14 
 
 42-19 
 
 44-15 
 
 15'1 
 
 350 
 
 36-40 
 
 38-73 
 
 40-92 
 
 43-02 
 
 45-01 
 
 15-6 
 
 Interpolated. 
 
 37-19 
 
 39-56 
 
 41-81 
 
 43-94 
 
 45-99 
 
 10-2 
 
 325 
 
 37-97 
 
 40-40 
 
 42-69 
 
 44-87 
 
 46-96 
 
 176 
 
 300 
 
 39-75 
 
 4229 
 
 44-69 
 
 46-97 
 
 49-16 
 
 19-2 
 
 275 
 
 41-78 
 
 44-45 
 
 46-97 
 
 49-38 
 
 51-67 
 
 21-1 
 
 250 
 
 4414 
 
 46-95 
 
 49-62 
 
 52-16 
 
 54-58 
 
 23-5 
 
 225 
 
 46'90 
 
 49-90 
 
 52-72 
 
 55-42 
 
 58-00 
 
 26-4 
 
 200 
 
 50-20 
 
 53-41 
 
 56-44 
 
 59-32 
 
 62-08 
 
 30-2 
 
 175 
 
 54-24 
 
 57-71 
 
 60-98 
 
 64-10 
 
 67-07 
 
 35-2 
 
 150 
 
 59-34 
 
 63-13 
 
 66-71 
 
 7012 
 
 73-37 
 
 37-7 
 
 140 
 
 61-78 
 
 65-72 
 
 69-45 
 
 73-00 
 
 76-39 
 
 42-2 
 
 125 
 
 66-02 
 
 70-23 
 
 74-22 
 
 78'Gl 
 
 81-64 
 
 48- 
 
 110 
 
 71-17 
 
 75-72 
 
 80-01 
 
 84-10 
 
 8800 
 
 52-8 
 
 100 
 
 75-29 
 
 80-09 
 
 84-64 
 
 sa-97 
 
 93-10 
 
 58-7 
 
 90 
 
 80 13 
 
 85-25 
 
 90-08 
 
 94*69 
 
 99-09 
 
 66- 
 
 80 
 
 85-93 
 
 91-42 
 
 96-61 
 
 101-54 
 
 106-26 
 
 75-4 
 
 70 
 
 93-04 
 
 98-98 
 
 104-60 
 
 109-95 
 
 115-05 
 
 88- 
 
 60 
 
 102-02 
 
 108-54 
 
 114-70 
 
 120-56 
 
 126-16 
 
 105-6 
 
 50 
 
 113-82 
 
 121-09 
 
 l-27'96 
 
 13450 
 
 140-74 
 
 117-3 
 
 45 
 
 121'27 
 
 129-01 
 
 136-34 
 
 143-30 
 
 14996 
 
 132- 
 
 40 
 
 130-20 
 
 138-51 
 
 146-38 
 
 153-86 
 
 161-00 
 
 150-8 
 
 35 
 
 141 14 
 
 150-16 
 
 158-68 
 
 166-79 
 
 17453 
 
 176- 
 
 30 
 
 154-95 
 
 164-84 
 
 174*20 
 
 183-10 
 
 191-61 
 
 211-2 
 
 25 
 
 173-05 
 
 184-10 
 
 194-56 
 
 204-50 
 
 214-UO 
 
 264- 
 
 20 
 
 198-12 
 
 210-77 
 
 222-73 
 
 234-11 
 
 244-98 
 
 352- 
 
 15 
 
 235-75 
 
 25080 
 
 265-04 
 
 278-58 
 
 291-52 
 
 528- 
 
 10 
 
 300-60 
 
 319-80 
 
 337-95 
 
 355-22 
 
 371-71 
 
 586-7 
 
 9 
 
 320-00 
 
 34043 
 
 359-76 
 
 378-14 
 
 395-70 
 
 660- 
 
 8 
 
 343-04 
 
 35965 
 
 385-67 
 
 40537 
 
 424-20 
 
 754-3 
 
 7 
 
 370-99 
 
 394-68 
 
 417-08 
 
 438-39 
 
 458-76 
 
 880- 
 
 6 
 
 405-79 
 
 431-70 
 
 456-21 
 
 47952 
 
 501-79 
 
 1056- 
 
 5 
 
 450-71 
 
 479-49 
 
 506-71 
 
 532-60 
 
 557-34 
 
 1320- 
 
 4 
 
 511-72 
 
 54439 
 
 575-30 
 
 604-69 
 
 63278 
 
 1760- 
 
 3 
 
 601-38 
 
 639-78 
 
 676-10 
 
 71064 
 
 743-65 
 
 2640- 
 
 2 
 
 753-29 
 
 80139 
 
 846 '89 
 
 890-16 
 
 931-50 
 
 5280- 
 
 1 
 
 1116-35 
 
 1187-62 
 
 1255-04 
 
 1319-17 
 
 1380 44 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 
 
 439 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing 
 in Pipes t Drains, Streams, and Rivers. 
 
 The hydraulic mean depth is found for all channels, by dividing 
 the wetted perimeter into the area. 
 
 Hydraulic mean depths 6 inches to 10 inches. Falls per mile 1 inch 
 
 to 12 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 6 inches 
 
 7 inches. 
 
 8 inches 
 
 9 inches 
 
 10 inches. 
 
 F. I. 
 
 1 
 
 63360 
 
 2-23 
 
 2-41 
 
 2-58 
 
 2-75 
 
 2-90 
 
 2 
 
 31680 
 
 3'47 
 
 3-76 
 
 4-03 
 
 4-28 
 
 4-52 
 
 3 
 
 21120 
 
 4-43 
 
 4-80 
 
 5-15 
 
 5-47 
 
 5'78 
 
 4 
 
 15840 
 
 5-26 
 
 5-69 
 
 6-10 
 
 6-49 
 
 6-85 
 
 5 
 
 12672 
 
 5-98 
 
 6-48 
 
 6-95 
 
 7-39 
 
 7-80 
 
 6 
 
 10560 
 
 665 
 
 7-20 
 
 7-72 
 
 8-20 
 
 8-66 
 
 7 
 
 9051 
 
 7-26 
 
 7-86 
 
 8-43 
 
 8-96 
 
 9-46 
 
 8 
 
 7920 
 
 7-83 
 
 8-48 
 
 909 
 
 9-67 
 
 10-21 
 
 9 
 
 7040 
 
 837 
 
 9-07 
 
 9-72 
 
 10-33 
 
 10-91 
 
 10 
 
 6336 
 
 8-88 
 
 9-63 
 
 10-32 
 
 10-97 
 
 11-58 
 
 11 
 
 5760 
 
 9-37 
 
 10-16 
 
 10-89 
 
 11-57 
 
 12-22 
 
 1 
 
 5280 
 
 9-84 
 
 1067 
 
 11-43 
 
 12-15 
 
 12-83 
 
 1 3 
 
 4224 
 
 11-16 
 
 1209 
 
 12'95 
 
 13-77 
 
 14-54 
 
 1 6 
 
 3520 
 
 12-35 
 
 13-38 
 
 14'34 
 
 15-25 
 
 16-10 
 
 1 9 
 
 3017 
 
 13-46 
 
 14-58 
 
 15'63 
 
 16-61 
 
 17-54 
 
 2 
 
 2640 
 
 14-50 
 
 1571 
 
 16'84 
 
 17-90 
 
 18-90 
 
 2 3 
 
 Interpolated. 
 
 15-45 
 
 16-75 
 
 18'24 
 
 19-08 
 
 20-15 
 
 2 6 
 
 2112 
 
 16-42 
 
 17-79 
 
 19'64 
 
 20-26 
 
 21-40 
 
 2 9 
 
 Interpolated. 
 
 17-29 
 
 18-74 
 
 20*37 
 
 2134 
 
 22-54 
 
 3 
 
 1760 
 
 18-17 
 
 19-69 
 
 21-10 
 
 22-42 
 
 23-fi8 
 
 3 3 
 
 Interpolated. 
 
 18-99 
 
 20-57 
 
 22-05 
 
 23-43 
 
 24-75 
 
 3 6 
 
 . 1508 
 
 19-80 
 
 21-46 
 
 23'00 
 
 24-44 
 
 2581 
 
 3 9 
 
 Interpolated. 
 
 20-56 
 
 22-28 
 
 23'88 
 
 2o'38 
 
 26-80 
 
 4 
 
 1320 
 
 21-33 
 
 2311 
 
 24-77 
 
 26-32 
 
 27-80 
 
 4 6 
 
 interpolated. 
 
 22-74 
 
 2464 
 
 26-41 
 
 28-07 
 
 29-64 
 
 5 
 
 1056 
 
 24-16 
 
 26-17 
 
 28-05 
 
 2981 
 
 31-48 
 
 5 6 
 
 interpolated. 
 
 25-45 
 
 27-58 
 
 29-56 
 
 31-42 
 
 33-17 
 
 6 
 
 880 
 
 26-75 
 
 28-98 
 
 31-06 
 
 33-02 
 
 34-86 
 
 6 6 
 
 nterpolated. 
 
 27-96 
 
 3029 
 
 32'47 
 
 34-51 
 
 36-44 
 
 7 
 
 754 
 
 29-17 
 
 31-60 
 
 3387 
 
 36-00 
 
 3802 
 
 7 6 
 
 Interpolated. 
 
 3030 
 
 32-83 
 
 35-19 
 
 37-40 
 
 39-50 
 
 8 
 
 660 
 
 31-43 
 
 34-06 
 
 3650 
 
 38-80 
 
 40-97 
 
 8 6 
 
 Interpolated. 
 
 32-51 
 
 35-22 
 
 37-75 
 
 4012 
 
 42-37 
 
 9 
 
 587 
 
 33-58 
 
 3639 
 
 38'99 
 
 41-45 
 
 43-77 
 
 9 6 
 
 nterpolated. 
 
 34-61 
 
 3750 
 
 40'20 
 
 4272 
 
 45-11 
 
 10 
 
 528 
 
 35-65 
 
 38-63 
 
 41'40 
 
 4400 
 
 46-46 
 
 10 6 
 
 'nterpolated. 
 
 36-63 
 
 39-69 
 
 42'54 
 
 45-22 
 
 47-75 
 
 11 
 
 480 
 
 37'62 
 
 40-76 
 
 43 '69 
 
 4644 
 
 49-03 
 
 11 6 
 
 nterpolated. 
 
 38-57 
 
 41-79 
 
 4479 
 
 47-61 
 
 50-27 
 
 12 
 
 440 
 
 39-52 
 
 42-82 
 
 45-90 
 
 48-78 
 
 51-51 
 
440 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing 
 in Pipes, Drains, Streams, and Rivers. 
 
 The hydraulic mean depth is found for all channels by dividing 
 the wetted perimeter into the area. 
 
 Hydraulic mean depths 11 inches to 91 inches. Falls per mile 1 inch 
 
 to 12 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," and 
 velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 11 inches. 
 
 12 inches. 
 
 15 inches. 
 
 18 inches. 
 
 21 inches. 
 
 F. I. 
 
 1 
 
 63360 
 
 3-05 
 
 3-19 
 
 3-57 
 
 3-92 
 
 425 
 
 2 
 
 31680 
 
 4-75 
 
 4-97 
 
 5-57 
 
 6-12 
 
 6-62 
 
 3 
 
 21120 
 
 6-07 
 
 6-35 
 
 7-12 
 
 7-82 
 
 8-46 
 
 4 
 
 15840 
 
 7-19 
 
 7-53 
 
 8-44 
 
 9-27 
 
 10-03 
 
 6 
 
 12672 
 
 8-19 
 
 8-57 
 
 9-61 
 
 10-55 
 
 11-42 
 
 6 
 
 10560 
 
 9 10 
 
 9-52 
 
 10-67 
 
 11-72 
 
 12-68 
 
 7 
 
 9051 
 
 9-94 
 
 10-39 
 
 11-66 
 
 12-80 
 
 13-85 
 
 8 
 
 7920 
 
 10-72 
 
 11-21 
 
 12-57 
 
 13-81 
 
 1494 
 
 9 
 
 7041 
 
 11-46 
 
 11-99 
 
 13'44 
 
 14-76 
 
 15-97 
 
 10 
 
 6336 
 
 12-16 
 
 12-72 
 
 14-27 
 
 15-66 
 
 16-95 
 
 11 
 
 5760 
 
 12-83 
 
 13-42 
 
 15-05 
 
 16-53 
 
 17*88 
 
 1 
 
 5280 
 
 13-48 
 
 14-09 
 
 15-81 
 
 17-36 
 
 18-78 
 
 1 3 
 
 4224 
 
 15-27 
 
 15-97 
 
 17-91 
 
 19-67 
 
 21-28 
 
 1 6 
 
 3520 
 
 1691 
 
 17-68 
 
 19-83 
 
 21-78 
 
 23-56 
 
 1 9 
 
 3017 
 
 18-23 
 
 19-27 
 
 21-62 
 
 23-73 
 
 25-68 
 
 2 
 
 2640 
 
 19-85 
 
 20-76 
 
 23-28 
 
 25-63 
 
 27-66 
 
 2 3 
 
 Interpolated. 
 
 21-16 
 
 22'13 
 
 24'82 
 
 27-29 
 
 2949 
 
 2 6 
 
 2112 
 
 22 -4-8 
 
 23-51 
 
 26'36 
 
 28-95 
 
 31-32 
 
 2 9 
 
 Interpolated. 
 
 23-68 
 
 24-76 
 
 2777 
 
 30-49 
 
 3299 
 
 3 
 
 1760 
 
 24-88 
 
 26-02 
 
 29*18 
 
 32-04 
 
 34-67 
 
 3 3 
 
 Interpolated. 
 
 25-99 
 
 27-18 
 
 30-47 
 
 33-48 
 
 36-22 
 
 3 6 
 
 1508 
 
 27-11 
 
 28-35 
 
 31-77 
 
 34-92 
 
 37-78 
 
 3 9 
 
 Interpolated. 
 
 28-15 
 
 29'45 
 
 33-01 
 
 36-26 
 
 39-23 
 
 4 
 
 1320 
 
 29-20 
 
 30-54 
 
 34-25 
 
 37-60 
 
 40-69 
 
 4 6 
 
 Interpolated. 
 
 31-13 
 
 32-56 
 
 36-52 
 
 40-10 
 
 43-39 
 
 5 
 
 1056 
 
 33-07 
 
 34-59 
 
 38-79 
 
 42-59 
 
 46*09 
 
 6 6 
 
 Interpolated. 
 
 34-85 
 
 36-44 
 
 40-87 
 
 44-88 
 
 48-56 
 
 6 
 
 880 
 
 36'62 
 
 38-30 
 
 42-95 
 
 47-16 
 
 51-03 
 
 6 6 
 
 Interpolated. 
 
 38-28 
 
 40-03 
 
 44'90 
 
 49-30 
 
 5334 
 
 7 
 
 754 
 
 3993 
 
 41-76 
 
 46"84 
 
 51-43 
 
 55-65 
 
 7 6 
 
 Interpolated. 
 
 41-48 
 
 43-39 
 
 46"66 
 
 53-43 
 
 57-81 
 
 8 
 
 660 
 
 43-04 
 
 45-01 
 
 50'48 
 
 55-42 
 
 59-97 
 
 8 6 
 
 Interpolated. 
 
 44-50 
 
 46-54 
 
 52'20 
 
 57-32 
 
 62-02 
 
 9 
 
 587 
 
 45-97 
 
 48-08 
 
 53 "92 
 
 59-21 
 
 64-06 
 
 9 6 
 
 Interpolated. 
 
 47-39 
 
 49-56 
 
 55'58 
 
 61-03 
 
 66-04 
 
 10 
 
 528 
 
 48-80 
 
 51-04 
 
 57'24 
 
 62-85 
 
 68-01 
 
 10 6 
 
 Interpolated. 
 
 50-15 
 
 52-45 
 
 58'83 
 
 64-59 
 
 69-89 
 
 11 
 
 480 
 
 51-51 
 
 53-87 
 
 60'41 
 
 66-33 
 
 71-78 
 
 11 6 
 
 Interpolated. 
 
 52'81 
 
 55-23 
 
 61'94 
 
 6&-01 
 
 73-59 
 
 12 
 
 440 
 
 54-11 
 
 56-59 
 
 63'47 
 
 69-68 
 
 75-40 
 
OEIFICES, WEIRS, PIPES, AND RIVERS. 
 
 441 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing 
 in Pipes, Drains, Streams, and Rivers. 
 
 The hydraulic mean depth is found for all channels by dividing 
 the wetted perimeter into the area. 
 
 Hydraulic mean depths 24 inches to 4 feet. Falls per mile 1 inch 
 
 to 12 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations, 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 24 inches. 
 
 30 inches. 
 
 36 inches. 
 
 42 inches. 
 
 48 inches. 
 
 F. I. 
 
 1 
 
 63360 
 
 4-54 
 
 5-09 
 
 5-59 
 
 6-04 
 
 6-47 
 
 2 
 
 31680 
 
 7-09 
 
 7-94 
 
 8-71 
 
 9-42 
 
 10-08 
 
 3 
 
 21120 
 
 9-06 
 
 10-15 
 
 11-14 
 
 12-04 
 
 12-89 
 
 4 
 
 15840 
 
 10-73 
 
 12-03 
 
 13-20 
 
 14-27 
 
 15-27 
 
 5 
 
 12672 
 
 1-2-22 
 
 13-69 
 
 15-03 
 
 16-25 
 
 17-39 
 
 6 
 
 10560 
 
 13-57 
 
 15-21 
 
 16-69 
 
 18-05 
 
 19-31 
 
 7 
 
 9051 
 
 14-83 
 
 16-61 
 
 1823 
 
 19-71 
 
 21-09 
 
 8 
 
 79-20 
 
 15-99 
 
 17-92 
 
 19-66 
 
 21-27 
 
 22-76 
 
 9 
 
 7041 
 
 17-10 
 
 19-16 
 
 21-02 
 
 22-73 
 
 24-33 
 
 10 
 
 6336 
 
 18-15 
 
 20-33 
 
 22-31 
 
 2413 
 
 25-82 
 
 11 
 
 5760 
 
 19-15 
 
 21-45 
 
 23-54 
 
 25-46 
 
 27-24 
 
 1 
 
 5280 
 
 20-11 
 
 22-53 
 
 24-72 
 
 2673 
 
 28-61 
 
 1 3 
 
 4224 
 
 22-78 
 
 25-53 
 
 28-01 
 
 3029 
 
 32-42 
 
 1 6 
 
 3520 
 
 25-23 
 
 28-27 
 
 31-02 
 
 33-54 
 
 35-90 
 
 1 9 
 
 3017 
 
 27-49 
 
 3081 
 
 3380 
 
 36-55 
 
 39-12 
 
 2 
 
 2640 
 
 29-62 
 
 33-18 
 
 36-41 
 
 3938 
 
 42-14 
 
 2 3 
 
 Interpolated. 
 
 31-57 
 
 35-38 
 
 38-82 
 
 41-98 
 
 44-92 
 
 2 6 
 
 2112 
 
 33-53 
 
 37-57 
 
 41-22 
 
 44-58 
 
 47-71 
 
 2 9 
 
 Interpolated. 
 
 3532 
 
 39-58 
 
 43-43 
 
 46-96 
 
 50-26 
 
 3 
 
 1760 
 
 37-11 
 
 41-58 
 
 45-63 
 
 49-34 
 
 5281 
 
 3 3 
 
 InterDolated. 
 
 38-78 
 
 43'45 
 
 47-68 
 
 51-56 
 
 55-18 
 
 3 6 
 
 1508 
 
 40-45 
 
 45-32 
 
 49-73 
 
 53*78 
 
 57-55 
 
 3 9 Interpolated. 
 4 13-20 
 
 42-00 
 4356 
 
 47-07 
 
 48-81 
 
 51-64 
 53-56 
 
 5585 
 57-92 
 
 5977 
 61-98 
 
 4 6 
 
 Interpolated. 
 
 46-45 
 
 5205 
 
 57-11 
 
 61 76 
 
 66-09 
 
 5 
 
 1056 
 
 4934 
 
 5528 
 
 60-66 
 
 65-60 
 
 70-20 
 
 5 6 
 
 Interpolated. 
 
 51-99 
 
 58-25 
 
 63-91 
 
 69-12 
 
 73-97 
 
 6 
 
 880 
 
 54-63 
 
 61-22 
 
 67-17 
 
 72-64 
 
 7774* 
 
 6 6 
 
 Interpolated. 
 
 57-11 
 
 6399 
 
 70-21 
 
 75-93* 
 
 81-25 
 
 7 
 
 754 
 
 59-58 
 
 66-76 
 
 73-25 
 
 7921 
 
 84-77 
 
 7 6 
 
 Interpolated. 
 
 6l-9 
 
 69-35 
 
 76-09* 
 
 87-29 
 
 88-06 
 
 8 
 
 660 
 
 64-21 
 
 71-94 
 
 78-94 
 
 85-37 
 
 91-35 
 
 8 6 
 
 Interpolated. 
 
 66-40 
 
 74-40 
 
 81-63 
 
 88-26 
 
 94-47 
 
 9 
 
 587 
 
 68-59 
 
 76-85* 
 
 84-32 
 
 91-19 
 
 97-59 
 
 9 6 
 
 Interpolated. 
 
 70-60 
 
 79-22 
 
 86-92 
 
 94'00 
 
 100-59 
 
 10 
 
 528 
 
 72-81 
 
 81-58 
 
 89-52 
 
 96-81 
 
 103-60 
 
 10 6 
 
 Interpolated. 
 
 74-83 
 
 83-84 
 
 91-99 
 
 99-49 
 
 106-47 
 
 11 
 
 480 
 
 76-84* 
 
 86-10 
 
 94-47 
 
 102-17 
 
 109-33 
 
 11 6 
 
 Interpolated. 
 
 78-78 
 
 88-28 
 
 96-86 
 
 104-75 
 
 112 10 
 
 12 
 
 440 
 
 80-72 
 
 90-45 
 
 99-25 
 
 107-33 
 
 114-86 
 
442 
 
 THE DISCHAEGE OP WATER FROM 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing 
 in Pipes, Drains, Streams, and Rivers. 
 
 The hydraulic mean depth is found for all channels hy dividing 
 the wetted perimeter into the area. 
 
 Hydraulic mean depths 4 feet 6 inches to 7 feet. Falls per milel inch 
 
 to 12 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or " mean radii," 
 and velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 54 inches. 
 
 50 inches. 
 
 66 inches. 
 
 72 inches. 
 
 34 inches. 
 
 F. I. 
 
 1 
 
 63360 
 
 6-86 
 
 7-24 
 
 7-60 
 
 7-94 
 
 8-58 
 
 2 
 
 31680 
 
 10-70 
 
 11-29 
 
 11-85 
 
 12-38 
 
 13-39 
 
 3 
 
 21120 
 
 13-68 
 
 14-62 
 
 15-14 
 
 15-83 
 
 17-11 
 
 4 
 
 15840 
 
 16-21 
 
 17-10 
 
 17-95 
 
 18-76 
 
 20-28 
 
 5 
 
 12672 
 
 18-46 
 
 19-47 
 
 20-43 
 
 21-35 
 
 23-13 
 
 6 
 
 10560 
 
 2050 
 
 21-63 
 
 22-70 
 
 23-72 
 
 25-64 
 
 7 
 
 9051 
 
 22-39 
 
 23-62 
 
 24-79 
 
 25-90 
 
 28-00 
 
 8 
 
 7920 
 
 24-16 
 
 25-48 
 
 26-74 
 
 27-95 
 
 30-21 
 
 9 
 
 7041 
 
 25-83 
 
 2724 
 
 28-59 
 
 29-88 
 
 32-30 
 
 10 
 
 6336 
 
 27-41 
 
 28-91 
 
 30-34 
 
 31-71 
 
 34-28 
 
 11 
 
 5760 
 
 28-92 
 
 30-51 
 
 32-01 
 
 3346 
 
 36-17 
 
 1 
 
 5280 
 
 30-37 
 
 32-03 
 
 33-62 
 
 35-13 
 
 37-98 
 
 1 3 
 
 4224 
 
 34-41 
 
 36-30 
 
 3-iO 
 
 39-81 
 
 43-04 
 
 1 6 
 
 3520 
 
 38-10 
 
 40-19 
 
 42-18 
 
 44-08 
 
 47-65 
 
 1 9 
 
 3017 
 
 41-5-2 
 
 43-80 
 
 45-97 
 
 48-04 
 
 51-93 
 
 2 
 
 2640 
 
 44-73 
 
 47-18 
 
 49-52 
 
 51-75 
 
 55-94 
 
 2 3 
 
 interpolated. 
 
 47-69 
 
 50-30 
 
 2-79 
 
 55-17 
 
 59-64 
 
 2 6 
 
 2112 
 
 50-65 
 
 53-42 
 
 56-07 
 
 58-59 
 
 63-34 
 
 2 9 
 
 interpolated. 
 
 53-35 
 
 56-28 
 
 59-06 
 
 61-72 
 
 66-72 
 
 3 
 
 1760 
 
 56-06 
 
 59-13 
 
 62-05 
 
 64-85 
 
 70-10 
 
 3 3 
 
 interpolated. 
 
 58-57 
 
 61-79 
 
 64-84 
 
 67-76 
 
 73-25 
 
 3 6 
 
 1508 
 
 61-09 
 
 64-44 
 
 67-63 
 
 7067 
 
 76-40* 
 
 3 9 
 
 [interpolated. 
 
 63-44 
 
 6692 
 
 70-^3 
 
 73-39 
 
 79-35 
 
 4 
 
 1320 
 
 6580 
 
 69-41 
 
 72-84 
 
 76-11* 
 
 8229 
 
 4 6 
 
 [interpolated. 
 
 70-16 
 
 7401 
 
 77-67* 
 
 81-16 
 
 87-74 
 
 5 
 
 1056 
 
 74-52 
 
 78-61* 
 
 82-50 
 
 86-21 
 
 93-20 
 
 5 6 
 
 Interpolated. 
 
 78-52* 
 
 82-83 
 
 86-92 
 
 90-84 
 
 98-20 
 
 6 
 
 880 
 
 82-5-2 
 
 87-05 
 
 91-35 
 
 95-46 
 
 103-20 
 
 6 6 
 
 Interpolated. 
 
 86-25 
 
 90-98 
 
 95-58 
 
 99-78 
 
 107-87 
 
 7 
 
 754 
 
 89-99 
 
 9492 
 
 99-62 
 
 104-10 
 
 11254 
 
 7 6 
 
 Interpolated. 
 
 93-48 
 
 98-61 
 
 103-48 
 
 108-14 
 
 116-91 
 
 8 
 
 660 
 
 96-98 
 
 102-30 
 
 107-35 112-19 
 
 121-28 
 
 8 6 
 
 Interpolated 
 
 100-29 
 
 105-79 
 
 111-02 
 
 116-01 
 
 125-42 
 
 9 
 
 587 
 
 103-59 
 
 109-27 
 
 114-68 
 
 119-84 
 
 129-56 
 
 9 6 
 
 Interpolated 
 
 10678 
 
 112-64 
 
 118-21 
 
 123-53 
 
 133-55 
 
 10 
 
 528 
 
 109-97 
 
 116-01 
 
 121-74 
 
 127-22 
 
 137-54 
 
 10 6 
 
 Interpolated 
 
 113-02 
 
 119-22 
 
 12511 
 
 130-74 
 
 141-34 
 
 11 
 
 480 
 
 116-06 
 
 12-2-43 
 
 128-48 
 
 134-27 
 
 145-15 
 
 11 6 
 
 Interpolated 
 
 119-00 
 
 125-52 
 
 131-73 
 
 137-66 
 
 148-82 
 
 12 
 
 440 
 
 121-93 
 
 128-61 
 
 134-97 
 
 141-05 
 
 152-49 
 
ORIFICES, WEIKS, PIPES, AND EIVEES. 
 
 443 
 
 TABLE VIII. For finding the Mean Velocities of Water flowing in 
 Pipes, Drains, Streams, and Rivers. 
 
 The hydraulic mean depth is found for all channels by dividing 
 the wetted perimeter into the area. 
 
 Hydraulic mean depths 8 feet to 12 feet. Falls per mile 1 inch to 
 
 19 feet. 
 
 Falls per mile in feet and 
 inches, and the 
 hydraulic inclinations. 
 
 " Hydraulic mean depths," or "mean radii," and 
 velocities in inches per second. 
 
 Falls. 
 
 Inclinations, 
 one in 
 
 96 inches. 
 
 108 
 
 inches. 
 
 120 
 
 inches. 
 
 132 
 
 inches. 
 
 14.4 
 
 inches. 
 
 F. I. 
 
 1 
 
 63360 
 
 9-18 
 
 9-75 
 
 10-28 
 
 10-79 
 
 11-27 
 
 2 
 
 31680 
 
 14-32 
 
 15-20 
 
 16-03 
 
 16-82 
 
 17-57 
 
 3 
 
 21120 
 
 18-30 
 
 19-43 
 
 20-49 
 
 21-50 
 
 22-46 
 
 4 
 
 15840 
 
 2169 
 
 23-02 
 
 24-28 
 
 25-47 
 
 26-62 
 
 5 
 
 12672 
 
 24-70 
 
 26-21 
 
 27-64 
 
 29-00 
 
 30-31 
 
 6 
 
 10560 
 
 27-43 
 
 29-11 
 
 30-70 
 
 32-21 
 
 33-66 
 
 7 
 
 9051 
 
 29-96 
 
 31-80 
 
 33-53 
 
 35-18 
 
 36-76 
 
 8 
 
 7920 
 
 32-32 
 
 34-30 
 
 36-18 
 
 37-96 
 
 39-66 
 
 9 
 
 7041 
 
 34-55 
 
 36-67 
 
 38-67 
 
 40-58 
 
 42-40 
 
 10 
 
 6336 
 
 36-67 
 
 38-92 
 
 41-04 
 
 43-07 
 
 45-00 
 
 11 
 
 5760 
 
 38-69 
 
 41-06 
 
 43-31 
 
 45-44 
 
 47-48 
 
 1 
 
 5280 
 
 40-63 
 
 43-12 
 
 45-48 
 
 47-72 
 
 4986 
 
 1 3 
 
 4224 
 
 46-04 
 
 48-87 
 
 51-54 
 
 54-07 
 
 56-50 
 
 1 6 
 
 3520 
 
 50-98 
 
 54-11 
 
 57-06 
 
 59-87 
 
 62-56 
 
 1 9 
 
 3017 
 
 55-60 
 
 58-96 
 
 62-18 
 
 65-25 
 
 68-17 
 
 2 
 
 2640 
 
 59-85 
 
 63-52 
 
 66-98 
 
 70-28 
 
 73'44* 
 
 2 3 
 
 Interpolated. 
 
 63-80 
 
 67-72 
 
 71-41 
 
 74-93* 
 
 78-29 
 
 2 6 
 
 2112 
 
 67-76 
 
 71-91. 
 
 75-84* 
 
 79-58 
 
 83-15 
 
 2 9 
 
 Interpolated. 
 
 71-38 
 
 75-75* 
 
 79-89 
 
 83-83 
 
 87-59 
 
 3 
 
 1760 
 
 75-00* 
 
 7959 
 
 83-94 
 
 88-08 
 
 92-03 
 
 3 3 
 
 Interpolated. 
 
 7837 
 
 83-17 
 
 87-71 
 
 92-03 
 
 06-16 
 
 3 6 
 
 1508 
 
 81-74 
 
 86-75 
 
 91-48 
 
 95-99 
 
 100-30 
 
 3 9 
 
 Interpolated. 
 
 84-88 
 
 90-09 
 
 9501 
 
 9969 
 
 104-16 
 
 4 
 
 1320 
 
 88-03 
 
 93-43 
 
 98-53 
 
 103-38 
 
 108-02 
 
 4 6 
 
 Interpolated. 
 
 93-87 
 
 99-62 
 
 105-06 
 
 110-24 
 
 115-18 
 
 5 
 
 1056 
 
 99-70 
 
 105-82 
 
 111-59 
 
 117-09 
 
 122-34 
 
 o 6 
 
 Interpolated. 
 
 105-06 
 
 111-49 
 
 11758 
 
 123-38 
 
 128-91 
 
 6 
 
 880 
 
 110-41 
 
 117-17 
 
 123-57 
 
 129-66 
 
 135-48 
 
 6 6 
 
 Interpolated. 
 
 115-40 
 
 122-47 
 
 129 16 
 
 135-53 
 
 141-61 
 
 7 . 
 
 754 
 
 12D -40 
 
 127-76 
 
 134-75 
 
 141-39 
 
 147-73 
 
 7 6 
 
 Interpolated. 
 
 125-07 
 
 132-74 
 
 139-99 
 
 146-88 
 
 153-47 
 
 8 
 
 660 
 
 129-75 
 
 137-70 
 
 145-22 
 
 152-38 
 
 15921 
 
 8 6 
 
 Interpolated. 
 
 134-18 
 
 142-40 
 
 15U-18 
 
 157-57 
 
 164-64 
 
 9 
 
 587 
 
 138-00 
 
 147-10 
 
 155-13 
 
 162-77 
 
 170-07 
 
 9 6 
 
 Interpolated. 
 
 14287 
 
 151-63 
 
 159-91 
 
 167-78 
 
 175-31 
 
 10 
 
 528 
 
 147-14 
 
 156-16 
 
 164-68 
 
 172-80 
 
 180-55 
 
 10 6 
 
 Interpolated. 
 
 151-21 
 
 160-48 
 
 169-24 
 
 177-58 
 
 18555 
 
 11 
 
 480 
 
 155-29 
 
 164-80 
 
 173-80 
 
 182-36 
 
 190-54 
 
 11 6 
 
 Interpolated. 
 
 15921 
 
 168-97 
 
 178-19 
 
 186-97 
 
 195-38 
 
 12 
 
 440 
 
 163-13 
 
 173-13 
 
 182-59 
 
 191-58 
 
 200-17 
 
444 
 
 THE DISCHARGE OF WATER FEOM 
 
 TABLE IX. For finding the Discharge in Cubic Feet per Minute, 
 when the Diameter of a Pipe, or Orifice, and the Velocity of 
 Discharge are known ; and vice versa. 
 
 sss 
 
 $ .,3 
 
 e 
 
 5.s 
 
 Discharge in cubic feet per minute, for different velocities. 
 
 Velocity of 
 100 inches 
 per second. 
 
 Velocity of 
 200 inches 
 per second. 
 
 Velocity of 
 300 inches 
 per second. 
 
 Velocity of 
 400 inches 
 per second. 
 
 Velocity of 
 500 inches 
 per second. 
 
 i 
 
 170442 
 
 3409 
 
 5113 
 
 6818 
 
 8522 
 
 1 
 
 68177 
 
 1-3635 
 
 2-0453 
 
 2-7271 
 
 3-4089 
 
 1 
 
 1-53398 
 
 3-0679 
 
 4-6019 
 
 6-1359 
 
 7-6699 
 
 
 2-727077 
 
 5-4541 
 
 8-1812 
 
 10-9083 
 
 13-6354 
 
 it 
 
 4-26106 
 
 8-5221 
 
 12-7832 
 
 17-0442 
 
 21-3053 
 
 1} 
 
 6-13593 
 
 12-2718 
 
 18-4080 
 
 24-5437 
 
 30-6797 
 
 if 
 
 8-35167 
 
 16-7033 
 
 25-0550 
 
 33-4067 
 
 41-7584 
 
 2 
 
 10-90831 
 
 21-1817 
 
 32-7249 
 
 43-6332 
 
 54-5415 
 
 2J 
 
 13-80583 
 
 27-6117 
 
 41-4175 
 
 55-2233 
 
 69-0291 
 
 &t 
 
 17-04423 
 
 34-0885 
 
 51-1327 
 
 68-1769 
 
 85-2212 
 
 2f 
 
 2062352 
 
 41 2470 
 
 61-8706 
 
 824941 
 
 103-1176 
 
 3 
 
 24-54369 
 
 49-0874 
 
 73-6311 
 
 98-1748 
 
 121-7185 
 
 3i 
 
 28-80475 
 
 57-6095 
 
 864143 
 
 115-2190 
 
 144-0238 
 
 3 
 
 33-40669 
 
 66-8134 
 
 100-2201 
 
 133-6268 
 
 167-0335 
 
 3| 
 
 38-34952 
 
 76-6990 
 
 115-0486 
 
 153 3981 
 
 191-7476 
 
 4 
 
 43-63323 
 
 87-2665 
 
 130-8997 
 
 174-5329 
 
 218-1662 
 
 4J 
 
 49-25783 
 
 98-5157 
 
 147 7735 
 
 197-0313 
 
 2462892 
 
 if 
 
 5522331 
 
 110-4466 
 
 165-6699 
 
 2208932 
 
 276-1166 
 
 4| 
 
 61*52968 
 
 123-0594 
 
 184-5890 
 
 246-1187 
 
 307-6484 
 
 5 
 
 68-17692 
 
 136-3539 
 
 204-53U8 
 
 2727077 
 
 340-8846 
 
 6* 
 
 75-16506 
 
 150-3301 
 
 225-4952 
 
 300 6603 
 
 375-8253 
 
 6* 
 
 82-49408 
 
 1649882 
 
 247-4822 
 
 329-9763 
 
 412-4704 
 
 6f 
 
 90-16399 
 
 180-3280 
 
 270-49-20 
 
 360-6560 
 
 450-8200 
 
 6 
 
 98-17478 
 
 196-3495 
 
 294-5243 
 
 392-6991 
 
 490-8739 
 
 6* 
 
 106-5-2645 
 
 213-0529 
 
 319-5794 
 
 426-1058 
 
 532-6323 
 
 6J 
 
 115-2190 
 
 230-4380 
 
 345-6570 
 
 460-8760 
 
 576-0950 
 
 6| 
 
 124-25245 
 
 248-5049 
 
 3727574 
 
 497-0098 
 
 621-2623 
 
 
 133-6268 
 
 267-2536 
 
 400-8804 
 
 534-5072 
 
 668-1340 
 
 7* 
 
 143-34199 
 
 286-6840 
 
 430-0260 
 
 573 3680 
 
 716-7100 
 
 7* 
 
 153-39809 
 
 306-7962 
 
 460-1943 
 
 613-5924 
 
 766-9905 
 
 1 
 
 163-79507 
 
 327-5901 
 
 491-3852 
 
 655-1803 
 
 818-9753 
 
 8 
 
 174-53293 
 
 349-0659 
 
 523-5988 
 
 698-1317 
 
 872-6647 
 
 & 
 
 197-03132 
 
 394-0626 
 
 591-0940 
 
 788-1253 
 
 985-1566 
 
 9 
 
 22089325 
 
 441-7865 
 
 6626798 
 
 883-5730 
 
 1104-4663 
 
 9* 
 
 246-11871 
 
 492-2374 
 
 738-3561 
 
 984-4748 
 
 1230-5936 
 
 10 
 
 272-70771 
 
 545-4154 
 
 818-1231 
 
 1090 8308 
 
 1363-5386 
 
 IQi 
 
 300-66025 
 
 601-3205 
 
 901-9808 
 
 1202-6410 
 
 1503-3013 
 
 11 
 
 329-97633 
 
 6599527 
 
 989-9290 
 
 13199053 
 
 1649-8817 
 
 MJ 
 
 360-65595 
 
 721-3119 
 
 1081-9679 
 
 1442-6238 
 
 1803-2798 
 
 12 
 
 392-6991 
 
 785-3982 
 
 11780973 
 
 1570-7964 
 
 1963-4955 
 
OEIFICES, WEIES, PIPES, AND EIVEES. 
 
 445 
 
 TABLE IX. For finding the Discharge in Cubic Feet, per Minute, 
 when the Diameter of a Pipe, or Orifice, and the Velocity of dis- 
 charge are known ; and vice versa. 
 
 Discharge in cubic feet per minute, for different velocities. 
 
 5s 
 1-S1 
 sXs 
 
 Velocity of 
 600 inches 
 per second. 
 
 Velocity of 
 700 inches 
 per second. 
 
 Velocity of 
 800 inches 
 per second. 
 
 Velocity of 
 900 inches 
 per second. 
 
 Velocity of 
 1000 inches 
 per second. 
 
 1-0227 
 
 1-1931 
 
 1-3635 
 
 1-5340 
 
 1-7044 
 
 i 
 
 4-0906 
 
 4-7724 
 
 5-4542 
 
 6-1359 
 
 6-8177 
 
 i 
 
 9-2039 
 
 10-7379 
 
 12-2718 
 
 13-8058 
 
 15-3398 
 
 1 
 
 16-3625 
 
 19-0895 
 
 21-8166 
 
 24-5437 
 
 27-2708 
 
 
 25-5664 
 
 29-8274 
 
 34-0885 
 
 38-3495 
 
 42-6106 
 
 H 
 
 36-8155 
 
 429515 
 
 49-0874 
 
 55-2234 
 
 61-3593 
 
 *! 
 
 50-1100 
 
 58-4617 
 
 66-8134 
 
 75-1650 
 
 83-5167 
 
 if 
 
 65-4499 
 
 76-3582 
 
 87-2665 
 
 98 1748 
 
 109-0831 
 
 2 
 
 828350 
 
 96-6408 
 
 110-4466 
 
 124-2525 
 
 138-0583 
 
 *! 
 
 102-2654 
 
 119-3096 
 
 136-3538 
 
 153-3981 
 
 170-4423 
 
 2 
 
 1237411 
 
 144-3646 
 
 164-9882 
 
 185-6117 
 
 206-2352 
 
 2f 
 
 147-2621 
 
 171-8059 
 
 196-3496 
 
 220-8933 
 
 245*4369 
 
 3 
 
 172-8285 
 
 201-6333 
 
 230-4380 
 
 259-2428 
 
 288-0475 
 
 31 
 
 200-4401 
 
 233-8468 
 
 267-2535 
 
 300-6602 
 
 334-0669 
 
 3$ 
 
 230-0971 
 
 268-4467 
 
 306-7962 
 
 345-1457 
 
 383-4952 
 
 8| 
 
 261-7994 
 
 305-4326 
 
 349-0659 
 
 392-6991 
 
 436-3323 
 
 4 
 
 295-5470 
 
 344-8048 
 
 394-0626 
 
 443-3205 
 
 492-5783 
 
 J 
 
 331-3399 
 
 386-5632 
 
 441-7865 
 
 497-0098 
 
 552-2331 
 
 4 
 
 369-1781 
 
 430-7077 
 
 492 2374 
 
 553-7671 
 
 615-2968 
 
 4f 
 
 409-0615 
 
 477-2384 
 
 545-4154 
 
 6135923 
 
 681-7692 
 
 5 
 
 450-9904 
 
 526-1554 
 
 601-3205 
 
 676-4855 
 
 751-6506 
 
 fij 
 
 494-9645 
 
 577-4586 
 
 659-9526 
 
 742-4467 
 
 824-9408 
 
 ty 
 
 540-9839 
 
 631-1479 
 
 721-3119 
 
 811-4759 
 
 901-6399 
 
 fif 
 
 589-0486 
 
 687-2235 
 
 7853982 
 
 883-5730 
 
 981*7478 
 
 6 
 
 639-1587 
 
 745-6852 
 
 852-2116 
 
 958-7381 
 
 1065*2645 
 
 *i 
 
 691-3141 
 
 806-5330 
 
 921-7520 
 
 10369710 
 
 1152-1900 
 
 6* 
 
 745-5147 
 
 869-7672 
 
 994-0196 
 
 11182721 
 
 1242-5245 
 
 6f 
 
 801-7608 
 
 935-3876 
 
 1069-0144 
 
 1202-6412 
 
 1336-2680 
 
 7 
 
 860-0519 
 
 1003-3939 
 
 1146-7359 
 
 1290-0779 
 
 1433-4199 
 
 7i 
 
 920-3885 
 
 1073-7866 
 
 1227-1847 
 
 1380-5828 
 
 1533-9809 
 
 7 
 
 982-7704 
 
 1146-5655 
 
 1310 3605 
 
 1474 1556 
 
 1637-9507 
 
 7f 
 
 1047-1976 
 
 1221-7305 
 
 1396-2634 
 
 1570-7964 
 
 1745-3293 
 
 8 
 
 1182-1879 
 
 1379-2192 
 
 1576-2506 
 
 1773-2819 
 
 1970-3132 
 
 8 
 
 1325-3595 
 
 1546-2528 
 
 1767-1460 
 
 1988-0393 
 
 2208-9325 
 
 9 
 
 1476-7123 
 
 1722-8310 
 
 1968-9497 
 
 2215-0684 
 
 2461-1871 
 
 $$ 
 
 1636-2463 
 
 1908-9540 
 
 2181-6617 
 
 2454-3694 
 
 2727-0771 
 
 10 
 
 1803-9615 
 
 2104-6218 
 
 2405-2820 
 
 2705-9423 
 
 3006 6025 
 
 10,} 
 
 1979-8580 
 
 2309-8343 
 
 2639-8106 
 
 2969-7870 
 
 3299-7633 
 
 11 
 
 2163-9357 
 
 2524-5917 
 
 2885 2476 
 
 3245-9936 
 
 3606-5595 
 
 Hi 
 
 2356-1946 
 
 2748-8937 
 
 3141-5928 
 
 3534-2919 
 
 3926-9910 
 
 12 
 
446 
 
 THE DISCHARGE OF "WATER FROM 
 
 TABLE X. For finding the depths of Weirs of different lengths, 
 the quantity discharged over each being supposed constant. 
 See pages 270 and 371. 
 
 Ratios of 
 lengths. 
 
 Coeffi. 
 cients. 
 
 Ratios of 
 lengths. 
 
 Coeffi- 
 cients. 
 
 Ratios of 
 lengths. 
 
 Coeffi- 
 cients. 
 
 Ratios of 
 lengths. 
 
 Coeffi- 
 cients. 
 
 01 
 
 0464 
 
 405 
 
 5474 
 
 605 
 
 7153 
 
 805 
 
 8654 
 
 02 
 
 0737 
 
 410 
 
 5519 
 
 610 
 
 7193 
 
 810 
 
 8689 
 
 03 
 
 0965 
 
 415 
 
 5564 
 
 615 
 
 7232 
 
 815 
 
 8725 
 
 04 
 
 1170 
 
 420 
 
 5608 
 
 620 
 
 7271 
 
 820 
 
 8761 
 
 05 
 
 1357 
 
 425 
 
 5653 
 
 625 
 
 7310 
 
 825 
 
 8796 
 
 06 
 
 1533 
 
 430 
 
 5697 
 
 630 
 
 7349 
 
 830 
 
 .8832 
 
 07 
 
 1699 
 
 435 
 
 5741 
 
 635 
 
 7388 
 
 835 
 
 8867 
 
 08 
 
 1857 
 
 440 
 
 5785 
 
 640 
 
 7427 
 
 840 
 
 8903 
 
 09 
 
 2008 
 
 445 
 
 5829 
 
 645 
 
 7465 
 
 845 
 
 8938 
 
 10 
 
 2154 
 
 450 
 
 5872 
 
 650 
 
 7504 
 
 850 
 
 8973 
 
 11 
 
 2296 
 
 455 
 
 5916 
 
 655 
 
 7542 
 
 855 
 
 9008 
 
 12 
 
 2433 
 
 460 
 
 5959 
 
 660 
 
 7580 
 
 860 
 
 9043 
 
 13 
 
 2566 
 
 465 
 
 6002 
 
 665 
 
 7619 
 
 865 
 
 9078 
 
 14 
 
 2696 
 
 470 
 
 6045 
 
 670 
 
 7657 
 
 870 
 
 9113 
 
 15 
 
 2823 
 
 475 
 
 6088 
 
 675 
 
 7695 
 
 875 
 
 9148 
 
 16 
 
 2947 
 
 480 
 
 6130 
 
 680 
 
 7733 
 
 880 
 
 9183 
 
 17 
 
 3069 
 
 485 
 
 6173 
 
 685 
 
 7771 
 
 885 
 
 9218 
 
 18 
 
 3188 
 
 490 
 
 6215 
 
 690 
 
 7808 
 
 890 
 
 9253 
 
 19 
 
 3305 
 
 495 
 
 6258 
 
 795 
 
 7846 
 
 895 
 
 9287 
 
 20 
 
 3420 
 
 500 
 
 6300 
 
 700 
 
 7884 
 
 900 
 
 9322 
 
 21 
 
 3533 
 
 505 
 
 6342 
 
 705 
 
 7921 
 
 905 
 
 9356 
 
 22 
 
 3644 
 
 510 
 
 6383 
 
 710 
 
 7959 
 
 910 
 
 9391 
 
 23 
 
 3754 
 
 5J5 
 
 6425 
 
 715 
 
 7996 
 
 915 
 
 9425 
 
 24 
 
 3862 
 
 520 
 
 6466 
 
 720 
 
 8033 
 
 920 
 
 9459 
 
 25 
 
 3969 
 
 525 
 
 6508 
 
 725 
 
 8070 
 
 925 
 
 9494 
 
 26 
 
 4074 
 
 530 
 
 6549 
 
 730 
 
 8107 
 
 930 
 
 9528 
 
 27 
 
 4177 
 
 535 
 
 6590 
 
 735 
 
 8144 
 
 935 
 
 9562 
 
 2.8 
 
 4280 
 
 540 
 
 6631 
 
 740 
 
 8181 
 
 940 
 
 9596 
 
 29 
 
 4381 
 
 545 
 
 -6672 
 
 745 
 
 8218 
 
 945 
 
 9630 
 
 30 
 
 4481 
 
 550 
 
 6713 
 
 750 
 
 8255 
 
 950 
 
 9664 
 
 31 
 
 4580 
 
 555 
 
 6754 
 
 755 
 
 8291 
 
 955 
 
 8698 
 
 32 
 
 4678 
 
 500 
 
 6794 
 
 760 
 
 8328 
 
 960 
 
 9732 
 
 33 
 
 4775 
 
 565 
 
 6834 
 
 765 
 
 8365 
 
 965 
 
 9762 
 
 34 
 
 4871 
 
 570 
 
 6875 
 
 770 
 
 8401 
 
 970 
 
 9799 
 
 35 
 
 4966 
 
 575 
 
 6915 
 
 775 
 
 8437 
 
 975 
 
 9833 
 
 36 
 
 5061 
 
 580 
 
 6955 
 
 780 
 
 8474 
 
 980 
 
 9866 
 
 37 
 
 5154 
 
 585 
 
 6995 
 
 785 
 
 8510 
 
 985 
 
 9900 
 
 38 
 
 5246 
 
 590 
 
 7035 
 
 790 
 
 9546 
 
 990 
 
 9933 
 
 39 
 
 5338 
 
 595 
 
 7074 
 
 795 
 
 8582 
 
 9D5 
 
 9967 
 
 40 
 
 5429 
 
 600 
 
 7114 
 
 SCO 
 
 8618 
 
 1-000 
 
 1-0000 
 
ORIFICES, TVEIRS, PIPES, AND RIVERS. 
 
 447 
 
 TABLE XL Mean relative Dimensions of equally Discharging Trape- 
 zoidal Channels, with Side Slopes varying from to 1, up to % to 1. 
 
 Half sum of the top and bottom is the mean width. The ratio of 
 the slope, multiplied by the depth, subtracted from the mean 
 width, will give the bottom ; and if added, will give the top. 
 
 TABLE XII. gives the discharge in cubic feet per minute from the 
 primary channel, 70 wide, and the corresponding depths taken 
 in feet. For lesser or greater channels and discharges, see 
 Eules, pp. 237, 229, 232, 249, and 252. 
 
 The mean widths are given in the top horizontal line, and the corresponding depths in 
 the other horizontal lines. They may be taken in inches, feet, yards, fathoms, or 
 any other measures whatever. 
 
 70 
 
 60 
 
 50 
 
 40 
 
 35 
 
 30 
 
 25 
 
 20 
 
 15 
 
 10 
 
 125 
 
 13 
 
 15 
 
 17 
 
 20 
 
 23 
 
 26 
 
 29 
 
 35 
 
 -48 
 
 25 
 
 27 
 
 30 
 
 35 
 
 40 
 
 45 
 
 52 
 
 58 
 
 71 
 
 98 
 
 375 
 
 41 
 
 46 
 
 54 
 
 60 
 
 67 
 
 76 
 
 88 
 
 1-09 
 
 1-51 
 
 5 
 
 55 
 
 62 
 
 73 
 
 80 
 
 89 
 
 1-02 
 
 1-19 
 
 1-48 
 
 2-04 
 
 625 
 
 68 
 
 78 
 
 91 
 
 1-00 
 
 1-12 
 
 1-29 
 
 150 
 
 1-88 
 
 2-62 
 
 75 
 
 82 
 
 94 
 
 1-10 
 
 1-20 
 
 1-35 
 
 1-56 
 
 1-82 
 
 2-28 
 
 3-22 
 
 875 
 
 96 
 
 MO 
 
 1-29 
 
 1-41 
 
 1-58 
 
 1-83 
 
 2-14 
 
 2-69 
 
 3-86 
 
 1- 
 
 1-10 
 
 1-26 
 
 1-48 
 
 1-62 
 
 1-81 
 
 2-10 
 
 2-46 
 
 3-11 
 
 4-50 
 
 1-125 
 
 1-24 
 
 1-42 
 
 1-67 
 
 1-83 
 
 2-04 
 
 2-37 
 
 2-79 
 
 3-54 
 
 5-19* 
 
 1-25 
 
 1-39 
 
 1'58 
 
 1-86 
 
 2-04 
 
 2-28 
 
 2-65 
 
 3-12 
 
 3-98 
 
 5'89 
 
 1-375 
 
 1-53 
 
 1-74 
 
 2-05 
 
 2-25 
 
 2-51 
 
 2-92 
 
 3-46 
 
 4-43 
 
 6-60 
 
 1-5 
 
 1'67 
 
 1-90 
 
 2-24 
 
 2-46 
 
 2-75 
 
 3-20 
 
 3-80 
 
 4-88 
 
 7-31 
 
 1-625 
 
 1-81 
 
 2-06 
 
 2-43 
 
 2-67 
 
 2-99 
 
 3-47 
 
 4-15 
 
 5-34 
 
 8-08 
 
 Kfi 
 
 1-95 
 
 2-22 
 
 2-62 
 
 2-88 
 
 3-23 
 
 3-75 
 
 4-50 
 
 5-80 
 
 8-86 
 
 1-875 
 
 2-09 
 
 2-38 
 
 2-81 
 
 3-09 
 
 3-47 
 
 4-03 
 
 4-86 
 
 6-29 
 
 9-68 
 
 2- 
 
 2-23 
 
 2-54 
 
 3-00 
 
 3-31 
 
 3-72 
 
 4-32 
 
 5-22 
 
 6-78 
 
 10-50 
 
 2-125 
 
 2-37 
 
 2-70 
 
 3-19 
 
 3-52 
 
 3-96 
 
 4-61 
 
 5-58 
 
 7-29 
 
 11-37 
 
 2-25 
 
 2-51 
 
 2-86 
 
 3-38 
 
 3-73 
 
 4-21 
 
 4-91 
 
 5-95 
 
 7-81* 
 
 12-25 
 
 2-375 
 
 2-65 
 
 3-02 
 
 357 
 
 3-94 
 
 4-45 
 
 5-20 
 
 6-31 
 
 8-32 
 
 13-12 
 
 2-5 
 
 2-79 
 
 3-18 
 
 3-76 
 
 4-16 
 
 4-70 
 
 5-50 
 
 6-68 
 
 8-84 
 
 14-00 
 
 2-625 
 
 2-93 
 
 3-34 
 
 3-95 
 
 4-38 
 
 4-95 
 
 5-79 
 
 7-06 
 
 9-38 
 
 14-92 
 
 2-75 
 
 3-07 
 
 3-51 
 
 4-15 
 
 4-60 
 
 5-21 
 
 6-09 
 
 7-45 
 
 9-93 
 
 15-84 
 
 2-875 
 
 3-21 
 
 3-67 
 
 4-34 
 
 4-82 
 
 5-46 
 
 6'39 
 
 7-83 
 
 10-48 
 
 16-76 
 
 3- 
 
 3-35 
 
 3-84 
 
 4-54 
 
 5-04 
 
 5-72 
 
 6-69 
 
 8-22 
 
 11-03 
 
 17-68 
 
 3-125 
 
 3-49 
 
 4-00 
 
 4-73 
 
 5'26 
 
 5-97 
 
 7-00 
 
 8-62 
 
 11-60 
 
 18-68 
 
 3-25 
 
 3-63 
 
 4-17 
 
 4-93 
 
 5-49 
 
 6-23 
 
 7-31 
 
 9-02 
 
 12-17 
 
 19-68 
 
 3-375 
 
 3-77 
 
 4-33 
 
 5-13 
 
 5-72 
 
 6-49 
 
 7-62 
 
 9-42 
 
 12-74 
 
 20-68 
 
 3-5 
 
 3<91 
 
 4-50 
 
 5.33 
 
 5-95 
 
 6-75 
 
 7-93 
 
 9-82 
 
 1332 
 
 21-68 
 
 3-625 
 
 4-05 
 
 4-66 
 
 5-53 
 
 6-17 
 
 7-01 
 
 S-25 
 
 1023* 
 
 13-92 
 
 2276 
 
 3-75 
 
 4-19 
 
 4-82 
 
 5-73 
 
 6-40 
 
 7-28 
 
 8-57 
 
 1065 
 
 14-53 
 
 23-84 
 
 3-875 
 
 4-33 
 
 4-98 
 
 5-93 
 
 6-62 
 
 7-54 
 
 8-89 
 
 ii-or. 
 
 15-14 
 
 24-92 
 
 4- 
 
 4-48 
 
 5-14 
 
 6-13 
 
 6-85 
 
 7-81 
 
 9-21 
 
 11-48 
 
 15-75 
 
 26-00 
 
 4-25 
 
 4-76 
 
 5-46 
 
 6-54 
 
 7-30 
 
 8-35 
 
 9-85 
 
 12-33 
 
 16-98 
 
 28-18 
 
 4-5 
 
 5-05 
 
 5-79 
 
 6-95 
 
 7-75 
 
 8-90 
 
 10-50 
 
 1319 
 
 18-22 
 
 30-36 
 
 4-75 
 
 533 
 
 6-12 
 
 7-35 
 
 8-20 
 
 9-45 
 
 11-14 
 
 14-07 
 
 19-50 
 
 32-68 
 
 5- 
 
 5-62 
 
 6-45 
 
 7*75 
 
 8-66 
 
 10-00 
 
 11-79 
 
 14-96 
 
 20-80 
 
 35-00 
 
 5-25 
 
 5-90 
 
 6'78 
 
 8-16 
 
 9-14 
 
 10-55 
 
 12-51* 
 
 15-86 
 
 22-13 
 
 37-40 
 
 5-5 
 
 6-18 
 
 7-12 
 
 8-57 
 
 9-62 
 
 11-10 
 
 13-24 
 
 16-77 
 
 23-47 
 
 39-81 
 
 5-75 
 
 6-46 
 
 7-46 
 
 8-98 
 
 10-11 
 
 11-66 
 
 13-94 
 
 17-71 
 
 24-86 
 
 42-33 
 
 6- 
 
 6 f 75 
 
 7-80 
 
 9-40 
 
 10-60 
 
 12-22 
 
 14-65 
 
 18-65 
 
 26-25 
 
 44-86 
 
448 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE XII. Discharges from the Primary Channel in the first 
 column of Table XI. 
 
 If the dimensions of the primary channel be in inches, divide 
 the discharges in this table by 500 ; if in yards, multiply by 
 15'6 ; if in quarters, multiply by 32 ; and if in fathoms, by 
 88-2, &c. : see pp. 233, 234. The final figures in the dis- 
 charges may be rejected when they do not exceed one-half 
 per cent., or 0-5 in 100. See pages 226 to 234. 
 
 Depths of a 
 channel 
 whose mean 
 width is 
 70: in feet. 
 
 Falls, inclinations, and discharges in cubic feet per minute. 
 Interpolate for intermediate falls ; divide greater falls by 4, 
 and double the corresponding discharges. 
 
 linch 
 per mile, 
 
 2 inches 
 per mile, 
 1 in 31680. 
 
 3 inches 
 per mile, 
 1 in 21120. 
 
 6 inches 
 per mile, 
 1 in 10560. 
 
 9 inches 
 per mile, 
 1 in 7040. 
 
 12 inches 
 per mile, 
 1 in 5280. 
 
 15 inches 
 per mile. 
 1 in 4224. 
 
 125 
 
 47 
 
 72 
 
 93 
 
 139 
 
 175 
 
 205 
 
 233 
 
 25 
 
 136 
 
 210 
 
 268 
 
 403 
 
 506 
 
 596 
 
 675 
 
 375 
 
 249 
 
 389 
 
 498 
 
 746 
 
 940 
 
 1105 
 
 1252 
 
 50 
 
 387 
 
 603 
 
 770 
 
 1155 
 
 1454 
 
 1709 
 
 1935 
 
 625 
 
 541 
 
 849 
 
 1078 
 
 1617 
 
 2036 
 
 2395 
 
 2714 
 
 75 
 
 714 
 
 1112 
 
 1420 
 
 2128 
 
 2681 
 
 3153 
 
 3573 
 
 875 
 
 900 
 
 1401 
 
 1791 
 
 2685 
 
 3382 
 
 3978 
 
 4507 
 
 1- 
 
 1100 
 
 1714 
 
 2190 
 
 3283 
 
 4134 
 
 4862 
 
 5507 
 
 1-125 
 
 1310 
 
 2042 
 
 2614 
 
 3909 
 
 4927 
 
 5792 
 
 6577 
 
 1-25 
 
 1534 
 
 2384 
 
 3058 
 
 4581 
 
 5766 
 
 6780 
 
 7690 
 
 1-375 
 
 1767 
 
 2757 
 
 3521 
 
 5279 
 
 6661 
 
 7823 
 
 8863 
 
 1-50 
 
 2013 
 
 3142 
 
 4006 
 
 6016 
 
 7588 
 
 8915 
 
 10099 
 
 1-625 
 
 2268 
 
 3540 
 
 4525 
 
 6781 
 
 8541 
 
 10044 
 
 11381 
 
 1-75 
 
 2534 
 
 3950 
 
 5053 
 
 7570 
 
 9537 
 
 11210 
 
 12703 
 
 1-875 
 
 2812 
 
 4384 
 
 5599 
 
 8386 
 
 10570 
 
 12429 
 
 14083 
 
 2- 
 
 3090 
 
 4821 
 
 6161 
 
 9230 
 
 11628 
 
 13675 
 
 15513 
 
 2-125 
 
 3377 
 
 5273 
 
 6738 
 
 10092 
 
 12718 
 
 14956 
 
 16943 
 
 2-25 
 
 3674 
 
 5736 
 
 7331 
 
 10981 
 
 13833 
 
 16281 
 
 18435 
 
 2-375 
 
 3977 
 
 6210 
 
 7937 
 
 11889 
 
 14981 
 
 17645 
 
 19960 
 
 2-50 
 
 4293 
 
 6699 
 
 8563 
 
 12829 
 
 16161 
 
 19045 
 
 21534 
 
 2-625 
 
 4616 
 
 7203 
 
 9204 
 
 13800 
 
 17380 
 
 20434 
 
 23135 
 
 2-75 
 
 4947 
 
 7716 
 
 9865 
 
 14782 
 
 18624 
 
 21886 
 
 24800 
 
 2-875 
 
 5280 
 
 8233 
 
 10525 
 
 15773 
 
 19887 
 
 23360 
 
 26473 
 
 3- 
 
 5621 
 
 8762 
 
 11204 
 
 16788 
 
 21165 
 
 24833 
 
 28176 
 
 3-125 
 
 5972 
 
 9310 
 
 11900 
 
 17b'30 
 
 22454 
 
 26410 
 
 29925 
 
 3-25 
 
 6329 
 
 9862 
 
 12614 
 
 18897 
 
 23780 
 
 27994 
 
 31714 
 
 3-375 
 
 6689 
 
 10420 
 
 13320 
 
 19963 
 
 25145 
 
 29570 
 
 33507 
 
 3-50 
 
 7049 
 
 10995 
 
 14048 
 
 21052 
 
 26509 
 
 31262 
 
 35329 
 
 3-625 
 
 7418 
 
 11574 
 
 14785 
 
 22153 
 
 27906 
 
 32860 
 
 37186 
 
 3-75 
 
 7794 
 
 12163 
 
 15526 
 
 23284 
 
 29321 
 
 34479 
 
 39080 
 
 3-875 
 
 8178 
 
 12753 
 
 16283 
 
 24416 
 
 30756 
 
 36170 
 
 41013 
 
 4- 
 
 8566 
 
 13354 
 
 17070 
 
 25592 
 
 32225 
 
 37898 
 
 42954 
 
 4-25 
 
 9355 
 
 14582 
 
 18643 
 
 27936 
 
 35191 
 
 41368 
 
 46916 
 
 4-50 
 
 10173 
 
 15849 
 
 20267 
 
 30366 
 
 38254 
 
 44982 
 
 50973 
 
 4-75 
 
 11001 
 
 17140 
 
 21908 
 
 32818 
 
 41356 
 
 48630 
 
 55102 
 
 5- 
 
 11833 
 
 18454 
 
 23595 
 
 35355 
 
 44546 
 
 52378 
 
 59346 
 
 5-25 
 
 12696 
 
 19802 
 
 25362 
 
 37939 
 
 47795 
 
 56209 
 
 63688 
 
 5-50 
 
 13576 
 
 21172 
 
 27248 
 
 40564 
 
 51097 
 
 60079 
 
 68U97 
 
 5-75 
 
 14478 
 
 22580 
 
 29160 
 
 43253 
 
 54478 
 
 64058 
 
 72591 
 
 6- 
 
 15393 
 
 23995 
 
 31122 
 
 45969 
 
 57897 
 
 68082 
 
 77154 
 
OftlFICES, WEIRS, PIPES, AND RIVERS. 
 
 449 
 
 TABLE XII. Discharges from the Primary Channel in the first 
 column of Table XI. 
 
 If the dimensions of the primary channel be in inches, divide the 
 discharges in this table by 500 ; if in yards, multiply by 15*6, 
 if in quarters, multiply by 32, and if in fathoms, by 88'2 etc. : 
 see pp. 233 and 234. The final figures in the discharges may 
 be rejected when they do not exceed one-half per cent., or 0*5 
 in 100. See pages 226 to 234. 
 
 Falls, inclinations, and discharges in cubic feet per minute. 
 Interpolate for intermediate falls; divide greater falls by 4, and 
 double the corresponding discharges. 
 
 Depths of a 
 channel 
 whose mean 
 width is 
 0: in feet. 
 
 18 inches 
 per mile, 
 1 in 3520. 
 
 21 inches 
 per mile, 
 1 in 3017. 
 
 24 inches 
 per mile, 
 1 in 2640. 
 
 27 inches 
 per mile, 
 1 in 2347, 
 
 30 inches 
 per mile, 
 1 in 2112. 
 
 33 inches 
 per mile, 
 1 in 1920. 
 
 36 inches 
 per mile, 
 1 in 1760. 
 
 258 
 
 281 
 
 303 
 
 323 
 
 343 
 
 362 
 
 380 
 
 125 
 
 748 
 
 815 
 
 877 
 
 936 
 
 993 
 
 1049 
 
 1100 
 
 25 
 
 1387 
 
 1511 
 
 1627 
 
 1736 
 
 1843 
 
 1952 
 
 2037 
 
 375 
 
 2145 
 
 2336 
 
 2515 
 
 2684 
 
 2852 
 
 3023 
 
 3155 
 
 50 
 
 3004 
 
 3274 
 
 3527 
 
 3753 
 
 4021 
 
 4207 
 
 4414 
 
 625 
 
 3957 
 
 4311 
 
 4645 
 
 4966 
 
 5287 
 
 5553 
 
 5817 
 
 75 
 
 4991 
 
 5422 
 
 5859 
 
 6274 
 
 6650 
 
 6992 
 
 7342 
 
 875 
 
 6097 
 
 6622 
 
 7159 
 
 7631 
 
 8107 
 
 8540 
 
 8974 
 
 1- 
 
 7266 
 
 7920 
 
 8531 
 
 9124 
 
 9660 
 
 10200 
 
 10693 
 
 1-125 
 
 8514 
 
 9284 
 
 9995 
 
 10658 
 
 11318 
 
 11923 
 
 12520 
 
 1-25 
 
 9816 
 
 10697 
 
 11539 
 
 12307 
 
 13045 
 
 13741 
 
 14479 
 
 1-375 
 
 11182 
 
 12185 
 
 13152 
 
 14007 
 
 14862 
 
 15656 
 
 16448 
 
 1-50 
 
 12601 
 
 13730 
 
 14821 
 
 15786 
 
 16750 
 
 17657 
 
 18552 
 
 1-625 
 
 14069 
 
 15331 
 
 16525 
 
 17616 
 
 18700 
 
 19698 
 
 20696 
 
 1-75 
 
 15593 
 
 16997 
 
 18306 
 
 19517 
 
 20728 
 
 21840 
 
 22944 
 
 1-875 
 
 17157 
 
 18697 
 
 20141 
 
 21469 
 
 22803 
 
 24017 
 
 25242 
 
 2- 
 
 18766 
 
 20446 
 
 22030 
 
 23480 
 
 24938 
 
 26269 
 
 27601 
 
 2-125 
 
 20410 
 
 22247 
 
 23965 
 
 25547 
 
 27129 
 
 28578 
 
 30027 
 
 2-25 
 
 22104 
 
 24087 
 
 25947 
 
 27662 
 
 29395 
 
 30934 
 
 32512 
 
 2-375 
 
 23848 
 
 25988 
 
 27992 
 
 29841 
 
 31701 
 
 33381 
 
 35096 
 
 2-50 
 
 25669 
 
 27953 
 
 30100 
 
 32069 
 
 34086 
 
 35910 
 
 37725 
 
 2-625 
 
 27479 
 
 29933 
 
 32247 
 
 34384 
 
 36512 
 
 38471 
 
 40415 
 
 2-75 
 
 29318 
 
 31947 
 
 34408 
 
 36697 
 
 38958 
 
 41055 
 
 43135 
 
 2-875 
 
 31206 
 
 34002 
 
 36624 
 
 39050 
 
 41464 
 
 43680 
 
 45896 
 
 3- 
 
 33141 
 
 36112 
 
 38897 
 
 41482 
 
 44048 
 
 46398 
 
 48747 
 
 3-125 
 
 35126 
 
 38266 
 
 41223 
 
 43954 
 
 46672 
 
 49174 
 
 51664 
 
 3-25 
 
 37109 
 
 40438 
 
 43556 
 
 46438 
 
 49330 
 
 51951 
 
 54586 
 
 3-375 
 
 39140 
 
 42631 
 
 45925 
 
 48963 
 
 51993 
 
 54775 
 
 57550 
 
 3-50 
 
 41184 
 
 44872 
 
 48343 
 
 51537 
 
 54728 
 
 57659 
 
 60580 
 
 3-625 
 
 43273 
 
 47158 
 
 50807 
 
 54162 
 
 57514 
 
 60585 
 
 63656 
 
 3-75 
 
 45407 
 
 49468 
 
 53300 
 
 56840 
 
 60341 
 
 63560 
 
 66784 
 
 3-875 
 
 47551 
 
 51818 
 
 55832 
 
 59514 
 
 63200 
 
 66576 
 
 69951 
 
 4- 
 
 51911 
 
 56586 
 
 60973 
 
 64974 
 
 69013 
 
 72694 
 
 76383 
 
 4-25 
 
 56448 
 
 61508 
 
 66176 
 
 70623 
 
 75017 
 
 79017 
 
 82994 
 
 4-50 
 
 61014 
 
 66500 
 
 71625 
 
 76408 
 
 81097 
 
 85426 
 
 89767 
 
 4-75 
 
 65713 
 
 71628 
 
 77140 
 
 82250 
 
 87351 
 
 92015 
 
 96653 
 
 5- 
 
 70509 
 
 76863 
 
 82779 
 
 88200 
 
 93731 
 
 98729 
 
 103745 
 
 5-25 
 
 75383 
 
 82159 
 
 88434 
 
 94344 
 
 100200 
 
 105550 
 
 110905 
 
 5-50 
 
 80379 
 
 87590 
 
 94348 
 
 100616 
 
 106823 
 
 112540 
 
 118254 
 
 5-75 
 
 85407 
 
 93093 
 
 100275 
 
 106911 
 
 113505 
 
 119616 
 
 125664 
 
 6- 
 
450 
 
 THE DISCHARGE OF WATER FROM 
 
 TABLE XIII. The Square Roots of the fifth powers of numbers for 
 
 finding the Diameter of a Pipe, or dimensions of a Channel from 
 
 the Discharge, or the Reverse; showing the relative Discharging 
 
 Powers of pipes of different Diameters, and of any similar Channels 
 
 whatever, closed or open. See pages 81, 230, 233, etc. 
 
 If d be the diameter of a pipe, in feet, and D the discharge in 
 
 cubic feet per minute, then for long straight pipes we shall ha,ve 
 
 for velocities of nearly 3 feet per second, D = 2400 (d 5 s) J , and 
 
 d = *044( \ ; or if D be the discharge per second, D = 40 (d 5 s)*, 
 
 /D 2 \i 
 
 228 I - ) . See pages 190 to 224, and pages 42 and 43. 
 V s ' 
 
 and d 
 
 Relative 
 dimen- 
 sions or 
 
 Relative 
 discharg- 
 
 Relative 
 dimen- 
 sions or 
 
 Relative 
 discharg- 
 
 Relative 
 dimen- 
 sions or 
 
 Relative 
 discharg- 
 
 Relative 
 dimen- 
 sions or 
 
 Relative 
 discharg- 
 
 diameters 
 of pipes. 
 
 ing 
 powers. 
 
 diameters 
 of pipes. 
 
 ing 
 
 powers. 
 
 diameters 
 of pipes. 
 
 ing 
 powers. 
 
 diameters 
 of pipes. 
 
 ing 
 powers. 
 
 25 
 
 031 
 
 10-5 
 
 357-2 
 
 30-5 
 
 5138- 
 
 61- 
 
 29062- 
 
 5 
 
 177 
 
 11- 
 
 401-3 
 
 31- 
 
 5351- 
 
 02- 
 
 30268- 
 
 75 
 
 485 
 
 11-5 
 
 448-5 
 
 31-5 
 
 5569- 
 
 63- 
 
 315U3- 
 
 1- 
 
 1- 
 
 12- 
 
 498-8 
 
 32- 
 
 5793- 
 
 64- 
 
 32768- 
 
 1-25 
 
 1-747 
 
 12-5 
 
 552-4 
 
 32-5 
 
 0022- 
 
 65- 
 
 34063- 
 
 1-5 
 
 2-756 
 
 13- 
 
 609-3 
 
 33- 
 
 (>256- 
 
 66- 
 
 35388- 
 
 1-75 
 
 4-051 
 
 13-5 
 
 669-6 
 
 33-5 
 
 6496- 
 
 67- 
 
 36744- 
 
 2- 
 
 5-657 
 
 14- 
 
 733-4 
 
 34- 
 
 6741- 
 
 68- 
 
 38131- 
 
 2-25 
 
 7-594 
 
 14-5 
 
 800-6 
 
 34-5 
 
 6991- 
 
 69- 
 
 39548- 
 
 2-5 
 
 9-882 
 
 15- 
 
 871-4 
 
 35- 
 
 7247- 
 
 70- 
 
 40996- 
 
 2-75 
 
 12-541 
 
 15-5 
 
 9459 
 
 35-5 
 
 7509- 
 
 71- 
 
 42476- 
 
 3- 
 
 15-588 
 
 16- 
 
 1024- 
 
 30- 
 
 7776- 
 
 72- 
 
 43988- 
 
 3-25 
 
 19-042 
 
 16-5 
 
 1105-9 
 
 36-5 
 
 8049- 
 
 73- 
 
 45531- 
 
 3-5 
 
 22-918 
 
 17- 
 
 1191-6 
 
 37- 
 
 8:327- 
 
 74- 
 
 47106- 
 
 3-75 
 
 27-232 
 
 17-5 
 
 1281-1 
 
 37-5 
 
 8611- 
 
 75- 
 
 48714- 
 
 4- 
 
 32- 
 
 18- 
 
 1374-6 
 
 38- 
 
 8901- 
 
 76- 
 
 50354- 
 
 4-25 
 
 37-24 
 
 18-5 
 
 1472-1 
 
 38-5 
 
 9197- 
 
 77- 
 
 52027- 
 
 4-5 
 
 42-96 
 
 19- 
 
 1573-6 
 
 39- 
 
 9498- 
 
 78- 
 
 53732- 
 
 4-75 
 
 49-17 
 
 19-5 
 
 1679-1 
 
 39-5 
 
 9806- 
 
 79- 
 
 55471- 
 
 5- 
 
 55-90 
 
 20- 
 
 1788-9 
 
 40- 
 
 10119- 
 
 80- 
 
 57243- 
 
 5-25 
 
 63-15 
 
 20-5 
 
 1902-8 
 
 41- 
 
 10764- 
 
 81- 
 
 59049- 
 
 5-5 
 
 70-94 
 
 21- 
 
 2020-9 
 
 42- 
 
 11432- 
 
 82- 
 
 60888- 
 
 5-75 
 
 79-28 
 
 21-5 
 
 2143-4 
 
 43- 
 
 12 125- 
 
 83- 
 
 62762- 
 
 6- 
 
 88-18 
 
 22- 
 
 2270-2 
 
 44- 
 
 12842- 
 
 84- 
 
 64669- 
 
 6-25 
 
 97-66 
 
 22-5 
 
 2401-4 
 
 45- 
 
 13584- 
 
 85- 
 
 66611- 
 
 6-5 
 
 107-72 
 
 23- 
 
 2537- 
 
 46- 
 
 14351- 
 
 86- 
 
 68588- 
 
 6-75 
 
 118-38 
 
 23-5 
 
 2677-1 
 
 47- 
 
 15144- 
 
 87- 
 
 70599- 
 
 7* 
 
 129-64 
 
 24- 
 
 2821-8 
 
 48- 
 
 15963- 
 
 88- 
 
 72645- 
 
 7-25 
 
 141-53 
 
 24-5 
 
 2971-1 
 
 49- 
 
 16807- 
 
 89- 
 
 74727- 
 
 7-5 
 
 154-05 
 
 25- 
 
 3125- 
 
 50- 
 
 17678- 
 
 90- 
 
 76843- 
 
 7-75 
 
 167-21 
 
 25-5 
 
 3283-6 
 
 51- 
 
 18575- 
 
 91- 
 
 78996- 
 
 8- 
 
 181-02 
 
 26- 
 
 3446-9 
 
 52- 
 
 19499- 
 
 92- 
 
 81184- 
 
 8-25 
 
 195-50 
 
 26-5 
 
 3615-1 
 
 53- 
 
 20450- 
 
 93- 
 
 83408- 
 
 8-5 
 
 210-64 
 
 27- 
 
 3788- 
 
 54- 
 
 21428- 
 
 94- 
 
 85668- 
 
 8-75 
 
 220-48 
 
 27-5 
 
 3965-8 
 
 55- 
 
 22434- 
 
 95- 
 
 87965- 
 
 9- 
 
 243- 
 
 28- 
 
 4148-5 
 
 56- 
 
 23468- 
 
 96- 
 
 90298- 
 
 9-25 
 
 260-23 
 
 28-5 
 
 4336-2 
 
 57- 
 
 24529- 
 
 97- 
 
 92668- 
 
 9-5 
 
 278-17 
 
 29- 
 
 4528-9 
 
 58- 
 
 25020- 
 
 98- 
 
 95075- 
 
 9-75 
 
 296-83 
 
 29-5 
 
 4726-7 
 
 59- 
 
 2(5738- 
 
 99- 
 
 97519- 
 
 10- 
 
 316-23 
 
 30- 
 
 4929-5 
 
 60- 
 
 27886- 
 
 100- 
 
 100000- 
 
ORIFICES, WEIRS, PIPES, AND RIVERS. 451 
 
 TABLE XIV. Weights and Measures, English and French, with 
 their relative values. 
 
 MEASUEES OF LENGTH. 
 
 12 inches 1 foot. 
 
 7-92 inches 1 link. 
 
 3 feet 1 yard. 
 
 5 \ yards = 16^ feet 1 pole or perch. 
 
 100 links = 22 yards 1 chain = 4 perches. 
 
 40 perches = 220 yards 1 furlong. 
 
 8 furlongs =17 60 yards 1 mile. 
 
 6 feet 1 fathom. 
 
 120 fathoms 1 cable's length. 
 
 1 Nautical mile 6082-7 feet. 
 
 69-12 miles 1 Geographical deg. 
 
 3 miles 1 league. 
 
 The Irish perch is 21 feet, or seven yards. Three inches make 
 a palm ; 4 inches a hand ; 5 feet a pace. In cloth measure 
 2 \ inches = 1 nail ; 4 nails = 1 quarter ; 4 quarters 1 yard. 
 11 Irish miles = 14 English. 
 
 MEASUEES OP SUEFACE. 
 
 144 square inches 1 square foot. 
 
 62-7264 1 square link. 
 
 9 square feet 1 square yard. 
 
 30 \ square yards = 272 J square feet ... 1 square perch. 
 
 10,000 square links =4,356 ... 1 square chain. 
 
 10 square chains = 160 square perches... 1 acre. 
 
 1 rood = 210 square yards ... 40 perches. 
 
 4 roods =4,840 ... 1 acre. 
 
 640 acres =3,097,600 ... 1 square mile. 
 
 The Irish perch is 49 square yards, or 441 square feet ; 1 Irish 
 acre=la. 2r. 19-17p. statute; and 1 statute acre=0a. 2r. 18-77p. 
 Irish. The Irish acre is to the English acre as 196 is to 121. 
 100 square feet, is a square of roofing, slating, or flooring. The 
 Cunningham acre is = la. Ir. 6-61p. English; and 1 English 
 acre is = Oa 3r. 3'904p. Cunningham measure. 
 
452 THE DISCHARGE OF WATER FROM 
 
 CUBIC MEASURES, AND MEASURES OF CAPACITY AND WEIGHT. 
 
 1728 cubic inches 1 cubic foot, 
 
 27 cubic feet 1 cubic yard. 
 
 16Jx lx 1 =24-75 cubic feet ... 1 perch of masonry 
 16Jxl6|x lf= 306 cubic feet ... 1 rod of brickwork, 
 21 x IJx 1 = 30 J cubic feet ... 1 Irish perch of masonry 
 The standard gallon, imperial measure, contains 10 Ibs. 
 avoirdupois, of distilled water at 62 Fahrenheit, the barometer 
 standing at 30 inches. 
 
 6-232 gallons 1 cubic foot. 
 
 8*665 cubic inches ... ... ... 1 gill. 
 
 4 gills 34-659 cubic inches 1 pint. 
 
 2 pints 69*318 cubic inches... ... 1 quart. 
 
 2 quarts 138-637 cubic inches ... 1 pottle. 
 
 2 pottles 277-274 cubic inches ... 1 gallon. 
 
 2 gallons 554*548 cubic inches ... 1 peck. 
 
 4 pecks 2218- 191 cubic inches ... 1 bushel. 
 The old Irish gallon contained 217-6 cubic inches, nearly, and 
 1 Irish gallon is therefore = -7850 imperial gallon. The Irish 
 barrel of lime still measures 40 Irish gallons, or 31*321 imperial 
 gallons, or 4 bushels, very nearly. It is measured by a cylindrical 
 measure 12 inches high, and about 21 J inches in diameter, con- 
 taining half the Irish barrel. In the old English liquid measures 
 for ale and beer, 36 gallons = 1 barrel = 36 gallons, 3J quarts, 
 imperial measure, nearly. 
 
 For old dry measures, 82 bushels = 1 chaldron = 31 bushels, 
 1 pint, imperial measure, nearly. 
 
 And 36 bushels of coal = 1 chaldron of coal = 34 bushels 3 
 pecks, and 1 gallon, imperial measure. The Irish barrel of 
 wheat is 20 stone ; barley 16 stone ; oats 14 stone. 
 
 TROY WEIGHT, 
 
 24 grains 1 pennyweight. 
 
 20 pennyweights ... ... ... 1 ounce 
 
 12 ounces ... 1 pound. 
 
 One pound Troy = 22-816 cubic inches of distilled water, baro- 
 meter 30 inches ; thermometer 62, 
 
ORIFICES, WEffiS, PIPES, AND RIVERS. 
 
 453 
 
 APOTHECARY'S WEIGHT. 
 
 90 Troy grains 1 scruple. 
 
 3 scruples 1 drachm. 
 
 8 drachms ... ... ... ... 1 ounce. 
 
 12 ounces .< 1 pound. 
 
 The ounce weighs 480 grains, and the pound 5760 grains, both 
 in Troy and Apothecary's weight. 
 
 AVOIRDUPOIS OR COMMERCIAL WEIGHT. 
 
 One pound Avoirdupois = 27*7274 cubic inches, when the 
 barometer stands at 30 inches, and Fahrenheit's thermometer 
 at 62. 
 
 16 drachms = 437-5 Troy grains 1 ounce 
 
 16 ounces = 7,000 Troy grains 1 pound 
 
 14 pounds =98,000 Troy grains 1 stone 
 
 8 stone = 112 pounds ... ... 1 cwt. 
 
 20 cwt. = 2,240 pounds 1 ton 
 
 One pound Troy= -82286 pounds Avoirdupois, and one pound 
 Avoirdupois, is equal to 1-2153 pounds Troy. One ton of water 
 contains about 36 cubic feet, equal to 224 imperial gallons, 
 nearly. Ten pounds of distilled water is equal to one gallon, 
 the Barometer and Thermometer being as above stated. 
 
 FRENCH MEASURES AND WEIGHTS COMPARED 
 WITH ENGLISH. 
 
 MEASURES OF LENGTH. 
 
 1 metre .. .. 3-2808992 feet 
 1 decimetre . . . . -3280899 
 1 centimetre . . . . -0328090 
 1 millimetre . . . . -0032809 
 1 Jdlometre(orlOOOmetres)-621383mile 
 
 1 foot English . . 
 1 inch 
 1 yard 
 1 perch, 5 yds. 
 1 mile 
 
 3047945 metre 
 0253995 
 9143835 
 5-0291092 
 1-60932 kilometre 
 
 1000 metres = 100 decametres = 10 hectometres = 1 kilometre =3280-849 
 feet. The metre is the 10,000,000th part of a quadrental arc of the meridian 
 or 39-3708 inches English. 
 
454 
 
 THE DISCHARGE OF WATER FROM 
 
 MEASURES OF SURFACE 
 
 1 centiare (one square ) 1n 7ftlq 
 
 metre) . . j U 
 1 declare.. .. 107-6430 
 
 1 are 1076-430 
 
 f . 
 * 
 
 119-6033 sq. yds. 
 11-9603 
 
 1-1960 
 
 1 are 
 1 declare 
 
 ( 1 centiare or sq. 
 \ metre. 
 
 17 
 
 100 ares = 10 declares = 1 hectare = 2-471143 English acres, and 
 hectares are nearly equal to 42 English acres. 
 
 The old Paris foot is equal 1-06578 English feet ; the French inch = 
 1-06578 English inches ; the French line -08882 of an English inch ; thetoise 
 is equal to 6 French feet = 76-736 English inches = 6-39468 feet. The 
 perches is 18 French feet ; and the perch royal 22 French feet. The French 
 square foot or inch = 1-13581 English square feet or inches, and the cubic 
 foot or inch = 1-21061 English. 
 
 MEASURES OF SOLIDITY AND CAPACITY. 
 
 1 millistere 
 
 .. -0353166 cubic ft. T ^.^.^ 
 
 0610279 
 
 1 centistere 
 
 .., -353166 
 
 
 
 1 decistere 
 
 .. 3-53166 
 
 1 centilitre 
 
 610279 
 
 1 stere (one cubic > Q n.oi cc 
 metre J 3 
 
 1 decilitre 
 1 litre 
 
 . . 6-10279 
 .. 61-0279 
 
 1 decastere 
 
 353-166 
 
 1 decalitre 
 
 ..610-279 
 
 1 hectostere 
 
 3531-66 
 
 1 hectolitre 
 
 6102-79 
 
 1 kilostere 
 
 35316-6 
 
 1 kilolitre 
 
 61027-9 
 
 The stere and kilolitre are each a cubic m^tre, and the litre is a cubic deci- 
 mStre ; 50 litres are nearly 11 English gallons, and 1 hectolitre 2-751207 
 English bushels. 
 
 MEASURES OF WEIGHT. 
 0648 gramme = 1 grain, and 7000 grains = 1 Ib. Avoirdupois. 
 
 1 milligramme 
 1 centigramme 
 1 decigramme 
 1 gramme 
 
 015432 grains 
 15432 
 1-5432 
 15-432 
 
 1 gramme . . 
 1 decagramme 
 1 hectogramme 
 1 killogramme 
 
 15-432 grains 
 02205 lb.avoir. 
 2205 
 2-2046 
 
 1-01605 tonnes = 1 ton ; and 1 tonne = -984206 ton. 
 A gramme is the weight of a cubic centimetre of water and its maxim, 
 density in vacuo, 1 kilogramme = 2-6795 Ibs. Troy=2-2046 Ibs. Avoirdupois. 
 1 metrical quintal 220-46 Ibs. Avoirdupois, and 10 quintals is equal to the 
 weight of a cubic metre of water. 
 

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ORIFICES, WEIRS, PIPES, AND RIVERS, 
 
 455 
 
 TABLE XV. Shewing the Weight, Specific Gravity, strength and elasticity 
 of various materials employed by the Physicist and Engineer. When 
 used bi/ the Engineer only about one-fourth of the ultimate strengths 
 here given should be calculated from. 
 
 MATERIALS. 
 
 Moduli 
 of 
 Rupture. 
 
 Moduli 
 of 
 Elasticity. 
 
 Crushing 
 forces per 
 sq. inch, 
 inlbs. 
 
 Tenacities 
 persq. in. 
 in Ibs. 
 
 Weights 
 of a cubic 
 foot in 
 Ibs. 
 
 specific 
 gravi- 
 ties. 
 
 Acacia, English Growth . . 
 Ash 
 
 11,200 
 12,000 
 
 1,150,000 
 1,600,000 
 
 9,000 
 
 16,000 
 17,000 
 
 44-3 
 
 48-0 
 
 71 
 
 77 
 
 
 
 8,900,000 
 
 10,300 
 
 18,000 
 
 525-0 
 
 8-40 
 
 Beech 
 
 9,300 
 
 1,350,000 
 
 8,500 
 
 16,000 
 
 48-0 
 
 77 
 
 Brick Red 
 
 
 
 800 
 
 280 
 
 135-5 
 
 2-20 
 
 
 
 
 
 
 112-5 
 
 1-80 
 
 Do. Pale Red 
 
 
 
 550 
 
 300 
 
 130-3 
 
 2-08 
 
 Cedar, American, Fresh . . 
 Do. do. Seasoned 
 
 
 
 490,000 
 
 5,600 
 4,900 
 
 11,400 
 19,000 
 
 56-8 
 47-0 
 538-0 
 
 0-91 
 0-75 
 8-61 
 
 Do. Sheet 
 
 
 
 
 30,000 
 
 549-0 
 
 8-80 
 
 Do. Wire-drawn 
 
 9,900 
 
 1,670,000 
 
 
 
 60,000 
 12,400 
 
 560-0 
 43-6 
 
 8-88 
 0-70 
 
 Do. Memel .... 
 
 10,400 
 
 1 530 000 
 
 
 
 37-0 
 
 0-60 
 
 Do. Norway Spruce 
 Elm Seasoned 
 
 6100 
 
 700000 
 
 10,300 
 
 17,600 
 13500 
 
 21-2 
 36-8 
 
 0-34 
 0-59 
 
 Fir, New England 
 
 6 600 
 
 2 190000 
 
 
 10,000 
 
 34-5 
 
 0-55 
 
 Do. Riga 
 
 7,600 
 
 1,100,000 
 
 6,100 
 
 12,000 
 
 47-0 
 
 0-75 
 
 Glass 
 
 
 8,000,000 
 
 33,000 
 
 2,400 
 
 153-3 
 
 2-45 
 
 Iron, Wrought, English . . 
 
 
 
 
 
 57,000 
 57,000 
 
 481-2 
 487-0 
 
 7-70 
 7-80 
 
 Do. rolled in Sheets and 
 Rivetted 
 Cast Iron Carron, cold blast 
 Do. Hot blast 
 
 38*500 
 37,500 
 
 17,270,000 
 16,080,000 
 
 106,000 
 108,000 
 
 31,000 
 16,700 
 13,500 
 
 487 
 441 
 440-0 
 
 7-8 
 7-07 
 7-04 
 
 Do. Buffery . 
 
 37 500 
 
 14 000 000 
 
 90,000 
 
 17,500 
 
 441-0 
 
 7-06 
 
 Larch, green 
 
 5,000 
 
 900,000 
 
 3,200 
 
 10,200 
 
 36-6 
 
 0-52 
 
 
 6,900 
 
 1,050,000 
 
 5,500 
 
 8,900 
 
 35-0 
 
 0-56 
 
 Lead, cast English 
 Do. milled sheet 
 Marble, white Italian .... 
 Do. black Gal way .... 
 Mortar, old, good 
 Oak, English 
 
 1,100 
 2,700 
 
 10,000 
 
 720,000 
 2,520,000 
 
 1,450,000 
 
 250 
 6,600 
 
 1,800 
 3,300 
 
 "so 
 
 17,300 
 
 717-4 
 712-9 
 165-0 
 168-4 
 107-1 
 58-3 
 
 11-44 
 11-40 
 2-64 
 2-70 
 1-75 
 0-93 
 
 Do. Canadian 
 
 10500 
 
 2 150 000 
 
 6 500 
 
 10200 
 
 54-5 
 
 0-87 
 
 
 8700 
 
 1 190 000 
 
 
 12 700 
 
 47-4 
 
 76 
 
 Do. African . . . 
 
 13600 
 
 2 280 000 
 
 
 
 60-7 
 
 0-97 
 
 Do. Adriatic 
 
 8,300 
 
 970000 
 
 
 
 62-0 
 
 0-99 
 
 
 9,800 
 
 1,230,000 
 
 
 7,800 
 
 41-2 
 
 0'66 
 
 Do. red 
 
 8900 
 
 1 840 000 
 
 5300 
 
 
 41-2 
 
 0-66 
 
 Silver, Standard 
 Slate, Welsh 
 
 11,800 
 
 15,800,000 
 
 
 40,900 
 12,800 
 
 644-5 
 180-5 
 
 10-31 
 
 2-89 
 
 Do. Westmoreland 
 Do. Valentia 
 
 5200 
 
 1,290,000 
 
 
 
 
 174-4 
 
 180-0 
 
 2-70 
 
 2-88 
 
 Steel, soft 
 
 
 
 
 120.000 
 
 486-2 
 
 7-80 
 
 Do, razor tempered .... 
 Stone, granite average .... 
 Do. Rochdale 
 
 5,500 
 2400 
 
 29,000,000 
 
 8,000 
 
 150,000 
 
 490-0 
 168-0 
 161-0 
 
 7-84 
 2-70 
 2-58 
 
 Teak, dry 
 
 14800 
 
 2 400 000 
 
 12101 
 
 15000 
 
 41-1 
 
 0-66 
 
 Tin, cast 
 
 
 4 600 000 
 
 
 5,300 
 
 455-7 
 
 7-30 
 
 
 
 
 
 
 
 
LONDON : 
 
 PRINTED BY GEORGE PHIPPS, 18 & 14, TOTHILL STREET, 
 WESTMINSTER. 
 
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