202 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



^September, 



» s ya— 7 



15 



Secondly^ x =: 0, and x ■=■ a: then 

 f" xdx\f{'2a—x) = a7|l(2«— 1)— ?(25— I)) =2.a3 

 Thirdly, x ■=. a, and ^ = 2u : then . 



Substitute these three values successively in equation (6), and we 

 have — 



I. For the quantity of discharge through an elliptical orifice, of 

 which the higliest point lies in the surface of the water, 



32 



c. Q ^ /c ^"(4^0) .ah. - s/'2. 



II. For the quantity of discharge when the orifice is a semi- 

 ellipse, its lowest edge being the horizontal axis, and the extremity 

 of the vertical axis 2a coinciding with the surface of the water, 



4 



d. Q = A;v'(*</o)-a*- 15 (8 V2— 7). 



III. For the quantity of discharge by a semi-ellipse in the re- 



\ ersed position, the horizontal edge being uppermost, and at a 



depth a below the surface, 



28 

 e. Q ^ A /s^{iga) . ah . -- . 



Tlie sum of the equations {d and e) gives the equation c. 



In order to find a more convenient function for the development 

 of the variable part of equation (a), by a series, let the origin of 

 co-ordinates be transferred to the centre of the orifice. We have 

 then the equation 



jr = — (o— jr). 



Let the altitude of pressure H' be put for H + a ; then the 

 equation (H § 9) becomes for these values 



/. Q = /c V(4</) ^^Jdx y/ { (H' -H ,r)(a-^-^") | . 



If, now, •j/(H' -f a?) be expanded in a series, we have 



yrfari/{(H'-l-aO(a— .r=)j = 



J ^ '' ». ^2 H' 2 4 H'- 2 4 6 H'» J 



Integrating the several parts of this expression, and taking the 

 integral 



1. Between limits j; = and x := a, 



/ rf.rV(a" — ar)= iCia^f) / xdx-/(cP—x-)= ia" 



x-dx./{a'-x^)= l.\{\a*Tr) I x^dx ■J{a^-x''-)= J'- *a = 



- D »/ 



I x*dx./{a--x''-)=\.\.^{la^ir) I x''dxV(a--x-) = i . ^ .^a^ 



and 



+ ''■(^J^+(4•}J)(^!)^y3+«•J■i•i•^i)(^i•^)^y.+ ■••■) 



= a=VH'{Si}. 



2. Between the limits x = and x =: —a, and we have 



y ''dx^({H' + x){a--x"-)\ 



= «-vn'{S,}. 



The first value assumed for the integral in equation /, gives for 

 the quantity of discharge through an orifice (as 1 ), 



g. Qz=Arv'(4irH').2ai{Si}. 



The second value gives for the quantity of discharge through an 

 orifice (as 2), 



A. q = k^^(igli').2ab{S,}- 



The altitude of pressure H' here = ec. 



By adding together equations (A and g) we have the quantity of 

 discharge for a complete ellipse — 



i. Q = A^(4j,H') .2«a^ { ._(J. J)(4 4)^^_(j . J . ».i)(4. i-i)^^ 



This equation shows that the formula Q = A: /y/(4(?H'), when 

 applied in the ordinary way, is nearly accurate only in the case 

 where the sum of the series in the parenthesis is nearly equal to ^ 

 — that is, when the altitude of pressure is so large that all the 

 terms with H' in the denominator disappear. 



For H'=0, H — a. Putting these values in equation (j), we have 

 for the quantity of discharge, in the case where the highest point 

 of the orifice is in the surface of the water, 



k. Q = fcV(4<;«)-2ai-{i-(i.i)(i.i)-(i.^S.|)(i.i.|)...}. 



e sf 

 itior 



^^v'2 = -(i-(4-i)U-i)-(M.t4)(i.|4)— -I. 



But the same quantity of discharge has been already determined 

 by equation c. Put, therefore, the two equal to one another, and 

 we find 



I. 



15 



( 



i 



Thus the sum of the series in the parenthesis ^ -— — = 0'48018. 



15jr 



Put in the equation (A) the altitude of pressure H' = a, and we 

 have the same quantity of discharge which the equation (d) has 

 already given. By the comparison of both we find 



m. l(8v/2-7) = 2{S.} 



=2|iT(4-(J-i)(i4)-(4.i-i.|)(i.*.i)-...-)-(i4 + (i.i-i)U-i) 

 + (^}•M.7,)(i. !■?) + . ...)}. 



Thus {SJ = ^5(8^2-7) = 0-5752. 



Substitute for the first series in ()7i) the value in equation (/), 

 and we have 



14-8V2 



15 



= i{i-(i 



•}.3)-i + (i4-i-S-7o)(I-;) + . 



■■}- 



01790" 



or, §(7-4y2) = J-(U.i).| + (J.}.g.f. A) (?•?)+.... 



Put now in the equation {y) the altitude of pressure H = o, and 

 compare it with ecjuation (e) : consequently, 



^r2(Si}=2{4^(j-(j.j)(M)_(i.j.MXi-J-i)-----) 



13 



+ 4-i+a-i •i)(M) + a- J •i-f.fi>)U-M) + ••••}. 



14 



and therefore { Si } = t-;. 



15 



