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so that P'I'=c/ir, and the curve AP is the tractrix; and the cotfiicient of contraction, or 4 breadth of the 'et 'ir . ', breadth of the orifice '1+2' (4) A change of fl and 0 into nfl and nb will give the solution for two walls converging symmetrically to the orifice'AA1 at an an le i/n. With 1l==§ , the re-entrant walls are given of Borda's moutiipiece, and the coefficient of contraction becomes Q. Generally, by making af = -oo, the line x'A' may be taken as a., straight stream line of infinite length, forming an axis of symmetry; and then by duplication the result can be ob- tained, with assigned n, a, g and b. of the efiiux from § a symmetrical converging f M i ' J' A


z 1- ~ Al 8 s B Fic.. 5. Fig. 6. mouthpiece, or of the flow of water through the arches of a bridge, with wedge-shaped piers to divide the stream., 42 Other arrangements of the constants n, a, b, a' will ive the results of special problems considered by j. M. Micheg, Phil. Trans. 1890. Thus with a'=o, a stream is split symmetrically by a wedge of angle vr/n as in Bobyleff's problem; and, by making a=oo, the wedge extends to infinity; then b 1| ch 1lQ= Q, Sh nQ= (I) Over the jet surface //=m, q=Q, ', u= -r1f¢/'"= -bw'/', /a ch fZ=cos n0= l;7§ ;-i, sh§ 'l=isinn0=i 55%, (2) =*""'=f““ '"'» ii=;%7r ' <9 For a jet impinging normally on an infinite plane, as in fig. 6, n = I, elf"/==tan 6, ch (hs/c) sin 20 = I, (4) sh évrx/c =cot 0, sh § 1ry/c =tan 6, Sh ifx/C Sh ivry/¢=1, el"<**">/°=e%"'/'+e¥"”/°+I- (5) With n = ~§ , the jet is reversed in direction, and the profile is the catenary of equal strength. In Bobyleff's problem of the wedge of finite breadth, ch nQ= dwg, sh n$2= /i?/é, (6) n - E sin mx= '£ £ () cos a- a, a, 7 and along the free surface AP ], q =Q, ¢=o, u=e-"¢/"'éae*'/°, j e'f'/°-1 C05 M=cos11a, le, " /C coszna sin'nB ' (8) = -|— -T»1s1n“n0 - sin'na. the intrinsic equation, the other free surface A'P']' being given by cos'na sin'n0 1rd/=, ... . .., . . 6 = sinfna - sin“n8 (9) Putting n = 1 gives the case of a stream of finite breadth disturbed by a transverse plane, a particular case of Fig. 7. A When a=b, a.=o, and the stream is very broad compared with the wedge or lamina; so, putting 'w=-w' (a-b)/a in the penultimate case, and, u=ac“"2:a-(a-b)'w', (Io) chnSZ==; %fl, shnt':= 7%-1, (Il) in which we may write w' =4>+¢i. (12) Along the stream line xABP], ¢=o; and along the jet surface APJ, -1 >¢»> -so; and putting 4>= *FS/C"I, the intrinsic equation isrs/c = wana, i ' (13) which for n= I is' the evolute of a catenary. xxv. 5 43. When the barrier AA' is “held oblique to the current, the stream line xB is curved to the branch point B on AA' (fig. 7), and so must, be excluded from the boundary of u; the conformal re- - ¢ presentation is made now with <§ Q ~, / (b-a.b-a') () 'du

  • Z'Tb), / (u-a.u' - )a I . A,

dwl 2; E'; .: aT. ' '1l'u-j 7ru-j m-l-m' u-b, " - “T , f 21 1;=' EJ lJ (2) » m+m ' si taking u=w at the source where FiG~7¢==oo, u=b at the branch point B, u=j, j' at the end of the two diverging streams where ¢=f-oo; while nl/=0 along the stream line which divides at B and passes through A, 'A'; and 'L- =m, m along the outside boundaries, so that m/Q, m'/ is the final breadth of the jets, and (m+m')/Q is the initial breaclgi, c, of the impinging stream. Then °*' *Q* sh iS2= (Sf 2b-0.-a' N °h“~Tr'm Q/(2.a-u.u-al) shS2-1/N-=1;3—, -b.b- ' ~ N =2'iT;f*l"' c (47 Along a jet surface, q=Q, and chSZ=cos0=cosu.-iesin'a.(a-a')/(u.-b), (5) if 0=a at the source x of the jet xB, where 1¢=°°; and supposing 0 =|S, B' at the end of the streams where u =j, j', u-b -§ sin'a. u-ez' . cos'0-cos 8a-a cos a.-cos 0' a-a é Smal (cos a-cos B) fcos a-cos 05 u-j' ' cos0-cos B a-a"'% Sm 2°'(cos a-cos ¢s'i (cos a.-cos 05' '(6) and il/ being constant along a stream line do dw ds do dw du Ein Qffw ='2irr>-1rQ ds 1rds ' (cos afcos B) (cos a-cos/S') sin 0, m+m @'c 35° (cos a.-cos 9f(cos 9-cos B)"(cos Q-cos i') sin 0 cos sz-cos B' sin 0

cos a.-cos 9+cos 5-cos /3"cos 0-cos B

cos a-cos'B sin 0 n cos B-cos B'°cos 0-cos B ' V (7) giving the, intrinsic equation of the surface of a jet, with proper attention to the sign. From A to B, a>u>b, 0=0, ch Sl=ch log %=cos a-5 sin 'ag sh f2==sh log%= sin a f I % (u-b) cos a-%(a-a'):in;q+{ (a.-u.u-g')sin;x (8), ds ds dd> A H Q¢F¢ Q¢f1»@' q du m+ng' (u-b) cos a.-%(a - Q/) sin 2a-I- V (a-u u»a') sin q ( 1r ' " j-ulu-jr ' 9 AB “(2b-o-a')(u-11)-2(a-b)(b-a')-l-2l(a-b.b-a'.a-u.u-a')du 0 7' 5 °' b' a-a'.j-u .u-j' ° (I) with a similar expression for BA'. The motion of a jet impinging on an infinite barrier is obtained by putting j =a, j'=a'; duplicated on the other side of the barrier, the motion reversed will represent the direct collision of two jets of unequal breadth and equal velocity. When the barrier is small compared with the jet, a.=B=B', and G. Kirchhoff's solution is obtained of a barrier placed obliquely in an infinite stream. Two corners B1 and B2 in the wall xA, with a' = -°°, and n=x, will give the solution, by duplication, of a jet issuing by a re entrant mouthpiece placed symmetrically in the end wall of the channel; or else of the channel blocked partially by a diaphragm across the middle, with edges. turned back symmetrically, problems discussed byj. H. Michel, ,A. Love and M. Réthy. I