194 



Mr Love, On the Theory of Discontinuous [May 4, 



to z = 0, and then the co-ordinates of a point on the free stream- 

 line are 



ac — 1 , c 



u ab — 1 , u — b 



c — b " & c — 1 

 y = Aj(l — a 2 ) cosh -1 it — 



c-b 



V(l - a 



log 



1-6 



c-b 



"^pzD^-M^.D^-DI 



V 



u — b 



y (26), 



+ V(c - 1) smh 1 — *-^ 



c — u 



in which w is a real parameter lying between 1 and c. 



To find the free stream-lines that bound the jet, we have to 

 integrate (24) up to values of u lying beyond u — c. We choose 

 the path of integration to start from u = 1 and proceed along u 

 real very nearly to u = c, then over a little semicircle whose centre 

 is u = c and then again along u real ; we thus find 



ab — 1 , u 



=-t^ 1o «t 



z = 



b ac 

 -7- + 



1, u — c 



— r- log — 



e — 6 ° c — 1 



i\ -u-i .\/(l-a 2 ) 



-V(o'-l)smh-j^^lM^l) 



I 11 C J 



• qc- 1 . V(l-a ! ')V(c'-iy 

 c — 6 c — 6 



'y^-Dsinh-j^-p^'- 1 ) 



where the last line is the part contributed to the integral by the 

 small semicircle. Hence on this stream-line 



ac — 1 



x = - — r log - 

 c—b ° c— 1 



— c at — 1 



o-6 



log 



-6. 7rV(l-OV(c 2 ~l) 



1-6 



+ 



c-b 



y = V(l - a 2 ) COsh" 1 U - ^L_|5 



^-Dsinh-f^-y^- 1 )! 



u — b 



^-D^ ^-^r- 1 ^ ]- 



ac 



(27). 



c — 6 , 



This stream-line will have an asymptote x = y cot a + oc which is 

 to be determined by making u infinite and 



a = cos a, V(l _ °0 = sm a - 



