1891.] Fluid Motions in two dimensions. 195 



When u is very great, we can expand x and y in the forms 

 x= a\ogu + A n -\ -+ ... 



o o u 



and, to find the asymptote, we must find A and B Q . We get 



x— a log u + 



y = V(1- a 2 ) log u + 



^zIlog(l-6)-^-ilog( C -l) 



c-b oV o-6 oV ' 



+! r_ya-«V(c 2 -i) 



c-6 



. 1 



+ terms in - , 



u 



V(l - a 2 ) log 2 



- ^L_^2 {^(6* _ i) cosh-^- 6) - V(c 2 - 1) cosh^c] - tt^— =J 



+ terms in - , 



u 



so that 



# n = — r-r- log (1-6) r- log (c — 1) H LA — \ y 



c-6 &v 7 c-6 oV y c— 6 



- a log 2 - -^=- [v / (6 2 - 1) cosh" 1 (- 6) - ^(c 2 - 1) cosh _1 c) 



C 



air ac-l 

 + V(l-a") o-6 (28) ' 



and x — ^tt cosec a is the distance to the right of the right-hand 

 end of the lamina of the point where the initial middle line of 

 the jet strikes the plane of the lamina. This is the point marked 

 P in the z figure. 



There are now sufficient relations to determine the unknown 

 constants a, 6, c. A jet of given breadth coming from oo in a 

 given direction strikes the lamina obliquely in such a way that 

 the middle line of the jet passes through a certain point in the 

 lamina. We choose the unit of length so that the given breadth 

 of the jet may be it. Then the constant a is determined by 

 making a = cos a, *J(1 — a' 2 ) = sin a, where a is the angle between 

 the direction of the impinging jet and the lamina; and there are 

 two relations to determine the constants 6 and c, viz., equation 

 (23) gives the breadth of the lamina in terms of 6 and c, and 

 equation (28) enables us to identify a point P whose distance from 

 the right of (1) is given with a point whose distance from the 

 right of (1) is a given function of 6 and c. 



VOL. VII. pt. iv. 16 



