1891.] Fluid Motions in two dimensions. 193 



we find for the breadth of the lamina 

 ac — 1 , c — 1 ab — 1 , 1 — 6 



c-6 ° c + 1 c — b & -1-6 



+ V(l - a 2 ) 7T - V( c 1 J 6 a2) W(¥ - 1) + V(C 2 " 1)] 7T. . .(23). 



•(24), 



When w is > 1 we must write (22) in the form 



dz _ an - 1 + i V(l - a 2 ) V(^ 2 - 1) 

 du (u — b)(u — c) 



where a = cos a and ^(1 — a 2 ) = sin a, a being the angle the im- 

 pinging jet makes with the lamina. The sign of the imaginary 

 term is determined by considering that when u = go the argument 

 of dz is a. 



When u lies between 1 and c we have 



dz _ ab-1 1 ac - 1 1 . V(l - a 2 ) 

 du c — bu — b c — bc — u »J{u 2 — 1) 



V (l - a 2 ) [b 2 -l c 2 - 1" 



(c — b) */(u 2 —l) \_U — b C — U_ 



and the argument of dz in the neighbourhood of u = c is the same 

 as that of 



ac — 1 . \/(l — a 2 ) 



c-b V(c 2 -l>(e-6)' 



or of - (ac - 1) - % V(l - 0/V(c 2 - 1). 



Thus the ultimate direction of this part of the stream is 0, where 



R cos 6=1 —ac, 



Rsmd = - V(l - a s )/V(c 8 - 1), 

 giving R = c — a, 



and tanfl = - n V( \~? n (2-5). 



(1 — ac) v(c — 1) 



In like manner the ultimate direction of the other part may be 

 determined. 



11. The form of the free stream-line from 1 to c would be 

 found by integrating (24). I shall suppose u = + 1 to correspond 



