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



where p is the density of the fluid, or 



u 2 — a 2 



h 



l- 



2 - d l - u 2 + 2 v /(l - a 2 ) V(l - u 2 )_ 



V(l-g 2 )+V(l-^) du 

 \Ju 2 — a 2 u 



f 1 { V(l - u 2 ) + V(l - a 2 )} V(l - u 2 ) du 

 p J a y(l-a 2 ) + ^/(l-u 2 )}^(u 2 -d 2 )- u ' 



or 



V(i-m'0 ck 



a V(w 2 — tt 2 ) w 

 The value of the integral is 



7T/1 



2 W 1 



Returning now to the problem of the lamina we see that 

 the pressure on it is 



pir(l-a)/a (33), 



and in the same case the velocity of the stream at infinity in 

 the direction from which it comes is 



o/{l + V(l - a 2 )} (34), 



the breadth of the stream is 



d = 7T [1 + V(l - a 2 )}/a (35), 



and the breadth of the lamina is 



7 1 - a V(l - « 2 ) / o ■ -i x M m 



l = 7r + — '-{it — 2 sin l a) (36). 



Now let the stream flow from oo with a given velocity V in 

 a canal of breadth d, and impinge symmetrically and directly on 

 a pier of breadth I. Then if a quantity a be determined from the 

 equation 



l 1 - a + V(l - a 2 ) (l - | sin -1 a] 



d = ~ V(l - a 2 ) ' 



the pressure on the pier will be 



n _ (1-g ) {1 + Va- a 2 )} 2 



a 2 [tt (1 - a) + V(l - a 2 ) (tt - 2 sin" 1 a)} ' 

 By writing a = cos a we find the convenient form 



