190 



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



e.g. if the rim be yg-th of the breadth the stream-line rises through 

 about ^jth of the breadth. Except quite close to the disc 

 where u l — 1 is small we may reject k altogether, and the form of 

 the free stream-line at a distance from the disc is the same as if 

 there were no rim. 



To find the pressure on the disc we have to find 



dz 



which is 



\p 



or \p 



hp 



= iP 



l— 



i-[ r *? 



dz 



u 2 (1-u 2 - F) 



du 



du, 



k + V(l — u 

 ~ 1(l (1 + cos e) 2 (k + cos e) 2 - sin 2 6 cos 2 e 



v^lg ti+va-v)]^ 



dd, rejecting & 2 , 



o 

 sin -1 a 







sin -± a 



(1 + COS 6) (k + COS 8) 



[(1 + cos 6)(k + cos 0) - (1 - cos 0) cos 6{l-k sec 0)] d<9 

 "(2 cos 2 9 + 2k) dd 



\p [sin -1 a 4- a y^l — a 2 ) + 2& sin x a] 



= p 



7T A; /C7T 



I + 2 + ~2~ 



to the first order in k. 



or 



The breadth of the disc, rejecting k 2 , is 



a + i [ srn_1 « + a V(l — °0] +k[a + sin" 1 a], 

 1 + -7- + - + &+-& to the first order in k. 



Thus the mean pressure, when the velocity on the free stream- 

 lines is V, is 



>F 



7T 7 8 + 4tt + 2tt 2 

 + A; 



4 + 7T 



(4 + 7T) 2 



or 



Trp 



F 2 



4 + 7T 



■ 8 + 4tt + 2tt 2 .. ' 

 ■ 1 + (4 + 7r) a V(26) 



.(20). 



Comparing this with the case where there is no rim we see that 

 the mean pressure is increased by about f ye of itself, e being the 

 ratio height of rim to breadth of disc. 



