432 Mr. J. H. Michell on the 



or, from the assumed form of -j^> 

 g r _i(a + « ie «» + «^ + ...)=-'^+«is; 1 i2</» + .... 



The function on the left is therefore real over the surface. 

 At yjr=vo , 



and the function will be finite there and equal to — ia . 



The only singular points to be considered are, then, the 

 summits of the waves. 



Suppose near the summit w = 0, 



and therefore 



cho~ Aw > 



(2 W 1_ n_ «_ 



-77= A »+l(?l+l) n+iz n+l, 



as 



and 



- 2 _ 2n 2n 



q 2 = A n+i(n + l)~n+ir »+i, 



where r is the distance from the summit. Now, since the 

 pressure is constant over the surface we have q 2 = Zgy ; and, 

 comparing, we see that n = — •§■ ; so that 



dz . , 

 aw 



near 10 = 0. From this it follows that the angle at the summit 

 is 120°, as was first shown by Stokes*. 

 Hence, also, 



dU_ _1_ 



dw " Zw 



near the summit 10 =0 ; and the summits are simple infinities 

 of the function considered. 



According, then, to the principles of Cauchy, the function 

 can only differ from the sum of the polar elements by a 

 constant, and we have 



.i(c h e^ + a 2 e^ + ...)=- it : 



dw 7 w—nir 3 



= — ^cotw— ~$l 

 * Collected Papers, vol. i. p. 227. 





