﻿Dr. J. R. Wilton on Rijyples. 691 



and the surface condition becomes * 



* + *S f =W + T//>B-iC (1) 



To determine the form of the term T/pR we expand 1/R 

 in a series of cosines of multiples of f by means of the 

 equations for x and y on p. 392 of " Waves/' Since on the 

 free surface ?' = £, we have 



R~ A \d£ d? djj dp)/\d%d£r 



The sign is determined by the fact that R is positive at 

 the trough of the wave where f =0. A l5 = —a, is negative, 

 and therefore the predominant term in 1/R is 27ra/\ 

 which is positive. 



On substituting in (1) we find, as in " Waves/' p. 387, 



0=4/* + { C + |A n cosnf| ^(l + |«A n cosnf) 2 + (f 72A ;i sin7if) 2 | 



+ k { | n 2 A n cos nf + 1 n 2 A 2 n 

 I i i 



00 oo I 



+ % % mn(m + n)A m A n cos (m — n)fj V 



m=l n=m+l J 



| (1 +SwA„ cos n%) 2 + (I nA n sin nf ) 2 ] "* . (2) 



where, as before, 



_2tto 2 _/2tt\ 2 T 



As a first approximation we see that 



Ai(l+20.+*)=0, 



so that Aj is arbitrary, 



Ai=-a, (3) 



and 



c 2 = •— (l+/e), 



as it should. 



* The explanation of any notation not given in this paper will be 

 found in " Waves/' pp. 385-0. 



2 Y 2 



