192 
MR. TV. M. HICKS ON THE STEADY MOTION AND 
Considering first the term £= M cos nv, the corresponding form for y will be 
2Y.fr Sin mv 
the coefficient Y being determined from the condition 
V/r KG-cf d\ 1 ^ R m . 1 
M cos »«= J v { 77 ^) 2Y. m sin mv j 
for all values of v when u=u. Therefore 
k(C — c) 3 fmeosm-r . sin v sin mv] „ 
M cos m = — 2 (C-c)> } Y * 
or 
„ Ma 3 S cos nv 
k (C -cf 
2M<x 3 S cos nv 
kC (C—c) f 
■ % {2mC cos mv— 2m cos mv cos v— sin mv sin v}Y„ 
«-> i ~ 2m —1 t ——r 2m+ 1 , 
z-j 2m cos mv -—— cos (m+ l)v -cos (m—l)v [• i r; 
2C 
From this we may obtain sequence equations to determine the Y„; but we require 
only the most important terms, hence 
2Ma 3 S 
2nY„= 
k& 
Y n =- 2 -^ a?i!LB 
n 
and 
2 v 7 2a 3 M/d R n . 
X= “ /Tri- TtW Sm UV 
^ ny/(C — c) R„ 
Since the cyclic motion due to this is zero, there is no correction to be introduced 
for it as in former cases. 
If cf> be the velocity potential, the condition for a free surface gives 
0=5-^,—l( ve l )*+/(«) 
r 
f(t) being an arbitrary function of the time. The velocity normal to the surface is of 
the first order of small quantities, and its square is to be neglected. 
The velocity along the surface is 
0_I \dk /o 
where U 0 is the velocity determined in § 10 and 
n 
—!TL 2 =0 
2 0 — u 
