250 Mr G. H. Bryan, On the waves on a [June 4, 
Also the equations (6) when transformed to polars lead to 
dy eae Bie Be ) 
wo (ap 2V") ¥ + 2a Buteag Orr 
Ee on Op da | 
eae ee =a | 
from which we find 
a 
—_ 
> 
wa 
w = een | (2m — 1a) Ar" + 2oBJ, (hr)}......... (11). 
4. The equation of the free surface is 
P= COU at cb inisn ete ce ee (12), 
where 
dee , Et Ca Op - 
Ao normal component velocity of liquid = Gam 
‘ o=—~e-*[Aa® + BJ, (ha)} iad (13). 
The gravitation potential of the deformed cylinder at any point 
(r, 6) is to the first power of ¢ 
V.=mpy (a —1°) + 2arpy = "en OMe 53: (14) 
if the point be inside the cylinder, or 
V,=— 2mpyat log” + 2epy  % emt C ‘cae (15) 
if the point be outside, y being the constant of gravitation and C 
some constant. Thus at any point of the deformed surface the 
potential is 
= — 2mpyca (1 — *) Aaa ie Oa AS 656 (16), 
and PaV+ tor —o 
— _ pat pené un { n 1 28 ( -:)t 
=—e-“e = i + BJ, (ha), \ 2arpy {1 al 
+ (20 — a) Aa" + 2oBJ, (ha | +04+40°Q ....0.... (Cpe 
If 1/R be the curvature at any point of the surface, then it is 
readily shown that 
