of Edinburgh, Session 1878-79. 
95 
however, we shall suppose D to he constant. Then (2) used in 
(5) or (2') in (5') gives after integration with respect to t 
. ( 6 ) 
( 6 ') 
dv du_ h 
dx dy W D 
or in polar coordinates 
r dr d£ _ h 
r + dr rdQ W D 
These equations (6) (6') are instructive and convenient though they 
contain nothing more than is contained in (2) or (2'), and (4) 
or (4'). 
Separating u and v in (4), or £ and r in (4'), we find 
d 2 u J „ / d dh _ dh' 
+ 9 , 0 — 
dy , 
dt* + 4 “ 2 “ 9 {dtdx + 2o> dy ) 
and 
d 2 v . _ /_ dh d dh\\ 
W + Uv= 9 V^-dtdy), 
(7) 
or in polar coordinates 
d 2 £ 
. 9S , (d dh _ dh' 
dfi + 4o> 1 - - 9 ( ^ 
(T) 
d 2 r . „ / _ dh d dh\ 
m +U t= 9 \ 2m dr- d t^e) l 
Using (7) (7'), in (2) (2'), with D constant, or in (6) (6') we find — 
and 
^fd 2 h d 2 h\ d 2 h . 97 
9B {d^ + di*r^ M 
^fd 2 h 1 dh d%\ d 2 h . 
9B {d? + rd-r + ^) = d? + 4,oA 
• ( 8 ) 
. (8') 
It is to be remarked that (8) and (8') are satisfied with u or v 
substituted for h. 
I. Solutions for Kectangular Coordinates. 
The general tpye-solution of (8) is h = e aX ^e yt where a, /3, y, are 
connected by the equation 
y 2 + 4o> 2 
a 2 + /3 2 = 
gv 
■ (9) 
For waves or oscillations we must have y = o- J - 1 where o- is real. 
