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 
dv du_ h 
dx~dy~ Zm I) 
or in polar coordinates 
r dr dt, h 
r + dr rd6 W D 
( 6 ) 
(6') 
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 
— + 4<A 
and 
— c 
d dh dh> 
Jtdx + Au> dy / 
d I. 2 * v , _ /. dh d dh 
+ 4™% = g [2^ - ^ — 
(?) 
or in polar coordinates 
d»£ 
dt 2 
dx dt dy, 
A o« (ddh dh\ 
+ ^ = - 9 {dtdP +2m ^e) 
dfr 
dt 2 
(7') 
. „ dh d dh\ 
+ 4 * * “ t= n^-di^e), 
Using (7) (7'), in (2) (2'), with D constant, or in (6) (6') we find- 
and 
^(d 2 h d 2 h\ d 2 h . 97 
T J(Ph 1 dh d 2 h\ d?h , „ 
g V{w> + rTr + ^ rW +Uh 
(8) 
(8') 
It is to he remarked that (8) and (S') are satisfied with u or v 
substituted for h. 
I. Solutions for Kectangular Coordinates. 
The general tpy e-solution of (8) is h = e ax e^ J e yt where a, (3, y, are 
connected by the equation 
r 2 + 4w 2 
a 2 + ,8 2 = 
9 D 
• ( 9 ) 
For waves or oscillations we must have 1 where a is real. 
