445 
of Edinburgh, Session 1879 - 80 . 
comparison with T. Substituting in (4) and (5) and neglecting 
squares and products of the infinitely small quantities we find, 
dp p ir n 
& + — + raw — U . 
dr r r 
Taking (7), eliminating w, and resolving for g, r, we find 
T\ (Y ,T \dw £/T . dT\ j 
• (8). 
mD \ r 
T\dw 
r) dr 
if T dT N 
r Vr 
'/T dT\ 
( . T\ dw i I 
~T 2 
(M 
H 
( . T V1 ) 
\r + dr) 
\ 1 r) dr r | 
[ r 2 
dS 
rv) _K 
where 
_ 2T /T dT 
D = — ( - + -v- 
r \r dr 
)-(•-# 
For the particular case of m = 0, or motion in two dimensions 
(r, 0), it is convenient to put 
— = </> .... ( 10 ). 
m 
In this case the motion which superimposed on r = 0 and rO = T 
gives the disturbed motion is irrotational, and <j> sin (nt - iO) is its 
velocity-potential. It is also to he remarked that when m does not 
vanish the superimposed motion is irrotational where if at all, and 
only where, T = const. /r, and that whenever it is irrotational </> as 
given by (10) is its velocity potential. 
Eliminating g and r from (8) by (9) we have a linear differential 
equation of the second order for w. The integration of this, and 
substitutions of the result in (9), give w t g, and r, in terms of r and 
the two arbitrary constants of integration which, with m, n, and i, 
are to be determined to fulfil whatever surface conditions, or initial 
conditions, or conditions of maintenance, are prescribed for any par- 
ticular problem. 
