444 
and 
Proceedings of the Royal Society 
dp dx . dx dx . dx 
~Tx-dt +X Tx +y dy + Z 7l z 
dp dy .dy 
dy dt dx dy dz 
dp 
dz 

dt 
dz . dz .dz . dz 
dy 
dt dx 
dx dy dz 
dx dy dz 
dz 
(i), 
( 2 )- 
To transform to the columnar co-ordinates we have 
x — r cos 6 , y = r sin 0 
x = f cos 6 — rd sin 0 
y = f sin 6 + rd cos 0 
d d . d 
j- = cos 0 ~7 sm 0 —tt\ 
dx dr rdO 
d . d d 
-j- = sm 0 -T- + cos 6 
dy dr rdd 
The transformed equations are 
( 3 ), 
and 
dp 
dr 
dr 
dt 
dr 
+ f Tr 
_w 
r 
, a dr . 
+ e Te + z 
dr 
dz 
dp 
dd 
= r— 
+f d ^ + te 
+ e*£p + 
& d(re) 
rdO 
dt 
dr 
dd 
dz 
dp 
dz 
dz 
. dz 
dz 
dz 
dt 
+ i "dr 
+ 6 Jo 
+ i dz 
dr 
dr 
r 
+ r + 
d(rd) dz 
rdr dz 
( 5 ). 
Now let the motion he approximately in circles round Oz, with 
velocity everywhere approximately equal to T, a function of r ; and 
to fulfil these conditions assume 
r — o cos mz sin ( nt - id) ; rd = T + r cos mz cos {nt - iO) 
z — w sin mz sin {nt - id) p = P + w cos mz cos {nt - id) 
/ T Hr 
r I 
with P 
U); 
where g, r } w f and «r are functions of r, each infinitely small, in 
