444 
Proceedings of the Poyal Society 
.dx 
dp dx .dx .dx . dx 
dx dt + * dx + ^ dy + Z dz 
d P_ = ty dy dy 
dy dt dx ^ dy dz 
dp dz dz dz . dz 
— 4 - or. — 4 - ?/ — 
dy 
~ + * j — v y ~r~ + 2 ~y 
dz dt dx dy dz 
and 
dx dy dz 
dx + dy + dz ® 
To transform to the columnar co-ordinates we have 
x = r cos 0 , y = r sin 6 
x = f cos 6 -rQ sin 6 
y = f sin O + rO cos 0 
d d . _ d 
Tx = ao&d Tr-^ d m 
d_ 
dy 
. J d 
sm^i- + cos 6 — 77 * 
dr rdO 
The transformed equations are 
dp _ dr _ dr (rO ) 2 a dr .dr 
dr dt **" r dr 
+ 0 + 
r dO dz 
dp dd d(r$) .7 a dir 6 ) . d(rO) 
de =r Tt +rA P + re + eA ^ +zA P f 
dr 
dQ 
dz 
dp dz 
dz 
dz 
dz 
dz - dt + *dr + d dd + i dz 
and 
dr r d(rO) +dz ^ 
dr + r + rdr 
dz 
(i). 
( 2 )- 
( 3 ), 
W, 
( 5 ). 
Now let the motion be approximately in circles round 0 z, with 
velocity everywhere approximately equal to T, a function of r ; and 
to fulfil these conditions assume 
r = £ cos mz sin ( nt - i6) ; rO = T + r cos mz cos (nt - iO) 
z — w sin mz sin (nt - i6) \ p = P + «r cos mz cos (nt - iO) 
J'THr 
with P = 
( 6 ); 
where g, r, w> and » are functions of r, each infinitely small, in 
