OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 377 
therefore 
But 
therefore 
v x — Uy=y (R sin $+0 cos 9) — f (R cos 9 —0 sin 9) 
CIX 
= R sin 0+0 cos 0)-\-0 x ^(R sin 0+© cos 6) 
—r y ^(R cos 0—0 sin 9) — 0^(R cos 0 — 0 sin 9) 
_d& 1 dR © 
dr r dO ~ r 
If d . . rfR 
V ‘~ U ’=r{ Jr^-TS 
To determine the current function A there are the equations 
1 dA 1 /t 2 F(A., t) b f rbr \ 
r d9 r e \ bXbt bt] (\ e )r) 
dA X t v /t 2 F(A, t) b T rbr \ 
dr V \ e '\ bXbt btj(\ e )r) 
From the first of these 
A= 
&F(\, t) 
bt 
where xp is an arbitrary function. 
Therefore 
But 
whence 
dA _b*F(\t) 
dr bXbt 
d_ 8 
dr bt 
+*>(>■> t) 
A 
dr 
d__ 
dd 
A . x — 
x A 
—-- + A 
b 
bX 
b d X t d 
bt dt X e d9 
3 c 
MDCCCLXXXIV, 
