378 PROFESSOR M. J. M. HILL OH THE MOTION OF FLUID, PART 
therefore 
dA t 2 F(X, l). b" f. . f rbr 
dr b\bt 
br b , , v 
—+!:#•> 0 
£> 2 F(X, £)x *>f ^X x b f rbr , b lf ^ 
-~^r K ~ (X ;,j ' w ’ * 
But 
b f rb\ 
^ (X<j)r 
f ,x b( 1 \ ( K Jd \ d\l _(rb\ d_f \ 
= L J \dd )\ 9 ~J X, d0\ X 9 
: £ 
where <F is an arbitrary function. 
Therefore 
dr 
= X, 
/t 2 F(X, t) 
b f rbr \ 
\ b\bt 
bt\ (X fl )r / 
Choosing the arbitrary function xfj(r, t) so that — t)=<$(r, t), the value of 
rees 1 
And since 
agrees with its known value 
© 
•Mr, t )=\i >,•*(,■, o={^{^+K(£)f = f rSr (*0; + sH§ +p k ° +Q(r ' t] 
A=^A+frir(^ +P(X, «)+QF. 0 
therefore 
and reasoning as in Art. 3 it follows that 
A=K(X, i) + jrdr 
X 
XJ t 
And since v T — u y is a function of X, g, it follows that 
f?A a 
dx^ dy~ 
d? A .d 2 A r 
S+^ = H ^ 9i 
therefore 
6. Now take the case of a vortex of invariable form rotating about the origin. 
Let its equation be 
X=L(r, 6 —<y) 
where a is a function of t only. 
a* I 
