396 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
These are the same equations as those obtained for the continuity of A if Jc=Q, 
whence 
1 _ ab [4/ . a. 3 + & 2 ] 
a&l 
C= n '“ ^- (o 
D: 
Therefore 
(cft — b 2 ) 2 [ ab 
ab 
\a?-b 2 f 
2/1^ 1-2- 
r ' 2 ?/ 2 m 
= const. -\-ab\ f a —cb 
a 2 b 2 
log (\/a 3 +e+\/6' 3 +e)+i.^y^r )(*'—y' 2 ) 
aft /' 
a5 
.c* /b+*)-*)vV+«K*+«) 
/o /o 
y 
a. 2 + e 5 2 + e 
V , 
To examine whether this will make the value of - +Y continuous at the surface. 
P 
Inside it is known that 
2+Y=x*(l<b 2 -^-+^)+y 2 [^~Hir+y^)+ arbitrary function of t 
P 
Outside 
b 2 a 2 b 2 
£ +v= 
-f ■-i((S)' + (<v)') + arbitrar - Y function of 
where <f> is the velocity potential. 
Now as cb is constant, if <£ be expressed as a function of x , y (e is a function of x, y') 
then t can only occur in </> through occurring in x, y . 
Therefore 
d(f) d<f> dx' d(p dy' 
dt dx' dt dy' dt 
u. (by — v'.tbx 
But it has already been shown that the velocities are continuous at the surface. 
Therefore at the surface 
dcj) 
dx' 
7>(b 
d(f> 
dy' 
Therefore at the surface 
=+ V= —2.x' 3 (^~—^dr\ — 2y'' 2 (y ^—arbitrary function of t. 
