376 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
r 
and now 
“=^rH( E(x ’ *>-*) 
, = y_ x /_SM) = -|( K (X, i )-.Y) 
therefore 
K(X, t)-x Y 
is the current function. 
Of course this could have been directly deduced from the form of the current func¬ 
tion in the preceding article by putting —Y. 
5. To obtain similar expressions in polar coordinates. 
Let R and © be the radial and tangential velocities. 
Therefore 
u —R cos 6 — © sin 6, v=R sin 0+© cos 6 
„ ^ sin 6 x • n\ i cos0 \ 
\ r = cos c/X r — ~ Kg, h-y— sm vh r -\-—^-k e 
'h-xtyy \/ X P-r - 
\ r xp e — \ e xjj r 
X i r + wX r -b'VX„—Y/—(— RX,+ Y 
J r 
In this case the equation corresponding to the equation in g of Art. 3 is \ r g 6 —X e g r =r. 
The auxiliary system of equations is 
<W dr dg 
X r —\ g r 
Let differentials of the variables r, t when regarded as independent be denoted by 
b\, hr, bt respectively. 
Therefore 
9= 
r rSr &F(\, t ) 
==G(X, r, f)+ 
8F(\, t) 
b\ 
Then since 
it follows that 
X/+ RX,.+ — \ g = 0, and pv+ R 9>~\~ r 9e — 0 
bt + m\ X 
t-V 
