384 
PROFESSOR M. J. M. HILL OR THE MOTIOH OF FLUID, PART 
Whence 
\ d^fr 
y- Y 
) dx 
" 2/ 
x* _ (2/-Y) 2 \4_ x 
^ V )dy 2/ 
Substituting for u and v in the dynamical equations 
p t 2 / 2 ■ • 
-j- (x 2 + (y —Y) 2 ) — yY + an arbitrary function of t which need not be considered. 
Now the equation determining y which is 
d X , 2/Qy-Y) ^ /y_^W 
(ft & 2 dx \ « 2 / dy 
= 2n0-|)g-'qY)-|A+(;/-Y)Y+ a function oft. 
becomes 
The integrals of the auxiliary system are 
, ,,/a : 2 , <J/-Y) 2 \ 
—j= const. =ra 
ah . _~.cc / f 
t — — sin A/ = const. = ?i 
2/ a v m 
in ~ (L—n)-{-b \JyY cos a function of t. 
O 7 0 
m a~—b~ 
X ~T~ab~ Sm 
Hence one value of the integral of the equation in y. which may be called y', is 
obtained by substituting for m and n their values, and is 
Whence 
But 
= 2 yY(!/-Y) + ’%-Y) 
u=f 
a 2 b°~ 
2-i 
a*b* 
2(y-Y) 
6 2 
T y 2/.« 
V = Y-^7T 
