390 
PROFESSOR M. J. M. HTLL ON THE MOTION OF FLUID, PART 
become 
dt 
,w 
i 
2 fx' 
i+ & 2 
dx 
" a 2 
dy') 
. W 
d 
2/Y 
d\ 
i -1- & 2 
dxd 
a. 2 
dy'r 
therefore 
sin )+ cos ^ tx ' 
(4-fl 4 \ / 
- cos cbt.y'l ^- + eu 3 j + sill (bt.x [ — 
therefore 
x 
y 
therefore 
VI, # 
« 2 4 a 2 
.Li + Vi 2 
= - cos &>£-4~+V 
dx\p 
= + sin +V 
dx\p 
a 2 Z> 2 ^ a 2 
- 3 
a 2 6 2+ a 2 
sin 
d y\p 
. d Ip , tt 
cos <y£— -4- V 
dy\p 
-I(f+ V ) 
-^ +v 
=-J^+ Y 
_ ^ +Y 
-+Y=a: ,2 ^ct> 2 —^^+^^^+2/' 3 ^a> 3 —an arbitrary function of the time. 
To find x 
The auxiliary system of equations to find x is 
dt dx dy dx 
1 u v 
i(M 2 +« 0 )---Y 
P 
V 
Substituting for u, v, —+Y their values, there may be substituted for these the 
equations 
One integral is 
Another is 
dt dx' 
dy' 
d x 
J 'W a? \b 2 cd )\ a? f i 2 J 
,J2 2 
r x . rV 
ccb . 
fx' / f 
