374 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
where P and Q are arbitrary functions introduced in consequence of the change in 
the order of integration. 
Therefore 
+P(X, t) + Q(x, t). 
•y! t 
Comparing now ~ and with their known values, it will follow that = 0, 
therefore Q(x, t) is a function of t only and may be considered to be included 
in P(\, t)> 
§F(A, t ) 
As 
St 
may also be considered to be included in it, it follows'"' that 
A=K(X, «)+(W^ 
[This form of A may be obtained much more conveniently thus. 
Since \t+v\ x +v\ !/ =0 and u=^, uy it follows that 
dy ’ 
dA dA _ 
h + dy K ~d^^-°' 
therefore 
A=K(X, t)+\Sx(^, 
The same way of obtaining A is applicable to Arts. 5 and 8.—August 30th, 1884.] 
* The form in which A appears does not appear to be related to y and —x in the same wav, as would 
be expected. 
But denoting differentials of X, y, t by cX, cy, St, it may be shown that 
The two forms of A will agree if 
Suppose that 
But 
therefore 
whence 
therefore 
j Sx + 1 cy = a function of X, t. 
f /XA* oR \t 
pwr =R - thcn ¥ = v 
d S d S S d S £ 6 d X y d 
dx =Xx ~$X’ dy = Sy + X ^SX’ dt~St + Xi! 6X’ whence E^ = dy~\x dx 
dR \ y dR_ \t 
dy \ x dx ~\ x 
R =-j te (v)A ,t(x ’' ) 
bCf+MM); 
so that the two forms agree. 
