372 PROFESSOR M. J. M. HIRE ON THE MOTION OF FLUID.. PART 
substituting this value of u in the equation 
\f -j- ViX x “t~ I'X^ == 0 
it may be shown that 
X, x fbG , t 3 F(X, t) 
v ~ \ y H + ^ 
To determine the current function A, there are the equations 
dA 
dy' 
b[ bx b~ F(X, t) 
bXbt 
-~Hx~-~Xj \ &J ( x y) r 
b j" bx f t'F(X, t) 
bXbt 
From the first of these equations 
A= 
bt 
where 0 is the symbol of an arbitrary function. 
Now - f— is a function of \ x, t; and y occurs in it only because it is contained 
in X, therefore 
\\ y dy 
' b r bx~ 
^ (Xy)f 
=f 
bX 
b j - bx 
btJ (\ y )x 
= t: f 6\ [x+arbitrary function of x and t 
bt 
(Xy)* 
but the arbitrary function of x and t may be supposed included in xp (x, t ), theiefore 
J (X y )j 
tF(X, 0 
bt 
to 
therefore 
But 
dA X 2 F(X, Ox d bx , d A 
&=^r A '“<fcd ;>x i ( y ) ; + s^ () 
A = A.\ A 
dx bx' x b\ 
d _ b 
dy y bX 
A—Ai\ A 
dt bt' 1 bX 
