OP WHICH IS MOVING ROTATION ALLY AND PART IRROTATIONALLY. 373 
whence 
therefore 
But 
b d X t d 
bt dt X y dy 
dA_ b 2 F(\, t) 
dx bXbt 
b 2 F(X, t) 
bXbt 
= f^-rY——■Warbitrary function of x and t 
J X y dy\ XyJ J 
Before the last integration can be performed 
1 d^ 
Xy dy 
X, 
must be expressed as a 
function of X, x, t. If — f be expressed as a function of X, x, t then y can occur in it 
Xy 
only through X. Therefore 
Therefore 
dy 
br bx _ r 
iJai - J 
bX 
bx 
-r =-r+*(*■ *) 
X, 
bX \ X, 
where <t> is the symbol of an arbitrary function, 
therefore 
dA 
dx 
h f bx 
, W(X, t) 
bXbt 
+ x~^ x> ^ 
Hence choosing the arbitrary function \jj (x, t) so that 
^\jj(x,.t) = <t>(x, t) 
dA 
this value of agrees with its known value. 
And 
xfj(x, i) = j(cc, = 
bx 
(X v )* 
t 
y/1 
te= f te (xj; + sh x I(^ +p ( x ’ ‘J+Qfo *) 
