OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 375 
But v x —u y =f(\, i {j)= some function of X, g, 
therefore 
dtf^~ dtf n ( X ’ ^ 
where H is an arbitrary function 
therefore 
dx( [ = H 
tF(\, t) 
r bx 
bx 
W 
The arbitrary functions K(X, t) and - ^ being implicitly contained in the integrals 
need not be expressed. 
4 . Now suppose that there exists a vortex of invariable form which moves with 
arbitrary velocity along the axis of y. Let the equation to its surface be a function of 
x, and y —Y only, where Y is an arbitrary function of t only. 
Let 
X=L(«, y—Y) 
then 
_dL 
f/ ~dy 
therefore 
Y __ __y 
Also y—Y can be expressed as a function of X, x only from the equation 
X=L(a:, y—Y). 
Therefore \ can be expressed as a function of X, x only, not t. 
Therefore G(X, x, £)= ——-7 does not contain t, since (X^)* does not contain t, and 
J (Y)f * 
may now be written (X^. 
Therefore 
SG 
St 
= 0 
In this case the equation takes the form 
±( - +4-/ x/%L =H( X, 
dx 
sx 
dy 
Sx 
f 
bF(X. t) 
J WJ 
^ b\ 
£+£)k ( m)=h(x, 
Sx 6F(X, t) 
bx 
therefore 
