OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 371 
Whence each of these 
\ x dx -f - \jdy 
0 
Since, in the differential equation, there are no differential coefficients with regard 
to t, X=const. is one integral of the auxiliary system. 
By means of the equation X — const., y can be expressed as a function of the constant, 
x, and t. As the constant will have to be replaced by X afterwards, it may be said 
that y can be expressed as a function of \,x,t; and when this value of y is substituted 
in —X y , let the result be denoted by (—X y )*. Let differentials of the variables X, x, t 
regarded as independent be denoted by b\, bx, bt respectively. 
Then g— — an arbitrary function of X, t. 
■ (Nb 
It will be convenient to write the arbitrary function in the form 
Therefore 
b: FQ, t) 
bX 
9 
= -.f 
bx , XF(\, t) 
bX 
Therefore g will appear in the form G(X, x, t) -}- * 
The equations of the vortex sheets are functions of r//, X and therefore of g, X. 
Now 
and 
therefore 
that is 
therefore 
But 
therefore 
X/+ >iX J --[-vX !/ — 0 
g t +ug,+vg y =Q 
bG , bG x , X 2 F(\, t ) , t 2 F(X, t ) 
\ 
bG . bG 
« + n x,+ 
btbX 
bX 2 
+ M, V + a 
x ->+^v M+w x '+“^ m=° 
bG b 2 F(A, t) bG (bG b 2 F(A, t)\ / \ i \ \ a 
bX bitir +«*+U+)^+^+A)= 0 
bt btbX 1 bx 
bG_ 
bx 
u=X 
bG b 2 F(A, t) 
A btbtbX 
3 b 2 
