1884.] which is moving Rotationally and part lrrotationally. 277 



from the equation which gives A, as a function of x, y, t. Let the 

 result be denoted hy(^^J* . 



Let differentials of X, x, t, regarded as independent variables, be BX, 

 Bx, Bt respectively. 



Thus / dX V- is an integral taken with regard to x, X and t being- 



considered constant. 



Bx 



Bt f d\\ K is the differential coefficient of the above integral with 



regard to t, X and x being considered constant. 

 ■dX-\ \ dX 

 dt . 



denotes that ^ is expressed as a function of X, x, t, and 



dy 



the result integrated with regard to x, X and t being considered 

 constant. 



F and H are the symbols of arbitrary functions. 

 Then the equation in X is shown to be — 



dX^ 



\dx^ dy*) 



Bt 



dt_ 

 dX 



Ay J 



=H 



hi"-). 



ex 



\dy)< 



and the current function is 



B¥(X,t) 



bt 



+ \Bx 



( dX\ 

 dt 



dy ) 



For a vortex of invariable form moving parallel to the axis of y 

 dY 



with velocity — (where Y is a function of t only) the equation in X 

 becomes — 



W dy*) Bt 



and the current function is 



X, 



B¥(X,t) 



sx 



Bt 1 dt' 



(2.) Plane motion, referred to polar co-ordinates r, 6. 



