1890.] A. Mukhopadliyay — Hydrohinetic Circulation. 59 



and 8 denotes particle differentiation. Equations (2; and (3) lead to 

 two similar equations, and we have 



dx dij dz 



leading to the subsidiary system 



dx dy dz 



which denote vortex lines. Hence, we see that it is possible to construct 

 a family of surfaces 



JB" = constant, 

 covered over by vortex lines, and the mode of integration shows imme- 

 diately tliat the constant is a function of the time alone. Therefore, for 

 steady rotational motion we have 



/ 



*+''+l+4M«-''(') 



along a vortex line. 



IV. — Note on StoJces's Theorem and HydroJcinetic Circulation. 

 By AsuTOSH MuKHOPADHTAY, M. A., F. R. A. S., F. R. S. E. 



[Received March 24th ;— Read April 3rd, 1889.] 



The object of this note is to give a new proof of Stokes's formula 

 for hydrokinetic circulation 



j (udx + vdy + wdz) = 2 j j {H ■{- m rj -\- n ^ d S, 



and to point out how it is an immediate consequence of the theory of 

 the change of the variables in a multiple integral. 

 Assume, after Clebsch, 



udx + vdy + wdz = d<p -{■ \ d-x, 



so that the integration being performed round a closed curve, we have 



I {tidx 4- vdy -\- ivdz) = ( ^ ^X* 

 But, the value of 



f 



