1890.] A. Mukhopadhyay — HydroKneiic Circulation. 59 



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

 two similar equations, and we have 



<ffl" dE dH _ 



dx dy dz 



leading to the subsidiary system 



dx dy dz 



J = V ~1 



which denote vortex lines. Hence, we see that it is p'ossible to construct 

 a family of surfaces 



H = constant, 

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

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

 steady rotational motion we have 



/ 



**"s+4M>-'» 



along a vortex line. 



IV. — Note on Stokes's Theorem and Hydrolcinetic Circulation. 

 By Asutosh Mukhopadhyay, M. A., F. R. A. S., F. R. S. E. 



[Received March 24tli ;— Eead April 3rd, 1889.] 



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

 for hydrokinetic circulation 



j (ndx + vdy + wdz) = 2 I j (I i + 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<f> + \ d-%, 



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



j (udx + vdy + ivdz) = | > d%. 



But, the value of 



\d X 



f 



