58 A. Mukhopadhyay — Hydrokinetic Equations. [No. 1, 



Thirdly, if the motion of the liquid is perfectly general, neither 

 steady nor irrotational, we may put, after Clebsch, 



udx + vdy + wdz = d<p + X dx- 



Observe for a moment that as this simply signifies that the differential 

 expression on the lefthand side, when not a perfect differential may be 

 resolved into two, one of which is so, and the other may be made so by 

 means of an integrating factor, the legitimacy of the transformation is 

 selfevident. We have then 



u = C ^+ A^ v = ^ + A^ 

 dx dx dy dy 



w = — + A — 

 dz dz' 



furnishing the known expressions 



dy dz dz dy 



<ft *X 



dX d x 



8 * -■*£." 



dx dz 



2£ = — -^ - 

 dx dy 



dX d x 



dy dx 



These lead to the equations 



■ dX dX dX _ 



dx dy dz 



dx dy dz 



both of which give the subsidiary system 

 dx dy dz 



the differential equation of vortex lines. Hence the vortex lines are 

 obtained as the intersection of the surfaces X = constant, x = constant. 

 Again, the value of u gives 



du _ d ( d<$> d\\ 

 dt ~~dx\~di ~dt) 



dX d\ dX d x 

 dt dx dx dt' 



Substituting in equation (1), we have at once 

 dE SA dx _ S x dX _ 

 dx 8t dx 8t dx 



where 



/dp 

 P 



'T+3+^+i* 



