60 A. Mukliopadhyay — Hydrokinetic Circulation. [No. 1, 



taken round the closed curve is clearly equal to tlie sum of the values of 



d\ dx 



ff 



taken round the projections of the closed curve on the coordinate planes. 

 Now, for the projected curve on tke coordinate plane of yz, we have at 

 once from the ordinary formulae for the transformation of multiple 

 integrals, 



If 



dX dx 



=ffm-m^ 



The projected curves on the other two coordinate planes lead to two 

 similar expressions. Hence, the circulation round the given closed 

 curve is furnished by 



(udx + vdy -\- wdz) 

 d\ dx d\ dx\ 



I 



J J \dydz dzdyj J 



But, as an immediate consequence of Clebsch's transformation, we have 



„ =. *$ + x dx 



dx dx 



dy dy 



d<t> d X 



dz dz 



■whence 



ot — *f _ ^1 — ^ *^X dXd\ 

 dy dz dy dz dz dy 



2 V 



du dw d\ dx dX dx 

 dz dx dz dx dx dz 



nj-_.dv^^du_d\dx dX dx 

 dx dy dx dy dy dx 



