1890.] A. Mukliopadliyay — HydroJrinetic Equations. 57 



In the first case, for irrotational motion, the components of mole- 

 cular rotation £, -q, £ vanish, implying the equations 



where 



Hence, the required first integral is 



u = 



d4> 

 dx 



dp 



dy 



w = 



d<p 

 dz 



otioi 



1 reduce to 







dU 

 dx 



= 0, 



£-* 



dU 



dz 



= 





U 



-2+* 





/ 



p A at 



where F is ordinarily a f auction of the time, but for steady motion an 

 absolute constant throughout the liquid. 



Secondly, if the motion is rotational but steady, we have 



*U,o, - = o, $ = 



dt ' dt ' dt 

 and the equations of motion lead to 



dB dB dB n 



u — — \r v —— + w — - = U 

 dx dy dz 



_dB dB „ cZZ2 rt 

 £ — + ??— + £ — = 0. 

 dx dy dz 



These linear differential equations lead, by Laplaces's method, to the 



subsidiary systems 



dx dy dz 



U V w 



dx dy dz 



1 = v " 7 



which denote respectively stream lines and vortex lines. Hence, it is 

 possible to construct a series of surfaces 



B = constant 

 each of which shall be covered over with a net work of stream lines 

 and vortex lines. Hence for steady rotational motion we have 



/ 



— + V + x qfi = constant, 

 P 2 



the constant being an absolute constant so long as we pass from point 

 to point on a stream line or vortex line, but which varies as we pass 

 from one stream line to another or from one vortex line to another. 

 8 



