1890.] A. Mukhopadhyay — HydroJdnetic 'Equations. 57 



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

 cular rotation |, rj, ^ vanish, implying the equations 

 _ dcj) _d<p _ d<t> 



dx^ dy^ dz 



and the equations of motion reduce to 



where 



fE=o, ^=0, !^=o 



dx dy dz 



--t*- 



Hence, the required first integral is 



/ 



p 2^ dt 



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

 absolute constant throughout the liquid. 



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



du _ ^^ _ r\ ^^ _ A 



11' ' Ti^ ' It' 

 and the equations of motion lead to 



dB dB , dB ^ 



U — 1- V -rr'-lr W —r- = 



dx dy dz 



.dB ^ dB ^ ^dB ^ 

 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 



T ~ ^ ~ T 



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 



/ 



dx> 1 



-L j^ Y j^ -gi z=: constant. 



P 



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 



