78 Mr Weatherhurn, On the Hydrodjjnarnics of Relatiinty 



K being given by (5'). On substitution of this vabio in (C) the 

 integral of the equation of motion, viz. 



becomes V + ~ \l uC^ + r-c-K.'- = (D). 



§ 7. FloLV and circulation. We define the flow from a point P 

 to another Q, along a path of which ds denotes an element, as the 

 quantity 



[Q 



kV • ds. 

 J p 



Whenever a velocity potential exists this is equal to <^y — (j)^. The 

 circulation round a closed curve is the line integral 



/= /cv.f/s (20) 



taken round that closed curve. This, by Stokes' theorem, is equal 

 to the surface integral 



7 = 1" curl («v).nrf;S' (20') 



taken over any surface drawn in the region and bounded by the 

 closed curve. When the motion is irrotational the integrand is 

 zero, and the circulation round the closed curve vanishes. It 

 follows that, for a simply-connected region, the velocity potential 

 is single- valued. 



III. Vortex Motion. 



I 8. When the vorticity w is not zero the motion will be 

 called vortical or vortex motion. A vortex tube is one bounded bj 

 vortex lines. Considering the portion of a vortex tube betAveei 

 any two cross sections, we find as usual on equating the volume 

 and surface integrals 



= div curl (/cv) dr = I 2w • nd,S, 



that the moment of the vortex tube 1 vj'XidS, Avhere the inte 



gration is extended over the cross section, is the same for all 

 sections. And hence, as in the classical theory, the vortex lines 

 either form closed lines, or else end in the surface of the fluid. 



