35-37] VELOCITY-POTENTIAL. 41 



at every point of the region. Hence </> is now subject to mathe- 

 matical conditions identical with those satisfied by the potential of 

 masses attracting or repelling according to the law of the inverse 

 square of the distance, at all points external to such masses; so 

 that many of the results proved in the theories of Attractions, 

 Electrostatics, Magnetism, and the Steady Flow of Heat, have also 

 a hydrodynamical application. We proceed to develope those 

 which are most important from this point of view. 



In any case of motion of an incompressible fluid the surface- 

 integral of the normal velocity taken over any surface, open or 

 closed, is conveniently called the 'flux' across that surface. It is 

 of course equal to the volume of fluid crossing the surface per unit 

 time. 



When the motion is irrotational, the flux is given by 



'd<t> 



-si 



, dS, 

 dn 



where SS is an element of the surface, and &n an element of the 

 normal to it, drawn in the proper direction. In any region 

 occupied wholly by liquid, the total flux across the boundary & 



A 



zero, i.e. 



the element &n of the normal being drawn always on one side (say 

 inwards), and the integration extending over the whole boundary. 

 This may be regarded as a generalized form of the equation of 

 continuity (1). 



The lines of motion drawn through the various points of an 

 infinitesimal circuit define a tube, which may be called a tube of 

 flow. The product of the velocity (q) into the cross-section (&, say) 

 is the same at all points of such a tube. 



We may, if we choose, regard the whole space occupied by the 

 fluid as made up of tubes of flow, and suppose the size of the tubes 

 so adjusted that the product qcr is the same for each. The flux 

 across any surface is then proportional to the number of tubes 

 which cross it. If the surface be closed, the equation (2) ex- 

 presses the fact that as many tubes cross the surface inwards as 

 outwards. Hence a line of motion cannot begin or end at a point 

 of the fluid. 



