18-19] VELOCITY-POTENTIAL. 19 



velocity-potential < exists be taken as the origin of time ; we 

 have then 



throughout the portion of the mass in question. Multiplying the 

 equations (2) of Art. 15 in order by da, db, dc, and adding, 



we get 



dx ^ dy -. dz 7 , , 7 . 



j2 dx + -? dy + -j, dz - (u^da + v db + w dc) = - dx, 



or, in the ' Eulerian ' notation, 



udx + vdy + wdz d (fa + %) = d<f>, say. 



Since the upper limit of t in Art. 15 (1) may be positive or negative, 

 this proves the theorem. 



It is to be particularly noticed that this continued existence 

 of a velocity-potential is predicated, not of regions of space, 

 but of portions of matter. A portion of matter for which a 

 velocity-potential exists moves about and carries this property 

 with it, but the part of space which it originally occupied may, 

 in the course of time, come to be occupied by matter which 

 did not originally possess the property, and which therefore 

 cannot have acquired it. 



The class of cases in which a velocity-potential exists in- 

 cludes all those where the motion has originated from rest under 

 the action of forces of the kind here supposed; for then we have, 

 initially, 



Uoda + v db + w dc = 0, 



or <f) const. 



The restrictions under which the above theorem has been proved must 

 be carefully remembered. It is assumed not only that the external forces 

 X, F, Z, estimated at per unit mass, have a potential, but that the density 

 p is either uniform or a function of p only. The latter condition is violated 

 for example, in the case of the convection currents generated by the unequal 

 application of heat to a fluid ; and again, in the wave-motion of a hetero- 

 geneous but incompressible fluid arranged originally in horizontal layers of 

 equal density. Another important case of exception is that of ' electro-magnetic 

 rotations.' 



19. A comparison of the formulae (1) with the equations 

 (2) of Art. 12 leads to a simple physical interpretation of <. 



Any actual state of motion of a liquid, for which a (single-valued) 

 velocity-potential exists, could be produced instantaneously from rest 



22 



