22 24.] VELOCITY-POTENTIAL. ] 9 



three equations (26) Art. 18 in order by da, db, dc, and adding, 

 we get 



or, with our present notation, 



udx + vdy + wdz = d (c/&amp;gt; + ^) = d(f&amp;gt;, say ; 

 which 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 portion of space which it originally occupied may, in the course 

 of the motion, come to be occupied by matter which did not 

 originally possess this property, and which therefore cannot have 

 acquired it. 



The above theorem, stated in an imperfect form by Lagrange 

 in Section xi. of the Mecanique Analytique, was first placed in its 

 proper light by Cauchy. Other proofs, to be reproduced further 

 on, have since been given by Stokes *, Helmholtz, and Thomson. 

 A careful criticism of Lagrange s and other proofs has been given 

 by Stokes *. 



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

 includes all those where the motion has originated from rest 

 under the action of forces of the kind here supposed; for then we 

 have, initially, 



udx + vdy + wdz = 0, 

 or (j) const. 



Again, if the motion be so slow that the squares and products 

 of u, v, w and their first differential coefficients may be neglected, 

 the equations (2) become 



du dV I dp s - 

 -7; = -j --- -ft &c., &c. : 

 dt dx p dx 



22 



* Camb. Phil Tram. Vol. vm. (1845), p. 305 et seq. 



