142-143] PERSISTENCE OF VORTICES. 225 



One or two independent proofs of the theorem may be briefly indicated. 



Of these perhaps the most conclusive is based upon a slight generalization 

 of some equations given originally by Cauchy in his great memoir on Waves*, 

 and employed by him to demonstrate Lagrange's velocity-potential theorem. 



The equations (2) of Art. 15, yield, on elimination of the function x by 

 cross-differentiation, 



du dx du dx t dv dy dv dy dw dz dw dz dw Q dv Q 



db dc dc db db dc ' 



(where u, v, w have been written in place of dxjdt, dyldt^ dz/dt, respectively), 

 with two symmetrical equations. If in these equations we replace the 

 differential coefficients of u, -y, w with respect to a, b, c, by their values in 

 terms of differential coefficients of the same quantities with respect to .r, y, 2, 

 we obtain 



.(i). 



d(a, b) d(a, b} 



If we multiply these by dx/da, dxjdb, dxjdc^ in order, and add, then, taking 

 account of the Lagrangian equation of continuity (Art. 14(1)) we deduce the 

 first of the following three symmetrical equations : 



p 



+ 

 da db do 



= + + 



p p da p db p Q dc ' J ' 



C = io dz_ + 50 ^ + & d?_ 

 p p da p Q db PQ dc ) 



In the particular case of an incompressible fluid (p = p ) these differ only in 

 the use of the notation , 77, from the equations given by Cauchy. They 

 shew at once that if the initial values , 7/ , of the component rotations 

 vanish for any particle of the fluid, then , 77, are always zero for that 

 particle. This constitutes in fact Cauchy's proof of Lagrange's theorem. 



To interpret (ii) in the general case, let us take at time t = B, linear 

 element coincident with a vortex-line, say 



where is infinitesimal. If we suppose this element to move with the fluid, 

 the equations (ii) shew that its projections on the coordinate axes at any other 

 time will be given by 



8tf = e/p, 8y = 07/p, 8z = e(/p, 



* L c. ante p. 18. 

 L. 15 



