226 



VORTEX MOTION. 



[CHAP. VII 



i.e. the element will still form part of a vortex-line, and its length (8s, say) will 

 vary as o>/p, where o> is the resultant angular velocity. But if a- be the cross- 

 section of a vortex-filament having 8s as axis, the product parSs is constant with 

 regard to the time. Hence the strength <ao- of the vortex is constant*. 



The proof given originally by von Helmholtz depends on a system of three 

 equations which, when generalized so as to apply to any fluid in which p is a 

 function of p only, become 



J>_(?\_tdu .vdu (du 

 Dt\p)~pdx pdy pcfe' 



Dt 



dv 77 dv f dv 

 p dx p dy p dz ' 



(dw 

 ~ 



(m)t. 



These may be obtained as follows. The dynamical equations of Art. 6 

 may be written, when a force-potential Q exists, in the forms 



provided 





where q 2 = u 2 + v 2 + w 2 . From the second and third of these we obtain, elimina- 

 ting x by cross-differentiation, 



du 



Remembering the relation 



and the equation of continuity 

 Dp 



du dv dw 



++ 



(vn) ' 



we easily deduce the first of equations (iii). 



To interpret these equations we take, at time t, a linear element whose 

 projections on the coordinate axes are 



&r = e/p, dy = 7/p, &z = c{/p .................. (viii), 



where e is infinitesimal. If this element be supposed to move with the fluid, 



* See Nanson, Mess, of Math. t. iii., p. 120 (1874); Kirchhoff, Mechanik, Leipzig. 

 1876 ..., c. xv.; Stokes, Math, and Phys. Papers, t. ii., p. 47 (1883). 

 t Nanson, I. c, 



