226 



VORTEX MOTION. 



[CHAP. VII 



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

 vary as eo/p, where o&amp;gt; is the resultant angular velocity. But if o- be the cross- 

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

 regard to the time. Hence the strength coo- 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 



Dt ... 



D (\ dw T) dw dw 

 Dt \p) p dx p dy p dz 



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* = u z + v 2 + w 2 . From the second and third of these we obtain, elimina 

 ting x by cross-differentiation, 



_,,,^^,_.&amp;lt; fa . ** - * 



dt dy dz 



Remembering the relation 



dz 



(vi), 



and the equation of continuity 



Dp fdu dv dw 

 Dt \dx dy 



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, 8y=(r)/p, 8z = f/p (viii)&amp;gt; 



where 6 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. 



