228 VORTEX MOTION. [CHAP. VII 



expansion (6, say) and of the component angular velocities f, 77, f, 

 at all points of the region. 



For, if possible, let there be two sets of values, u ly v lt w l} 

 and u 2 , v. 2 , w 2 , of the component velocities, each satisfying 



the equations 



du dv dw fi 



1 h i VT- &quot; ........................ (1), 



dx dy dz 



dw dv .. du dw dv du 



dy~Tz = ^&amp;gt; dz~Tx = l7] &amp;gt; dx dy-^ ......... (2) &amp;gt; 



throughout infinite space, and vanishing at infinity. The quantities 



will satisfy (1) and (2) with 0, , 77, each put = 0, and will vanish 

 at infinity. Hence, in virtue of the result above stated, they will 

 everywhere vanish, and there is only one possible motion satisfying 

 the given conditions. 



In the same way we can shew that the motion of a fluid occupying any 

 limited simply-connected region is determinate when we know the values of 

 the expansion, and of the component rotations, at every point of the region, 

 and the value of the normal velocity at every point of the boundary. In the 

 case of a multiply-connected region we must add to the above data the values 

 of the circulations in the several independent circuits of the region. 



145. If, in the case of infinite space, the quantities 6, f, 77, f 

 all vanish beyond some finite distance of the origin, the complete 

 determination of u, v, w in terms of them can be effected as 

 follows*. 



The component velocities (u lt v l} w lt say) due to the ex 

 pansion can be written down at once from Art. 56 (1), it being 

 evident that the expansion 6 in an element bx &y Sz is equivalent 

 to a simple source of strength l/4?r . Sw Sy Bz . We thus obtain 



where &amp;lt;D =-dx dy dz .................. (2), 



* The investigation which follows is substantially that given by von Helmholtz. 

 The kinematical problem in question was first solved, in a slightly different 

 manner, by Stokes, &quot;On the Dynamical Theory of Diffraction,&quot; Camb. Trans., 

 t. ix. (1849), Math, and Phys. Papers, t. ii., pp. 254.... 



