150 VORTEX MOTION. [CHAP. VI. 



will satisfy (3), (4), and (5) with 6, f, 77, f, X put each = ; that 

 is to say u, v, w are the components of the irrotational motion 

 of an incompressible fluid occupying a simply-connected region 

 whose boundary is at rest. Hence (Art. 47) these quantities 

 all vanish ; and there is- only- one possible motion satisfying the 

 given conditions. 



The above theorem an extension of one given in Art. 49 is 

 equally true when the region extends to infinity, and (5) is re 

 placed by the condition that the fluid is there at rest. 



129. If, .in the last-mentioned case, all the vortices present 

 are within a finite distance of the origin, the complete determina 

 tion of u, v, w in terms of 0, g, ??, can be effected, as follows*. 



Let us assume 



_dP dN_dM -} 



dx dy dz 



dP dL dN 



ry L 



dy dz dx 

 _ dP dM dL 



W 1 j 7 , 



dz dx dy 



and seek to determine P, L, M, N so as to satisfy (3) and (4) and 

 make u, v, w zero at infinity. We must have in the first place 



^p = e (7). 



Again, 



dw dv d dL dM dN\ . 2J . 



Lr = = -j 1 j -j- I Y Lt. 



(6), 



dy dz dx \dx dy dz J 

 Hence, provided - 



dx dy dz 

 we have 



Now (7) and (9) are satisfied by making P, L } M, N equal to 

 the potentials of distributions of matter whose densities at the 



f) * 



point (x,y,z) are - T- , j~ , g- , s res P ectivel y- This gives 



* See Stokes, Camb. Trans. Vol. ix. (1849), and Helmholtz, C relic, t. LV. (1858). 



