Prof. Helmholtz on Integrals expressing Vortex-motion. 495 



mined, the preceding theorems enable us to determine f, 77, f com- 

 pletely. We shall now consider the problem of finding u 9 v, w 

 from f, 17, £ 



Thus, let there be given within a mass of fluid which includes 

 the space S the values of f, tj, f, which latter satisfy the condition 



doc dy dz ' 



(2a) 



u } v, and w must be found so as to satisfy within the whole space 

 S the conditions 



du dv dw 

 dx dy dz 



o, 



(1), 



dv dw _ ,. 1 

 dz~~dy~~ "*« 

 dw du 



du __dv^ _ j, 

 dy dx 



m 



the bounding 



We require also the necessary conditions for 

 surface of S according to the particular problem. 



According to the given values of %, rj, £, we may have some 

 vortex-filaments which are reentrant within the space S, and also 

 some which reach the boundary of S and then break off. If the 

 latter be the case, we can always continue these filaments along the 

 surface of S or without it till they return into themselves, so 

 that a greater space Sj exists which contains only reentrant vor- 

 tex-filaments. And at the whole of the surface of Sj either f, 

 7j, f and their resultant q are each = 0, or at all events 



f cos a + 1) cos /3 + ?cos y = q cos 9 = 0, 



(2 b) 



where a } /?, y, 6 have the same values as before. We find values 

 of u, v, iv which satisfy (1) 4 and (2) if we put 



_dY t 



dx 



dy 



dM 

 "dz' 3 



_d? 



" dl J 



dL 



dz 



dx 



dV 



dM 



dL 



w = 



dz dx dy 



J 



(4) 



and determine the functions, L, M, N, and P so as to satisfy 

 within the space S x the conditions 



