Prof. Helmholtz on Integrals expressing Vortex-motion. 497 



Equations (2) are thus satisfied if it can be shown that in the 

 complete space S x 



dx \dy dz 

 That this is the case is seen from equations (5 a), 



or by partial integration 



4N 



dz 



rMS7**-*SB'%***- 



Adding these three equations, and again putting dco for the 

 surface-element of S, we have 



dL dM dN 1 C ty Q „ 



ZvJJJ r\da db ,dc) 

 But throughout the entire space 



* + *£+*«0 (2a) 



da do dc 



And over the whole surface, 



f a cosa + ?; ft cos/3 + f a cos7=0. . . (2 b) 



Both integrals therefore vanish, and equations (5 b) are satisfied 

 as well as (2). (4) and (5) or (5 a) are thus integrals of (1) 4 

 and (2). 



The analogy, mentioned in the introduction, between the dis- 

 tance-action of vortex-filaments and the electromagnetic action of 

 current- conducting wires, which gives a very good means of ex- 

 hibiting the form of vortex-motions, is deducible from these 

 theorems. 



If we put in (4) the values of L, M, N from (5 a), and denote 

 by Aw, At', Aw the indefinitely small elements of u, v, w which 

 in the integrals result from the element da db dc, also their 

 resultant by A/?, we have 



