PAPER BY PROF, HELMHOLTZ. 43 



arbitrary function A: must be so determined that the boundary con- 

 ditions are satisfied, a problem whose difficulty is similar to those 

 [difficulties that are met with iu problems] on the distribution of elec- 

 tricity and magnetism. 



That the values of u^ v, and w. given iu equation (4), satisfy the 

 condition (14), is seen at once by differentiation and by considering the 

 fourth of equations (5). 



Further, we find by differentiation of equations (4), and considering 

 the first three of equations (5) that : 



clx dz ~ ""'^ :)y [_\\v '^Jy'^ Jz J 



?y ?s - ' ^zl M- ^ ?y "^ ?z J 



The equations (2) are also equally satisfied when it can be shown that 

 throughout the whole region S] we have 



•f+f+f;=« (-) 



That this is the case results from the equations {5a) which give 



or after partial integration 



;)x 27r \ \ r 2;r r ^a 



^ = TV- \ — da dc — i^ \ - . ~ da db dc 

 dy 27r\ \ r 2;r r J)& 



^^= L f [^dadb - ^ [ [ [ ~ . '^^-^ da db dc. 

 Jz 27t \ \ r 2/H I r Jc 



If we add these three equations aud again indicate by dcj the element 

 of the surface of S, we obtain : 



^ + i;; + -^ = <t;: (^„ cos « + ;/„ cos /i + ^„ cos r),7<^^ 

 cix oy i}~ -i7i \ I 



27r rV, M ^b Jc / 



But since throughout the whole interior of the space >Siwe have 



^,lVa,^Ja^Q (2a) 



?a ^ ^b ^ ^c ^ 



and since for the whole surface we have 



$a COS a + ?/„ COS /i + Ca COS ;' = (If/;) 



