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 in problems] on the distribution of elec- 

 tricity and magnetism. 



That the values of u, v, and w, given in equation (4), satisfy the 

 condition (1 4 ), is seen at once by differentiation and by considering the 

 fourth of equations (5). 



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

 the first three of equations (5) that : 



dz " " dy ' " dx L dx dy dz J 

 dx dz v dy L dx "*" dy dz J 

 dy dz dz L dx + dy dz J 



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

 throughout the whole region Si we have 



^ + £^4.^ = 



dx ~*~ dy 'dz 



That this is the case results from the equations (5a) 



(56) 



d_L 

 dx 



2tz 



[ Za(x-a) dadbd ^ 



t.' «.' t' 



or after partial integration 



dh _ 1 

 dx 



M 



dy 



dz' 



~2tT 



M 



2tt 



- a dbdc 



Z7T 



r 



3s <fo rfc - ^ 



r 27r 



a «dadb- 



2? ? 



n ds, 



)r- da 



fl 3% 

 r ' ^)6 



r ° Jc 



rfa (76 dc 

 da db dc 



da db dc. 



If we add these three equations and again indicate by doo the element 

 of the surface of 8, we obtain : 



& + ^ + *E = Jl f (£ C os a + 7a cos /? + C a cos y)\doo 



drh , cKa W db dCt 



db T <)c/ 



2?r 





But since throughout the whole interior of the space 8 we have 



M? _i_ ^ _i_ ^=? = 



Ja " db dc 



and since for the whole surface we have 



£ o cosa + v« cos / ? + , *« cos > /==0 ■■••••■ 



(2a) 

 (26) 



