398 BELL SYSTEM TECHNICAL JOURNAL 



Ec]uation (8) is a vector integral equation '^ in U. The nucleus or 



kernel of the equation, exp( — r) /r, is s\'mmetrical with respect toanv 

 ^ c ' ' 



two points {xijiZx) and {x^y^i^-i}, the distance between which is r. By 



virtue of this s\mmetr\- the following recijirocal relation is easily 



established." 



If u' — u' (.v,_y,s) is a function satisfying equation (8) zi'licn G = G' = 



G' {x,y,z) and u" = u" {x, y, z) a second function satisfying, (8) when 



G = G" = G" (.V, _v, s), then 



)\u'-G"hW = }(u"-G')iW. (9) 



Coiise(|ucMUl\' since G = F— V<i> 



)"( u' • F")fW - }\u" ■ F')dv = J" -j {u' ■ r*") - (u" ■ v*')(iv. (10) 



Tlu- prodf (if ilic theorem is now reduced to showing that 



)'] (u'-v*")-(""-V*') |dv = (). 



Now integrating 1)\' parts 



( itt' ■ V*")ilv = — I *" 'I'v "' <1^ • 



= — I <!> p (Iv. 



c • 



since, from the eiitiatioiis of conlimiitN', dix' U= p. Hut from 



c 



tin- fimdamc'iital Ticld ciiiiations: 



4;^p'=-v-*'+('^)"'^' 

 whence 



J"-{ (u'-r'i'")-(w"-v'i''i [<i\ = ,'- (""').! i 'i>'v''*"-*"r'-'*'{-<iv, 



and 1)>- ( ireens Theorem, the right hand xolimu- integral is equal to 

 till' surface integral 



AttXc/J I dn dn 



dn dn 



the surface being any surface which totally encloses the volume, and 

 9 '9m denoting differentiation along the normal to the surface. 



'The forimilution of the clertroni.ignctic field equations in this form is ol lon- 

 sidcralile importance. The inlegral equation furnishes a basis for developing eleetrir 

 circuit theory from the fumianienlal field equations. In addition it leads to the 

 solution of problems in wave propagation which can not be directly solved from the 

 \va\e equation itself. 



■ Perhaps the easiest way to prove this proposition is to regard the integral equa- 

 tion as the limit of a set of simultaneous e(|uations, a point of view which forms the 

 basis of Fredholni's researches on integral equations. 



