528 Mr. J. Larmor. [May 14, 



where 



F = ( dr. G = (L dr' t H = tl rfr". 



K, d0 , , d<t> , ,d0\ , , 

 tt ?.+ V Z--I.V} !- \(ir ; 

 dx'^ dy dz ) 



in these formulae the accents may now be dropped, as the integrals 

 are extended over the whole system. 



It is through this function x that the indeterminateness enters 

 into the equations of electrodynamics. In a certain class of cases the 

 function may be expressed in another form, which is useful in the 

 subsequent analysis. By integration by parts throughout space, we 

 obtain 



(d A 



= J ' *' 



provided we can neglect* tte surface integral over the infinite sphere ; 

 and this we can do, if the system is confined to a finite region and <p 

 contains only inverse powers of r, or it may be direct powers of r 

 when there is no total current flow to infinity. Thus 



if, , d 

 X= -- H T 0V 2 -TT 

 4jr J at 



dV 

 TT 

 at 



dV 



by Green's theorem, provided the surface integrals vanish as before. 

 In this equation, 



so that, on von Helmholtz'a assumption 



0(r) = Cr, 

 we hare V 2 = 2C/r ; 



. 1 (*, 20 dV 



and therefore v = -- I dt - 



4rj r dt 



so that V ? X = 20-^, 



dt 



a result to be used immediately. 



