Mr. A. Schuster's Electrical Notes. 15 



therefore led to find the vector potential for a system of 

 currents, 



_ clef) _ d(f> _ def) 

 dx' ~ dy' ~~ dz> 



when </> is a homogeneous function of degree n. 

 The following transformation will be useful : — 



v d4 = d4 vV + 2 twdfy + ^r 2 <P<ft 



d# c?a? \<i^ 6?^ u2 <i// dx dy 



c/2 c/^' c/* / rf* 



a# V akr J dydx dzdx] 



dydi 

 Writing 



we may satisfy the condition V 2 F' = — 47r -~ by 



2w + 1 tfa? ' 



(13) 



with similar expressions for G' and H'. 



Butthese expressions do not satisfy the characteristic equation 

 for the vector potential, and it is easily seen that the problem 

 is indeterminate as long as we do not fix the manner in which 

 the system of currents is completed at the origin. This will 

 have to be determined in each case from the nature of the 

 problem. From (13) we easily obtain 



d¥' dG' dE! 



+ -r- + 



-n<t>. 



dx drj dz tin 



If F 1? Gi, Hi are quantities satisfying the condition 



and 



d¥ 1 dG, dH, 4tt 



— , 1- -j h -7- = -> 7" • n 9' 



dx dy dz In -\- 1 



The components F+F 1? G' + G 15 H' + H 15 will satisfy all 

 necessary conditions for the vector potential. The solution 



