300 ELECTROSTATICS. [PT. II. CH. VII. 



Forming the derivatives for y and z, writing 



9 2 a 2 a 2 



and comparing with the expression for A V above, we get 



But I / 1' is harmonic except at 0, and therefore 

 (8) L 



If we put 



(9) 



e/ ' 



and we get the proposition that if V is a harmonic function of the 

 point M, then V is a harmonic function of the corresponding 

 point M'. If the distribution causing F is distributed continu- 

 ously in three dimensions, the density is p A F/4?r and in the 

 image the density is p = A 7 F'/47r so that 



> * -*-* " 



If ds and ds be corresponding infinitesimal arcs, expressing ds 

 in terms of the curvilinear coordinates x, y', z 



dx' z dv' 2 dzf* I 4 , l*ds* 



so that we have for the ratios of corresponding infinitesimal arcs, 

 surfaces, and volumes 



^_E 2 _Z^ dS'_&_l^ dT^_It? = l^ 

 Ts~l*~R*' dS~l*~R*' dr~l~W 



The ratio of charges of corresponding infinitesimal volumes is 



de' _p'dr' _R_l f 

 de~ pdr == I ~ R" 



and of the surface densities 



v__<M_ dS_]^ = R? 

 <r~ d$ de~R 3 ~ I'*' 



