191, 192] DIELECTRICS AND MAGNETIZABLE BODIES. 369 



field F Q . We shall now examine in what cases the introduction 

 of a polarizable body into a uniform field will produce such a 

 resultant field that the polarization of the body will be uniform. 

 Let the potential of the undisturbed field be F and the potential 

 of the forces due to the induced polarization be F^, so that the 

 total potential of the field is 



(i) F=F,+ F i . 



If X Q , F , ZQ denote the constant components of the force of 

 the undisturbed field, we have 



(2) V=C 



Let a, 13, 7, be the constant direction cosines of the uniform 

 polarization, so that 



(3) ^ = /> B = I/3, C=Iy. 



Then since / = icF we must have for the total potential 



(4) V=C'-Xx-Yy-Zz = C'-- K (<w + py + vz). 



But we have seen in 123 (6), that if fl be the potential of a 

 single distribution of density unity occupying the space filled by 

 the polarized body we have for the potential due to the polariza- 

 tion 



Consequently if we put for II 

 (6) n = C" - i [Lx* + Mf + N*} , 



so that 

 (;) V t = / (Laos + Mfty + Nyz) = LAx + MBy + NCz, 



(8) V=V+V i = C+(LA-X )x + (MB-Y )y + (NC-Z )z, 

 all our conditions will be fulfilled by taking 



(9) Z^-Z = --~, Z5-F = --, LC-Z, = --.-- 



1C 1C tC 



Now the only body for which H has the form of a quadratic 



function of the sort given is an ellipsoid. The values of the 



constants L y M, N in terms of the axes are given in 113. We 



have accordingly found that an ellipsoid introduced into a uniform 



w. E. 24 



