JEolotropic Potential. 431 



Therefore 



+=£r*=*tr*£ (89) 



9 Bt J a v /J 6j A v/j' v ; 



the second, with e = pr , written for comparison with a 

 volume-distribution. The field-integral J £SXX'fl?T, extend- 

 ing to the whole space outside the conductor depends on the 

 surface value of cf>, and is 



^-^•=i27 fl J„ vJ XJ.TJ 5, • ( " 0) 



§24. For the volume-distribution write 



^ = <£ — SLa? 2 — 'iSlj'yz, 



in which for external values </> has the functional value just 

 used, and L .. are functions of X which make 



agree with the form found above for an seolian of the ellipsoid, 

 i.e. L=a^, L' = *'<j>, ... Then 



era? 2 ^ v ^ ' r <£& d# 



d*^r _ _2V — d> — — s= — 2N' — 6 — — 

 <:/. b e /// dy"b x dx~&y 



Use the multipliers ^ . . . r' and add, then 



= -22(pL+2p'L')-4^ by (78). 



Thus for the external solution V!^ = 0, 



if 2 (pL+2p'L') + 2(/> = (91) 



If we differentiate (91) with regard to X, and use L = «<£, ... 

 we get %(p*+2p'a') + 2#/<j>=0, which is (79); thus (91) 

 will be satisfied if a proper arrangement as to the constant 



of integration is made. Write then i/r = C I — -^=(1— w a ) for 



J* v J 

 the external potential, and for the internal potential put 



