JEolotropic Potential. 439 



There is a maximum question relating to the amount of 

 energy when different shapes of ellipsoid with the same 

 volume are considered. The condition of no variation is 



2,- da—K). subject to 2, — =0. u e. a 7 =l> ,, =c y-, or 



da ' a da df> etc 



assumed that the translation must be along a principal axis 



2 say, the equation (95) makes ^ = Loa 2 =M & 8 =N c 9 . The 



other relation of (95) gives L 4- M + N (l — /•*) = 2, and when 

 these are combined 6 = cf) {a-' 2 + l>- 2 +i-~ 2 (l — r 2 )). Thus the 

 transcendental character has disappeared from the integral 

 JdX x J', and therefore J' has the form ^l + A/) 3 (l-r 2 ). 

 The conditions are satisfied by a = b and c 2 = « 2 (l — >' 2 ). and 



these conditions make 6 = 2a 2 , L U = M = X U (1 — r 2 )= -. 



o 



This i- the spheroid which has in other respects the status of 



the sphere in statical potential. 



Again, reference was made above § 27) to the fraction of 

 the whole energy which, for a case of volume distribution, 

 belongs to the field-integral within the ellipsoid. When 

 the translation is along a principal axis this fraction is 

 L u 2 rt 2 + M ? /' 2 + X 2 c 2 (l — r Q ) : -l$rt [the quadric is more compli- 

 cated in the general case] . This ratio is also a maximum for 

 a=b and c~ 2 = a <2 (l — r 2 ), and has then the value 1/6. 



The condition for these two maximum questions is not 

 consistent with that for stability as regards the effect of 

 translation. They may be of moment when losses or 

 accretion- of charge arc in view; or the conflict of conditions 

 on the electrical and mechanical sides may lead to a volume 

 distribution which is not uniform. 



§29. We may now deal briefly with special cases, the 

 simpler exact cases and the general approximation. J 7 i< at 

 one divisible into factors (1) for the >phere, viz. 



J'=(a« + \)*{a*+X(l-Sp*)}, 



(2) for any ellipsoid with translation in the direction of a 

 principal axis : — Thus for p=zq=zO, 



The integrations for (2) can be effected when a = l>. and as 

 this case has been treated, it is sufficient to schedule the 



