'0\ 



JEolotropic Potential. 435 



i.e. J(a y Xj+Xg) is the product of the discriminants required 

 for the two steps. [In terms of A(X) . . . the property is 

 A;, A(X 1 + X 2 ) =A(X 1 ) A ai (X,).] 



Apply this to the potential of a conductor at an external 

 point on \ x . which is 



. _ he j d\ x _ ke f" tl\. 2 



Av f* </X 2 _ke P" c/Xo _. 



'3lMi)Jo y/J(*i,\j ~ 6r i Jo \7J(«iA2)' 



The potential is the same whether the charge is on the 

 original surface, or on the a?olian on which the point lies, or 

 as at once follows on any intermediate seolian. In the 

 application to volume distribution we make the same change 



£romj A to J by writing a as dependent on Xi+X 2 instead 



of \i, and integrating with regard to X 2 . Then a 2 is expressed 

 in terms of a x and X 2 , and through \/J (the form for L being 

 \*d\/\/J) we have the alteration from t to Tj as above. 

 Thus for a volume distribution also, the potential at any point, 

 on an a?olian surface is the same whether the charge is evenly 

 distributed through the volume of the original ellipsoid, or 

 through that of the aeolian in question or any intermediate 

 aeolian, with one total charge for all. cases. 



For conductors the whole meaning is simple, viz. if a 

 conductor is supposed to shrink following a3olian forms and 

 retaining its charge, the potential is not thereby altered at 

 any point external to the forms considered ; and each point 

 in -pace as it passes from the inside to the outside by the 

 shrinking of the conductor attains a potential which then 

 remains unaltered. Thus the energy of any part of the field 

 outside a given ceolian is not altered by the shrinkage, and 

 the whole addition to energy is the energy of the volume 

 which becomes external in the shrinkage. The element 

 added during a small shrinkage is 



I gSXX' . dn . <JS=- [ItX^.dn . tfS 



J 2 J 2 dx 



dn is an element of the normal, + rf<j> is the increment in 

 moving outwards, i. e. —d(f> the positive increment in moving 

 inward-. 



