Ellipsoidal Shells and of Solid Ellipsoids. 405 



potential equal to that which existed at S 2 before the 

 transference. But the surface S 2 is by hypothesis one over 

 which a distribution can be made of density inversely pro- 

 portional at the different points to the normal step to an 

 adjacent 'surface S 3 , and producing uniform potential in the 

 interior. If the mass in this latter distribution be made the 

 same as that which has been transferred from S„ the potentials 

 at S 2 and within it produced in the two cases will be the 

 same. For it can be proved * that there cannot be two 

 distributions of a given charge of matter over a surface so 



© © 



as to produce uniform potential at all parts on or within the 

 surface. 



The distribution therefore transferred from Si must have 

 the same density as in the other case supposed, that is <r 

 must be inversely proportional to the step from S 3 to S 3 . 

 S 3 is a level surface and the charge can now be transferred 

 to S 3 , when S 4 will be found to be a level surface and so on. 

 Hence Bertrand's theorem is proved. It is easy to construct 

 (see Picard, Traite d' Analyse, tome i.) an analytical proof of 

 the theorem, founding on Lamc ; s theorem of the integration 

 of Laplace's equation for a given system of level surfaces. 



26. In the transference of matter imagined in the last 

 article the particles may be regarded as carried out along 

 trajectories, cutting the successive surfaces at right angles. 

 Thus, if we draw these trajectories from points in the peri- 

 phery of rfs l3 they will mark out elements ch 2 , ds dy etc., on 

 the successive surfaces. The matter first on ds x will be 

 carried to rfs 2 , then to ^s 3 , and so on ; and this law will hold 

 however small (/s 1? rfs 2 , etc. may be made. 



Let now this process be applied to the elliptic homoeoid 

 discussed above. It is plain that we may take as cr the 

 value ^ppdk/kj which gives us the result 



dV _ dk /0 ~x 



-^ = 2 ^T (3o) 



But if m denote the total mass of the homoeoid 



m=^p~r \pd& = 2irpabc$dk, . . . (36) 



since fjxh is three times the rolume of the homoeoid, that is, 

 kirabck*. Thus we obtain dk/'k=m/27rpabck* and 



dV _m p (31) 



an m aoc 



* This is one of a set of theorems as to the uniqueness of solutions of 

 potential problems. The proof will be found in treatises on Electricity 

 or on Gravitational Attraction. 



