1886.] A General Theorem in Electrostatic Induction. 



421 



Remark also that since ^ must in general be positive, the quantity 

 a, and therefore also jr, must be negative, by (7). 



It is not difficult to prove that the condition = const, leads to 



aK 



the conclusion that the whole electric field must be occupied by an 

 electrically homogeneous dielectric. 



The following proof seems to be convenient. Let us imagine the 

 assumed heterogeneous medium to consist of shells of dielectric 

 material whose boundaries are equipotential surfaces ; each shell is 

 supposed to be itself homogeneous. If the bounding equipotential 

 surfaces consisted of excessively thin conducting shells, the distri- 

 bution of electric force in the field would be unaltered. Each 

 consecutive pair of conducting equipotential surfaces with the (homo- 

 geneous) shell of dielectric between, would then form a condenser. 

 And since the same quantity of electric induction crosses all the 

 equipotential surfaces in the field, the capacity of the whole system 

 would be simply that of a series of condensers in " cascade." 



When air is the dielectric, denote the capacity of the condenser 

 which consists of the primary, the first conducting equipotential 

 surface, and the space between, by C x ; the capacity of the condenser 

 formed by the first and second equipotentials by C 2 , and so on. If. 

 instead of air, the spaces be respectively filled up by dielectrics of 

 specific inductive capacities K x , K 2 , and so on, we have for the capacity 

 ot the whole system C, the relation — 



i=J_ + J_ + ... 



C I£^Ci KgCo 



Replacing the shells of dielectric by others of different specific induc- 

 tive capacity, and denoting the changed quantities by dashes, we have — • 



C K^'C^ K 2 Co 



Let now Ky-K^Kj' — K 2 = . . . = BK, 



so that the alteration of specific inductive capacity is the same for all 

 the dielectrics. 



It is evident that unless — 



K 1 =K 2 =K 3 = . . . 



and K 1 ' = K 2 / = . . . 



it is impossible that — 



m 



should be independent of SK. 



