1886.] A General Theorem in Electrostatic Induction. 419 



the quantity h need not be zero, unless under very special circum- 

 stances. 



As it appears in (1), In . £K is clearly the quantity of electricity that 

 must be given to the primary in order to maintain the potential 

 constant whilst the specific inductive capacity is altered by £K, and 

 this in addition to the influence of the mere change of capacity of the 

 system. 



We may assume as a well-known result that for a closed cycle of 

 operations 



and Bq is a perfect differential.* 



Expressed in words, this is equivalent to stating that, when after 

 undergoing a series of changes, the potential is brought back to any 

 given value V", and the molecular condition of the dielectric in the 

 field of force is brought back to its initial state, then the charge of the 

 primary is the same as at first. 



The analytical statement of the condition that 8q in (1) is a perfect 

 differential gives us — 



dC_ d / v d0 , A 

 ~dK~ dV\ dS/ 



^L+V.A.(^\=0 . (2.) 



dV dV\dKj K } 



In order to obtain another relation between the quantities, let us 

 denote by £e the increment of electrical energy of the system during 

 the series of operations described above as leading to (1). This is 

 expressed by an equation of the form — 



Se-—YZq + 7rSK (3) 



The meaning of the first term of the right hand number of (3) is 

 obvious ; the other term, 7r^K, denotes the work done against electrical 

 forces when the specific inductive capacity of the dielectric is increased 

 by SK. 



By the principle of the conservation of energy, for a closed cycle of 

 operations 



h°> 



and £e is a perfect differential. Hence, if we express the analytical 

 condition of this, after putting for Sq its value from (1), we get — 



* I would here acknowledge my great indebtedness to a paper on the " Consen ft- 

 tion of Electricity," by M. Gc. JLippmann, "Annates de Chimie et de Physique,'' 

 5 me Ser., T. 24 (1881) p. 145. 



