148, 149] INDUCTION MACHINES. 287 



conductors they are respectively connected with, their charges are 



p (n) _ IT Y (> f> M K V < n) 

 61 v ' -ft-!*! 1 , er 2 -**2 K 2 



If q^ and q 2 be the absolute values of the coefficients of induction 

 of A and B respectively upon the carriers when they are connected 

 to earth, the charges induced on the carriers are 



- ftF^ on C and - g 2 F 2 (n) on D. 



These are given up after a half-turn to B and A respectively, 

 so that 



or in terms of the potentials, 



If we write q 2 and p 2 for the positive constants ~ and ~- , we have 



Multiplying these equations respectively by p and q, adding and 

 subtracting, gives 



l) + qV 2 (n+ * = (pV* + qV) (1 -pq), 



i) _ q y 2 (n+i) = ( ^ Fi ( 



Consequently the linear functions 



pV 1 + qV 2 arid 



decrease and increase respectively in constant ratios for each half- 

 turn, and 



- gr F 2 (rl) = ( p F! (O) - 5 F 2 (0) ) ( 1 + pq) n . 



As n increases, no matter what the original signs or values of V 1 

 and F 2 , pV l + qV 2 tends toward zero, so that ultimately Fi and F 2 

 become opposite in sign, and since pV 1 qV 2 increases, the values 

 of the potentials may be made as large as we please, and increase 

 very rapidly. If the replenisher be turned in the reverse direction 

 Fj and F 2 are rapidly reduced to equality. 



