68 Mr. 0. Heaviside on the 



the two coils. Then l x — m and l 2 —m are the inductions 

 through them, and these must be proportional to the resist- 

 ances, making therefore the actual inductions through them 

 always the same. 



Similarly, if any number of coils be in parallel, exposed to 

 the same impressed force V, with the equations 



Y = (r 1 + l x p) d + m 12 pC 2 + m 13 p<J 3 + . . . 



V = mnpGi + (r 2 + l 2 p)C 2 + m 23 pC 3 + . . . , V . . (M) 



} 



we have, by solution, 



DC I /V = N 11 + N 21 + N 31 + . 



DC 2 /V=N 21 + N 22 + N 23 + ...,}• . . (KM) 



r 22 +N 23 +;..,| 



if D be the determinant of the coefficients of the C's in (9d), 

 and ~N rs the coefficient of m rs in D. So, if C = C 1 + C 2 + ... 

 be the total current, we have 



C=Y(^N)/D; therefore Z = D/2N, . . (lid) 



where the summation includes all the N's. To reduce Z to 

 the single coil form, we require the satisfaction of a set of 

 conditions whose number is one less than the number of coils. 

 The simplest way to obtain these conditions is to take 

 advantage of the fact that, if any number of coils in parallel 

 behave as one, the currents in them must at any moment be 

 in the ratio of their conductances. Then, since, by (9d) 9 



V-r 1 C 1 = j p(Z 1 C 1 + 7n 12 C 2 + m 13 C 3 + . . .),] 



Y-r 2 C 2 = p(m 2l C 1 + l 2 C 2 + m 23 C 3 + ...),l. . (12d) 



V — r 3 G 3 =p (m^Cx + m 32 G 2 + l 3 G 3 + ...),} 



are the equations of electromotive forces, when we introduce 



r 1 C 1 =r 2 C 2 =r 3 C 3 = (13d) 



into them, we obtain the required conditions : — 



l[ y m \2 j ^13 { = m Sl j k | ^23 } 



r t r 2 r 3 ' r x r 2 r 3 



fflM Wo 2 h 



= _31 + _32 + _3 + s Ud , 



rj r 2 r 3 v ' 



The induction through every coil at any moment is the same in 

 amount; also the electromotive force due to its variation,and 

 the electromotive force supporting current, and the impressed 

 force. 



