206 Mr. 0. Heaviside on the 



the beginning, but it may be as well to give the values of all 

 the X's, although we want but four of them. Thus 



X n = E, + E 3 + R 6 + (L a + L 3 + L 6 + 2M 61 + 2M 63 + 2M 31 >, 



X 12 = R 6 + (L 6 + M 62 + M 64 + M 12 + M 14 + M 16 + M 32 + M 34 + M 36 )jt>, 



X 13 = R x + (Li + M 61 - M 62 + M 65 - M 12 + M 15 + M 31 - M 32 + M 35 )^, 



X 21 = Ba + (L x + M 13 + M 15 + M 16 + M 53 + M 56 - M 21 - M 23 - M 26 )j9, 



X 22 =-R 2 +(-L 2 + M 12 + M 14 + M 16 + M 52 + M 54 + M 56 -M 24 -M 26 )j9 ; 



X 23 - Rl + R 2 + R 5 + (Li + L 2 + L 5 + 2M 15 - 2M 25 - 2M 12 >, 



X 31 = R 3 + (L 3 + M 31 + M 36 - M 41 - M 43 - M 46 - M 51 - M 53 - M 56 >, 



X 32 =-R 4 + (-L 4 +M 32 + M 34 + M 36 -M 42 -M 46 -M 52 -M 54 -M 56 ) P? 



X 33 =-R 5 +(-L 5 + M 31 -M 32 + M 35 -M 41 + M 42 -M 45 + M 52 -M 51 )jt>. 



Now, using the required four of these in (68 c),and arranging 

 in powers of p, it becomes 



= A + A lP + A 2 p 2 (70c) 



So A = gives the resistance-balance; A 1 = 0, in addition, 

 makes the integral transient current vanish ; and A 2 = 0, in 

 addition, wipes out all trace of current. 

 There is also the periodic balance, 



A 1 = 0, A = A 2 n 2 , (71c) 



if the frequency is n/27r. 



The values of A and A! are 



A = R 2 R 3 -R,R 4 , (72c) 



A 1 = R 2 L 3 + R 3 L 2 — R X L 4 — R 4 L , 



+ R 2 (M 31 + M 36 -M 41 -M 43 -M 46 -M 51 -M 53 -M 56 ) 

 + R 3 (M 24 + M 26 -M 12 -M 14 -M 16 -M 52 -M 54 -M 56 ) 



+ R 1 (M 32 + M 34 + M 36 -M 42 -M 46 -M 52 -M 54 -M 56 ) 



+ R 4 (M 21 + M 23 + M 26 -M 13 -M 16 ~M 15 -M 53 -M 56 ). . (73 c ) 



In this last, let the coefficients of R 2 , R 3 , Rj, R 4 in the 



brackets be q 2 , £ 3 , q x , q±. Then the value of A 2 is 



A 2 = L 2 L 3 — L 1 L 4 + L 2 g 2 + L 3 5' 3 + L 1 §'j + L 4 5' 4 + ^ 2 $' 3 — q x q±. . (74c) 



It is with the object of substituting one investigation for a 

 large number of simpler ones that the above full expressions 

 for A, and A 2 are written out. 



If we take all the M's as zero, we fall back upon the self- 

 induction balance (25 c) to (27c). Next, by taking all the 

 M's as zero except one, we arrive at the fifteen sets of three 

 conditions. Of these we may write out three sets, or, rather, 



