66 



Mr. 0. Heaviside on the 



When the systems are not independent the above simplicity 

 is lost ; and I have not formulated the necessary conditions of 

 similarity in an extended sense except in some simple cases, 

 of which a very simple one will occur later in connexion with 

 another matter. 



(3) The Christie Balance of Resistance, Self and Mutual 

 Induction. — The three general conditions of this are given in 

 equations (72c) to (74c). If, now, we introduce the following 

 abbreviations, 



= L x + L 2 +L 5 + 2(M 15 -M 12 -M 25 ), 



m 1 



m 3 = L 3 -f-L 4 +L 5 



m 6 = L 2 + L 4 +L 6 



™ 13 =-L 5 -fM 13 -M 14 



,-™lA, f 



J 



m 



-f2(M 45 -M S4 -M 35 ), 

 + 2(M 62 + M 64 + M 34 ), 

 - M 15 - M 23 + M 24 + M 25 -+ M 35 - M 45 , 

 m 36 =-L 2 + M 12 + M 14 + M 16 -M 24 -M 26 + M 25 + M 45 + M 56 , 

 W61 =-L 4 + M 32 + M 34 + M 39 -M 24 -M 46 ~M 25 -M 45 -M 56 , 

 the conditions mentioned reduce simply to 



E 1 R 4 =E 2 B 3 , 

 (?n 1 + ?72 1 3 + m l6 )R 4 — 772 36 E 1 = (?7? 31 + w 3 + m 36 )E 2 

 (m l + m 13 )m 16 — (m 3 + w 13 )m 36 . 



The interpretation is, that as there are only three indepen- 

 dent currents in the Christie arrangement, there can be only 

 six independent inductances, viz. three self and three mutuai; 

 and these may be chosen to be the above m's, whose mean- 

 ings are as follows. Let the three circuits be ABiBjA., CB 2 B 1 C, 

 and AB 2 CA in the figure, so B 



that the currents in them are 

 Ci, C 3 , and C 6 . Then m 

 m 3 , and m 6 are the self, and 

 m 13 , m 36) m 6 i the mutual in- 

 ductances of the three cir- 

 cuits. 



Now if the four sides of the 

 quadrilateral consist merely 

 of short pieces of wire, which 

 are not bentinto nearly closed 

 curves, it is clear that {<6d) 

 are the true conditions, to 

 which alone can definite meaning be attached; the inductance 

 of a short wire being an indefinite quantity, depending upon 

 the position of other wires. We may therefore start ab initio 

 with only these six inductances, and immediately deduce * the 

 * ' Electrician,' April 1, 1887, p. 457. 



I 

 I 



