Self-induction of Wires. 179 



the paper referred to cores are placed in the coils, giving a 

 special form to K. 



When K is conductance merely, the characteristic function 

 contains within itself expressions for the resistance between 

 every two points in the combination, which can therefore be 

 written down quite mechanically. For it is the sum of pro- 

 ducts each containing first powers of the K's, and therefore 

 mav be written 



F=K 12 X 12 + Y 12 =K 23 X 23 + Y 23 =...,. . (14c) 

 where X 23 , Y 23 do not contain K 23 , and X 12 , Y 12 do not contain 

 K 12 . (It is to be understood that the diagonal K n , K 22 , . . . , 

 is got rid of.) 



Then 



R'^ = X 12 /Y 12 = resistance between points 1 and 2, 

 R/ 23 = X 23 /Y 23 = „ „ „ 2 and 3 



;}(15 ( 



&c, it being understood that these resistances are not R 12 , 

 R 23 , &c, but the resistances complementary to them, the com- 

 bined resistance of the rest of the combination ; thus, if c 12 be 

 the impressed force in the conductor 1, 2, the current (steady) 

 in it is e ^ 



K 12 + X 12 / Y 12 K 12 + It 12 



The proof by determinants is rather troublesome, using the 

 K's, but, in terms of their reciprocals, and extending the 

 problem, it becomes simple enough. Thus if we turn K to 

 R -1 in F, and then clear of fractions, we may write F = as 

 R ]2 X' 12 + Y' 12 =0, R 23 X' 23 + Y' 23 = 0, &c, . . (17c) 

 where X' 12 , Y' 12 , do not contain R 12 ; &c. From this we see 

 that the differential equation of the current C 12 in 1, 2, sub- 

 ject to c 12 only, is 



(R 12 + R' 2 i)C 12 = c 12 , (18c) 



if R' 2 i=Y' 12 /X' 12 . For this make the dimensions correct, 

 and that is the only additional thing required, when we 

 observe that it makes the fixed steady current 



Ci 2 =%/(Ri2 + H' 21 ), (19c) 



so that R' 21 is the resistance complementary to R 12 . 



Although it is generally best to work in terms of resist- 

 ances, yet there are times when conductances are preferable, 

 and, to say nothing of conductors in parallel arc, the above 

 is a case in point, as will be seen by the way the characteristic 

 function is made up out of the K's. There is also less work 

 in another way. Thus, ^n(n— 1) conductors uniting n points 

 give i(n — l)(w — 2) degrees of freedom to the currents. It 

 is the least number of branches in which, when the currents 

 in them are given, those in all the rest follow. Thus, if 10 



