Self-induction of Wires. 77 



When this is done, we know that the terminal arbitraries are 

 F^SA/^o, F 2 =2A/> , F 3 =2A/> , . . (16.) 

 and that 



Y = ic {a 1 F 1 f+a 2 ¥ 2 f 2 + a,¥ 3 f 3 + . . .} . . (17*) 



in the case (14?) of sums of squares, wherein the F's may 

 have any values, assuming that we have satisfied ourselves 

 that they are all independent ; with the identities 



0=2A/i« 3 = %Af 2 n, &c. ... (18c) 



Thus, in the case (9*), the first, second, and fourth terms 

 are of the proper form for reduction to (He), but the third is 

 not. We are certain, therefore, that there cannot be more 

 than three arbitraries, if there be so many. Now, if we do 

 not recognize the connexion between the third term and those 

 which precede and follow it (as may easily happen in some 

 other case), we should rearrange the terms to bring it to the 

 form (14**) ; for instance, thus : — 



tfZ _ 8 ( r _W\_ M»R,»/L, no . 



dp - (Ko + Sop)' K^ Lj (K 2 + L 2jP y <- 1J * J 



which is what we require. We may then take 



/^(Ko + So^- 1 , / 2 =1, / 3 -(B 2 + L 2 ^)- 

 a 1= So, %= - (L^M'L,), a 3 = -M 2 R 2 2 /L S 



Further, we can certainly conclude, provided a, is positive, 

 and a 2 and a 3 are negative, that the first term on the right 

 of (19e) stands for electric (or potential) energy, and the 

 remainder for magnetic (or kinetic). It is clear that we may 

 assume any form of Z that we please of an admissible kind 

 (e. g., there must be no such thing as p') y find the arbitraries, 

 and fully solve the problem that our data represent, whether 

 it be or be not capable of a real physical interpretation on 

 electrical principles. I have pursued this subject in some 

 detail for the sake of verifications ; it is an enormous and 

 endless subject, admitting of infinite development. Owing, 

 however, to the abstractly mathematical nature of the investi- 

 gations — to say nothing of the length to which they expand, 

 although when carried on upon electrical principles they are 

 much simplified, and made to have meaning — I merely propose 

 to give later one or two examples in which circular functions 

 of p are taken to represent Z. 



Although, however, the state of the line at any moment is 

 fully determinable for any form of the terminal Z's, when they 

 alone are given, from the initial state of the line, provided the 



