Self-induction of Wires. 21 



The following are the expansions of the quantities occurring 

 in the denominator of (136) : — 



Let 



I 2 = R' 2 + L' 2 n 2 , I 2 =R ' 2 + L 'V, I 1 2 =R 1 /2 + L l 'V. . (156) 

 Then 

 A 2 + B 2 = I 2 + (E? + &tfJL*I* + 2 (E 'E/ - L 'L> 3 XKR' + L'f 



+ 2(E/L ' + E 'L/>2(KL' -E'S), 



« 2 +& 2 = (P 2 + Q 2 ){W + R/) 2 + (V + L/)2n^, 

 U + B6 = (E ' + R/XE'P + L'nQ) + (L ' + L/>(L'nP - R'Q) ^ (16 b) 



+ (K'l* + E/VXKP -I- SnQ) + (L 'V + L/I >(KQ - SnP), 

 L6 - oB = (E ' + E/)(E'Q - L'raP) + ( V + L,> (B/P + L'nQ) 



+ (E^ + B/I *)(KQ - SnP) - (L^ + L, 'I >(KP + SnQ). ^ 



These may be used direct in the denominator of (146), which 

 is the same as that of (136). But G and H may be each 

 resolved into the product of two factors, each containing the 

 apparatus-constants of one end only. Noting therefore that 

 the 6 in (146) is given by 



*»*'- J^Zf-v ■ ■ • ("*) 



whose numerator and denominator are given in (166), it will 

 clearly be of advantage to develop these factors. First 

 observe that the expansion of H is to be got from that of G, 

 using (166), by merely turning P to — P and Q to — Q. We 

 have therefore merely to split up one of them, say G. If we 

 put ^^=0^ ^' = in G it becomes 



F+ (P 2 + Q 2 )I 2 + 2P(R 'R' + L 'W) +2Q(L'nK '-E'«L '). (186) 



If, on the other hand, we put B ' = 0, L ' = in G it becomes 

 the same function of E/ IV as (186) is of E ', L '. It is then 

 suggested that G is really the product of (186) into the 

 similar function of IV, L/; when the result is divided by I 2 . 

 This may be verified by carrying out the operation described. 

 But I should mention that it is not immediately evident, and 

 requires some laborious transformations to establish it, making 

 use of the three equations (106). When done, the final 

 result is that (146) becomes 



p _ rK^+SW-|i 



u °- 2Vo Lk«+j/vJ 



*■ [G G 1 6 2Pi +H H 1 6- 2K -2(G G 1 H H 1 ) i cos2(Q?+e)] i , (194) 



