﻿•462 Mr. W. M. Hicks on the Self-induction 



If (n 2 — £)Q„Q» + i — Ql»Qn+i be written X n , it can be shown 

 that X n =X n _ 1 + (2n-l)Q n Q w _ 1 , 



which is more convenient fur calculation. 

 So also 



R,X-RX=87r^V2[SPL{(n-l)QL_ 1 -(n + l)QUi} 

 -(^-i)SP.{(n-l)Q»_ 1 -(n+l)Q lH . 1 }]. 



As in the former case, it may be shown that 



(^-i)P w Q«-i- P LQ'n-i = 2n7r+ (n 2 -l)P H Q 7i+1 -P' n Q' n+ i, 

 and -s . 



b.= =Sr- 2 { »(-D-(* f - iH + , +PLQU. } if 



= 166aV2A (say) 

 with B =8Z>aV2(P Qi + 4F Q'i). J 



Also, if the expression in the bracket for B„ be denoted by Y n , 



Y n = Y n _ 1 + 2(n-l)7r+(2n-l)P n _ 1 Q n . 



Thus yjr and ^ are now completely determined. It must 

 be remembered that the functions P, Q in formulae (6) and 

 (7) are the values at the surface of the tore u . 



5. In dealing with the self-induction of such a ring as we 

 are now considering, it is necessary to particularize. Its 

 value will differ for alternating currents of different periods. 

 The time-constant for a constant E.M.F. applied to the ring 

 will not be the same as on the assumption here made as to 

 distribution of current-density. In order to be definite we 

 will find the coefficient for the energy on the assumption of 

 steady current — in other words, the value of L in the expres- 

 sion -JLI 2 . This will very approximately give the self- 

 induction except for rapid alternations. The energy is 

 given by 



E=±$i]r<rdpdz, 



the integral being taken over a cross section. 



Putting in the value of yjr and transferring to curvilinear 

 coordinates, 



tt. a 9,2 CC 2 ? j j , Ik PP B„T n cos nv 1 dn dn' _ , 



E= — 2tt 2 6 2 — dpdz+ ^bZ —rjn % 3~ * -j~dudv 



JJ P 2 JJ V(C-c) pdu dv 



= - 1 tt 3 6 2 R 2 (1 - cos «)'(! + 2 cos a) + ab%B n f" T ^^^ di 



! 



Now 



J n COS 7115 2s/2 , 



