JJ{ 



Field of an Electrical Current, 217 



and since x is the same for all the filaments, in taking Yx we 

 may put x under the sign of integration, so that 



Yx=~i{(l + l J2 )K-2E\ P ~d8, . . . (62) 



As we require only the terms to be added to (59) by the 

 introduction of the term -, we may keep only terms of the 

 first order in the expressions for (l + k f2 )K — 2E and p. Thus 



(l + ^)K-2E = L-2 + BcMg-g""*"^ . ( 63) 



La 



T 1 , Rcos0 — 2mcos6\ '* 



P = 2a i 1 + Ya -/• (64) 



Hence, neglecting the factor -r- the correction introduced is 



T 9 rT -v Ecos^-2mcos^"| J __ Rcos#— mcos(f> 

 * ' 2a ) \ a 



(Rcos<9 — racos(/>) 2 \ 

 + - 2 ^-fUdUdx. . (65) 



In this the term of the first order is 



--Jj(L-2)(Rcos0-mcos0)B<£EU;ft . . {Q6) 



COS (D f 



which, by putting cos 0= cos c£ cos ^, is simply A — • ^— . 



The term of the second order is a " m 



JL.JJ{(2L-4)(R cos 6-m cos 0) 2 



-(L-l)(R 2 cos 2 6>-3?72Ecos0cos^) + 2w 2 cos 2 ^)}R^R^, (67), 



= ^2-jj{(L-3)R 2 cos 2 6>-(L-5)mRcos0cos0 



-2m 2 cos 2 ^)}R^R^ (68) 

 which, by putting cos 2 6 = J + \ cos 2<£ cos 2^, &c, reduces to 



_|l_|_L(2m2-cS)-7 C s + cos2<£(- C 2+gQ}. (69) 



[To prevent error, it may be observed that it is not allow- 

 able to put cos 6— cos <£ cos % in any of the equations (63), 

 (64), or (65) until, as in (66) or (67), all multiplications 

 introducing powers of cos 6 have been performed.] 



The terms (6Q) and (69), after neglecting the factor A, 

 Phil, Mag. S. 5. Vol. 36. No. 219. Aug. 1893. Q 



