498 Lord Bayleigh on the Self-induction and 



first and second. If we suppose the second and third to be 

 correct, the first would have to be 1" 20 instead of 1*11. Such 

 an error as this may easily occur in estimating the equality 

 of sounds heard successively ; and there can be no doubt 

 that the smaller branch current largely exceeded the main 

 current*. 



Appendix. — The Induction- Compensators (p. 473). 



For the mutual induction-coefficient between two circular 

 circuits, subtending angles a l7 a 2 at the point of intersection 

 of their axes (lines through their centres and perpendicular 

 to their planes), and distant c u c 2 from that point, Maxwell 

 gives f 



M=4tt 2 sin 2 ^ sm 2 * 2 cAi C Z Q , i(« 1 )Q , i(« s )Q 1 (0) 



i. C\ 



' + ■■• +^$W ( VMQ,(«) + • • • }' (17) 



the angle between the axes being denoted by 6. Q ( - . . • 

 denote Legendre's coefficients (more usually represented by 

 P { ), and the dash indicates differentiation with respect to /j,. 

 In our present application the circuits are concentric, so that 

 « 1 = « 2 = Jtt, and c 1} c 2 are equal to their radii. Moreover 

 {Q/Ci 73 ")} 2 vanishes if i be even; while if i be odd (2n + l) 

 we have 



( A/ /i \,2 3.5.7 . . (2n + l ) / 1ft v 



\H2n+i{2 7r )\ = — 2 2 . 4 2 . 6 2 . . (2nV ' ' ^ ' 



so that 



XjAl-WLto+ififflim 



+ 



+ 2U 2 \5/ Q5W + 



1 3 2 .5 2 .7 2 . . (2n + l)- 



(2n + l)(2n + 2) 2 2 . 4 2 . 6 2 . . (2n) 2 



(j)V+i(0), (19) 



which is what we have to calculate for various values of 6 on 

 the supposition that c 2 — ^c 1 . 



The following are the values of Q2n+i($) at intervals of 10°. 

 It is unnecessary for our purpose to go further than Q 7 . 



* These experiments were described before the British Association at 

 Birmingham, September 3. 

 t 'Electricity and Magnetism/ § 697. 



