47*2 REPORT— 1880. 



10. Nute on the Theory of the Induction Balance. By Lord Rayleigh, 

 F.It.8., Professor of Experimental Physics in the University of Cambridge. 



This subject lias been treated by Dr. Lodge in the ' PMl. Mag.' for February, 

 1880, who has arrived at several interesting results. The investigation may be 

 considerably simplified by taking the case of pure tones, as is usual in acoustics. 

 We may also suppose, for distinctness of conception, that the current in the primary 

 circuit (.I'j) is sensibly unaffected by the reaction of derived currents, though our 

 results will be independent of this hypothesis. 



If .f] .r^ .... be the currents, JR^Bo . ■ . the resistances, ilfj, ilf^, ^12 



the coefficients of self-induction, and of mutual induction, the equations for three 

 circuits are 



M,, ^ + M^. -^ + JR^T^ = - i»fi2^ 

 ^- dt ^^ dt * ^ '^ dt 



We now assume that .i-j .v^ .... are proportional to e"", where WH-27r is the 

 frequency of vibration. Thus : — 



in (Mr,2 .1'o + ^23 '^'s) + ^2 '^'3 = ~ ™ -^12 •■''1 

 in (.¥„3 .r„ + M^^.v^) + i?, .r^ = - in M,^ .v^ 



whence by elimination of .v^ 



^- ^ m .W33 + -R3 i m M^^ + R^ 



From tliis it appears that a want of balance depending on M^^ cannot compen- 

 sate for the action of the tertiary circuit, so as to produce silence in the secondary 

 (telephone) circuit, unless J?, be negligible in comparison with w ilf33, that is unless 

 the time-constant of the tertiary circuit be very great in comparison with the period 

 of the vibration. Otherwise the effects are of different phases, and therefore in- 

 capable of balancing. 



We will now introduce a fourth circuit, and suppose that the primary and 

 secondary circuits are accurately conjugate, so that 3/jj = 0, and also that the 

 mutual induction between the third and fourth circuits {M^^ may be neglected. 

 Thus 



in {M22 ,r„ + M„^ .r^ + M^^ .v^) + R^ .v„ = 

 . in (^32 .r, + M^^ .v^) + R^ .r^ = - in M,^ .r, 

 in (ir,2 -^'a + M^i -"i^) + ^4 ■'^\ = - «'« ^14 -^'i 



whence 



.rJ in M„„ + R„+ -. — ,^ ^^ -„ + -. — T^f- — w 

 ^\ ^^ ^ tn M^, + R^ m M^^ + RJ 



33 ^ -"-3 "'" -"^ii 



= — w* .r, I - 



MM MM 



■''^13 ■'"as ■ ■'"u ■'"21 



in M^^ + i?3 in M^^ + R^) 



Two conditions must be satisfied to secure a balance, since both the phases and 

 the intensities of the separate effects must be the same. The first condition 

 requires that the time-constants of the third and fourth circuits be equal, unless 

 both be either very great or very small in comparison with the period. If this 

 condition be satisfied, a balance may be obtained by shifting the circuits so as to 

 bring M^^ M^^ into equality with M^, M^^. 



For a coil of mean radius a, and radius of section equal to a -s-3'22, the coefficient 

 of self-induction (i) is * 12 tt n" a, n being the number of turns. Also, if >• be the 

 specific resistance, 



^^ 2jrwa_ ^ ... 2 (■3-22)^ w" r 



trr. rA ft 



n (3-22 )'^ 

 * Maxwell, ElcctHcity and Magnetism, § 707. 



