26 PROCEEDINGS OF SECTION A. 



72 1 



If 7 = 2\,-p7 = fx^, this equation can be written in the form 



of Ohm's law as 



rC = E (I) 



where r represents the differential operator L =r — 



li E = 0, and if the condenser be charged and then allowed to 

 discharge through the circuit, Lord Kelvin first showed that, provided 

 yu^ > \2, damped harmonic electric oscillations ensue whose period is 



27r ,.,.,. „ . . , 



eqiial to — p=:3 which, if X is small is very approximately 



la the sequel — will be called the natural period, and fx the natural 



./' . . 



frequency of the circuit ; \ ^?ill be called the damping co-eflicient of 



the circuit, and a circuit such as the one here dealt Avith will be called 

 an oscillating circuit. 



2. If two inductively coupled circuits, that is, two oscillating 

 circuits so placed that they act inductively on each other, have 

 vibratory currents Ci and C2 circulating in them, and if M is the 

 mutual inductance between the circuits whose other characteristics 

 are identified by the subscripts 1 or 2, then the e.m.f. in circuit 1 is 

 equal to - MDC%, and in circuit 2 is equal to - MDCi. 

 Hence by (I) § 1 



riCi= - MDC2 \ .Ts 



r2 C2 = - MDGi / ^^' 



As Ci = KiDVi, and C2 — K2DV2, where Vi and F? are the values 

 at any instant of the P.D.s of the condenser, we find, after substi- 

 tuting for Ci and C? in the above equations and integrating each once, 

 that 



ra Z2F3 = - MDKiVi / ^^^' 



as the constant to be added in either case is obviously zero. 



Eliminating Ci or C2 from equations (I), or Fi or F2 from equations 

 (II), we find that Cy, O2, Vi, or F^ will satisfy the differential equation. 



{rxr2 - M^D^) ((, = 0, 

 that is, 



j {D^ + 2\iD+fx{'){D^ + 2\2D + ^2^) - -^l~ D^\,p=0 (III) 



which, when damping is neglected, reduces to 



{ (2)2 + ^,2)(Z)2 + ^^2) - ^~~ /)^ } = 0. (IV) 



