568 Prof. T. R. Lyle on Mechanical Analogy to the 



If E = 0^ and if the condenser be charged and then allowed 

 to discharge through the circuit, Lord Kelvin first showed, 

 that provided ^ 2 >\ 2 , damped harmonic electric oscillations 

 ensue whose period is equal to 



2 



IT 



S/fJL 2 -X 2 



which, if X is small, is very approximately 



= — - 2tt v'KL. 



In the sequel 2ir\^ will be called the natural period, and 

 fjb the natural frequency of the circuit ; X will be called the 

 damping coefficient of the circuit, and a circuit such as the 

 one here dealt with will be called an oscillating circuit. 



2. If two inductively coupled circuits, that is, two oscil- 

 lating circuits so placed that they act inductively on each 

 other, have vibratory currents Ci and C 2 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 2 , and in 

 circuit 2 is equal to — MDOi- 



Hence by I., § 1, 



rA = -M-DC, | T 



r 2 C 8 = -MDCVj 



As C 1 = K 1 DY 1 and 2 = K 2 DV 2 , where V, and V 2 are the 

 values at any instant of the P.D.s of the condensers, we find, 

 after substituting for Ci and C 2 in the above equations and 

 integrating each once, that 



f^KiVx- -MDK 2 V 2 | 

 r 2 K 2 V 2 = -MDKiVi, J 



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



Eliminating Ci or C 2 from equations I., or Y, or V 2 from 

 equations II., we find that 0„ C 2 , V 1? or V 2 will satisfy the 

 differential equation 



(?V' 2 -M 2 D 2 )c/> = 0, 

 that is 



|(D 2 + 2X 1 D + Ml 2 )(D 2 + 2X 2 D-f ^/;- jy^- D4 } </> = °> ( m 



