PROCEEDINGS OF SECTION A. 29 



It is evident that the " coupling " of the pendulum system, or the 

 value of sin v// for the latter is equal to 



, _ / Wi m^ ^ 



vpi V-2 - V (i/ + m,) {M + Wo) * 



Hence all the variables, either of the mechanical system or of the 

 electrical system, satisfy the differential equation 



{C0S2 ^ 7)* + (^1-2 + ytia^) 1)2 + /Xi2 ^2^ j = (IV) 



wliere fxi and /<2 are the natural frequencies and ;// the coupling angle 

 in either case. 



4. In order to follow up this analogy it is necessary to know fully 

 the details of the motion in either system arising from analogous initial 

 conditions. We shall therefore obtain the solution when for the 

 electrical system the initial conditions are Vi = E, Fg = 0, Cj = 0, 

 Ci = when ^ = 0, the usual initial conditions to a disturbance in a 

 Marconi transmitter, and when for the mechanical system the initial 



conditions are 0i = E,B2 = 0, Ji = 0, O2 = when « = 0. 



Let us put a for ni^ and h for f.ii^ in equation (IV) § 3 and it 

 becomes 



{cos2 ^ 7)^ + (a + &)D2 + ah}<p = 0. 



Proceeding with the solution of this differential equation in the 

 usual way by obtaining the factors of the operator considered as a 

 simple quadratic function we fuid that 



D2 



_ - [a + h) ± V(a + 6)2 - 4a6 cos2 4/ 

 2 cos2 i^ 



= - (<t + h) ± N/g2 -H 62 - 2ah cos 2 ;// 

 2 cos2 \p 



If now a and 6 be taken as the two sides of a triangle whose included 

 angle is 2 v//, then as 



, /s {s - c) 



where c is the third side and a + b + c = 2s, the roots of the 

 quadratic are 



s - c ah 



and - 



C0S2 4/ S 



8 __ ab 



