28 PROCEEDINGS OF SECTION A. 



where <p is any one of the co-ordinates, we find that 



{M + mi + m^) X - milidi + m2h.d2 = 



- X + liBi + gdi = 



X + hOz + g02 = 



Eliminating x from the first and second, and also from the first and 

 third of these equations, we obtain 



and ^^' + ^i')^i = - ^i^'^2 \ ;r. 



^''^ (Z)2 + ^^^)d2 = - P2D% / ('^ 



where /.j^ = M + m, + m^ g_. ^^^ ^ M + m^ + m2 g 



"'^ ■ ' M + mi h 



Pi = 



mi 



(11) 



M + mi 



Equations (II) § 2 connecting the P.D.s of the condensers in the 

 coupled circuits can, when damping is neglected, be written in the 

 identical form of equations (I) of this paragraph and the values of 

 the constants for the electrical case are given by 

 1 . o 1 



2 = -" . „„2 



(III) 



K2 M . Ki M 



KiLi K2L2 



^' Ki Li' ^2 ^^-^^ 



Hence the angular displacements of the two pendulums in the 

 mechanical system are mutually connected by equations identical in 

 form to those which connect the P.D.s of the condensers in the electrical 

 system, and as Ci = KiDVi, O2 = K2DV2 the angular velocities of 

 the pendulums are similarly analogous to the currents in the two 

 electric circuits. 



If, in the proposed system, the strings of the second pendulum 

 become rigid and be rigidly attached to the beam, then D^O.^ = ^' ^^^ 

 the equation of motion of the first pendulum becomes 



(i)2+y)0i =0 

 that is, the motion is simple harmonic and of frequency /^j. 



Hence to determine the quantity relating to the mechanical 

 system that is analogous to the "natural period" of the primary electric 

 circuit we have simply to place the bob of the second pendulum on the 

 beam, and then measure the period of the first by observation in the 

 usual way. Similarly the natural period, as we shall call it, of the 

 second pendulum is determined. 



Keturning to equations (I) above, if we eliminate Q^ or 62 between 

 them we find that either i)^ or d^ will satisfy the differential equation 



{(2)2 + ^^2)(D2 + }X2^) - pi f2 D"} e = 

 which is identical in form with equation (IV) § 2, which is satisfied by 

 the variables in the coupled circuit system. 



