32 PROCEEDINGS OF SECTION A. 



The complete solution for the pendulum system for analogous 

 initial conditions is obviously given by the same equations. Thus if 



when t = 0,Oi = E,e2 = 0, ^i = 0, «2 = 0, 



E \ ) 



di = ~- \ {s - a) cos Wji + (s - 6) cos u)2t > 



62 = P2CI' - ]cOS Wit - cos W2^[ 



where pz = - — — (See Eqns. (II.) § 3.) 



I2 M + i>ii. 

 The motion of the beam is also the resultant of two superposed 

 harmonic motions of frequencies wj and W2» ^"^^ can eatily be obtained 

 from the consideration that during the motion, when the initial condi- 

 tions are those considered above, the centre of mass of the system 

 must remain fixed, hence 



{M + mi + mg) X - li nil di + I2 M2 O2 = constant. 



6. It is interesting to consider the analogies between the constants 

 of the two systems. Taking as a starting point the correspondence 

 between F and 6, then as the energy expressions must be equivalent, 

 that is 



hKiV\ and hniiglidi^ 

 therefore Ki ^ milig 



K2 (^ ^2^29' 



where the symbol " ~ " means " analogous to." 

 Again as Ci = KiDVi 



Ci ~ f>hh9 ^i> 

 C2 '^ nuihg O2. 

 As the frequency must be the same thing in both systems 

 1 M + nil + ''^2 9 



Kill M + m2 ' h 



hence 



M + m2 



J- o 



</2 mi [M + nil + ^'-2) 

 r 1 M + mi 



'>«j 



g- ni2 {M + mi + mz). 

 The coupling must be the same for both systems, so that 

 M- mim2 



L1L2 ^ {M + mi) [M + m2) 

 hence 



M ^ 



g"^ {M + m■^ + m^). 



