372 Lord Rayleigh on Forced Harmonic 



which the influence of the second function may be got rid of. 

 The investigation is almost the same whether it be T or V that 

 enters ; for the sake of definiteness I will take the first 

 alternative. 



Consider then a system, devoid of potential energy, in 

 which the coordinate fa is made to vary by the operation of 

 the harmonic force M/^, proportional to e ipt . The other coor- 

 dinates fa, fa, . . . may be chosen arbitrarily, and it will be very 

 convenient to choose them (as may always be done) so that 

 no product of them enters into the expressions for T and V. 

 They would be in fact the principal or normal coordinates of 

 the system on the supposition that fa is constrained (by a 

 suitable force of its own type) to remain zero. The expres- 

 sions for T and F thus take the following forms : — 



T= ia n fa 2 + i«22^ 2 2 + iass^s 2 + • • • 



+ «l2-f 1^2 + «13^1^ r 3 + «l4^ 1^4+ • • • (1) 



F = Ifojr* + i^f 2 2 + hh d fa* + . . . 



+ b 12 fafa + b 13 fafa + O u fafa + ... (2) 



from which we get the equations of motion 



«ii^i + «12^2 + <%^3 + • • • + 11 ^r l + b n fa + . . . = ^1, 

 a 12 fa + a 2 2^2 + ^12^1 + ^22^2 = 0, 



«13^1 + a 33^3 + ^13^1 + ^33^3 = 0, 



since there are no forces other than M^. We now introduce 

 the supposition that the whole motion is harmonic in response 

 to M^. Thus the above equations may be replaced by 



(ip % + b n )fa + (ip a 12 + b 12 )fa + 0> «i3 + b lS )fa + . . . = ¥1, 



(ip a 12 + b 12 )fa + (ip a 22 + b 22 )^ 2 = 0, 



(ip a n + b u )fa + (ip «33 + 633)^3 = 0. 

 • •••••• 



By means of the second and following equations, fa, fa, . . . 

 are expressed in terms of fa. Introducing these values into 

 the first, we get 



fe-fron + ftn- ^-y - (t^' + t 10 ' -... . (3) 



fa ipa 22 + b 22 ipa 33 + b B3 



The ratio ^ : fa is a complex quantity, of which the real 

 part corresponds to the work done by the force in a complete 

 period, and dissipated in the system. By an extension of 



