﻿20 Mr. C. V. Raman on Motion in a 



Substituting now the odd terms alone left on the right- 

 hand side of (4), for IT in equation (2), we have the following 

 series of equations : — 



— (fro— p 2 ) Ai + kj?B x = — by A 3 + «iB 3 + b x A 5 — a x B 5 - &c. + &c. 

 -(b - 1 i 2 )B 1 -kpA l = a 1 A d + b 1 B 3 + a 1 A- o + b 1 R- > + &c.+&c. 



— (& -V)A 8 + 3^B 3 =-6 1 A 1 + fl 1 B 1 -^A 5 + a 2 B5 + &c.-Ac. 

 -(b -9p 2 )B 3 -3kpA z =a 1 A 1 + b 1 B 1 + a 2 A 5 + b 2 B b + &c. + &c.{6) 



and so on. 



Evidently, the possibility of this being a consistent set of 

 convergent equations depends upon the suitability of the 

 values assigned to the constants k, p, b , a ly b^ &c. 



It is not possible here to enter into a complete discussion 

 of the solution of these equations. One point is, however, 

 noteworthy. From the first two of the set of equations given 

 above, it will be seen that such of the harmonics in the steady 

 motion of the system as are present serve as the vehicles for 

 the supply of the energy requisite for the maintenance of the 

 fundamental part of the motion. Paradoxically enough, the 

 frequency of none of these harmonics is the same as that of 

 the field. 



We now proceed to consider the odd types of vibration, 

 i. e. the 1st, the 3rd, &c. Taking the 3rd as a typical case, 

 we put n = 3p and get 



af(t) = a x sin 3pt + a 2 sin 6pt + a B sin 9pt + &c. 



+ b + b x cos Spt + b 2 cos 6/tf + b 3 cos 9pt + &c. . (7) 



Substituting (4) and (7) in equation (2) and equating the 

 coefficients of sine and cosine terms of various periodicities 

 to zero, we find that the quantities A 3 , A 6 , A 9 , &c. and B , 

 B 3 , B 6 , B 9 , &c, do not enter into the equations containing 

 Ai and B x . We therefore write them all equal to zero. The 

 significance of this is that the maintained motion contains no 

 harmonics the frequency of which is the same as, or any 

 multiple of the frequency of the periodic field of force. This 

 remarkable result is verified by a reference to fig. 3, PI. II. 

 from which it is seen, that the vibration curve is roughly 

 similar to that of the motion of a trisection point of a string 

 bowled near the end, the 3rd component, the 6th, the 9th, &c, 

 being absent at the point of observation. 



