232 Lord Rayleigh on Maintained Vibrations, 



the differential equation becomes 



S +K f +(» 2 -2asm2 i >0*=0, • • • (5) 



in which k and « are supposed to be small. A similar equa- 

 tion would apply approximately in the case of a periodic 

 variation in the effective mass of the body. The motion ex- 

 pressed by the solution of (5) can only be regular when it 

 keeps perfect time with the imposed variations. It will 

 appear that the necessary conditions cannot be satisfied rigo- 

 rously by any simple harmonic vibration; but we may assume 



6 = A 1 sin pt + B x cos pt + A 3 sin Spt + JB 3 cos Spt 



+ A 5 sin5p£ + ..., ... (6) 



in which it is not necessary to provide for sines and cosines of 

 even multiples of pt. If the assumption is justifiable, the 

 series in (6) must be convergent. Substituting in the diffe- 

 rential equation, and equating to zero the coefficients of sin pt, 

 cospt, &c. ; we find 



Ai(n 2 -p 2 ) - K pB x - aBx - aB 3 = 0, 

 B x (n 2 —p 2 ) + tcp A x — a Aj — a A 3 = 0, 



A 8 (w 8 -9p 2 ) -3^?B 8 -«B 1 + «B 6 = 0, 

 B 8 (n 2 - V) + 3«pA 8 + aA 1 ~A 5 = 0, 



A 5 (/i 2 -25p 2 )-5^>B 5 -aB 3 + «B 7 =0. 

 B 5 (> 2 — 25p 2 ) + 5/cpA 5 + «A 3 — «A 7 =0, 



These equations show that relatively to A 1; B x , A 3 , B 3 are of 

 the order u; that relatively to A 3 , B 3 , A 5 , B 5 are of the order a, 

 and so on. If we omit A 3 , B 3 in the first pair of equations, 

 we find as a first approximation, 



AiO 2 -p 2 ) - ( K p + «)B! = 0, 



A 1 («p-«) + (n 2 -p 2 )B 1 -0; 

 whence 



Bi _ n 2 —p 2 _ u — /cp __ s/{u — Kp) 

 l^~Kp + ^~ n 2 --p 2 ~ ^/(ct + fcp)' ' * ^'' 



and 



(n 2 -p 2 ) 2 = a 2 -K 2 p 2 . (8) 



Thus, if a be given, the value of p necessary for a regular 

 motion is definite ; and p having this value, the regular mo- 

 tion is 



&-,? sin (pt + e), 



