Oscillations by Disturbances of Different Frequencies. 119 

 and therefore 



z -f yur(l + 2a cos nt)z = —b sin \(q—mn)t + e\. 

 On putting q — mn= fi—p, where p is small, we have 



z= SB r sin \(jj,—p — rn)t + e\, 

 in which -x 



B r (rn + p)(2fj,-p-rn) + ^(B,_! + B r+1 ) = 0, 



but for r = 0, 



B oP (2/x -/,) + */* 2 (B-i + B : ) = - 6. 



In order that B may be large compared with a, p must 

 be of order a 2 and be adjusted correctly to the order a lm \+ 1 . 

 If p has the value obtained by putting &=0 in the con- 

 ditional equations and eliminating the B's, the solution for 

 the forced oscillations passes over into the form 



oo ltox 



s=t%BrCOs{(fi—p—rn)t + e\+ % G r sm\(fi— p— m)t-\- e\, 



-co ._xtO-l 



where 



B (rn +p)(2/* + p - rn) + ^ 2 (B,_ X + B r+1 ) = 

 and 



-2B r (fi-p-rn)+Cr(rn+p)(2/i--p— rn)+*fju 2 (Cr-i + C r+ i) = Q, 



except for r=0, when 



-2B (/j,-p) +«/i«(C-i+C 1 )= -b. 

 Hence 



( -\ r: 

 B =+h/2fi, an( l B^Bq — — — approximately. 



IIm(2/z,— rra) 



i 



Thus the rate of increase of amplitude is proportional 

 to a 7 ", a result which shows how rapidly the intensity of the 

 magnifying effect diminishes as m increases. 



In practice, the equation of motion is not linear in x, the 

 force of restitution being in general an odd function of x 

 containing x z and higher powers ; nevertheless the above 

 investigation holds good so long as x z is negligible compared 

 with b, i. <?., if a 2 is small compared with 



m 



m 



0^) w nV-(?-™)-'}. 



1 



If this condition is not satisfied the linear equation does not 

 furnish a sound approximation, and higher powers of x 



