Mr Harrison, Notes on the Theory of Vibrations 85 



as /3 increases. The results of calculation, with n -= 1, a = 1, are 

 as follows: 



n'^x + ^ax^ = 0, 



(b) The Asymmetric System. 

 The equation of motion is 

 d^x 



where a may be assumed to be positive, as changing the sign of a 

 is equivalent to reversing the direction of the axis of x. 



Let the scale of time be such that n = 1, and the scale of 

 length chosen so that the amplitude of the motion measured from 

 aj ^ in the direction of x positive is unity. Then 



f , j =-■ {I — x) {I + a -\- x + ax -\- ax^) 

 ^{l-x){b + x) (c + x), 

 where 6 = i {1 + « - (1 - 2a - ^a^f}la, 



c=\{l + a+{l--2a- 3a2)^}/a. 



The limits of the vibration are x = \ and x = — b. It is 

 necessary that a should be less than \, so that the greatest value 

 of b is 2. 



Writing I — x = {b + I) y^, we, have 



|)'=i«(c+l)(l-?/2)(l 



where F= (6+ l)/(c + 1). 

 Hence 



y= sn {|a^(c+ lft,l}, 



and x^l-{b-^l) sn^ {^a^ (c + l)^i, k} 



b+ I {, E 27r2 * mf " 



Fy^),. 



= 1 - 



/C2 



1 ^2772* 



2Z 



The results of calculation are as follows : 



