25 



213 



whence, after dividing by Zp^R^^/'ip, 



rJ y - 1 ^Rx> ^rJ y-l ^rJ 



[50] 



The relation between the radius and the time t may be found by 



solving Equation [28] for dt and integrating: 



J J iR^ 3 y - 1 p ^ R ' 3pJ 



or, in terms of 



^ = f • ^= If 1^ f51a. b] 



For infinitesimal amplitudes of oscillation about R = R,, or x = 1 , 

 the Integration is easily effected by writing 



X = 1 + w, C = 1 + -^^ + b 

 y - 1 



where 6 is a small positive constant, expanding powers of (1 + w) by the bi- 

 nomial theorem, and dropping all terms whose effect on t becomes negligibly 

 small as 6 ■>• 0. Then Equation [52] becomes 



* = l^^o/(b + &«'- ^y^') ' dw = 



9 



Po ""^ \" ■ ""- 2 



[33] 



' 9y ^►'62 + 18 yb ' 



2Po ^7 •'0--riOyl 



For a half osclllatioii, the limits for w are the roots of the quantity In 

 parentheses in the integral; these roots give to the sine of the angle the 

 values +1 and -1, respectively, as is easily verified. Thus for a complete 

 oscillation the sin"^ contributes a factor 2n, and the period Is 



n=i^^o ^^ 2.= 2.i^„/^ [3'^] 



2po "« ''9y '" " '""»»' 3yp„ 

 or, for y = 4/3, 



To = nR, ff [5lta] 



Po 



For larger amplitudes numerical integration is necessary. This Is 

 simplest when y = V3, so that Equation [32] becomes 



