122 



KINEMATICS. 



[221. 



Integrating from = o to = and multiplying by 2 we find 

 for the time ^ of one oscillation : 



dO 



o / . 



\/sm 

 \ 



. 2 <9 



2 







sm 2 - 



(43) 



As cannot become greater than we may put sin 

 = sin (0 /2) sin </>, thus introducing a new variable <f> for which 

 the limits are o and ir/2. Differentiating the equation of sub- 

 stitution, we have 



6 







or, as cos(0/2)= Vi sin 2 (0 /2) sin 2 </>, 

 ,2 sin cos <f) d<f> 



de= 



Substituting these values and putting for shortness 



sin = K, 



(44) 



we find for the time ^ of one oscillation : 



ti = 2\l- i ' , -r (45) : 



Vi-* 2 sin 2 < 



The integral in this expression is called the complete elliptic I 

 integral of the first species, and is usually denoted by K. Its 

 value can be found from tables of elliptic integrals or by ex- | 

 panding the argument into an infinite series by the binomial I 

 theorem (since K sin < is less than i), and then performing the 

 integration. We have 



1 '3 4 



K sin > 



hence 



(46) 





