46 II- PARTICULAR MOTIONS OF A POINT. 



The time taken for a single swing is one half this. Here we have 

 an example of a small vibration, which is harmonic, as stated in 19. 

 In order to find the degree of approximation of this solution 

 we must examine the exact equation. Determining the constant of 

 integration Ji by putting for the velocity when # = # , 



we have 



Subtracting this from 55) we obtain 



from which it follows, 



57) 



There are three cases according as ~ o?"" 2 " * 8 ^ ess ^ nan ? equal 

 to, or greater than 1. We shall consider the first. 



(<v *4* \ 



sin 2 Y sin 2 - j 



58) *-- ld 



sn -- - g sn 



If we put 



59) 

 we have 



60) 



Let us now introduce a new variable t^ such that ft sin ^ = sin? then 



Jc cos iff dty = cos -=rd(-= 



2 \ I 



