21, 22] PLANE PENDULUM. 45 



from which 



52) dt 



1/2 Z 2 sin 2 -9- (g I cos # -f fc) - c 2 

 or reinserting # and integrating, 



53) t = C 



Idz 



]/2 (?-*) ( 



Since the integral contains the square root of a polynomial of the 

 third degree in z, the time is given as an elliptic integral in z, or g 

 is an elliptic function of the time. 



Inserting the value of dt from 52) in 50) we have 



, cd& 



dw = - 



sin # }/2 1* sin 2 # (# Z cos # + ^) - c 2 



or in terms of # 



* 



54) - ^ = + r 



~J (* 2 - 



cldz 



22. Plane Pendulum. If c = 0, hy 50) 9 = cow^. and we 

 have plane motion of a pendulum. The integral 49) then reduces to 



55) 



Differentiating this gives 



da- 2a . <J 



^' or 



the differential equation of plane pendular motion, which might have 

 been directly obtained for this particular case. 



If now during the motion # always remains so small that its 

 square may be neglected in comparison with unity, we may put 



so that 



The integral of this is (cf. 19, 8) 



# = asinM/^ a), 



representing a harmonic motion with period 

 56) r-2 



