190-192] FINITE MOTION OF PENDULUM. 181 



where the square root is always to be taken positive. The com- 

 plete period is 



With the above relation between t and ty, sin ty is said to be an 

 Elliptic Function of A?> an< ^ the relation is written 



sn 



= sn (t A /?) ( mod sin ^ ) . 

 V v * / \ / 



The function has a real period, and the integral 



is one quarter of this period. 



The position of the pendulum at any time t is determined by 



the equation 



sin 



6 . a ( /g\ f , . a\ 

 2 = sin 2 SI TVU ^modsm-J. 



*192. Complete Revolution. If the constant in the energy 

 equation of Article 188 is such that 6 never vanishes it must be 

 greater than g, and the velocity at the lowest point is greater than 

 that due to falling from the highest point. Hence there will be 

 some velocity at the highest point. Let us suppose the velocity 

 at the highest point is that due to falling through a height h ; 

 then, when 6 TT 



and for any other value of 





/ 



V 



i0 2 g(h+2l 



iv - vrr~ 



giving sin | = sn ( A /f) (mod A?), where A; 2 = 21 /(h + 21). 

 A \K 'y t / 



