22] TIME OF OSCILLATION. 47 



V 



The integral I - is an elliptic integral in Legendre's normal 



1 - 2 * 



form and is denoted by F(fy, &). In this notation 

 61) *- 



and the upper or lower sign is to be taken according as the particle is 

 rising or falling at # . For call t m the time of reaching the lowest 

 point. When # = 0, ^ = and since -F(0, &) = 0, we have 



62) t m = : 



Since this is to be positive, we see that the lower sign is to be taken 

 in 61) if the particle is falling at # . Subtracting 61) from 62) 

 we have 



63) C - 1 



as the time of falling from any inclination # to the lowest point. 

 The particle swings by the lowest point and continues with nega- 



j f\ (* 



tive & until = 0. that is until W sin 2 - = 0. 

 dt 2 



-9* -, . 7t 



If the time on reaching the highest point is t h we have by 63) 



t m _ f ==~[/^Lpi ^_ f h\ 



or 



The integral 



is called the complete elliptic integral, and depending only on the 

 parameter & is denoted by KQc). Tables of values of F and K are 

 given in Legendre's The'orie des Fonctions Elliptiques. The period of 

 a double oscillation is 4(4 ), 



64) T- 4 }/-!*(*). 



