554 
THE TIME OF OSCILLATION OF 
Since the values of elliptic functions of the first and second orders, having given 
amplitudes and moduli, are given by the tables of Legendre, it follows that the value 
of t is given by this formula for all possible values of c and d/. 
If the angle of oscillation 0^ be very small c is very small, so that its square may be 
neglected in comparison with unity. In this case 
Fc-4. = Ec-4.=-4. and f4=Ec~, 
f4Ec-v//-EcJFc-4/=0. 
For small oscillations therefore 
t=: 
\/gh[k^ + l‘^)'2' 
If the pendulum oscillate on knife-edges a=0, l=h, and we obtain the well-known 
theorem of Legendre (Fonctions Elliptiques, vol. i. chap, viii.) 
( 2 «-) 
where (18.) 
vers ^,=sin^^ 
.-. c =sin (21.) 
In the case of the small oscillations of a pendulum resting on knife-edge, equation 
20. becomes 

which is the well-known formula applicable to that case. 
If the pendulum be one which for small arcs beats seconds (21.), 
/FTT^ 
(20.) 2/=—;;—, (23.) 
by which equation the time of the oscillation through any arc, of a pendulum which 
oscillates through a small arc in one second, may be determined. I have caused the 
following Table to be calculated from it. 
