20 MEASUREMENT OF OSCILLATIONS. 



The motion we are considering is susceptible of a very simple 

 geometrical representation. If we imagine the arc 20, expressed as a 

 straight line, and taken as the diameter of a circle, the position of the 

 movable body on the diameter is given at each instant by the projec- 

 tion of a second movable body starting at the same time as the first 

 from the end of the diameter, and moving uniformly along the 

 circumference with the velocity 



o> n = no. = a 



677. PENDULUM MOTION. Equation (3) cannot be completely 

 integrated in a finite form. If it be multiplied by dt, and 



integrated for a first time, we get (observing that x = a, for = 0), 



t**\ 



w 



(7) i-jj] = 2 2 (cos x - cos a). 



We d.educe from this, for the time T of an oscillation, 



dx 



'p' _ v 



n J o ij cos x - cos a ' 



Expanding the expression under the root, we get, by a well- 

 known calculation, 



(8) 



1 - ' - \ 2 '4 



7T 



or, representing the series by i + /3, and still calling T the time - , 



The conditions of the problem will here be determined by the 

 equations 



2 



