Vibrations of Heavy Bodies. 93 



T := 2aJ'2, X circular Arc to radius unity and cos. ' 



a 



When X = a the equation vanishes 



When X = 



T — '2,^J'2 X quadrantial Arc to radius unity. 



3d. The Semi-vibration in the Arc of a Circle. Fig. III. 



Let CP the length of the pendulum = 4, and from C with 

 CP as a radius, describe the vibratory circle. 



Let C, as before, be the centre of a cycloid, and DP =^ 2. 

 The diameter of the generating circle. 



a = the length of the chord in the generating circle, cor- 

 responding with the Arc Pp in the vibratory circle, through 

 which the pendulum is supposed to descend. 



X = the length of the chord in the generating circle corre- 

 sponding with the Arc Ptt remaining to be described. 



Then will the velocity of the point tt — aP — 6P = - — 



jj "Z 



as before. 



To find the fluxion of the space in relation to — x. 



The absciss P6 in the generative circle corresponding to the 



chord Px will be — 

 2 



But this absciss being common to both circles, the ordinate bTr 



in the vibrating circle will be /q _ ^ x / — = 2x . 



A - ^ 



While X the chord in the generative circle diminishes by — x 

 the decrement of the abscis common to both circles will be — xx 



and this multiplied by — of the vi- 



ordinale „ / r^ 



2 X- . / 1 _ -^ 



4 

 bratory circle, will give — xx X = 



2x.y I _i:! 



