98 Mr. Davies Gilbert on the 



Moreover, the chord and arc in the vibrating circle may be 

 taken as equal ; for the arc in terms of the chord being z :=^ c 



+ 1 X 1 ^ + 1^ X -L ^ + i:M X 15: &c. 



2 3 2^ 2,4 52-5 2.4.6 7 2" 



when c is the chord of 9°4 the second term will be verv 



10000 ^ 



nearly, and consequently the cycloidal arc, equal to the chord of 



the circle, will blend itself with circular arc. 



The circular excess may therefore be taken in terms of the 



chord of the arc of semi-vibration, of the sine of one iialf this 



arc, and of its verse sine. 



1 „ 



— c- 

 16 



4 



1 



— V 

 8 



which last corresponds with the expression given by Eulcr. 



When a free or detached pendulum vibrates, the arc must 

 continually diminish, and with it the circular excess. To ascer- 

 tain the amount of this quantity, which may be termed the 

 variable circular excess, from the incipient and final arcs, 

 together with the elapsed time ; it is obvious that the law govern- 

 ing the rate of decrement in the arcs must previously be known. 

 Two causes contribute towards producing this diminution of 

 the arc, resistance of the medium, in which the pendulum 

 moves ; and friction on its axis of suspension. These must 

 be considered separately ; and in doing so, it is perfectly ob- 

 vious that the minute difference between cycloidical and cir- 

 cular vibrations in small arcs, cannot produce any sensible 

 effect on the rate of decrement ; so that whatever law is esta- 

 blished in regard to the cycloid, it may, without error, be ex- 

 tended to the circle, where no change takes place, in the centre 

 of oscillation, during the semi-vibration, when a ball of finite 

 magnitude is used, as would be the case in a cycloid. 



First, with respect to the resistance of the medium considered 



