358 Professor Airy on a Correction requisite 



and by solving the two simple equations we find that d and 

 (p are expressed by the forms 



C cos {t s/gp' + B') + C" cos {t J'^" + B"), 



and c' cos {t ^/gp' + E) + c" cos {I ^ gp" + B") , 



where two of the constants C, C", c', c" are arbitrary. The motion 

 of the wire, and the rotation of the ball in alternate directions, 

 may therefore be represented by the superposition of two vibra- 

 tions, each of whicli follows the law of the cycloidal pendulum. 



Now it is easily seen that one of these vibrations is of that 

 kind which will take jdace if, without giving any motion to the 

 center of gravity of the ball, it be turned round a horizontal 

 axis and then be set at liberty : and this is performed in a 

 short period. The other is of the kind which is commonly con- 

 sidered, and except the disturbance of the ball be very great, 

 is the only one that catches the eye in vibration. This then 

 is the only one which is used in the observations of the pen- 

 dulum : it is therefore the only one which concerns us here. 

 It may be distinguished from the other by observing that it is 

 performed in a longer period, and therefore that value of p must 

 be taken which requires the greatest value of t to make the 

 term J cos (< Vp + ^) S^ through all its periodical values: that 

 is, we must take the smaller value of p. Let this be p'. Then 



the apparent time of a double vibration will be ■ . — : or the 



length of the simjde pendulum which would vibrate in the same 



time will be — = — p". Call this I. 

 p r 



