862 Pendulum swinging through an Arc of Finite Magnitude. 

 Turning to the second figure of last Section, Ave see that 

 B'O B'A ;„ . Q , 



therefore 



so that 



WA__ 2 



/ = B'G(l-« L 'sin 2 c/>). 



From the above we may deduce that the time occupied by 

 the bead in passing from the bottom of the circle to any other 

 position varies as 



■<P dx 



1 



\/± 



We see from this result that the time is proportional to the 

 area outside the original sphero-conic. The motion of the 

 bead may therefore be described as follows : — Let the circle 

 and the straight line joining the bead to the point A be pro- 

 jected centrally onto the sphere. Let the arc which arises 

 from the projection of the radius vector be produced so as to 

 meet the polar circle of A. Then the portion of the great 

 circle lying outside the sphero-conic will describe equal areas 

 in equal times. 



The theorem at the end of the second Section may be 

 interpreted as follows : — Suppose two beads are making 

 complete revolutions round the circle in such a way that 

 when one is at the top the other is at the bottom, and vice 

 versa ; then, if the straight lines joining the beads to the 

 point A make angles <j> and <$>' respectively with B'A, we 

 shall have 



tan cj> tan cj>' = , 



A/1— K 2 



Prof. Greenhill has shown that the addition-theorem can 

 be thrown into the following form : — If two beads are sent off 

 one after the other from the lowest point of the circle with the 

 same velocity, so as to make complete revolutions, then the 

 straight line joining the two beads will always touch a circle 

 lying within the original circle. 



The connexion between the motion of a bead making 

 complete revolutions, as considered above, and the motion of 

 a pendulum swinging through a finite arc is well known, and 

 need not be given here. 



