860 Mr. J. Rose-Innes on the Motion of a Pendulum 



In our previous investigation we considered an arc of 

 a great circle which moved about so as to cut off a segment 

 of constant area from the sphero-conic. It is well known 

 that such an arc will always touch an inner sphero-conic 

 concyclic with the original one. 



If we project the smaller sphero-conic as above, we shall 

 obtain a 'circle which lies within the circle having AB' as 

 diameter ; by symmetry we shall have the centre of this 

 inner circle somewhere on AB'. Let the points in which 

 the inner circle cuts the straight line AB' be denoted by 



L and M, and consider the plane figure obtained as the 

 result of cutting the solid figure by the plane through the 

 line GB' and the centre of the sphere. Let us denote 

 the centre of the sphere by O ; and let OC be drawn 

 bisecting the angle AOB' and meeting the straight line AB' 

 in C. Then we have the folio win g relations among the 

 angles of the figure : — 



COB' = COA, 

 CB'O = AOG. 



But 



therefore 



OCG = COB' + CB'0, 

 COG- COA + AOG; 



OCG = (JOG. 



Because 00 is the bisector of the angle AOB', it must 

 also bisect the angle LOM ; hence 



Again, 



therefore 



MOC = LOC. 



0LG = OCG + LOC, 

 G0M = Gr0C + M0C; 

 OLG= GOM. 



