732 Sir G-. Greenliill on Pendulum 



cuts off a constant area UJU' = 7r — 2c ; so that the angles 

 JUU', JU'U are equal to the angle UJU 7 , and then 



JQ=QU=QU', JM=MU, JM'=M'U'. 



Since AU is a quadrant and AYU a right angle, YU is 

 a quadrant, and so also is XU' ; and XU'= YU, QX = QY. 



If DX cuts AM in W, the spherical triangles XQW, 

 YQA are equal, and 



(10) DWQ=QAY=DAQ=£', DW=DA, 



(11) DW=DX+XW=DX+AY=DA=c. 



4. So much for the geometry o£ the sphero-conic AQD, as 

 developed in Salmon's ' Solid Geometry ' and by Mr. Rose- 

 Innes; returning to the vertical circle AQD, draw another 

 interior circle aqd, centre c, with the same limiting point 

 L and radical axis HEK, cutting QLQ' in q, q' ; then (fig. 3) 



(1) R q . H^=HL 2 =HQ . HQ', 



(2) Q 7 .Q 7 '=(HQ-H 7 )(HQ-H 7 ') 



=HQ 2 - (H 7 + H 7 ')HQ + HQ . HQ' 

 = (HQ + HQ'-H 7 -H 7 ')HQ 

 =2Mm.HQ=2Cc.KQ. 



If QT is the tangent to this inner circle, cutting the outer 

 circle again in Q u 



(3) QT 2 = 2Cc.KQ, QL 2 = 2CL.KQ, 

 m QVP_Cc_Q,T 2 



w QL* ~ CL ~" QiL 2 ' 



and LT bisects the angle QLQ } ; also IT = IL, if QQ X cuts 

 HE in I, giving a simple construction of the inner circle for 

 a given QQ X . 



As the tangent QTQ X moves round, cutting the outer circle 

 at equal angles, 



vel. of Q 1 _ QiT _ / KiQj 

 vel. of Q ~~ QT ~V KQ' 



and this is the ratio of the velocity under gravity of two 

 particles, Q and Q l5 describing the circle in the same manner, 

 so that Q and Q 1 will remain simultaneous positions of the 

 particles if QQ X is a tangent of the inner circle ; and putting 

 ADQ 1 = ^r = amM 1 , then 



(6) u 1 — u=iv, a constant. 



5. Draw gcf in fig. 2 through c perpendicular to the plane 

 AQD, cutting OA in /, and OL in g ; the circle, centre/ and 



