Motion and Spherical Trigonometry. 737 



And with 



(11) p=XQ 1 = XSQ 1 = i7r-DSZ 1 = i7r-ASZ / , 



/- 10 \ . Qr7 or/ , cosSAZ' cosi/r' 



(12) sin SZ = cos SZ = - — . j.^, = J-, 



v J sm ASZ' cosp ' 



ry sin SAZ' sin AZ' sin -dr'Acb 



cos SZ = . — 1 — ii 



sin ASZ' cos ^ ' 



/ 1 0\ nry COS SDZi COS (f) 



(13) cos SZi= . p.gy = ^ , 



sm DSZi cosp ' 



. ar7 sin SDZ X sin DZ X sin <f>Ai// 



sin bZ 1 = . rwr/ s = — r r , 



sin UbZi cosp ' 



(14) sin (SZ - SZQ = C05 ^ C ° S *' ~ Si " * Si " *^**g 

 v ' y cos^p 



= en (u + K— Mj) = en (K — to), 



(15) SZ — SZ 1 = j7r— am(K — 1«), 

 Z 1 Z/=j7r= am (K + io) + am (K — w) — \tt, 



(16) SZ + SZ^amCK+w). 



Thus the constant in (6) is am (K + w) — i7r — fc'w. 



The sphero-conic ERE' of fig. 4 is the projection on the 

 sphere o£ the polar reciprocal o£ the circle AQD o£ fig. 1 

 with respect to D, and this is the parabola EP ; while the 

 sphero-conic S in fig. 5 will be the projection of the 

 hyperbola o£ S in fig. 3, polar reciprocal of the circle aTd. 



10. The motion of P at the same level as Q in fig. 1, 

 oscillating on the arc BAB' o£ the circle on the diameter 

 AE, will represent the associated motion of a pendulum, 

 swinging through an angle 4c, and then if 6 denotes the 

 inclination of the pendulum OP to the vertical 



m • 21 . AN AD AN 2 . 



cos-i<9 = A<2> = cosDQ, £0 = DQ, in fig. 4. 



Draw a circle through B and B', centre o. in fig. 6; draw 

 PB, and PpB' crossing this circle at p\ and draw PH and 

 Qq perpendicular to BB' and PB'. 



Phil. Mag. S. 6. Vol. 20. No. 118. Oct. 1910. 3 C 



