Motion and Spherical Trigonometry. 7ol 



Fig. 4. 



With EJ the polar circle o£ A, and E'J of D, and from 

 (3) § 1, when Q makes a small advance to q, and M, M' to 

 m, m\ 



d<f> Afidxf) cos DQ 



(5) dU = -t^t = — i-f-z = d<f> 



v y A<£ /c cos c r 



_ spherical area QM'w'^ 

 cos c 



( 6) 2 du =*£, = W£ = c -!lAQd4>> 



v ' A</> /c cos c T 



_ spherical area QMmq 



cos c 



(7) 



area AE'M'Q _ j 



irea DEMQ 



cos c 



cos c 



(8) K = 



area AE'JDQA 



cos c 



The point Q describes a sphero-conic, with EJ, E'J the 

 cyclic arcs, since 



(9) cos AQ cos DQ = sin QM sin QM' = cos c, 



(Salmon, ' Solid Geometry,' Chap. X.), and the tangent 

 XJQTJ' intercepted by the cyclic arcs is bisected at Q, and 



