734 Sir G. Greenhill on Pendul 



um 



to that described by Q, and with the same cyclic arcs, so 

 that its tangent QQ L cuts off from the cyclic arcs a triangle 

 UJU' of constant area, aud UU' is bisected at T (Salmon, 

 'Solid Geometry,' §§247, 248). 

 Bat since, in figs. 2, 3, 



m QT_ _ QL_ QO 

 [ - } TQ 2 ~ LQ X ~ OQ; 



OT bisects the angle QOQi, and T in fig. 5 is the midpoint of 

 QQi, so that (Salmon, § 252) QQ X cuts off a constant area 

 from the outer sphero-conic. 



With constant u x —u=w, the area QMM 1 Q l is constant, 

 so that the spherical quadrilateral QMMjQj^ is constant, and 

 this implies that the sum of the angles DQQ 1? DQ X Q or DQV 

 is constant, and this is found in § 8 to be am(K— w)-j-^7r. 



As before in fig. 4, XCJ' and YU are quadrants, 

 QY = Q l K=p suppose, Q l Y~QX = g suppose ; 



DX=YV, DX+AY=AV=DW=c\ 



7. The angle ADX = (£-f^ in fig. 5, as in fig. 3; 

 QDX = ^, Q 1 DX = 0. 



Similarly DAY=f + ^» QiAY = ^, QAY=^'. 



Then in the spherical triangle AQV in fig. 5, 



(1) sin QV = sin DQ l = k sin yjr' , sin AQ = /c sin $, 



sin QV sin AQ tc sin $ sin AY 



(2) 



sin QA V sin QVA sin QVA sin AQV 



so that QYA = (j) ; this is seen also from the equality of the 

 triangles QYV, Q X DX, in which QY = Q X X, YV=DX, so 

 that QVY = Q L DX = <f> ; this is the equivalent of Mr. Rose- 

 Innes's theorem (III.)- 

 Also 



sin c' 



(3) sin AQV = =^-, sin AQ V = sin MQV = - cos DQV, 



since DQM is a right angle ; and so we put, as in (8) § 8, 



(4) AQV=am(K + u>), MQV=am (K-w), 



sin c' = k sn (K 4- w) = k sn (K — w), 

 cosc / = dn(K + iy) =dn(K— w). 



