30 On Equations of Condition for a Quardrilateral. [No. 121. 



angle, and the sines of the two consecutive left hand angles going round 

 by the left, is equal to the product of the sine of the opposite whole 

 angle, and the sines of the two corresponding right hand angles returning 

 by the right. 



Sin A3 sin Bi sin Ci = sin C3 sin B2 sin A2 } These (for rea- 

 Sin B3 sin Ci sin Di = sin D3 sin C2 sin B2 f sons afterwards 



^ ' Sin C3 sin Di sin Ai = sin A3 sin D2 sin C2 C to be shewn) may 

 Sin D3 sin Ai sin Bi = sin B3 sin A 2 sin D2 J be termed exter- 

 nal alternate equations. 



Prop. II. — In a quadrilateral, the continued product of the sines of 

 two adjacent whole angles, and the sines of the angles between the 

 diagonals, and the opposite sides, is equal to the continued product of 

 the two pairs of opposite angles. 



,p. Sin A3 sin B3 sin Ci sin D2 = sin C3 sin D3 sin Ai sin B2 1 

 Sin B3 sin C3 sin Di sin A2 = sin D3 sin A3 sin Bi sin C2 J 

 These may be termed opposite alternate equations. 



Also the known property, that the product of the sines of all the left 

 hand angles is equal to that of the sines of all the right hand angles. 



(G) Sin Ai sin Bi sin Ci sin Di = sin A2 sin B2 sin C2 sin D2 which 

 may be termed the internal alternate equation. 



Demonstration. 



BD = CD — — — - by common Trignometry, 



sin Bi ' J 



also AD = CD$^=BD^ = CD sinC3 ' si ° B! 



sin A2 sin A3 sin Bi sin A3 



sin Ci sin C3 . sin B2 



sin A2 sin Bi sin A 3 



Sin A3 sin Bi sin Ci = sin C3 sin B2 sin A2 . . fej 



a • t> t^ a ta sm As _ „ sin C3 

 Again B D = A D . _ = B C ~— - 

 sin B2 sm D2 



AC=BC S !^=AD sinDs 



and multiplying vertically, 



sin A 1 sin Ci 



sin A3 . sin B3 sin C3 . sin Ds 



sin B2 sin Ai sin Ds sin C 1 

 .'. Sin A3 ' sin B3 ' sin Ci * sin D2 = sin C3 * sin D3 * sin Ai sin B2 (f) 



