1842.] On Equations of Condition for a Quardrilateral. 29 



Ai -1- B 3 + C 2 — 180 = 6l \ 



(B) B, + C 3 + D 2 — 180 = g : ^ Thege may be termed Tr . an _ 

 C] + D3 + A2 — 180 = £ 3 r gular equations. 



Di + A 3 + B 2 — 180 = e4 \ 



(C) A 3 + B 3 + C 3 + D 3 — 360 = g, + f3 = £2 + £4 This may 

 be termed the Quadrangular equation. 



In the Triangular equations substitute for the middle angle its totopar- 

 tial value as above, and subtract each expression from the one immediately- 

 above it, and we have 



Ai + B 2 — Ci — D 2 = £ j — £2 = £4 — fc3 1 These may be 

 '■^ t> > m n a r termed Vertical 



Bi + C 2 — Di - A 2 = e 2 — H ~ e 1 — e,^ equations. 



In like manner in the 2d figure we have 



Ai + A 2 = A 3 

 Bj + B 2 = B 3 



( a ) n 1 + cf = C 3 I Totopartial equations. 



Di + D 2 + D 3 = 316 ) 



Ai + B 2 + Di— 180 = H 



Bi + C 2 -f D 2 — 180 = H f Triangular equations. 



(J3) Ci + A 2 + D 3 — 180 = f3 



A 3 + B 3 + C 3 -180 = £i + £2 + £2 



Also Di = 360 — D 2 — D 3 and substituting the values of D 2 

 and D 3 from the triangular equations 

 = 360 + Bi + C 2 — 180 — £2 -f Ci + A 2 — 180 — 

 t 3 = Bi -f C 3 + A 2 — £2 — £3 



.;. Bi + C 3 -j- As — Di = £ + E ) These may be term- 



/csv n 1 a 1 u t\ 1 I e d Cuneal equations, 



(8) C, + A 3 + B 2 - D 2 = £3 + £l \ fmm ^ wed 4 g shape 



ed Cuneal equations, 



.from the wedge shape 



Ai + B 2 + C 2 — D 3 = £i + ^ J f thL eangles concerned. 



If we examine in detail, these angular equations, we shall find that in 

 both cases, besides the totopartial equations, there are two separate 

 equations of condition, whereby to determine the error on each of the 

 12 angles in the figure. As these, with the exception perhaps of the 

 Vertical and Cuneal equations contain nothing new, I shall proceed to 

 the investigation of the sinal equations, which indeed is the main object 

 of the present communication. These I shall give in the form, in which 

 they first occurred to me. 



Prop. I. — In a quadrilateral the product of the sine of any whole 



