of Edinburgh, Session 1884 - 85 . 
345 
which, on completing, so as to have the denominator which is com- 
mon to all the terms, is 
p$Sin(2, 3, 4) 
sin (1,2, 3, 4) 5 
where sin (2, 3, 4) denotes the product of the sines of a 2 - a 3 , 
a 2 - a 4 , and a 4 — a 3 , that is, the product of the sines of the angles 
of the figure formed by the lines 2, 3, 4 ; while the denominator 
denotes the product of the sines of the angles of the quadrilateral. 
Similar expressions holding for the coefficients of p 2 2 , p 2 , and pf, 
we have 
2 Area = 
]Sy» 1 2 sin (2, 3, 4) 
sin (1, 2, 3,4) 
-22 
PlP-2 
sin (1,2) 
> 
each summation consisting of 4 terms. 
And generally, for a figure of n sides, we shall get in like 
manner 
2 Area (1, 2, 3 , n) 
L Sp/sin (2 , 3, 4, , ■ . m) _ 
sin (1, 2, 3, 4, ... n) . sin (1, 2) 5 
each summation consisting of n terms, the figure (2, 3, 4, ... n) 
being formed by the sides 3, 4, ... n - 1 and the sides 2, n pro- 
duced. 
With regard to the signs of the terms in these series, we observe, 
in respect of the angles a l5 a 2 , a 3 , . . . a n (see fig. 2), that cq is the 
greatest and a 3 the least; also a 4 , a 5 , . . . a n are in ascending order 
of magnitude, while a 2 may be ^ or ^ any of these latter angles. 
Thus the actual coefficients of pf, p 4 2 , . . . p n 2 are all negative, that 
of p 2 2 is positive, while the coefficient of pf is positive or negative 
according as the last drawn line of any figure makes with BC 
an angle ^ or ^iC. But the figure (1, 3, 4, ... n) has re-entrant 
angles at the points (4, 5), (5, 6), <fcc., so that the expression 
sin (1, 3, 4, . . n), the apparent coefficient of p 0 2 , is negative. Also 
if p n make with BC an angle the figure (2, 3, 4, ... n) has re- 
entrant angles at the points (34), (45), &c., so that the expression 
sin (2, 3, 4, . . n) is also negative. With regard to the signs of the 
products, they are all positive except those oip 1 p 2 and £> 2 y? 3 , that 
is, except those in which p 2 occur. 
