342 
Proceedings of the Royal Society 
three similar terms involving the same letters, of which any one of 
the above three expressions for (12) is a type, is the same, that sum 
denoting the area of the triangle ABC and being equal to the 
expression (A). 
Eurther we may notice that if the sides of the triangle ABC he 
given by equations of the form Ix + my + nz = 0, where x, y, z denote 
the areal co-ordinates of a point, corresponding results will he 
obtained by writing la, mb, nc, for l, m , n respectively ; or, which 
comes to the same thing, by putting a = b — c— 1 in any expression 
denoting an area. 
Thus, in areal co-ordinates, the value of (12) is 
A(m 1 7Z 2 - m 2 ?q) 2 
( m l - n l)( m 2 ~ n 2) 
h 
m x 
% 
h 
m 2 
n 2 
l, 
1, 
1 
and the area of triangle ABC is 
m 1 
U 1 
m 2 
n 2 
m 3 
n 3 
h m 2 '""I 
/ 3 m B n B 
l\ m i n i 
h m s n % 
Zj m x n 1 
l 2 m 2 n 2 
1 , 1 , 1 
1 , 1 , 1 
1, 1, 1 
2. The area of a quadrilateral, and generally of a figure of n sides, 
may be deduced in the same manner as in Case I. We thus get, 
A„ denoting the area of a figure of n sides, 
A n = (12) + (23) + (34) + (45) +...+(»-!,*) + (nl ) , 
in which (12) denotes any one of the three expressions obtained 
above as its value, with similar corresponding expressions for the 
other n- 1 terms of the series. 
And we may notice, as an algebraical result, in addition to that 
already stated for the case of three expressions, that the sums of 
4, 5, 6, . . . , n expressions (selected and having signs as indicated 
above), of which any one of the above three expressions for (12) is 
a type, are equal to the sums of the corresponding number of 
expressions of each of the other types, each sum denoting the area 
of the polygon of the corresponding number of sides — the truth of 
