340 
Proceedings of the Royal Society 
series of three-termed expressions, each of which denotes the area of 
A 2 
a triangle, and is of the form , — the quadrilateral having two 
such expressions, the pentagon having three, &c., and the figure of 
n sides having n - 2 ; and such an expression, as we have seen, is 
obtained whether it is composed of terms of the first or second type. 
The same remarks will apply to the pair of expressions 
(Vs- Vi ) 2 and JVrViL. 
b 1 b 2 (a 1 b 2 - ajbft a 1 a 2 (a 1 b 2 - a 2 bft 
II. When the straight lines forming the sides of the figures are 
referred to three given intersecting straight lines in the plane of the 
hgure . 
The method employed is uniform with the first method. 
1. Let XYZ (fig. 3), the triangle 
formed by the three given straight 
lines, be taken as the triangle of 
reference, the position of any point 
being determined by its distances 
(a, ft , y) from the sides. 
With the usual notation, let the 
As A) z equations to the sides of the triangle 
ABC, whose area we seek, be 
l Y a + m ^ft + n Y y = 0 , l 2 a + m 2 ft + n 2 y = 0 , l^a + mfft -+- n 2 y = 0 , 
and suppose the sides when produced to meet YZ (a = 0) in the 
points A x , A 2 , A 3 . Then 
AABC = AAA 2 A 3 + ACA 1 A 2 - ABA X A 3 . 
Now 2 ACA x A 2 = A x A 2 x the a of the point C, that is, 
of the point of intersection of the lines l x a + mfft + n Y y = 0 and 
l 2 a + m 2 ft + n 2 y = 0, which is given by 
a ft 
m Y n 2 - m 2 n-± nf 2 - nf^ 
y 
2A 
l l m 2 - l 2 m 1 
a, 
&, 
c 
h> 
mi, 
n i 
1% 5 
m 2 , 
n 2 
A denoting the area of the triangle of reference, and a, b, c its sides. 
And, Af), A 2 E being perpendiculars on XZ, 
