382 Proceedings of Royal Society of Edinburgh, [june 18 , 
the theorems already obtained are only particular cases of a series of 
more general ones, which we now propose to investigate. 
X 
In the case of Trilinear co-ordinates, taking XYZ as the triangle 
of reference, let ABC be any triangle the equations to whose sides 
are 
l-^a -f + n-^y = 0, = 0, = 0 ; 
and suppose the sides, when produced, to meet YZ{a = 0) in the 
points Aj, A 2 , Ag. Then it may be shown that 
area CA^A 2 = 
Aabc 
1 i ■ 
2 
a b c 
b c 
6 c 
1112 U2 
) 
A denoting the area of the triangle of reference, and a, h, c its sides. 
In like manner we shall obtain for the areas of the triangles 
having their vertices at C and bases in ZX and XY, the two 
corresponding expressions 
I^abc 1 l-gi.2 
2 
and 
^ahc 
1 1 ® 
a b c 
a c 
h 
a c 
I 2 1^2 
a b c 
Zj m-^ 
Zg n%2 ^'2 
a b 
Zj 
a b 
h ^2 
respectively ; with similar expressions for the triangles whose 
vertices are at A and B, and sides intercepted in like manner. 
But, from the geometry of the figure, we have 
ABC = AA2A3 - BAjAg + CAj Ag , 
with two like expressions for ABC involving the triangles whose 
bases are in ZX and XY. 
