338 
Proceedings of the Royal Society 
each of which expressions may he shown to be equal to 
{<>i( a 2 b 3 a sh) + c 2( a A ~ a A ) + c s( a A ~ a 2 h iS 2 ' 
( a 2 b 3 ~ a A)( a 3 b l ~ a l b §)'( a i b 2 ~ U 2 b l) 
2. We shall now deduce corresponding expressions for the area 
of a quadrilateral, and generally for the area of a rectilineal figure of 
any number of sides. 
The triangle ABC is formed by three lines which we call 1, 2, 3. 
Denoting the area of the triangle formed by any two lines (1, 2) and 
either axis, by (12), we have seen that 
A(ABC) = (12) + (33) + (31) , 
where (12) stands for either 
[fa ~ c 2) 2 ( m i c 2 ~ m 2 c^) 2 
2 (m 1 - m 2 ) 2 - m 2 ) 
with similar expressions for (23) and (31). 
Now (fig. 2), let a fourth line 
be drawn cutting the sides 3 
and 1 of the triangle, and so in- 
tercepting a quadrilateral formed 
by the lines 1, 2, 3, 4 in order 
opposite to that of the hands of 
a clock. Then 
Area of quadrilateral (1234) 
L A(123- A(143) 
= (12) + (23) + (31) -(14) -(43) -(31) 
= (12) + (23) + (34) + (41); 
that is, consists of the four triangles formed by either axis, and the 
lines of the figure taken two at a time in order round the figure from 
1 to 1 in the same direction as before stated. 
Again, let a fifth line be drawn cutting the sides 4 and 1 of the 
quadrilateral, and thus intercepting a pentagon formed by the lines 
1, 2, 3, 4, 5 in order as before. Then 
Area of pentagon (12345) 
= quadrilateral (1234) - A (154) 
= (12) + (23) + (34) + (45) + (51), 
