of Edinburgh, Session 1884 - 85 . 
341 
A, A, = ZA, - ZA, = A2E ^ lP 
12 2 1 smZ 
2A 
sin Z\bn 2 - cm 2 bn^ - 
abcigiitfi-L — m 1 ?i 2 ) 
(6?^ - cmf(j)n 2 — cm 2 ) 
■ cm. 
since A 2 D is the /3 of the point of intersection of the lines a = 0 
and a + mj/3 + n Y y = 0, A 2 E being obtained in like manner. 
<1 
r-O 
e 
X ( 
) (m ± n 2 - m 2 n^) 2 
a, 
b, 
c 
(5?z x — cm-^)(bn 2 - cm 2 ) 
h 
m 1 
n i 
l 2 
m 2 
n 2 
Thus ACA-.A, 
(c -c) 2 
which is the corresponding expression to — — inCasel. Similar 
expressions holding for the triangles AA 2 A 3 and BA 3 A 1 , we shall 
get 
AABC = (23) + (12) + (31), 
which may be shown to he equal to the expression 
a , b , c 
a, b , c 
a, b, c 
If, m 2 Tie, 
l 3 m s n 3 
h m i n \ 
l 3 m 3 n 3 
Zj m ± 
l 2 m 2 n 2 
We may observe (as possessing an algebraical interest), that by 
considering in like manner the triangles intercepted by the other 
sides ZX (/3 = 0) and XY (y = 0) of the triangle of reference, we shall 
obtain two other expressions for the area of the triangle ABC in 
the form (12) + (23) + (31), namely, those in which (12) denotes 
respectively 
fr(V 2 - -Vi) 2 and Wi m 2 - h m iY 
(cI-l — an^){cl 2 - an 2 ) — bl l )(am 2 - bl 2 ) 
&abc 
in which k stands for the common factor 
b. 
'2 1 2 
with similar expressions for (23) and (31); and thus the sum of 
