of Edinburgh, Session 1884 - 85 . 
343 
which may also be shown, as in Case I., without reference to the 
areas of figures. For, completing the series of terms in any 
instance, the expression for the area of a figure will consist of a 
series of symmetrical expressions similar to [A ) ; and such an 
expression, as we have seen, is obtained whether it be composed of 
terms of the first, second, or third type. 
III. When the straight lines forming the sides of the figures are 
referred to a given point in the plane of the figure . 
1. Let 0 (fig. 1) be the given point; and suppose, for the pre- 
sent, OX to be a fixed straight line through 0. We shall adopt a 
method uniform with the foregoing methods. 
With the usual notation, let the equations to the sides of the 
triangle ABC be 
» COS cq + y sin cq -p x = 0, (cq, pf) = 0, (a 3 ,p 3 ) = 0 , 
where cq, a 2 , a s are in descending order of magnitude. Then 
2ACA : A 2 
( AM 
cot A 1 + cot A 2 
( p 1 sec oq —p 2 sec a 2 ) 2 
tan cq — tan cq 
(p 1 cos a 2 — p 2 cos oq ) 2 
cos cq cos cq sin (oq — oq) ? 
with similar expressions for the triangles BAgAj and AA 2 A 3 . 
Also, by reference to the triangles intercepted by OY, we shall 
get 
o.pp t> _ ■ ( B i B 2 ) 2 fa cosec g, -p t cosec a 2 ) 2 
1 2 - t cot B x + cot B 2 cot a 2 — cot a x 
(p 1 sin a 2 — p 2 sin cq ) 2 
sin cq sin a 2 sin (cq - a 2 ) J 
similar expressions holding for the triangles BB 8 B 1 and AB 2 B 3 . 
Thus, denoting either of the two expressions obtained by (12), 
we have 
2AABC = (12) - (13) + (23) ; 
and each of these expressions for 2 A ABC may be shown to be 
equal to 
{Pi sin (« 2 ~ a s) +P 2 sin ( a 3 ~ a i ) +Pb sin ( a i - a 2) } 
sin (a 2 - a 3 ) sin (a 3 - cq) sin (a x - a 2 ) 
