344 
Proceedings of the Eoyal Society 
The corresponding expression for a figure of n sides is 
(12) + (23) + (34) + (45) + ... + (»- 1, ») + (nl) , 
the terms of which may he of either form of expression above 
obtained for (12). And, as an algebraical result, it may he shown, 
as in Case I., that the sum of n expressions in the form of this series 
is the same, whether the terms of the series he of the form 
(p 1 sec cq - p 2 sec a 2 ) 2 (p 1 cosec oq — p 2 cosec a 2 ) 2 
tan cq — tan a 2 cot a 2 — cot oq 
2. We can now obtain a very interesting form of expression for 
the area of a figure involving the distances of its sides from the 
given point, and without reference to the lines OX and OY. 
Since a 2 — a 3 is the angle between the perpendiculars on the lines 
2, 3 it is equal to the angle between the lines themselves, that is 
the A of the triangle ABC. The expression for 2AABC thus 
becomes 
(p l sin A - p 9 sin B + p 3 sin C) 2 
sin A sin B sin C 
A A A 
which may also be appropriately written with 23, 31, 12 for 
A,B, C respectively. 
When the point 0 is within the triangle, the expression for twice 
its area is (p 1 sin A +p 2 sin B +p. d sin C) 2 sin A sin B sin C ; and 
we observe that the necessary change in the sign of p, when the 
point is outside the figure, is made by the same rule as when the 
area is given in the form p Y a +pf) +y> 3 c (a, b, c being the sides), and 
can be seen by inspection. 
In the case of a quadrilateral we have 
2 Area = (12) + (23) + (34) + (41) . 
Using for (12), &c., the form of expression 
(p 1 cosec oq — p 2 cosec a 2 ) 2 
cot a 2 — cot cq 
expanding and collecting the terms in jo 2 , we find the expression in- 
volving pf (which consists of the difference of two terms), is 
jt? 1 2 cosec 2 a 1 (cot cq — cot a 2 ) 
(cot a 2 — COt a 1 )(cot a 4 - COt oq) ’ 
Pi sin (a 2 -a 4 ) 
sin (oq - a 2 ) sin (cq - a 4 ) ’ 
that is, 
