346 Proceedings of the Royal Society 
Hence, the expression for twice the area of figure (1, 2, 3, ... n) 
may be written 
PiP 2 _ ^P\ s in (2, 3, 4, . . . n) 
sin (1, 2) sin (1, 2, 3, 4, . . . n) ’ 
where all the terms in %p 2 are positive, while all those in the first 
series are positive except those in p x p 2 and p 2 p%. In the cor- 
responding expression, when the point 0 is within the figure, all the 
signs in both series are positive ; and we observe that the necessary 
change of sign, for an external point, is made according to the rule 
stated above (in this particular instance by writing - p 2 for p 2 ), the 
truth of which is evident on inspection of the figure. 
3. Lastly, we may deduce the results for regidar polygons. 
Denoting the angle of a regular polygon of n sides by 7 r - 0 , 
27T 
so that 6 = — - , we have 
n 
sin (2, 3, 4, . . . n) = (sin 0 ) n ~ 2 sin 2 <9 
= 2 sin M_1 0 cos 0 . 
Thus the expression for the area of the polygon becomes 
cosec 6 . %p x p 2 - cot 0 . S^q 2 , 
the signs of the terms in each series being the same as before. 
If the point 0 be the centre of the figure, all the p's are equal to 
one another, each bfeing equal to r the radius of the inscribed circle 
of the polygon ; and thus the area becomes 
wr 2 (cosec 0 - cot 6) 
= nr 2 tan 
A 
= nr 2 tan — , 
n 
the usual expression for the area of a regular polygon of n sides in 
terms of the radius of the inscribed circle. 
Note . — In the case of an equilateral triangle and a square, the 
above expression for area reduces to particular forms : in the former 
case to J J^(p 1 ± ^ 2 +y> 3 ) 2 , and in the latter to {p l +p 3 )(p 4: 
according as the point 0 is within or without the line. 2, the truth 
of which is obvious on inspection of the figures. 
