370 
ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
the before-mentioned result for the polygon : for the case 2 the several separate iden- 
tities a—c , a—e, c—e are of course equivalent to each other ; and selecting one of them, 
a~c=x, the number of triangles is given as — 2x(x— l)eBDF. There is a convenience 
in thus writing down the several forms a=c, a=e, c~e of the identity or identities 
which constitute the 52 distinct cases of the Table; and I have accordingly done so 
throughout the Table, the expression for the number of triangles being however in each 
case given under one form only. It only remains to mention that for the curve x the 
Greek letter g denotes what may be termed the “ stativity” of the curve, viz. this is = 
number of cusps -(-3 times the class, or, what is the same thing, = number of inflec- 
tions + 3 times the order ; viz. the curve is determined by its order x, class X, and | ; 
and similarly for q and 
Observe that, in the column “ Specification,” each line is to be read separately from 
the others, and, where the word “or” occurs, the two parts of the line are to be read 
separately; thus case 5, the six forms are a = B, a — F, c=D, c— B, e—~F, e=D : the 
letter x (or, as the case may be, x , y , or x, y , z) accompanies the first of the given forms ; 
in the present instance a — V>—x, and it is to this first form that the number of triangles, 
here 2(X&'— X— x) ce DF, applies. 
I remark that what is primarily determined is the number of positions of a particular 
angle of the triangle, and that in some cases, on account of the symmetry of the figure, 
the number of triangles is a submultiple of this number ; viz. the number of positions 
of the angle is to be divided by 2 or 6 ; this is expressly shown, by means of a separate 
column, in the Table. 
