514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
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respectively. And I use the same letters a, c, e, B, D, F to denote the curves con 
taining the angles and touched by the sides respectively; viz. the angle a is situate 
in the curve a, the side B touches the curve B, and so for the other angles and 
sides respectively. An equation such as a = c or a = B denotes that the curves a, c or, 
as the case may be, the curves a, B are one and the same curve: it is in general 
convenient to use a new letter for denoting these identical curves; viz. I write, for 
instance, a = c = x or a = B = x, to denote that the curves a, c or, as the case may be, 
the curves a, B are one and the same curve x; the new letters thus introduced are 
x, y, z, there being in regard to them no distinction of small letters and capitals. 
The expression “ no identities ” denotes that the curves are all distinct. But I use 
also the letters a, c, e, b, d, f, x, y, z, and A, G, E, B, D, F, X, Y, Z quantitatively, 
to denote the orders and classes of the curves a, c, e, B, D, F, x, y, z respectively; 
thus, in the Table, for the case 1 “ no identities ” the number of triangles is given 
as = 2aceBDF, which agrees with the before-mentioned result for the polygon : for the 
case 2 the several separate identities 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 — 1) eBJDF. 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 £ 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 inflections + 3 times the order; 
the curve being determined by its order x, class X, and f; and similarly for rj 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 — B = x, and it is to this first form that the number 
of triangles, here 2 (Xx — X — x) ceDF, 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.