252 ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. [514 
23. Thus the first case is that of the united points (a, B), viz. we have here a 
point a on the curve, and from it we draw to the curve a tangent aB touching it 
at B; the points a and B are to coincide together. Observe that from a point in 
general a of the curve we have X — 2 tangents (X the class as heretofore), viz. we 
disregard altogether the tangent at the point, counting as 2 of the X tangents from 
a point not on the curve, and attend exclusively to the X — 2 tangents from the point. 
Now if the point a is an inflection, or if it is a cusp, there are only X — 3 tangents, 
or, to speak more accurately, one of the X — 2 tangents has come to coincide with 
the tangent at the point; such tangent is a tangent of three-pointic intersection, viz. 
we have the point a and the point B (counting, as a point of contact, twice) all three 
coinciding; that is, we have a position of the united point (a, B); and the number 
of these united points is = i + k. 
24. It is important to notice that neither a point of contact of a double tangent, 
nor a double point, is a united point. In the case of the point of contact of a double 
tangent, one of the tangents from the point coincides with the double tangent; but 
the point B is here the other point of contact of this tangent, so that the points 
a, B are not coincident. In the case of a double point, regarding the assumed 
position of a at the double point as belonging to one of the two branches, then of 
the X — 2 tangents there are two, each coinciding with the tangent to the other 
branch; hence, attending to either of these, the point B belongs to the other branch, 
and thus, though a and B are each of them at the double point, the two do not 
constitute a united point. (In illustration remark that for a unicursal curve, the 
position of a answers to a value = A, and that of B to a value = ya of the parameter 0, 
viz. A, ya are the two values of 6 at the double point; contrariwise in the foregoing 
case of a cusp, where there is a single value A = ya. Hence the whole number of the 
united points (a, E) is = i + k, and this is in fact the value given, as will presently 
appear, by the theory of correspondence.) 
I recall that I use A, = 2D, to denote twice the deficiency of the curve, viz. that 
we have A = X — 2x + 2 + k, — — 2x— 2X + 2 + £. 
25. The several cases are 
United points. 
(a, B) b-/3-/3'=2A, 
(a, c) c - 7 - 7' + 2 (b - /3 - /3') = (X - 2) A, 
(B, D) c 0 — 7o — 7o' by reciprocity, 
(a, D) d -8 -8' +2(c 0 —7„ —7 0 ') + (X-3)(b-/3-y3') = 0, 
(a , e) e —e — e + 2(d—8 — S' ) + (X — 3) (c — 7 — 7') = 0, 
(B, F) e 0 — e 0 - 6 0 ' by reciprocity, 
(a, F) f -f + 2 (e„ — e 0 — e 0 ") + (X — 3) (d — 8 — S') = 0, 
(a, g) g - % - X + 2 (f - <f> - </>') + (X - 3) (e - e - e') = 0, 
(B, II) go - %o ~ Xo by reciprocity, 
and so on.