514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 253 
26. The mode of obtaining these equations appears ante, Nos. 5 and 6, but for 
greater clearness I will explain it in regard to a pair of the equations, say those for 
(a, e), (a, D). Regarding a as given, we draw from a the tangents aBc, touching at 
B and besides intersecting at c (viz. the number of tangents is — X — 2, and the 
number of the points c is = (X — 2){x — 3)); from each of the positions of c we draw 
to the curve the (X — 3) tangents cDe touching at D and intersecting at e; the 
whole number of these tangents is = (X — 2) (x — 3) (X — 3); and this is also the number 
of the points D, but the number of the points e is = (X — 2) (oc — 3) (X — 3) (x — 3). 
Now this system of the (X — 2){x — 3) (X — 3) tangents is the curve © of the general 
theory {ante, Nos. 3, 4), viz. the curve © (which does not pass through a) intersects 
the given curve in the three classes of points c, D, e, the number of intersections at 
a point e being =1, at a point D being = 2, and at a point c being = X — 3. And 
we have thus the equation 
e — e — e + 2 (d — 6 — S') + (X — 3) (c — y — y) = 0, 
where e, d, c are the numbers of united points and (e, e'), (8, 8'), (y, y') the corre 
spondences in the three cases respectively. 
27. Observe that we cannot, starting from a, obtain in this manner the equation 
for the number of the united points {a, D); for we introduce per force the points e, 
and thus obtain the foregoing equation for {a, e). But starting from D, the tangent 
at this point besides intersects the curve in {x — 2) points, each of which is a position 
of c; and from each of these drawing a tangent cBa to the curve, we have the 
curve © consisting of these {x — 2) (X — 3) tangents, not passing through D, but inter 
secting the given curve in the three classes of points c, B, a, viz. the number of 
intersections at each point c is = X — 3, at each point B it is = 2, and at each point 
a it is = 1; and we have thus the equation 
(d-S— S')+2(c 0 — 7o — 7o ') + (X—3)(b —/3 —/3') = 0, 
where the numbers (d, 8, 8'), (c 0 , y 0 , y,,'), (b, ft) refer to the correspondences {D, a), 
(D, B), and (D, c) (or what is the same thing (a, B)) respectively. 
28. Correspondence {a, B). 
We have 
/3 = X- 2, ft=x-2, 
and thence 
b = # + X — 4+2A 
= - Sx - 3X + 2£, 
which is the solution: the value obtained above was b = i + k, and we in fact have 
identically 
i + k — — Sx — 3X + 2£. 
It was in this manner that I originally applied the principle of correspondence to 
investigating the number of inflections of a curve, regarding, however, the term re as 
a special solution; it is better to put the cusp and inflection on the same footing 
as above.