224 ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. [514
points Q', &c.; then the relation between the several quantities is as stated above:
see my “Second Memoir on the Curves which satisfy given conditions,” Philosophical
Transactions, vol. 159 (1868), pp. 145—172, [407]. I omit for the present purpose the
term “ Supp.,” treating it as included in the other terms.
4. In the present case we consider, as already mentioned, the unclosed trilateral
aBcDeFg, where the angles a, g are on one and the same curve a (= g) (the curve
in the general theorem); and the curve © is the system of lines eFg which by their
intersection with the curve a determine the points g. Considering these as the points
(P, P') of the general theorem we have p — 1: I change the notation, and instead of
a — a — a.' write g — % — X ■> I take (g) for the number of the united points {a, g),
and suppose that the points (a, g) have a (%, correspondence. The most simple
case is when the curve a is distinct from each of the curves e, F; here all the
intersections of the line-system eFg with the curve a are points g, that is we have
only the correspondence (a, g)\ and since the line-system eFg does not pass through
the point a, we have simply
5. But suppose that the curves a, e, F are one and the same curve, say that
a = e = F\ understanding by the point F the point of contact of a line eFg with the
curve a, then the intersections of the line-system eFg with the curve a are the points
g each once, the points F each twice, and the points e each as many times as there
are lines eFg through the point e, say each M times. (In the present case, where the
curves e, F are identical, we have M= F — 2 or P—3 according as the curve D is
or is not distinct from the curve F; in the cases afterwards referred to, the values
may be P or P—1; that is, we have always M = P, P—1, P—2, P—3, as the case
may be.) We have to consider the several correspondences (a, g), {a, P), (a, e)\ k is
as before = 0; and the form of the theorem is
(g “% - X) + 2 ( f - <f> ~ 0') + M ( e - e - «0 = 0,
where the symbols denote as follows, viz.
(a, g ) have a (%, %') correspondence, and No. of united points = g,
(a, F) „ (<£, <f>) „ „ „ =f,
{a, e) ,, (e, e ) ,, „ ,, e,
so that the determination of g here depends upon that of f —</> — </>' and e — e — e'.
6. The curve a might however have been identical with only one of the curves
e, F; viz. if a = P, but e is a distinct curve, then the equation will contain the term
2 (f —(f) — </>'), but not the term M (e — e - e); and so if a = e, but P is a distinct
curve, then the equation will not contain 2 (f — <f> — <£'), but will contain M (e — e — e'):
it is to be noticed that in this last case we have M= F or M = F — 1, according as
the curve D is not, or is, one and the same curve with P. The determination of (g)
here depends upon that of f — 0 — 0' or e — e — e, as the case may be. These sub-
