514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 251
where the constants 2, X have to be determined ; I have in the first instance written
2 (X + x) + X%, instead of LX + lx+X%, thus introducing two constants only, since it is
clear from the symmetry in regard to «, X that we must have 2 = L. We must have
<fix = 0, when the curve is a conic or cubic. Writing « = 2, we have
<f>x = X* + 2X 3 - 115X 2 + 144Z + 532 + £ (- 9X 2 + 111X + 234) + l (2 + X) +
and then for the conic, X = 2, g = 6.
Writing x = 3, we have
<f>x = X* + 2X 3 - 67X 2 - 264X + 828 + f (- 9X 2 + 99X + 324) + 2 (3 + X) +
and then for the three cases of the cubic X = 6, f = 18 ; X=4, £ = 12; and X = 3, f=10.
We have thus the four equations
2912 + 4fl+ 6\=0,
9252 + 9/-F 18X = 0,
5796 + 72+ 12\ = 0,
4968 + 62 + 10\ = 0,
all satisfied by 2=+ 172, X = — 600. Hence finally
X 4
(
+ 1)
+
X s
(
2x?
— 18« 2 + 52« —
46)
+
X 2
(
- 18«»
+ 162« 2 — 420« + 221)
+
X
(
52« 3
- 420« 2 + 704« +172)
+
p
— 46« 3
+ 221« 2 + 172«
+
£
X 2 (
- 9)'
j
1
+
* (
- 12«+135)
-
— 9x- + 135« — 600
J
but on account of the symmetry the number of triangles is = one-sixth of this expression.
Article Nos. 22 to 36. The Case 52, as belonging to a different series of Problems.
22. In the foregoing Case 52, where all the curves are one and the same curve,
we have the unclosed trilateral aBcDeFg, and we seek for the number of the united
points (a, g). But we may consider this as belonging to a series of questions, viz. we
may seek for the number of the united points (a, B), (a, c), (a, D), (a, e), (a, F), (the
last four of these giving by reciprocity the numbers of the united points (B, D), (B, e),
(B, F), (B, gfj, and finally the number of the united points (a, g). It is very instructive
to consider this series of questions, and the more so that in those which precede
(a, F) there are only special solutions having reference to the singular points and
tangents of the curve, and that the solutions thus explain themselves.
32—2
