236
ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE.
[514
Case 33. B = F = y, a = e — D — x. By reciprocation of 34.
No. = 2 (# — 1) (xX — x — X) Y(Y — l)c.
Case 34. c — e=y, B = D — a = x.
X = {X-2)y(X-l)(y - 1 )Fx, x=FyX(y- 1) (X - \) {x - 2),
g = y(y-l)(X-l) {(X- 2)a;+X(X-2)} F
= 2 (X — 1) (xX — x — X) y {y — 1) F.
Case 35. a — B — y, c — e = B = x.
X = X(x-2) Y(x-1 )Fy, rf = FxY(x- l)(X-2)y,
g = yY(x- 1) {X (x - 2) + (X - 2) x} F
= 2(x- 1 )(xX-x-X)yYF.
Case 36. a = D = y, B = F = e = x. By reciprocation of 35.
No. = 2 (X — 1) {Xx — x — X)yYc.
Case 37. a = e = D = x, c = B = y. By reciprocation of 38.
No. — 2{x—l) [xyXY — xy (X + Y) — XY(x + y)-h 2xy + 2XF}F.
Case 38. B = D = a — x, F — e — y.
X = (X-2) C (X-l)y(Y-2)x, x = Y(y— 2) Xc (X — 1) (# — 2),
g =(X-i)c [xy(X-2)(Y-2) + XY(x- 2) (y- 2)}
= 2 (X — 1) [xyXY — xy (X + F) — XY(x + y) + 2xy + 2XF} c.
Case 39. a = c = e = B = x.
Functional process; the curve is assumed to be the aggregate of two curves, say
a = c = e = B = x + x'. Forming the enumeration
Case
x X x DxF
x'X'x'Dx'F 39
x'X x . x .
&c. 10
x X'x . x.
6
x'X'x . x.
14
x X x’. x .
10
x'X x'. x .
12
x X'x'. x .
14
x'X'x' .x.
8
(where the second column is derived from the first by a mere interchange of the
accented and unaccented letters), I annex to each line the number of the case to