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