514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE.
249
Case 52. a=c = e = B = D = F=x.
Functional process, by taking the curve to be the aggregate of two curves, say
= x + x'. The enumeration of the cases is conveniently made in a somewhat different
manner from that heretofore employed, viz. we may write
x or x'
x' or X
Case
times
II
all
II
none
(52)
1
a
residue
(50)
3
B
99
(51)
3
a, c
99
(46)
3
B, D
99
(47)
3
a, D
99
(48)
3
a, B
99
(49)
6
a, c , e
B, D, F
(43)
1
a, B, F
c , e , D
(44)
3
a, B, D
c , e , F
(45)
6
and the functional equation then is
<f)(x + x) — <f)X — (fix'
= 3a;' /x 3 (
+
1) + . .
3ÎT;
x 3 { 2X 3 — 14X 2 + 28X —11)
J x (— 10X 3 + 70X 2 — 116X — 8) y
+ 12X 3 — 76X 2 + 64X
+ £( — 6x — 4X + 42) (
+ 3X'
X 3 ( + 1)\
X 2 ( 2ar* — 14a; 2 + 28®-11)
■I X (- 10a; 3 + 70a? - 116a; - 8) \
+ 12a? — 76a; 2 + 64a;
4- £■ ( — 6X — 4a; 4- 42) t
(51) x 3
+ 3 (x' 2 — x')
. + x (2X 3 — 10X 2 + 12X — 1)
- 4X 3 + 20X 2 -16X-3f
V
4..
(46) x 3
C. VIII.
32