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