THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
393 
Case. 
x X x IDF 
x'Xx'Bx'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 1 . x . 
12 
x XV . 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 which it belongs ; 
thus #'XVD#F is B =c=e=x, which is Case 10, and so in the other instances. Observing 
that cases 10 and 14 occur each twice, we have thus 
<p(x+x')— <px— <p4=DF multiplied into 
4(x-l)(Xx-X-xW + .. (10) x 2 
+ {2ff(a?-l)(ff-2)+X}X' + *. (6) 
+4a(a-1)(XV-X , -a j ) + .. (14) x 2 
-\-2x{x—l)FXl + . . (12) 
+ 2x(x-3)(X-2)x J + . . (8) 
where the (. .)’s refer to the like functions with the two sets of letters interchanged. 
Developing and collecting, this is 
<p(x-\-rf) — Ox — <}V=DF multiplied into 
2XX' 
+ 2X( 3x 2 x' + 3xx' 2 + x' 3 - 1 Oxx' - 5x' 2 + 6x') 
+ 2X'(x 3 4 3x 2 x' + 3xx' 2 — 5x 2 — 1 Oxx' + Gx) 
— 12(x 2 x' + xx' 2 ) + 40xx', 
and thence 
<p#=DF multiplied into 
X 2 
+ X(2x 3 - 1 Ox 2 -|- 1 2x) - LX 
~4x 3 +20x 2 -Ix-a%, 
where the constants L, /, X have to be determined. Now for a cubic curve the number 
of triangles vanishes ; that is, we have <px—0 in each of the three cases, 
x=3, X=6, §=18, 
„ X=4, §=12, 
„ X=3, §=10, 
3 i 
MDCCCLXXI. 
