THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
403 
= x\ + 1 ) 
+x\ 2X 3 -14X 2 + 28X-11) 
+^(-10X 3 + 70X 2 -116X- 8) 
+ 12X 3 — 76X 2 + 64X 
+£(-6^-4X+42 ). 
As a verification, observe that for a conic, x—X — 2, £=6, this is =0. 
Second process, by correspondence: form c= 0 :=B=:D=:F=,£. 
We have . , ^ , 
“Red., 
% = X(x - 2)(X - 3)(ar - 3)(X - 3)a, 
% '=X(^~2)(X-3)(^-3)(X-3K = x . 
%+%'=« into 
2(^-2)(^-3)X(X-3) 2 . 
There is a first-mode reduction, which is 
=a{2i(X-4)(X-5) + 3*(X-3)(X-4)+*(X-3) + 2r(X-3)}, 
where the term a. 2r(X— 3) arises, as shown in the figure, 
and a second-mode reduction, which is 
= a { 2 r(x— 4)(#— 5) + 3i(x— o)(x — 4) } ; 
and the two together are —a into 
(X - 4)(X - 5)(x 2 - x + 8X - 3|) 
+ (X-3)(X-4)( 9X + 3|) 
+ (X— 3)/ — 3X + |\ 
\+X 2 - X + 8^-3£/ 
+( a?-4)( tf-5)(X 2 -X+8tf — 3£) 
+ ( a? -3X*-4)( -9*+3£); 
that is, = « into 
— x 3 
+tf 2 .2X 2 -10X+ll 
+^.-10X 2 +26X + 8 
4-4X 2 + 44X 
+|(6ar+4X-42); 
and subtracting this from the foregoing value of which is —a into 
x\ 2X 3 -12X 2 + 18X) 
-\-x ( — 10X 3 -f-60X 2 — 90X) 
-f- 12X 3 — 72X 2 +108X, 
the result is as before. 
3 k 2 
