THE IN- AND- CIRCUMSCRIBED TBI ANGLE. 
395 
Case 41. c=e=T)=F—x. 
X =B#(X— 2)(.r— 3)(X— 3)«, 
Z '=X(#-2)(X- 3)(x— 3)B a. 
g =(*— 3)(X— 3)aB{a:(X-2)+X(ar— 2)} 
=2(^— 3)(X— 3)(^X— -a;— X)«B. 
Case 42. «=c=D=F=;r. 
Functionally, viz. the curve is supposed to be the aggregate of two curves, say 
a=c=B=F=x J r x J . 
The enumeration is 
Case. 
x Bx X e X 
x' Bx'X! e X', (42) 
x' . x X . X 
&c. (11) 
x .afX .X 
(11) 
x' . at X . X 
(IT) 
x . x X! . X 
(10) 
x' . x X' . X 
(19) 
a; . a*' X' . X 
(21) 
X' . X 
(10) 
whence 
<p(x-\-x') — (px—<pF — eB multiplied into 
4(X-l)(X^-X-^y 
(11) x 2 
+ 2x(x— 1)X'(X'— 1) 
(IT) 
4- 4(#— 1)(X#— ^)X f 
+ •• 
(10) x 2 
+2aX.:i'X' 
(19) 
+ 2xx'XX! - 2(x+x J )XX t - 2(X+X>F+ 4(Xr' 
+ X'a’)+ . . 
(21) 
where the (. .)’s refer to the like functions with the two sets of letters interchanged. 
Developing and collecting, we have 
®(x-\-x') — <px— <px'=eB multiplied into 
X 2 ( ixx’ + 2x' 2 — 6x') 
+XX'(4.z 2 + 8^+4^ 2 - 12x-12x J + 8) 
+X' 2 (2x 2 ixx 1 — 6x) 
+ X (-I2.m'-Gx' 2 +lSx') 
-J-X' (— 6x 2 — 12xx ! -F\8x) 
+ 8xx', 
