THE IN-AND-CIRCITMSCRIBED TRIANGLE. 
401 
aXx'X'x'X' 
Sec. 
Case. 
50 
41 
40 
32 
42 
33 
38 
32 
40 
36 
28 
3 0 
O 
38 
36 
29 
41 
ciX x X x X 
. Xx X x . 
. X x'X x . 
.Xx'Xx . 
.XxX'x . 
.X'xX'x . 
.Xx'X'x . 
. Xx'Xx . 
.XxXx! . 
.X'xXx' . 
.XxXx' . 
.Xx'Xx' . 
.XxX'x' . 
.XxXx! . 
.Xx'Xx' . 
. X x'Xx'X 
and we thus have 
<p(x-\-x') — <px — ( px'=a multiplied into 
= 4 (# — 3 ) (X — 3)(xX — # — X)X' + 
+ 2x'[x 2 +x(2X 3 - 1 OX 3 + 1 2X - 1 ) - 4X 3 + 20X 2 - 1 6X - 3£] + 
+ 4X(X— 3)(x— 2)(x'X -X-x!) 
+ [X 2 (2^ 2 - 6^-f 4) +X( - 6 a’ 2 + 1 8x - 4) + 4^ 2 - 4x- 4fjX' 
+ 4(x-1)(xX-x-X)(X 2 -X) 
+ 4(X-l)[XX'xx'-XX'(x+x')-xx'(X+X)+2XX' + 2xx , ]+ 
+ 4£X(X' - 1)(X V - X' - x') 
+ (x 2 -x)(2X' 3 - 6X' 2 + 4X' + x') 
+ 2x(x - 3)(X - 2)(X' 2 - X') 
where as before the (. .)’s refer to the like functions with the two sets of letters inter- 
changed. Developing and collecting, this is 
<p(x-\- x') — <px— (px’—a multiplied into 
3x?x'-\-3xx 12 
+x 2 . 6X 2 X'+6XX' 2 +2X' S 
— 28XX'— 14X' 2 
+28X' 
3 K 
2(41) 
+ . 
. 2(40) 
+ • 
. 2(32) 
+ • 
. (42) 
+ . 
. 2(33) 
+ . 
. 2(38) 
+ . 
. 2(36) 
+ . 
. (28) 
+ • 
. (29) 
MDCCCLXXI. 
