THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
405 
+ 3X' 
-X 3 ( + 1)- 
X 2 ( 2a 3 - 14a 2 + 28a— 11) 
, X(-10a 3 +70a 2 -116a- 8) - 
+ 12a 3 — 76a 2 + 64a 
. +£(-6X-4a +42) . 
(51) x 3 
+ 3(a' 2 — a') 
- 
A 2 
+a(2X 3 — 10X 2 +12X 
- 4X 3 +20X 2 — 16X- 
i 
-1) - 
+ • • 
(46) x 3 
+ 3(X' 2 — X') 
- 
X 2 j 
+ X(2 A 3 — 1 0 A 2 + 1 2 A — 1 ) • 
— 4a 3 +20a 2 — 16a— 3| 
(47) x 3 
+ 3a'X' 
r X 2 ( 2 a 2 - 6a+4) 
+ X ( — 6a 2 +18a— 4) 
+ 4 a 2 — 4a— 4g 
+ • • 
(48) x 3 
+ 12(a'-3)(X'-3){aa'XX'-aa'(X+X')-XX'(a+a') + 2aa'+2XX'} + .. (49) X 6 
+ {2a'(a'— 1)(a / — 2)X(X— 1)(X— 2)+aa'(a' — 1)(a'— 2) + X'X(X— 1)(X — 2)} + . . 
+ 6(a'-2)X'(X'-3)(X-2)(a- 3) + . . (44) x 3 
+ 12(X'-1)(a-1){aa'XX'-aa'(X+X , )-XX'(a+a')+2aa') + 2XX^ +.. (45) x 6 
where as before the (. .)’s refer to the like functions with the two sets of letters inter- 
changed. Developing and collecting, this is found to be 
= 4X 3 X' + 6X 2 X' 2 + 4XX' 3 
+X 3 6a 2 a' + 6aa' 2 +2a' 3 
• — 36aa' — 1 8 a' 2 
+ 52a' 
+ (X 2 X'+XX' 2 ) ' 6a 3 + 18aV+18aa' 2 +6a' 3 
— 54a 2 — 108 AA f — 54a' 2 
+156a +156a' 
— 138 
+X' 3 f 2 a 3 + 6 a 2 a' + 6aa' 2 
j — 18 a 2 — 36 aa' 
^+52a 
+&c. &c. 
I abstain from writing down the remaining terms, as they can at once be obtained 
backwards from the value of <pA ; they were in fact found directly, and the integration 
of the functional equation then gives 
