250 
ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
[514 
+ 3(X' 2 -X / W X 2 ) +.. (47) x 3 
+ X (2« 3 — 10« 2 + 12« - 1) 
-4«3+ 20« 2 - 16« - 3£ 
+ 3x'X' X 2 ( 2« 2 — 6« + 4) ) + .. (48) x 3 
+ X (— 6« 2 + 18« — 4) 
+ 4« 2 — 4« — 4£ 
+ 12 («' - 3) (X'- 3) {««'XX' - xx (X + X') -XX' (« + x) + 2««' + 2XX'} + .. (49) x 6 
+ {2«' («' - l)(«'-2) X(X - 1)(X - 2) + xx' (x' - l)(x' - 2) + X'X (X - 1)(X - 2)} + .. (43) 
+ 6 («' — 2) X' (X' — 3) (X — 2) (« — 3) +.. (44) x 3 
+ 12 (X' - 1) (« - 1) [xx'XX' - xx' (X + X') - XX' (« + «') + 2««' + 2XX'} + .. (45) x 6 
where as before the (. .)’s refer to the like functions with the two sets of letters 
interchanged. Developing and collecting, this is found to be 
= 4X 3 X / + GX 2 X' 2 + 4XX' 3 
+ X s 6« 2 «' + 6««' 2 + 2« /3 ) 
— 36««'— 18«' 2 
+ 52«' 
6« 3 + 18« 2 «' + 18«« /2 + 6«' 3 
— 54« 2 — 108««' — 54«' 2 
+ 156« +156«' 
. - 138 
-f- X' 3 2« 3 + 6« 2 «' + 6«« /2 ^ 
— 18« 2 — 36««' 
+ 52« 
+ &c. &c. 
I abstain from writing down the remaining terms, as they can at once be obtained 
backwards from the value of 0«; they were in fact found directly, and the integration 
of the functional equation then gives 
<f>x — 
+ (X 2 X' + XX /2 ) 
X 4 ( 
+ 
1) 
+ X 3 ( 
2a? — 18« 2 + 
52« — 
46) 
+ X 2 ( 
— 18« 3 + 162« 2 — 
420«+ 221) 
+ * ( 
52« 3 — 420« 2 + 704« +1 
) 
+ « 4 
— 46« 3 + 221« 2 + 
lx 
+£ ( 
X 2 ( 
9 )) 
+ X ( - 12« + 135) 
— 9« 2 + 135« + A