412 ON THE PROBLEM OE THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
36. Correspondence ( a , g) : from the values of f— < p — <p' and e— s — e', we have 
g - * -*'= (X 3 - 20X 2 - 8X#— 4x? + 125X + 44#— 486) A, 
and then 
;£ = ^ = (X — 2) (# — 3)(X — - 3)(# — 3) (X — • 3)(# — 3), 
wherefore 
g=2(X-2)(X-3)>( ; t-3)> 
+ (X S -20X J — 8X* + 125X+44i'-486)(-2X-2x+2+§), 
viz. this is 
g= - 2) 
4-X 3 ( 2# 3 — 18# 2 + 52#— 12) 
+X 2 ( -16# 3 +144# 2 -376#+142) 
+ X( 42# 3 -362# 2 +780#+ 88) 
— 36# 3 +236# 2 + 88# 
+£ r 
-i 
X 3 ( 
+X 2 ( 
+X( 
!)j 
- 20 )| 
- 8# 4-125) ’ 
+ 44#— 486. ; 
Comparing with the expression of <p#, Case 52, we have 
g -<px= X 4 ( - 3) 
+X 3 ( +34) 
+ X 2 ( 2# 3 — 18# 2 + 44#— 79) 
+ X( — 10# 3 + 58# 2 +76#— 84) 
+$‘ 
— # 4 +10# 3 + 15# 2 — 84# 
X 3 ( 
1) 
+X 2 ( 
- 11) 
+X( 
o' 
pH 
1 
+ 9# 2 — 
• 91#+114, 
which difference must be the number of heterotypic solutions having relation to the 
singularities of the curve ; but I have not further considered this. 
