514] ON THE PROBLEM OF THE 1N-AND-C1RCUMSCRIBED TRIANGLE. 257 
This result includes proper solutions of the problem of finding the number of 
the triangles aBcDeF, which are such that the side ea touches the curve at a; and 
also heterotypic solutions having reference to the singular points of the curve; but 
I have not determined the number of solutions of each kind. 
36. Correspondence (a, g) : from the values of f — <£ — <£' and e — e — e, we have 
g~X~X = (X 3 — 20X 2 — 8Xa? — 4a? 2 + 125 X + 44a? — 486) A, 
and then 
x - X '-(X-8)(«-3)(Z-8)(«-8)(JT-8)(»-8). 
wherefore 
g = 2 (X —2) (X — 3) 2 (a? — 3) 3 
+ (X s - 20X 2 - 8Xx + 125X + 44a - 486) (- 2X - 2x + 2 + f), 
viz. this is 
g = X 4 ( ~ 2) 
+ X 3 ( 2x 3 — 18a; 2 + 52.-»— 12) 
+ X 2 (- 16a? 3 + 144a; 2 - 376a; + 142) 
+ X ( 42a? 3 — 362a? 2 + 780a? + 88) 
— 36a? 3 + 236a? 2 + 88a? 
/ X 3 ( 
+ X 2 ( 
1) 
- 20) 
- 8a? + 125) f 
+ 44a? — 486. j 
Comparing with the expression of cpx, Case 52, we have 
g - 0a? = X 4 ( - 3) 
+ X 3 ( +34) 
+ X 2 ( 2a; 3 - 18a; 2 + 44a;-79) 
+ X ( - 10a; 3 + 58a; 2 + 76a? - 84) 
— a? 4 + 10a? 3 + 15a? 2 — 84a? 
+ 1 
^ 3 ( 1)\ 
+ X 2 ( - 11) 
+ X ( 4a?- 10) f 
( + 9a? 2 — 91a? + 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. 
C. VIII. 
33