THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
411 
The united points (a, e) are in fact, 1°, each of the x— 4 intersections of a double 
tangent with the curve, in respect of the two contacts and of the remaining x—6 inter- 
sections ; 2°, each double point in respect of the two branches and of the pairs of 
tangents from it to the curve; 3°, each of the x—o intersections of each of the tangents 
at a double point with the curve; 4°, each of the a-— 3 
intersections of a tangent at an inflection (stationary tan- 
gent) with the curve, in respect of the (x— 4) remaining 
intersections; 5°, each inflection in respect of the x — 3 
intersections of the tangent with the curve ; and 6°, each 
cusp in respect of the pairs of tangents from it to the curve. 
Thus (2°), the double point in respect of the branch which 
contains c, and of the two tangents from it to the curve, 
is a position of the united point ( a , e ), as appearing in the 
figure. 
34. Correspondence (B, F). By reciprocation of («, e). 
e 0 — & 0 — £ o=(— 1 # 2 +13a;-|-4X— 54)A, 
e 0 = (X - 4)(X - 5)2c> + 2(x— 3) (a- - 4)/ 
+ O- 4)(ar - 5) +X- 3]2r + [3(X - 3)(X - 4) + (X- 3)>. 
35. Correspondence (a, F). By means of the values of e 0 — s — s' and d — h — W, we have 
Fig. 9. 
and then 
whence 
which is 
f_^-^z=(2X 2 +2X^+2x 2 -32X-32a- + 162)A, 
<p = (X — 2)(a;— 3)(X —3)(x— 3)(X — 3), 
<p'=(x—2)(X—3)(x—3)(X— 3)(x—3), 
f=(X-j-x—4)(x — 3) 2 (X—3) 2 
+ (2X 2 +2Xx+2x 2 -32X-32x+162)(-2X-2x + 2+g), 
= X 3 ( a- 2 - 6x+ 5) 
+ X 2 ( x 3 — 16^ 2 + 61a;— 22) 
fl-X (— 6x 3 + 61a; 2 — 120a;— 91) 
+ 
5a; 3 — 22a; 2 — 91a* 
+1 
X 2 ( 
^ +X( 
2a;- 
2 ) 
32) - 
+ 2a- 2 - 32a’ + 132). 
This result includes proper solutions of the problem of finding the number of the 
triangles «BcDcF, 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. 
