256 
ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. [514 
tangents at a double point with the curve; 4°, each of the # — 3 intersections of a 
tangent at an inflection (stationary tangent) with the curve, in respect of the (x — 4) 
remaining intersections; 5°, each inflection in respect of the x — 3 intersections of the 
Fig. 9. 
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) y as appearing in the figure. 
34. Correspondence (B, F). By reciprocation of (a, e) 
e 0 — e 0 — e 0 ' = (— # 2 + 13# + 4X — 54) A, 
e 0 = (X - 4) (X - 5) 28 + 2 (x - 3) (x - 4) t 
+ [(# — 4) (# — 5) + X — 3] 2r + [3 (X-3) (X — 4) + (X — 3)] «. 
35. Correspondence (a, F). By means of the values of e 0 — e — e' and d — 8 — 8\ 
we have 
f _ ^ _ 0' = (2X 2 + 2Xx + 2x? - 32X - 32# + 162) A, 
and then 
</> —(X—2){x— 3) (X — 3) (x - 3) (X - 3), 
4>' = (x- 2) (X-3) (a;-3)(X-3) (#-3), 
whence 
which is 
f = (X + x- 4<)(x — 3) 2 (X — 3) 2 
+ (2X 2 + 2Xx + 2x 2 - 32X - S2x + 162) (- 2X - 2x + 2 + £) 
= X s ( a? — 6x + 5) 
-f X 2 ( X s — 16x 2 + 61#— 22) 
+ X (— 6# 3 + 61# 2 — 120# — 91) 
+ ox 3 — 22# 2 — 91# 
+ £ X 2 ( 2)\ 
+ X ( 2# - 32) 
+ 2# 2 — 32# + 132