514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
243 
the reduction on this account is = xy (x — 1) (F — 1). But interchanging in the figure 
the letters c, e, B, F, there is an equal reduction belonging to the coincidence of a, e\ 
and the whole reduction in this manner is = 2x (x — i)y(r-i). 
Fig. 7. 
3°. If the side cDe intersects the curve a= c = e in two coincident points, then 
taking these in either order for the points c, e, and from the two points respectively 
drawing two other tangents to the curve D = B = F, we have a triangle wherein the 
angles a, c, e all coincide. The side cDe may be a proper tangent to the curve 
ci = c — e, or it may pass through a node or a cusp of this curve, viz. it is either a 
common tangent of the curves B = JD = F and a=c = e (as in the figure, except that for 
greater distinctness the points c and e are there drawn nearly instead of actually 
coincident), or it may be a tangent to the curve B = D= F from a node or a cusp of 
the curve a = c = e; we have thus the numbers 
Common tangent XY (F — 1)(F — 2), 
Tangent from node 2SF(F — 1) (F — 2), 
Tangent from cusp 2kY(F— 1)(F — 2); 
but (as we are counting intersections with the curve a = c = e) the second of these, as 
being at a node of this curve, is to be taken 2 times; and the third, as being at a 
cusp, 3 times; and the three together are thus 
(X + 4S + 6* ) Y(Y — 1) (F— 2), 
= {2x(x-l)-X} F(F—1)(F—2). 
The reductions 1°, 2°, 3° altogether are 
which is 
2x(x-l) F(F —1)(F—2) 
— 2x (x — 1) y(Y — 2) 
+ 2x(x-l)y(Y - 1) 
+ 2x(x-l) F(F— 1) (F— 2) 
-IF(F-l)(F-2), 
= 4x(x — 1) F (F— 1)(F —2) 
+ 2x(x — 1) y 
-IF(F-1)(F—2); 
31—2