398 
PEOFESSOE CAYLEY ON THE PEOBLEM OF 
The reduction is due first and secondly to triangles wherein the angle a coincides with 
an angle c or e, and thirdly to triangles wherein the angles a, c, e all coincide. 
1°. Take for the side cDe a double tangent of the curve B=D=F, this meets the 
Fig. 5. 
e 
curve a=c=e in x points, and selecting any one of them for e and any other for c, we 
have from the last-mentioned point Y — 2 tangents to the curve B = D=F ; and in respect 
of each of these a position of a coincident with c. The reduction on this account is 
2rx(x— 1)(Y — 2) ; hut since we may in the figure interchange c and e, B and F, we 
have the same number belonging to the coincidence of the angles #, e, or together the 
reduction is = 4rx(x— 1)(Y— 2). 
But instead of a double tangent we may have cDe a sta- ^ g 
tionary tangent ; we have thus reductions 3/ x(x— 1)(Y— 2) 
and 3< x(x— 1)(Y— 2), together 6/ x(x — 1)(Y — 2) ; and for 
the double and stationary tangents together we have 
=2{Y(Y— 1 )—?/}#(#— 1)(Y— 2), 
that is, 
= 2x(x— 1)Y ( Y — 1 )(Y — 2) — 2xfx— l)y(Y — 2). 
2°. The side c T)e may be taken to be a tangent to the 
curve B = ] ) = F at any one of its intersections with the curve ci=c=e. Taking then 
the point e at the intersection in question, and the point c at any other of the intersec- 
tions of the tangent with the curve a=c=e, and from c drawing any other tangent to 
the curve B=D=F, there is in respect of each of these tangents a position of a at c; 
and the reduction on this account is =xy(x— 1)(Y — 1). But interchanging in the figure 
the letters c, e, B, F, there is an equal reduc- Fig. 7. 
tion belonging to the coincidence of a, e; / 
-p * 
and the whole reduction in this manner is 
= 2x(x —l)y{Y—l). 
3°. If the side c De intersects the curve 
a—G=e in two coincident points, then taking 
these in either order for the points c, e, and V f h 
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 a—C=e , or it may 
