514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 223 
the curves a, B, c, D, e, F are all of them distinct curves, then the number of the 
tangents aBc is = B; there are on each of them c points c; through each of these 
we have D tangents cDe; on each of these e points e; through each of these F 
tangents cFg; and on each of these a points g; that is, ^ = BcDeFa. But if some 
of the curves become one and the same curve—if, for instance, a = B = c,—the line 
aBc is here a tangent from a point a on the curve, we exclude the tangent at the 
point a, and the number of the remaining tangents is = (A — 2); each tangent meets 
the curve in the point a counting once, the point B counting twice, and in (a —3) 
other points; that is, the number of the points c is = (A — 2) (a — 3), and so in other 
cases; the calculation is always immediate, and the only difference is that, instead of 
a factor a or A, we have such factor in its original form or diminished by 1, 2, 
or 3, as the case may be. Similarly starting from g, considered as a given point on 
the curve g (=a), we find the number of the corresponding points a; thus in the 
case where the curves are all distinct curves, we have = FeDcBa (— %); and so in 
other cases we find the value of The points (a, g) have thus a (%, ^') corre 
spondence, where the values of % are found as above. 
2. There will be occasion to consider the case where in the triangle aBcDeF (or 
say the triangle aBcDeFa) the point a is not subjected to any condition whatever, 
but is a free point. There is in this case a “locus of a,” which is at once con 
structed as follows: viz. starting with an arbitrary tangent aBc of the curve B, 
touching it at B and intersecting the curve c in a point c; through c we draw to 
the curve D the tangent cDe, touching it at D and intersecting the curve e in a 
point e; and finally from e to the curve F the tangent eFa, touching it at F and 
intersecting the original arbitrary tangent aBc in a point a, which is a point on the 
locus in question. We can, it is clear, at once determine how many points of the 
locus lie on an arbitrary tangent of the curve B (or of the curve F). 
3. The general form of the equation of correspondence is 
p (a - a - a!) + q (b - /3 - /3') + ... = M ( x ); 
viz. if on a curve for which twice the deficiency is = A we have a point P corre 
sponding to certain other points P', Q', ... in such wise that P, P' have an (a, a!) 
correspondence, P, Q' a (/3, /3') correspondence, &c.; and if (a) be the number of the 
united points (P, P'), (b) the number of the united points (P, Q'), &c.; and if more 
over for a given position of P on the curve the points P', Q', ... are obtained as the 
intersections of the curve with a curve © (depending on the point P) which meets 
the curve k times at P, p times at each of the points P', q times at each of the 
1 To avoid confusion with the notation of the present memoir, I abstain in the text from the use of D 
as denoting the deficiency, and there is a convenience in the use of a single symbol for twice the deficiency 
but writing for the moment D to denote the deficiency, I remark, in passing, that perhaps the true theoretical 
form of the equation is 
k(0-D-D)+p (&-a-a') + q (b — /3 - /S') +... =0 ; 
viz. the point P is here considered as having with itself a (Z), D) correspondence, the number of the united 
points therein being =0.