222 
ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. [514 
The foregoing results are chiefly obtained by means of the theory of correspondence; 
viz. if instead of the triangle aBcDeF we consider the unclosed trilateral aBcDeFg, 
where the points a and g are situate on one and the same curve, say the curve 
a=g, then the points a and g have a certain correspondence, say a (%, ^') corre 
spondence with each other; and when a, g are a “ united point ” of the correspondence, 
the trilateral in question becomes an in-and-circumscribed triangle aBcDeF; that is, 
the number of triangles is equal to that of the united points of the correspondence, 
subject however (in many of the cases) to a reduction on account of special solutions. 
It may be remarked that by the theory of correspondence the number of the united 
points is, in several of the cases, but not in all of them, = % + But i n some 
instances I employ a functional method, by assuming that the identical curves are each 
of them the aggregate of the two curves x, x': we here obtain for the number (f)x 
of the triangles belonging to the curve x a functional equation (f> (x + x') — <px — <f)x' = 
given function; viz. the expression on the right-hand side depends on the solution of 
the preceding cases, wherein the number of identities between the several curves is 
less than in the case under consideration; and taking it to be known, the functional 
equation gives (j)x = particular solution + linear function of (x, X, f). The particular 
solution is always easily obtainable, and the constants of the linear function can be 
determined by means of particular forms of the curve x. 
Article Nos. 1 to 6. The Principle of Correspondence as applied to the present Problem. 
1. Consider the unclosed trilateral aBcDeFg, where the points a and g are on 
one and the same curve, a=g. Starting from an arbitrary point a on the curve a, 
we have aBc any one of the tangents from a to the curve B, touching this curve, 
say at the point B, and intersecting the curve c in a point c; viz. c is any one of 
the intersections of aBc with the curve c; we have then similarly cDe any one of 
the tangents from c to the curve D, touching it, say at D, and intersecting the curve e 
in a point e ; viz. the point e is any one of the intersections in question ; and then 
in like manner we have eFg any one of the tangents from e to the curve F, touching 
it, say at F, and intersecting the curve g (= a) in a point g ; viz. g is any one of 
the intersections in question. Suppose that to a given position of a there correspond 
% positions of g; it is easy to find the value of viz. if (as above tacitly supposed)