ON THE PEOBLEM OF THE IN - AND-CIBCUMS CEIBED TEI ANGLE. 
379 
The foregoing results are chiefly obtained by means of the theory of correspondence ; 
viz. if instead of the triangle aFcFeF we consider the unclosed trilateral aFcDeFg, 
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 (%, f) correspondence with 
each other ; and when «, g are a “ united point” of the correspondence, the trilateral in 
question becomes an in-and-circumscribed triangle «BcDcF ; 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 in some instances I employ a functional 
method, by assuming that the identical curves are each of them the aggregate of the 
two curves x , cd : we here obtain for the number <px of the triangles belonging to the 
curve a 1 a functional equation <p(x -C x' ) — <px — <px' = 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 consider- 
ation; and taking it to be known, the functional equation gives <px= particular solution 
+ linear function of (x, X, |). 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. 
The Principle of Correspondence as applied to the present Problem. — Article Nos. 1 to 6. 
1. Consider the unclosed trilateral aFcDeFg, 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 «Bc 
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 aF>c 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 inter- 
sections in question; and then in like manner we have eF g 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 y positions of g ; it is easy to find 
the value of y ; viz. if (as above tacitly supposed) the curves a , B, c, D, e , F are all of them 
distinct curves, then the number of the tangents «Bcis=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, %=BcDcF«. 
But if some of the curves become one and the same curve — if, for instance, a=B=c, the 
line aFc is here a tangent from a point a on the curve, we exclude the tangent at the 
point «, 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 
3 G 2 
Fig. 1. 
