514] ON THE PROBLEM OF THE 1N-AND-CIRCUMSCRIBED TRIANGLE. 227 
in succession may be taken for the point c, and the other of them will be the 
point e; so that the node counts twice. It requires more consideration to perceive, 
but it will be readily accepted that the cusp counts three times. Hence if for the 
curve c = e the number of nodes be =8 and that of cusps = k, the value of the 
first-mode reduction is = (28 + 3k + C) BD, or, what is the same thing, it is = {f — c)BD. 
As regards the second-mode figure, the only difference is that c, e will be here 
any pair of intersections (each pair twice) of the tangent with the curve c = e; the 
value is thus = (c 1 2 — c) BD. 
It would be by no means uninteresting to enumerate the different cases, and indeed 
there might be a propriety in doing so here; but I have (instead of this) considered 
the several cases, when and as they arise in connexion with any of the cases of the 
in-and-circumscribed triangle. 
14. Observe that the general result is, that in the case B—F of the identity 
of the curves B and F, but not otherwise, the locus of a includes as part of itself 
a system of lines; or, say, that it is made up of these lines, and of a residual curve 
of the order co + co' — Red., which is the proper locus. 
Article Nos. 15 to 17. Application of the foregoing Theory as to the locus of (a). 
15. Reverting now to the case where the angle a is not a free angle but is 
situate on a given curve a, then if the curve a is distinct from the curves e, F, 
the number of positions of a is, as was seen, g = % + But the points in question 
are the intersections of the curve a with the locus of a considered as a free angle; 
and hence in the case B = F, but not otherwise, they are made up of the intersections 
of the curve a with the system of lines, and of its intersections with the proper 
locus of a. But the intersections with the system of lines are improper solutions of 
the problem (or, to use a locution which may be convenient, they are “heterotypic” 
solutions): the true solutions are the intersections with the proper locus of a; and 
the number of these is not ^ + f, = a (co + co'), but it is = a (co + co' — Red.); say it is 
= % + X ~ Bed., w h ere the symbol “ Red.” is now used to signify a times the number 
of lines, or reduction in the expression co + co' — Red. of the order of the proper 
locus of a. 
16. It is however to be noticed that if the curve a, being as is assumed distinct 
from the curves e, and F=B, is identical with one or both of the remaining curves 
c, D, the foregoing expression % + x ~~ Bed. may include positions which are not true 
solutions of the problem, viz. the curve a may pass through special points on the 
proper locus of a, giving intersections which are a new kind of heterotypic solutions ( J ). 
1 More generally, if the curve a be a curve identical with any of the other curves, then if treating in 
the first instance the angle a as free we find in any manner the locus of a, the required positions of the 
angle a are the intersections of this locus and of the curve a; but these intersections will in general 
include intersections which give heterotypic solutions. The determination of these is a matter of some 
delicacy, and I have in general treated the problems in such manner that the question does not arise; but 
as an example see post, Case 43. 
29—2