384 
PEOFESSOE CAYLEY ON THE PEOBLEM OF 
Application of the foregoing Theory as to the locus of (a). Art. Nos. 15 to IT. 
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 posi- 
tions of a is, as was seen, g=xAx!‘ 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 inter- 
sections with the proper locus of a\ and the number of these is not x + xj, =a(u-\-d), 
but it is =a{u-\-d— Red.) ; say it is — Red., where the symbol “Red.” is now 
used to signify a times the number of lines, or reduction in the expression v-\-d — 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 xAf — Red. 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*. 
17. But this cannot happen if the curve a is distinct also from the curves c , D ; or, say, 
simply when a is a distinct curve. The conclusion is, that in the case where a is a 
distinct curve we have 
g=X+%'- R ed., 
where the term “Red.” vanishes except in the case of the identity B = F of the curves 
B, F ; and that when this identity subsists it is =a times the reduction in the order of 
the locus of a considered as a free angle ; viz. this consists of a first-mode and a second- 
mode reduction as above explained. 
Remarks in regard to the Solutions for the 52 Cases. Art. Nos. 18 to 23. 
18. Before going further I remark that the principle of correspondence applies to 
corresponding and united tangents in like manner as to corresponding and united points, 
and that all the investigations in regard to the in-and-circumscribed triangle might thus 
be presented in the reciprocal form, where, instead of points and lines, we have lines and 
points respectively. But there is no occasion to employ any such reciprocal process ; the 
result to which it would lead is the reciprocal of a result given by the original process, 
and as such it can always be obtained by reciprocation of the original result, without any 
performance of the reciprocal process. 
* 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 sec post, Case 43. 
