THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
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all ce . B tangents), and each of these counts once in respect of each of the I) tangents to 
the curve D, that is, it counts D times. We have thus, as part of the locus of a, ce . B 
lines each D times, or, say, first-mode reduction =ce.B.D. 
12. The second mode is that shown in the annexed “ second-mode figure.” The 
Fig. 3. Second-mode figure. 
tangent from D is here a common tangent of the curves D, and B = F. This meets the 
curve c in c points, and the curve e in e points; and attending to any pair of points c, e, these 
give the tangents cB«, e¥a, coinciding with the common tangent in question, and forming 
part of the locus of a. The number of the common tangents is =BD ; but each of these 
counts once in respect of each combination of the points c, e, that is in all ce times. 
And we have thus as part of the locus BD lines each c.e times, or, say, second-mode 
reduction = BD.c. 0 . This is (as it happens) the same number as for the first mode ; but 
to distinguish the different origins I have written as above ce. B . D and BD .c.e respectively. 
13. It is important to remark that each of the two modes arises whatever relations of 
identity subsist between the curves c, e, D, and B=F, but with considerable modification 
of form. Thus if the curves c, e are identical (c=e) but distinct from D, then in the 
first-mode figure ce may be a node or a cusp of the curve c=e, or it may be a point of 
contact of a common tangent of the curves D, and c=e. As regards the node, remark that 
if we consider a tangent of D meeting the curve c=e in the neighbourhood of the node, 
then of the two points of intersection each 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 =h and that of cusps 
=*, the value of the first-mode reduction is =(2&-f-3/f-(-C)BD, or, what is the same 
thing, it is = (c 2 — 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 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 
— Bed., which is the proper locus. 
