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ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. [514 
here a tangent at D passing through a point ce of the intersection of the curves 
c, e, and from this point a tangent drawn to the curve B=F. For the position in 
question of the tangent of D, the points c, e coincide with each other, and we have 
thus the coincident tangents cBa and eFa to the identical curves B = F. It is further 
Fig. 2. First-mode figure. 
to be remarked that the number of the points of intersection is = ce; from each of 
these there are B tangents to the curve B = F (in all ce. B tangents), and each of 
these counts once in respect of each of the D 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 
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 cBa, eFa, coinciding with the common tangent in 
Fig. 3. Second-mode figure. 
BF 
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.e. 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 consider 
able 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