514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 225 
sidiary values f — <£ — </>' and e —e —e' are obtained by means of a more simple 
application of the principle of correspondence, as will appear in the sequel 0, but for 
the moment I do not pursue the question. 
Article Nos. 7 to 14. Locus of a free angle (a). 
7. I consider the case where a is a distinct curve e, F, and where, as was 
seen, the equation is simply 
g~X~X =0 - 
I suppose further that a is distinct from all the other curves, or say, simpliciter, that 
a is a distinct curve. The values of y, y' will here each of them contain the factor a, 
say we have y = aw, y' = aw'; and therefore the equation gives g = a (« -f w'). It is 
obvious that w, w are the values assumed by y, y' respectively in the particular case 
where the curve a is an arbitrary line (a = 1); and w + w is the number of the 
united points on this line. 
8. Suppose now that in the triangle aBcDeFa the point a is a free point, we 
have, as above-mentioned, a locus of a, and the united points on the arbitrary line 
are the intersections of the line with this locus; that is, the locus meets the arbitrary 
line in &> + &>' points; or, what is the same thing, the order of the locus is = w -f w. 
9. I stop for a moment to remark that in the particular case where the curve 
B is a point (B= 1), then in the construction of the locus of a the arbitrary tangent 
aBc is an arbitrary line through B, and the construction gives on this line w positions 
of the point a. But drawing from B a tangent to the curve F, and thus constructing 
in order the points F, e, D, c, a, the construction shows that B is an «'-tuple point 
on the locus; and (by what precedes) an arbitrary line through B meets the locus in 
w other points; that is, in the particular case where the curve B is a point, the 
order of the locus of a is = w + w, which agrees with the foregoing result. 
10. The construction for the locus of a may be presented in the following form: 
viz. drawing to the curve D a tangent cDe, meeting the curves c, e in the points 
c, e respectively; then if from any point c we draw to the curve B a tangent cBa, 
and from any point e to the curve F a tangent eFa, the tangents cBa, eFa intersect 
in a point on the required locus. Hence if in any particular case (that is for any 
particular position of the tangent cDe) the lines cBa, eFa become one and the same 
line, the point a will be an indeterminate point on this line; that is, the line in 
question will be part of the locus of a. 
11. The case cannot in general arise so long as the curves B, F are distinct 
from each other; but when these are one and the same curve, say when B — F, it 
will arise, and that in two distinct ways. To show how this is, suppose, to fix the 
ideas, that the curves c, D, e are distinct from each other and from the curve B = F. 
Then the first mode is that shown in the annexed “first-mode figure,” viz. we have 
C. VIII. 
1 See post, Nos. 24 et seq. 
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