382 
PROFESSOR CAYLEY ON THE PROBLEM OF 
I suppose further that a is distinct from all the other curves, or say, simjpliciter , that a is 
a distinct curve. The values of %, will here each of them contain the factor a , say 
we have x—aco, %' = aoJ ; and therefore the equation gives g=a(a + <*>'). It is obvious 
that a, cJ are the values assumed by %, %' respectively in the particular case where the 
curve a is an arbitrary line (a= 1) ; and is the number of the united points on 
this line. 
8. Suppose now that in the triangle aBcF)eFa 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 inter- 
sections of the line with this locus; that is, the locus meets the arbitrary line in u-\-J 
points; or, what is the same thing, the order of the locus is 
9. I stop for a moment to remark that in the particular case where the curve B is a 
point (B = l), 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 co 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 a/-tuple point on the locus ; 
and (by what precedes) an arbitrary line through B meets the locus in u other points; 
that is, in the particular case where the curve B is a point, the order of the locus of a is 
=ou-\-co , 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 cF>e, meeting the curves c, e in the points c, e respec- 
tively ; 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 eF a, 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 cB«, 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 Fig. 2. First-mode figure. 
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 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 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 
