CURVES WHICH SATISFY GIVEN CONDITIONS. 
171 
133. These are, I think, the true theoretical forms of the Supplements, viz. (attending 
to the signification of the Capitals) the expressions actually exhibit how the Supplement 
arises, whether from proper conics passing through or touching at a cusp, or from point- 
pairs (coincident line-pairs) or line-pairs (including of course in these terms line-pair- 
points). Thus, for instance, Supp. (5) = N -}- O. Referring to the explanations, First Me- 
moir, Nos. 41 to 47, N(=/) is the number of the line-pair-points described as “inflexion 
tangent terminated each way at inflexion,” and 0(=%) the number of the line-pair-points 
described as “ cuspidal tangent terminated each way at cusp,” or in what is here the 
appropriate point of view, we have as a coincident line-pair each inflexion tangent 
and each cuspidal tangent. Reverting to the generation of the first equation, when the 
point P is a point in general of the given curve, the curve 0 is the conic (5), having 
with the curve 5 intersections at P, and besides meeting it in the 2m— 5 points P'. When 
the point P is at an inflexion, the curve © becomes the coincident line-pair formed by 
the tangent taken twice, the number of intersections at P is therefore =6, and the 
inflexion is therefore (specially) a united point. Similarly, when the point P is at a 
cusp, the curve 0 becomes the coincident line-pair formed by the tangent taken twice, 
the number of intersections at P is therefore =6, and the cusp is thus (specially) a united 
point: we have thus the total number of special united points agreeing with the 
foregoing a posteriori result, Supp. (5)=N+0. 
134. Or to take another example ; for the fifth equation we have 
Supp. (2, 3)=a(2^I, 3) + Q; 
Q( = 2r) is the number of the line-pair-points described as “double tangent terminated 
each way at point of contact,” or, in the point of view appropriate for the present purpose, 
we have each double tangent as a coincident line-pair in respect to the one of its points 
of contact, and also as a coincident line-pair in respect to the other of its points of 
contact. Reverting to the generation of the equation, when the point P is a point in 
general on the given curve, the curve 0 is the system of conics (2, 3) touching the curve 
at P, and having besides with it a contact of the third order ; since for each conic the 
number of intersections at P is =2, the total number of intersections at P is =2(2, 3), 
and the remaining (2m — 2)(2, 3) intersections are the points P'. Suppose that the point 
P is taken at the point of contact of a double tangent ; of the (2, 3) conics, 1 (I assume 
this is so) becomes the coincident line-pair formed by the double tangent taken twice, 
and gives therefore 4 intersections at P, the remaining (2, 3)— 1 conics are proper 
conics, giving therefore 2(2, 3) — 2 intersections at P, or the total number of intersections 
at P is 2(2, 3)-j-2 intersections; or there is a gain of 2 intersections. As remarked 
(No. 96), this does not of necessity imply that the point in question is to be considered as 
being (specially) 2 united points ; I do not know how to decide a priori whether it is to 
be regarded as being 2 united points or as 1 united point, but it is in fact to be regarded 
as being (specially) only 1 united point ; and as the points in question are the 2r points 
of contact of the double tangents, we have thus the number 2 r of special united points. 
mdccclxviii. 2 B 
