290 
SECOND MEMOIR ON THE 
[407 
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) = iY + 0. Referring to the ex 
planations, First Memoir, Nos. 41 to 47, N (= i) is the number of the line-pair-points 
described as “inflexion tangent terminated each way at inflexion,” and 0 (= /c) 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 © 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 © 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 = k + i, agreeing with the foregoing & posteriori 
result, Supp. (5) = N + 0. 
134. Or to take another example; for the fifth equation we have 
Supp. (2, 3) = k (2*1, 3) + Q; 
Q (= 2t) 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 © 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-paii 
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) -I- 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 2t of special united points. Again, when 
the point P is at a cusp, all the (2, 3) conics remain proper conics ((2/cl, 3) = (2, 3),