CURVES WHICH SATISFY GIVEN CONDITIONS. 
291 
407] 
First Memoir, No. 73), but each of these (qua conic touching the cuspidal tangent) has 
with the given curve at the cusp not 2 but 3 intersections, so that the total number 
of intersections at P is 3(2/cl, 3), =3(2, 3), and there is a gain of (2, 3) = (2«1, 3) 
intersections. Each cusp counts (specially) as (2/cl, 3) united points, and together the 
cusps count as k (2/cl, 3) united points; we have thus the total number /c(2/cl, 3) + 2t 
of special united points, agreeing with the expression, Supp. (2, 3) = /c (2/cl, 3) + Q. 
135. As appears from the preceding example, or generally from the remark, ante, 
No. 96, I have not at present any a priori method of determining the proper numerical 
multipliers of the Capitals contained in the expressions of the several Supplements. 
I will only further remark, that the reason is obvious why (while in the first seven 
equations the multipliers are mere numbers) in the eighth and following equations 
the multipliers are linear functions of to; in fact in these last equations the barred 
symbol is 1, that is, when P is a point in general on the given curve, each of the 
conics which make up the curve ® has with the given curve not a contact of any 
order, but an ordinary intersection at P. Imagine a position of P for which one of 
these conics becomes a coincident line-pair; this regarded as a single line has with 
the given curve (m — a) ordinary intersections (a a number, = 4 at most, depending on 
the contacts which the line may have with the curve); for each of the m — a points, 
taken as a position of P, one of the conics which make up the curve © becomes the 
coincident line-pair, and there are in respect of this conic two intersections at P 
instead of one intersection only. We have thus in respect of the particular coincident 
line-pair a group of (m — a) special united points, viz. these are the m — a ordinary 
intersections of the coincident line-pair regarded as a single line with the given curve, 
and we thus understand in a general way how it is that the order m of the given 
curve enters into the expressions of the multipliers of the several Capitals in the last 
five equations. The object of the present Memoir was, however, the d posteriori 
derivation of the expressions {ante, No. 132) of the twelve Supplements; and having 
accomplished this, but being unable to discuss the results with any degree of com 
pleteness, I abstain from a further discussion of them. 
37—2