172 
ON THE CUKVES WHICH SATISFY GTVEN CONDITIONS. 
Again, when the point P is at a cnsp, all the (2, 3) conics remain proper conics 
((2*1, 3)=(2, 3), 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*1, 3), =3 (2, 3), and there is a gain of (2 3) =(2*1, 3) 
intersections. Each cusp counts (specially) as (2*1, 3) united points, and together the 
cusps count as *(2*1, 3) united points; we have thus the total number *(2*1, 3)+2r of 
special united points, agreeing with the expression Supp. (2, 3)=*(2*1, 3)+Q. 
135. As appears from the preceding example, or generally from the remark, ante , 
No. 96, 1 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 m; 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 be- 
comes 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—u points, taken as a posi- 
tion 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 inter- 
section 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 coin- 
cident 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 expres- 
sions of the multipliers of the several Capitals in the last five equations. The object of 
the present Memoir was, however, the a, 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 completeness, I abstain from a further discussion 
of them. 
