WHICH SATISFY GIVEN CONDITIONS. 
113 
the values 1, 2, ... up to a limiting value of a, the formula gives the number of the 
proper curves C r which have with the given curve U m a contact of the required order a : 
beyond this limiting value the formula no longer gives the number of the proper curves 
C r which satisfy the required condition, and it thus ceases to be applicable ; but there 
is no algebraic function of a which would give the number of the proper curves C r as 
well beyond as up to the foregoing limiting value of a. 
80. The formulae are applicable provided only the conditions include the conditions 
of passing through a sufficient number of arbitrary points ; viz. when the number of 
arbitrary points is sufficiently great, it is not possible to satisfy the conditions specially 
by means of improper curves C r , being or comprising a pair of coincident curves. Thus 
to take a simple example, suppose it is required to find the number of the conics which 
touch a given curve t times and besides pass through 5 — t given points: if the number 
of the given points be 4 or 3 there is no coincident line-pair through the given points, 
and therefore no coincident line-pair satisfying the given conditions ; if the number of the 
given points is =2, then the line joining these points gives a coincident line-pair having 
at each of its to intersections with the given curve a special contact therewith, that is, 
having in \m(m— 1)(to — 2) ways three special contacts with the given curve ; if the number 
of the given points is 1 or 0, then in the first case any line whatever through the given 
point, and in the second case any line whatever, regarded as a coincident line-pair, has 
to special contacts with the given curve ; and so in general there is a certain value for 
the number of given points, for which value the conditions of contact may be satisfied by 
a determinate number of improper curves C r , and for values inferior to it the conditions 
may be satisfied by infinite series of improper curves C r . It is by such considerations 
as these that De Jonqujeees has determined the minimum value T of the number of 
arbitrary points to which the conditions should relate in order that the formulae may 
be applicable : I refer for his investigation and results to paragraphs XVII and XVIII of 
his memoir. I remark that in the case where the number of improper solutions is 
finite, the formula can be corrected so as to give the number of proper solutions by 
simply subtracting the number of the improper solutions : but this is not so when the 
improper solutions are infinite in number ; the mode of obtaining the approximate 
formula is here to be sought in the considerations contained in the first part of the pre- 
sent Memoir; see in particular ante. Nos. 8, 9 & 10. 
81. The expressions for [a), (a, b), & c. may be considered as functions of rm, 1 + A, 
and K, and they vanish upon writing therein rm=0, A=0, x=0 ; they are consequently 
of the form (rm, A ,z) l -\-(rm, A, «) 2 + &c., and I represent by [«], [a, b~]. See. the several 
terms (rm, A, *)', which are the portions of (a), (a, b), &c. respectively, linear in rm, 
A, and z. The terms in question are obtained with great facility ; thus, to fix the ideas, 
considering the expressions for (a, b, c, d ), — 
1°. To obtain the term in rm, we may at once write D=l, «=0, the expression is 
thus reduced to 
(a-fl)(5-l- l)(c-J-l)(^ -f 1) { [rm— a] 4 -b [rm— a— l] 3 a} , 
