230 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 
78. In all these formulae there is, as before, a numerical divisor in the case of 
any equalities among the numbers a, b, c, &c. And D denotes, as before, the deficiency, 
viz. its value now is D = \ (m — 1) (m— 2)— B — k ; or observing that the class n is 
= m 2 — m — 28 — 3k, we have D = \n — m +1 4- ^k, or say D — 1 —m + fyn + ^k, = 1 4- A it 
A = -m4|»4 
79. It is to be observed with reference to the applicability of these formulae 
within certain limits only, that the formulae are the only formulae which are generally 
true; thus taking the simplest case, that of a single contact a, the only algebraical 
expression for the number of the curves G r which have with a given curve U m a con 
tact of the order a, and besides pass through the requisite number \r (r 4- 3) — a of 
arbitrary points, is that given by the formula, viz. 
(a) = (a 4-1) (rm — a 4- aJD) — uk. 
Considering the curve U m and the order r of the curve G r as given, if a has 
successively the values 1, 2,... up to a limiting value of a, the formula gives the 
number of the proper curves G r which have with the given curve JJ m a contact of the 
required order a: beyond this limiting value the formula no longer gives the number 
of the proper curves G r which satisfy the required condition, and it thus ceases to be 
applicable; but there is no algebraic function of or. which would give the number of 
the proper curves G 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 m intersections with the given curve a special 
contact therewith, that is, having in (m — 1) (m — 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 m 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 1 '. It is by such considerations as these that 
De Jonquieres 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