406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
201 
26. A single condition (X) imposed upon a conic has two representative numbers, 
or simply representatives, (a, /3); viz. if (4Z) be an arbitrary system of four conditions, 
and (fi, v) the characteristics of (4Z), then the number of the conics which satisfy 
the five conditions (X, 4Z) is = a/j, + /3v. 
27. As an instance of the use of the characteristics, if X, X', X", X'", X"" be 
any five independent conditions, and (a, /3), ... (a"", /3'"') the representatives of these 
conditions respectively, then the number of the conics which satisfy the five conditions 
(X, X', X", X'", X"") is 
= (1, 2, 4, 4, 2, l$a, /3)« /3') (a", /3")(a'", /3'") (a"", /3"") 
viz. this notation stands for laaW'a"" + 2tacta"a."'/3"" ... + l/3/3'/3"/3'"/3"". 
28. In particular if X be the condition that a conic shall touch a given curve 
of the order to and class n, then the representatives of this condition are (n, to), 
whence the number of the conics which touch each of five given curves (to, n), ... 
(to"", n"") is 
= (1, 2, 4, 4, 2, l$w, to)(n' } m)(n", to"')(%"", to""). 
29. A system of conics (42”) having the characteristics (y, v), contains 
2v — fi line-pairs, that is, conics each of them a pair of lines; and 
2/u, — v point-pairs, that is, conics each of them a pair of points (coniques 
infiniment aplaties). 
80. I stop to further explain these notions of the line-pair and the point-pair; 
and also the notion of the line-pair-point. 
A conic is a curve of the second order and second class; qua curve of the second 
order it may degenerate into a pair of lines, or line-pair (but the class is then = 0): 
qua curve of the second class it may degenerate into a pair of points, or point-pair 
(but the order is then = 0). The two lines of a line-pair may be coincident, and 
we have then a coincident line-pair; such a line-pair (it must I think be postulated) 
ordinarily arises, not from a line-pair the two lines of which become coincident, but 
from a proper conic, flattening by the gradual diminution of its conjugate axis, while 
its transverse axis remains constant or approaches a limit different from zero; the 
conic thus tends (not to an indefinitely extended but) to a terminated line ( x ); in other 
words, the tangents of the conic become more and more nearly lines through two fixed 
points, the terminations of the terminated line; and these terminating points, which 
continue to exist up to the instant when the conjugate axis takes its limiting 
value = 0, are regarded as still existing at this instant, and the coincident line-pair 
as being in fact the point-pair formed by the two terminating points. Similarly the 
two points of a point-pair may be coincident, and we have then a coincident point- 
1 A line is regarded as extending from any point A thereof to B, and then in the same direction, from B 
through infinity to d ; it thus consists of two portions separated by these points; and considering either portion 
as removed, the remaining portion is a terminated line. 
C. VI. 
26