WHICH SATISFY GIVEN CONDITIONS. 
85 
27. As an instance of the use of the characteristics, if X, X', X", X'", X"" be any five 
independent conditions, and (a, j3), . . . (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, 1X«, (3)(cJ, (3')(*'\ (3")(u"', /3"')K", ff) 
viz. this notation stands for laM'aV + . .+1(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, 1 ~Jn, to""). 
29. A system of conics (4X) having the characteristics (/a, v), contains 
2 v—fjj line-pairs, that is, conics each of them a pair of lines; and 
2/a — v point-pairs, that is, conics each of them a pair of points ( coniques infini- 
ment aplaties). 
30. 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*; 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-pair ; such a point-pair (it must in like manner be 
postulated) ordinarily arises, not from a point-pair the two points of which become coin- 
cident, but from a proper conic sharpening itself to coincide with its asymptotes, and so 
becoming ultimately a pair of lines through the coincident point-pair ; and the coincident 
point-pair is regarded as being in fact the line-pair formed by some two lines through the 
coincident point-pair. 
31. In accordance with the foregoing notions we may with propriety, and it will in 
* A line is regarded as extending from any point A thereof to B, and then in the same direction, from B 
through infinity to A ; it thus consists of two portions separated by these points ; and considering either portion 
as removed, the remaining portion is a terminated line. 
o 2 
