86 
PEOFESSOE CAYLEY ON THE CURVES 
the sequel be found convenient to speak of a point-pair as a line terminated by two 
points on this line, and similarly to speak of a line-pair as a point terminated (that is, 
the pencil of lines through the point is terminated) by two lines through the point. 
32. If in a point-pair thus considered as a line terminated by two points the two 
points become coincident (the line continuing to exist as a definite line), or, what is the 
same thing, if in a line-pair thus considered as a point terminated by two lines, the two 
lines become coincident (the point continuing to exist as a definite point), we have a 
“ line-pair-point viz. this is at once a coincident line-pair and a coincident point-pair ; it 
may also be regarded as the limit of a conic the axes of which, and the ratio of the con- 
jugate to the transverse axis, all ultimately vanish : it may be described as a line termi- 
nated each way at a point thereof, or as a point terminated each way at a line 
through it. The notion of a line-pair-point first presents itself in Zeuthen’s researches, 
as will presently appear ; but it may be noticed here that line-pair-points, and these the 
same line-pair-points, may present themselves among the 2v — p line-pairs, and among 
the 2(a — v point-pairs of the system of conics 4X. 
33. Returning to the foregoing theory of characteristics, I remark that the funda- 
mental notion may be taken to be, not the characteristics (p, v) of the conics which 
satisfy four conditions, but in every case the number of the conics which satisfy five con- 
ditions. Thus. for the conics not subjected to any condition, we may consider the 
symbols 
(:•:). (■■■■/), {■■III (■/III {■Hill {II III) 
denoting the number of the conics which pass through five given points, or which pass 
through four given points and touch a given line, &c. . . ., or which touch five given lines ; 
these numbers are respectively 
= 1, 2, 4, 4, 2, 1. 
So for the conics which satisfy a given condition X, or two conditions 2X, . . ., or five 
conditions 5X, we have respectively the numbers 
X, (::), (.-./), (://), (•///), (////) 
2X, (.-.), (:/), (•//), ( //I ) 
3X, (:), (•/), ( //) 
4X, (•)> ( /) 
5X, 
where the X, 2X, &c. belong to the symbols which follow: read (X:: ), (X.*./), &c., 
or, as we may for shortness represent them, 
(jJ\ f, o’", r 1 " 
