90 
PROFESSOR CAYLEY ON THE CURVES 
And by means of this formula he investigates the characteristics of the several systems 
of conics which satisfy four conditions (4X) of contact with a given curve or curves, viz. 
these are the conics 
(1) (1)(1)(1), (1,1)(1)(1), (i, 1)(1, 1), (1,1, 1X1). (1.1. 1.1). 
(2) (1)(1) , (2X1,1) , (2,1)(1) , (2,1,1), 
( 2 )( 2 ) , ( 2 , 2 ) , 
(3) (1) ,(3,1) , 
( 4 ) 
where (1) denotes contact of the first order, (2) of the second order, (3) of the third 
order, (4) of the fourth order, with a given curve ; (1)(1) denotes contacts of the first 
order with each of two given curves, (1,1) two such contacts with the same given curve, 
and so on. A given curve is in every case taken to be of the order m and class n, with & 
nodes, k cusps, r double tangents, and i inflexions (m 15 n x , r 1? m 2 , n 2 , See., as the 
case may be). The symbols (1) &c. might be referred to the corresponding curves by a 
suffix ; thus (l) m would denote that the contact is with a given curve of the order m 
(class n, See .) ; but this is in general unnecessary. 
40. In a system of conics satisfying four conditions of contact, as above, it is compa- 
ratively easy to see what are the point-pairs and line-pairs in these several systems 
respectively ; but in order to find the values of X and vj, each of these point-pairs and 
line-pairs has to be counted not once, but a proper number of times ; and it is in the 
determination of these multiplicities that the difficulty of the problem consists. I do 
not enter into this question, but give merely the results. 
41. For the statement of these I introduce what I call the notation of Zeuthen’s 
Capitals. We have to consider several classes of point-pairs and the reciprocal classes of 
line-pairs. A point-pair may be described [ante. No. 31) as a terminated line, and a 
line-pair as a terminated point ; and we have first the following point-pairs, viz. : — 
A, line terminated each way in the intersection of two curves or of a curve with itself 
(node). 
B, tangent to a curve, terminated in a curve, and in the intersection of two curves or 
of a curve with itself. 
C, common tangent of two curves, or double tangent of a curve, terminated each way 
in a curve. 
D, inflexion tangent of a curve terminated each way in a curve : 
and the corresponding line-pairs, viz. : — 
A', point terminated each way in the common tangent of two curves or the double 
tangent of a curve. 
B', point of a curve terminated by the tangent of a curve, and by the common tangent 
of two curves or double tangent of a curve. 
C', intersection of two curves, or of a curve with itself (node), terminated each way by 
the tangent to a curve. 
D', cusp of a curve terminated each way by the tangent to a curve : 
