84 
PROFESSOR CAYLEY ON THE CURVES 
above referred to, where the conics considered being the conics which pass through three 
given points, the equation is taken to he fyz-\-gzx-\-hxy—h, and we have only the three 
parameters (f, g, h) ; and also in Annex No. 3, where the conics pass through two given 
points, and are represented by an equation containing the four parameters ( a , h, c, h ) : I 
give this Annex as a somewhat more elaborate example than any which is previously 
considered, of the application of the foregoing principles, and as an investigation which 
is interesting for its own sake. See also Annexes 4 and 5, which contain other exam- 
ples of the theory. The remark as to the number of parameters is of course applicable 
to the case where the curve which satisfies the given conditions is a curve of any given 
order r; the number of the parameters is here at most =-^(r+l)(r+2), and the space 
therefore at most \r(r-\- 3)dimensional ; but we may in particular cases have u-\-\ para- 
meters, the coordinates of a point in ^-dimensional space, where a is any number less 
than -^r(r+3). 
23. I do not at present consider the case of a curve of the order r , or further pursue 
these investigations; my object has been, not the development of the foregoing quasi- 
geometrical theory, so as to obtain thereby a series of results, but only to sketch out the 
general theory, and in particular to establish the notion of the order of condition, and 
to show that, as a rule (though as a rule subject to very frequent exceptions), the order 
of a compound condition is equal to the product of the orders of the component condi- 
tions. The last-mentioned theorem seems to me the true basis of the results contained 
in a subsequent part of this paper in connexion with the formulae of 1)e Jonquieres, 
post, No. 74 et seg. But I now proceed to a different part of the general subject. 
Reproduction and Development of the Researches of Chasles and Zeuthen. 
Article Nos. 24 tb 72. 
24. The leading points of Chasles’s theory are as follows : he considers the conics 
which satisfy four conditions (4X), and establishes the notion of the characteristics 
(g>, v) of such a system, viz. g,, =(4X • ), denotes the number of conics in the system 
which pass through a given (arbitrary) point, and i/, = (4 X/), the number of conics in 
the system which touch a given (arbitrary) line. We may say that gj is the parametric 
order, and v the parametric class of the system. 
25. The conics 
oo. (•■•/). (■■//), (•///). (////) 
which pass through four given points, or which pass through three given points and 
touch a given line, &c., ... or touch four given lines, have respectively the characteristics 
(1,2), (2,4), (4,4), (4,2), (2,1). 
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 (gj, v) the characteristics of (4Z), then the number of the conics which satisfy the 
five conditions (X, 4Z) is=ugj+pt>. 
