200 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
can be brought to the form in question, then in the latter case we may employ the 
equation in such reduced form, attending only to the remaining conditions; and in 
either case we have the equation of a conic containing linearly 2, 3, 4, or 5 parameters, 
which parameters are taken as the coordinates of a point in 1-, 2-, 3-, or 4-dimensional 
space, and the discussion relates to loci in such dimensional ’space. This is in fact 
what is done in Annex No. 2 above referred to, where the conics considered being 
the conics which pass through three given points, the equation is taken to be 
fyz + gzx + hxy = 0, 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 bv 
an equation containing the four parameters (a, b, c, li): I give this Annex as a some 
what 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 examples 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 +1) (r + 2), and the space therefore 
at most ^ r (r + 3) dimensional; but we may in particular cases have to +1 parameters, 
the coordinates of a point in co-dimensional space, where to 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 con 
dition, 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 conditions. 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 De Jonquieres, post, No. 74 et seq. But I now proceed to a different part of the 
general subject. 
Article Nos. 24 to 72.—Reprodiiction and Development of the Researches of 
Chasles and Zeuthen. 
24. The leading points of Chasles’s theory are as follows: he considers the conics 
which satisfy four conditions (4A), and establishes the notion of the characteristics 
(g, v) of such a system, viz. /x, = (4X •), denotes the number of conics in the system 
which pass through a given (arbitrary) point, and v, =(4Xj), the number of conics in 
the system which touch a given (arbitrary) line. We may say that g is the parametric 
order, and v the parametric class of the system. 
25. The conics 
(://), (•///), (////) 
which pass through four given points, or which pass through three given poiuts 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).