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A SIXTH MEMOIR UPON QUANTICS. 
563 
On Geometry of One Dimension, Nos. 149 to 168. 
149. In geometry of one dimension we have the line as a space or locus in 
quo, which is considered as made up of points. The several points of the line are 
determined by the coordinates (x, y), viz. attributing to these any specific values, or 
writing x, y = a, b, we have a particular point of the line. And we may say also 
that the line is the locus in quo of the coordinates (x, y). 
150. A linear equation, 
(*$>, y'y = °> 
is obviously equivalent to an equation of the before-mentioned form x,y = a, b, and 
represents therefore a point. An equation such as 
(*$#, y) m = 0 
breaks up into m linear equations, and represents therefore a system of m points, or 
point-system of the order m. The component points of the system, or the linear 
factors, or the values thereby given for the coordinates, are termed roots. When 
m = 1 we have of course a single point, when m — 2 we have a quadric or point- 
pair, when m = 3 a cubic or point-triplet, and so on. The point-system is the only 
figure or locus occurring in the geometry of one dimension. The quantic (*$#, y) m , 
when it is convenient to do so, may be represented by a single letter U, and we 
then have 17 = 0 for the equation of the point-system. 
151. The equation 
(*]$>, y) m = o 
may have two or more of its roots equal to each other, or generally there may exist 
any systems of equalities between the roots of the equation, or what is the same 
thing, the system may comprise two or more coincident points, or any systems of 
coincident points. In particular, when the discriminant vanishes the equation will have 
a pair of equal roots, or the system will comprise a pair of coincident points; in 
the case of the quadric (a, b, c$x, y) 2 = 0, the condition is ac—b 2 = 0, or as it may 
be written, a, b = b, c; in the case of the cubic 
the condition is 
(a, b, c, d\x, y 3 ) = 0, 
a 2 d 2 — 6abcd + 4ac 3 + 4<b 3 d — Sb 2 c 2 = 0. 
The preceding is the only special case for a quadric: for a cubic we have besides 
the special case where the three roots are equal, or the cubic reduces itself to three 
coincident points; the conditions for this are 
ac — b 2 — 0, ad — be = 0, bd — c 2 = 0, 
equivalent to the two conditions 
a : b — b : c — c : d. 
71—2