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AN INTRODUCTORY MEMOIR UPON QUANTICS. 
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3. Aii equation or system of equations represents, or is represented by a locus. 
This assumes that the facients depend upon quantities x, y, ... the coordinates of a 
point in space; the entire series of points, the coordinates of which satisfy the equation 
or system of equations, constitutes the locus. To avoid complexity, it is proper to take 
the facients themselves as coordinates, or at all events to consider these facients as 
linear functions of the coordinates; this being the case, the order of the locus will be 
the order of the equation, or system of equations. 
A I have spoken of the coordinates of a point in space. I consider that there is 
an ideal space of any number of dimensions, but of course, in the ordinary acceptation 
of the word, space is of three dimensions; however, the plane (the space of ordinary 
plane geometry) is a space of two dimensions, and we may consider the line as a space 
of one dimension. I do not, it should be observed, say that the only idea which can 
be formed of a space of two dimensions is the plane, or the only idea which can be 
formed of space of one dimension is the line; this is not the case. To avoid complexity, 
I will take the case of plane geometry rather than geometry of three dimensions; it 
will be unnecessary to speak of space, or of the number of its dimensions, or of the 
plane, since we are only concerned with space of two dimensions, viz. the plane ; I say, 
therefore, simply that x, y, z are the coordinates of a point (strictly speaking, it is the 
ratios of these quantities which are the coordinates, and the quantities x, y, z themselves 
are in determinates, i.e. they are only determinate to a common factor pres, so that in 
assuming that the coordinates of a point are a, /9, 7, we mean only that x : y : z = a : ¡3 :y, 
and we never as a result obtain x, y, z = a, /3, 7, but only x : y : z = a : /3 : y, but 
this being once understood, there is no objection to speaking of x, y, z as coordinates). 
Now the notions of coordinates and of a point are merely relative ; we may, if we 
please, consider x : y : z as the parameters of a curve containing two variable para 
meters ; such curve becomes of course determinate when we assume x : y : z = a : (3 : 7, 
and this very curve is nothing else than the point whose coordinates are a, /3, 7, or 
as we may for shortness call it, the point (a, /3, 7). And if the coordinates (x, y, z) are 
connected by an equation, then giving to these coordinates the entire system of values 
which satisfy the equation, the locus of the points corresponding to these values is the 
locus representing or represented by the equation; this of course fixes the notion of a 
curve of any order, and in particular the notion of a line as the curve of the first 
order. 
The theory includes, as a very particular case, the ordinary theory of reciprocity in 
plane geometry; we have only to say that the word “point” shall mean “line,” and the 
word “ line ” shall mean “ point,” and that expressions properly or primarily applicable 
to a point and a line respectively shall be construed to apply to a line and a point 
respectively, and any theorem (assumed of course to be a purely descriptive one) relating 
to points and lines will become a corresponding theorem relating to lines and points; 
and similarly with regard to curves of a higher order, when the ideas of reciprocity 
applicable to these curves are properly developed. 
5. A quantic of the degrees m, m'... in the sets (x,y...), (x', y'...) &c. will for the 
most part be represented by a notation such as 
m 
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