these latter remaining perfectly independent of each other. Thus there will be a certain equation between the three variable co-ordinates, which will be permanent, and which will be the only one, in order to correspond to the precise degree of indetermination in the position of the point.
Expression of Surfaces by Equations. This equation, more or less easy to be discovered, but always possible, will be the analytical definition of the proposed surface, since it must be verified for all the points of that surface, and for them alone. If the surface undergoes any change whatever, even a simple change of place, the equation must undergo a more or less serious corresponding modification. In a word, all geometrical phenomena relating to surfaces will admit of being translated by certain equivalent analytical conditions appropriate to equations of three variables; and in the establishment and interpretation of this general and necessary harmony will essentially consist the science of analytical geometry of three dimensions.
Expression of Equations by Surfaces. Considering next this fundamental conception in the inverse point of view, we see in the same manner that every equation of three variables may, in general, be represented geometrically by a determinate surface, primitively defined by the very characteristic property, that the co-ordinates of all its points always retain the mutual relation enunciated in this equation. This geometrical locus will evidently change, for the same equation, according to the system of co-ordinates which may serve for the construction of this representation. In adopting, for example, the rectilinear system, it is clear that in the equation between the three variables, x, y, z, every particular value attributed to z will give an equation between at x and y, the geometrical locus of which will be a certain line situated in a plane parallel to the plane of x and y, and at a distance from this last equal to the value of z; so that the complete geometrical locus will present itself as composed of an infinite series of lines superimposed in a series of parallel planes (excepting the interruptions which may exist), and will consequently form a veritable surface. It would be the same in considering any other system of co-ordinates, although the geometrical construction of the equation becomes more difficult to follow.