CHAPTER III.

curve with the surface which results from this co-ordinate. Thus, in this case, the two other co-ordinates of the point must be conceived as functions necessarily determinate and distinct from the first. It follows that every line, considered in space, is then represented analytically, no longer by a single equation, but by the system of two equations between the three co-ordinates of any one of its points. It is clear, indeed, from another point of view, that since each of these equations, considered separately, expresses a certain surface, their combination presents the proposed line as the intersection of two determinate surfaces. Such is the most general manner of conceiving the algebraic representation of a line in analytical geometry of three dimensions. This conception is commonly considered in too restricted a manner, when we confine ourselves to considering a line as determined by the system of its two projections upon two of the co-ordinate planes; a system characterized, analytically, by this peculiarity, that each of the two equations of the line then contains only two of the three co-ordinates, instead of simultaneously including the three variables. This consideration, which consists in regarding the line as the intersection of two cylindrical surfaces parallel to two of the three axes of the co-ordinates, besides the inconvenience of being confined to the ordinary rectilinear system, has the fault, if we strictly confine ourselves to it, of introducing useless difficulties into the analytical representation of lines, since the combination of these two cylinders would evidently not be always the most suitable for forming the equations of a line. Thus, considering this fundamental notion in its entire generality, it will be necessary in each case to choose, from among the infinite number of couples of surfaces, the intersection of which might produce the proposed curve, that one which will lend itself the best to the establishment of equations, as being composed of the best known surfaces. Thus, if the problem is to express analytically a circle in space, it will evidently be preferable to consider it as the intersection of a sphere and a plane, rather than as proceeding from any other combination of surfaces which could equally produce it.