The general equation to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only curve which intersects the line infinity in the circular points.
The equation to a parabola in triangular co-ordinates is generally derived by expressing the condition that the line at infinity is a tangent in the equation to the general conic. For example, in trilinear co-ordinates, the equation to the general conic circumscribing the triangle of reference is 113y+mya+naf3=o; for this to be a parabola the line as + b/ + cy = o must be a tangent.
Hence in trilinear co-ordinates, with ABC as fundamental triangle, its equation is Pa+Q/1+R7=o.
Trilinear and Tangential Co-ordinates.---The Geometrie descriptive, by Gaspard Monge, was written in the year 1794 or 1 795 (7th edition, Paris, 1847), and in it we have stated, in piano with regard to the circle, and in three dimensions with regard to a surface of the second order, the fundamental theorem of reciprocal polars, viz.
Trilinear co-ordinates (see Geometry: Analytical) were first used by E.
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