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In varied methods of triangular co-ordinates the equations to circles specifically related to the triangle of reference assume comparatively simple kinds; consequently they supply elegant algebraical demonstrations of properties concerning a triangle and the circles intimately related to its geometry. ????2-????)1/2. The Cartesian co-ordinates. √(???? cot 1/2C)＝0, with centre sin A, sin B, sin C; the Areal co-ordinates. 2, the accents being dropped, and ????, ????, ???? regarded as current co-ordinates. The corresponding equations in areal co-ordinates are readily derived by substituting ????/????, ????/????, ????/???? for α, β, γ respectively in the trilinear equations. The overall equation to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only curve which intersects the road infinity in the circular points. ????γ＝0 is the road infinity, it is obvious that equation (1) represents a conic passing through the circular points, i.e. a circle. 2＝0 as the equation to the circular points. ????2 happen in the equation of every circle, it is seen that all circles cross through two mounted points at infinity. Each grain is a crystal, the elementary parts of which are all oriented a technique, but the orientation modifications as we go from grain to grain.

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