An Elementary Treatise on Modern Pure Geometry - download pdf or read online

By Robert Lachlan

This quantity is made out of electronic pictures from the Cornell college Library old arithmetic Monographs assortment.

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Infer from the above construction that a pencil in involution has in general one and only one pair of conjugate rays at right angles. the centre of the K is The exceptional case occurs when the point that is when the two given pairs of conju- circle, gate rays are at right angles. will In this case every line through A' be a diameter, and hence every pair of conjugate rays will be at right angles. }, in which each of the angles AOA', involution, of BOB, ... is a right angle, is a pencil in which any pair of conjugate rays may be considered as the principal rays.

Intersect in 0'. the last article, the line joining the harmonic conjugates with respect to A, B; and G, D; must pass through 0'. Hence, 00' meet if conjugate of P AB in A with respect to Q, Q must be the harmonic and B. {AB, PQ), 0' {A'B', P'Q) have one ray and O'A' are coincident, the three points in which the rays OB, OP, OQ intersect the rays O'B', O'P',O'Q' respectively, are collinear and likewise the three points in which the rays OB, OP, OQ intersect the rays O'B', O'Q', O'P' If the pencils 62.

Cx. Ax. NL+Ax Bx LM=Q. T . ] Relations connecting the angles of a pencil. 29. If several straight lines be drawn in the same plane through a point 0, they are said to form a pencil. rays of the pencil. is The point is and the straight lines are called the The pencil formed by the rays OA, OB, 00,... }. AND PENCILS. 17 The six angles of any pencil of four rays 30. \ABCD) art connected by the relation sin BOC . sin A OD + sin CO A . sin 110 1> + sin AOB. sin COD -0. O Let any straight in line the points A, B, C, D.

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An Elementary Treatise on Modern Pure Geometry by Robert Lachlan


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