. The principles of projective geometry applied to the straight line and conic . that all algebraic curves of the second degree or of the 314 Principles of Projective Geometry second class are conies. The correspondence here defined will be termedan algebraic one to one correspondence or shortly when there is no fear ofconfusion a one to one correspondence. In this case Art. 44 applies andthe ranges and pencils are projective. Although the point does notat present arise the theorem now enunciated may be extended bysubstituting any algebraic curve for conic in the statement of it. The theorem t
. The principles of projective geometry applied to the straight line and conic . that all algebraic curves of the second degree or of the 314 Principles of Projective Geometry second class are conies. The correspondence here defined will be termedan algebraic one to one correspondence or shortly when there is no fear ofconfusion a one to one correspondence. In this case Art. 44 applies andthe ranges and pencils are projective. Although the point does notat present arise the theorem now enunciated may be extended bysubstituting any algebraic curve for conic in the statement of it. The theorem that two ranges on linear bases between which thereis an algebraic one to one correspondence are projective may be extendedto ranges on conies between which there is an algebraic one to onecorrespondence. The proof excludes the case of ranges obtained bymeasuring lengths along the curve. Such ranges have not generally a-one to one correspondence. If there is an algebiaic one to one correspondence between two reon the same or on two different conies, the ranges are Let A, B, C, D and A, B, C, D be corresponding points ofranges situated on conies S and 8. Join A, B, C, D to any pointP on S and A, B, C, D to any point P on S. Let the pencils soformed be cut by any line s in Ai, i?,, Cj, A and , 5/, Cj, D/respectively. Since there is one and one algebraic correspondencebetween A BCD and ABCD, there is a similar correspondence betweenA^, -Bi, (7], Dj and Ai, 5/, C/, Z)/- Hence (Art. 44) these ranges areprojective. Therefore the ranges A, B, C, D and A, B, C, D are alsoprojective. If the corresponding elements of two ranges, or pencils, havingalgebraic one to one correspondence mutually correspond, it follows fromArt. 51 that they form an involution. Correspondence 315
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