Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1903 Excerpt: ...of xf + yf2, and as x and y vary, the locus of the transformed point is a line which we shall call the satellite of a. The satellites generate the complex (frlpffWf'qff'l) = o (sis), and the form of this equation should be compared with (296) and (298). There is also the complex of conjugate satellites obtained by replacing f and f, by their conjugates, but when the functions are self-conjugate, or when one is the conjugate of the other, the two complexes combine into one. For the functions f and f this is (f-W-1Pf-llf-1l) = 0 (316). The four points fa, fa, fa, fa form a harmonic range on the satellite of the point a. There are also harmonic properties connecting pencils of planes Sqfa = 0, Sqfa = 0, Sqfa = 0, Sqf/t = 0; and it may be verified that these four planes intersect in a satellite for the inverse functions. This we shall prove for the general case. The reciprocal of the complex of satellites is the complex of the conjugate satellites for the inverse functions. If p and q are any two points on the reciprocal of the satellite of a, Sp/Xa = Sp/3a = 0, Sqfya = S?/2a = 0 (317), and on taking conjugates we see that the four points ftp, f/p, f'q, fjq are co-planar, so that The locus of points whose satellites meet the line ab is the quadric surface (compare (297)) (/?/&) = 0 (319). 77. The satellite of a point which describes a line q = a + tb constructs one system of generators of the quadric q = (fi + sf)(a + tb) (320), but the regulus degrades into a system of lines enveloping a conic whenever (/.«/««#/.&) = o (32l), that is, whenever the line belongs to the reciprocal of the complex of conjugate satellites (318). The conic is co-planar with the line when the further conditions (abfafb) = 0, (abf2af2b) = 0 (322), are satisfied (...
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