Collected Mathematical Papers of Arthur Cayley Volume 7
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. 1894 edition. Excerpt: ...equation (3). I believe the More...
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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. 1894 edition. Excerpt: ...equation (3). I believe the better course is to assume (1) and (3) as the fundamental equations, from them deducing (2); and we thus also get over a difficulty presently referred to, but which did not occur to me when the memoir was written. In fact, starting with the equations x' y" / = X: Y: Z (which are to give x: y: z = X' Y' Z'), we have in the first instance the equation (1). Moreover, establishing for x', y, z' a linear equation ax + by' + cz' = 0, we have corresponding hereto a curve aX + bY+cZ=0, and the coordinates x, y, z of a point on this curve are proportional to X' Y: Z1; that is, substituting for z' the value---(ax' + by'), they c are proportional to rational and integral (homogeneous) functions of (x y), that is, to rational and integral functions of the single parameter x' y'; wherefore the curve aX + b Y + cZ = 0 is unicursal; whence the equation (3). The like change may be made in the theory of the rational transformation between two spaces; and it is in this case a more important one. The difficulty is as follows: It is not self-evident that we are at liberty to assume a, + 3a2 + 6a, ... (n? + Sn)-2; for imagine that we had a system of (a, a2, a, ...) points, such that a, + 4a, j+... being = ) -'--1, and a, + 3a2+... being (ns + 3n)--2, the points were such that the conditions in question (viz., the condition that the curve passes once through each of the points an twice through each of the points aj...) should be less than a, + 3a2 +..., and in fact = or (?i-+ 3n)--2; then the functions X, Y, Z would not of necessity be connected by a linear relation X + fiY+ vZ= 0, and the ground for the assumption in question, a, + 3a +... (n3 + 3n)--2, would no longer exist. And except by the process...