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. 1880 Excerpt: ...to follow the asterisk. In connexion with this it should be observed that the determinant in 2 is bound to vanish, from the mere fact that on putting x = 0, it becomes an invariant of (o„ a„ a,,-.aM)(£, vYei and that its several terms are what the a elements become when X becomes when £ 4-Jctj. We are thus led to view the whole subject of invariance under a somewhat broader aspect, as a theory not directly concerned with quantics, but with homogeneous functions in general. VOL. IX. G f being a perfectly general operand, or as we may phrase it, an operand absolute. This enables me to express the idea which was struggling into light when I wrote the antecedent footnote. It is this: Let 0 now be made to do duty for any given homogeneous function of given order X in x, y. The value of F will remain unaltered when we write X--/t + 1...---in place of Y, y (x-/ + a)(x-/ + i)?" x' This is an immediate consequence of the invariantive property of F combined with the fact that previously shown. The numerators in the above expressions are the first terms in the expression for y Y' as a function of xX modified by writing successively X--/ + 1, X--(i + 2,...X in place of X on account of the powers of X which precede Yf Yf... FM in F and lower the degrees of the operands in respect to these powers by /-1, /--2,...0 units respectively. Thus ex. gr. the pure invariant (X4:) (Y:)-4 Xs Y:) (XY':) + 3 (X2 Y:) where the colon (:) does duty for an operand absolute is equivalent to i (X-3)(X-2)(X-1)X(X«:)(X:)-4 (X-3)' (X-2) (X-1) (X3:) (X:) + 3(X-3)"(X-2j2 (Xs:) (X':), the colon now representing a homogeneous function of order X is x, y. So in general we may say that a pure invariant, or it might be more correct to say the Schema of an invaria...
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