The Messenger of Mathematics Volume 9

The Messenger of Mathematics Volume 9

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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. 1880 edition. Excerpt: ...2 = 0. 3. If we take for our 22+ 1 distinct elements of the persymmetrical matrix, the quantities where cents is of 7t dimensions in x, y, we shall find by virtue of 1 that they will be represented by X1 Ylcents, X5-1YE+lcents, '.. Y'lEcents, This is true for any function gb homogeneous in:1: and y. If;l is a rational integral function of w, y, say A (ao, an an Fl 1 the last written determinant becomes a covariant whose leading coeflicient is the persymmetrical determinant formed with the elements a0, al, a', ...a2, multiplied by trical determinant formed with the elements ao, a, ...a, ', as was to be proved. Sclzoliitm. The theory of hyperdeterminants teaches us that every in-and co-variant has its source of being in a higher existence, viz. in a pure form typified by It seems to me very likely or almost certain that every covariant of f (as' y), or what becomes such when f is a quantic, may in like manner be converted into a function of f and of its derivatives in respect to one of the variables alone divided by an appropriate power of the other; and, if true, as it can hardly help being, the proof ought not to be far to seek. It is indeed virtually contained in a formula obvious from inspection of the expression for y'Y' in 1, viz. whatever homogeneous function is supposed 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:0: 0, it becomes an invariant of (a0, al, 11, ...a; )(E, n)," and that its several terms are what the a elements become when X becomes when E +xr. We thus led to view the whole subject of invariance under a somewhat broader aspect, as a theory not directly concerned with quant
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Product details

  • Paperback | 32 pages
  • 189 x 246 x 2mm | 77g
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 123657592X
  • 9781236575920