Lessons Introductory to the Modern Higher Algebra

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Hodges, Figgis, and Company, 1876 - Algebra, Abstract - 318 pages
 

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Page 104 - Any function of the coefficients of a quartic is called an invariant, if, when the quadric is linearly transformed, the same function of the new coefficients is equal to the old function multiplied by some power of the modulus of transformation ; that is to say, when we have Ф И? Д QI •••) — Д"Ф (a) & c) •••)•" It is, I think, better to replace the words in Italics by others less exacting, viz.
Page 103 - Cayley set himself the problem to determine a priori what functions of the coefficients of a given equation possess this property of invariance, and found, to begin with, in 1845, that the so-called " hyper-determinants
Page 219 - Article ; for the product of the squares ot the differences of all the roots is made up of the product of the squares of the differences of the roots of...
Page 29 - The general theorem of which these are particular cases can he proved in a similar manner, and may be expressed as follows : — A minor of the order m formed out of the inverse constituents is equal to the complementary of the corresponding minor of the original determinant A multiplied by the (m — 1)'* power q/"A.
Page 55 - Put for x in succession a, b, c,••. and add; thus The terms on the right-hand side which are equidistant from the beginning and the end are equal ; therefore by rearranging and dividing by 2 we obtain ...+^(-ir v ]p ;yNow Sl, S3,...
Page 294 - Cayley, nearly three years later (1845), to propose to himself the problem to determine a priori what functions of the coefficients of an equation possess this property of invariance, and...
Page 10 - ... The process of evaluation of a numerical determinant is dependent on four principles : — (1) That the value of a determinant is not altered if rows are changed into columns. (2) The interchange of two rows or two columns reverses the sign of the determinant. (3) If every constituent in any row or column be multiplied by the same factor, then the determinant is multiplied by that factor. (4) A determinant is not altered if we add to each constituent of any row or column the corresponding constituents...
Page 260 - Art. 5, by the formula 8 + 2f = a' (I + m + n 4. r} - ff, using which values for 8 and 8' we arrive at the same result as in the last article. 12. We next suppose the quantics to have common two surfaces having i points of intersection. The method would be the same if there were several surfaces. Let the last quantic be a complex one, consisting of Z which passes through the first surface and Z' which passes through the second.

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