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The Theory of Determinants in the Historical Order of its Development

Published online by Cambridge University Press:  15 September 2014

Thomas Muir
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Extract

It is next pointed out that the transformation of the primitive permutation into any other may be accomplished by interchanges only, because by this means any given letter may be made to occupy the first place, then any other given letter to occupy the second place, and so on. From this also it follows that any system of circular substitutions may be replaced by a system of interchanges. Should the transformation of one permutation into another be effected by interchanges, the number of these will be even or odd according as the two permutations belong to the same or different classes; for, by the above theorem, every interchange makes only one group more or one group less, and consequently the total number of interchanges, and the net increase or diminution of the number of groups, must be both even or both odd. The counting of interchanges may thus be substituted for the counting of cycles.

Type
Proceedings 1888-89
Information
Proceedings of the Royal Society of Edinburgh , Volume 16 , 1890 , pp. 748 - 772
Copyright
Copyright © Royal Society of Edinburgh 1889

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References

note * page 754 Liouville, in a paper published in the same year as Cauchy's memoirs, uses resultant, but adds in a footnote, “Au lieu du mot résultante, les géomètres emploient souvent le mot déterminant” (Liouville's Journ., vi. p. 348).

note * page 759 In a strictly chronological arrangement Cayley's paper would not follow, but precede the papers of Craufurd, Cauchy, and Jacobi of the same year. It was published in February : Cauchy's note was presented to the Academy on 8th March, and Jacobi's memoir bears the date 17th March, though not published for more than two months afterwards. As Cayley's first appearance, however, marks the beginning of a new epoch, and as the other papers referred to belong by their character to the preceding epoch, a slight deviation from the chronological order seems warranted.

note * page 763 The first factor being 16 times the second, and the w's unnecessary.

note * page 764 The continuation intimated at the close (p. 131) was never made.

note * page 765 The passage in question, which we quoted under Cramer, is to be found in the Annales de Math., xx. p. 45.

note * page 766 The commas which Cayley prints after the elements in a determinant we omit here and henceforth.

note * page 769 See art. “Quaternions,” by Professor Tait, in Encyclopedia Britannica; or Hamilton's Lectures on Quaternions.