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The Theory of Alternants in the Historical Order of its Development up to 1841

Published online by Cambridge University Press:  15 September 2014

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Extract

The first traces of the special functions now known as alternating functions are said by Cauchy to be discernible in certain work of Vandermonde's; and if we view the functions as originating in the study of the number of values which a function can assume through permutation of its variables, such an early date may in a certain sense be justifiable. To all intents and purposes, however, the theory is a creation of Cauchy's, and it is almost absolutely certain that its connection with determinants was never thought of until his time.

Type
Proceedings
Copyright
Copyright © Royal Society of Edinburgh 1902

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References

page 93 note * The history of this subject is referred to in Serret, M. J.-A.: “Sur le nombre de valeurs qui peut prendre une fonction quand on y permute les lettres qu'elle renferme,” Liouville's Journ. de Math., xv. pp. 170 (1849).Google Scholar

page 95 note * See Proc. Roy. Soc. Edinb., xiv. pp. 499–502.

page 110 note * See Muir, , “Theory of Determinants,” p. 176 (1882).Google Scholar

page 110 note † See Crelle's Journal, lxxxix. pp. 82–85.

page 130 note * The form is such that the result of any interchange among x, y, z, … is attainable by a corresponding interchange among a, b, c, ….

page 132 note * Since V = F(x). F(y). F(z) …., the first term of the alternating aggregate may be written

which, on the substitution being made, becomes F′(a). F′(b). F′(c) ….; and it is this form which in Jacobi is replaced by (-1)tn(n-1)P2.