1909-10.] The Theory of Persymmetric Determinants. 
407 
XXV. — The Theory of Persymmetric Determinants in the Historical 
Order of Development up to 1860. By Thomas Muir, LL.D. 
(MS. received January 22, 1910. Read February 7, 1910.) 
As has already been pointed out {Hist., i. pp. 485-487 *), the special form of 
determinant named “ persymmetric ” in 1853 by Sylvester came first to 
light in 1835 in a paper of Jacobi’s on the elimination of the unknown 
from two equations of the n tYl degree, the fact being that the adjugate of 
Bezout’s condensed eliminant — in other words, the adjugate of the determi- 
nant resulting from Bezout’s “ abridged method ” of elimination — is there 
shown to be such that the elements of it whose place-numbers have the 
same sum are equal. 
Rosenhain, G. (1844). 
[Exercitationes analyticse in theorema Abelianum de integralibus 
functionum algebraicarum. Crelle’s Journ., xxviii. pp. 249-278.] 
What concerns our subject here is a digression (§§ 5-10, pp. 263-278) 
on the elimination of the unknown from two equations of the n th degree. 
The first two sections (pp. 263-268) are little else than a reproduction of 
part of Jacobi’s paper of 1835 dealing with Bezout’s so-called “abridged 
method,” and the remainder contains a discussion of other methods. In 
subject, therefore, the digression resembles Cauchy’s paper of 1840. 
At this point we have to recall the fact already reported, f that in 
Borchardt’s paper of 1845 (January) a determinant of the special form we 
are now considering appeared as an expression for the square of the 
difference-product, and that a generalisation of this result was given by 
Cayley the year following. 
Jacobi, C. G. J. (1845, August). 
[Ueber die Darstellung einer Reihe gegebner Werthe durch eine 
gebrochne rationale Function. Crelle’s Journ., xxx. pp. 127-156 : 
or Gesammelte Werke, iii. pp. 479-511.] 
The subject here dealt with by Jacobi is that first considered by Cauchy 
in the fifth note to the Analyse Algebrique of 1821, namely, the extension 
* The 7th and 8th lines of p. 486 have unfortunately been transposed by the printer. 
Also, in the first determinant of the footnote on the same page the first b x should be b 0 . 
t Proc. Roy. Soc. Edin., xxvi. p. 362, pp. 364-366. 
VOL. XXX. 
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