1902 - 3 .] Dr Muir on Axisymmetric Determinants. 
561 
values in question, is obtainable on eliminating x, y, z, . . . from 
the set of n equations 
(A**, — s)x + A xy y + A xz z + 0 
Ay x X + (Ayy Z/ "f" AygZ 4" 6 
A zx x + A zy y + ( A zz - s)z + •■•• = 0 
where A yx = A xy , .... Remembering Cauchy’s great paper of 
1812, we are quite prepared to find him at this stage proceeding^ 
to say : — 
“ S sera une fonction alternee des quantites comprises dans 
le Tableau 
-^-XX S A x y 
A A 
■^xy -n-yy 
Kz A y z 
\ 
savoir celle dont les differents termes sont representees, aux 
signes pres, par les produits qu’on obtient, lorsqu’on multiplie 
ces quantites, n a n, de toutes les manieres possibles, en ayant 
soin de faire entrer dans chaque produit un facteur pris dans 
chacune des lignes horizontales du Tableau et un facteur pris 
dans chacune des lignes verticales.” 
The lengthy discussion of the character of the roots of S = 0 
which thereupon follows, and in which the properties of “fonctions 
alternees ” are freely used, belongs almost entirely to a different 
portion of our subject : for the present there concerns us only one 
theorem subsidiary to the said discussion. In modern phraseology 
this lemma is — S being any axisymmetric determinant , R the deter- 
minant got by deleting the first row and first column of S, Y the 
determinant got by deleting the first row and second column of S, 
and Q the determinant got from R as R from S, then , if R = 0, 
SQ = - Y 2 . The mode adopted for testing the truth of this is 
applicable to any determinant S, whether axisymmetric or not ; 
and when the second condition, viz., the vanishing of R, is also 
removed, there emerges the simplest case of Jacobi’s theorem of 
1833 regarding a minor of the adjugate.* 
* v. “The Theory of Orthogonants ” . . . Proceedings Roy. Soc. Edin- 
burgh , xxiv. p. 266. 
