1899-1900.] Dr Muir on the Theory of Skew Determinants. 209 
the theorem and notation introduced at the outset, he writes 
Cayley’s propositions in the form — 
n even, P P 0 + -b • • ’"b ^ 11^22 • • • ^Wj 
71 odd, P = Q'rf^Vii) 0 "b • • • "b ^ 11^22 * * * 5 
which, he says, when the principal elements are all unity become 
n even, P = P 0 + ^ .( 2 Tn)o + + 1, 
» odd, p = 2/^)0 + 2 /Wo + • • • + 1 > 
the development now being in each case a sum of squares, as all 
the minors appearing in it are even-ordered. 
CAYLEY (1857). 
[Theoreme sur les determinants gauches. Crelle’s Journ., Iv. 
pp 277, 278; or Collected Math. Payers , iv. pp. 72, 73.] 
This is practically a note to rectify the oversight made in the 
paper of 1854, where, as has been pointed out, he omitted to draw 
attention to the case in which the skew determinant submitted to 
the operation of 1 bordering * has zeros for the elements of the 
principal diagonal. 
“Un determinant,” he now says, “de cette espece se 
reduit toujours au produit de deux Pfaffiens. En effet en 
ecrivant dans les exemples 11 = 22 = 33 = 44 = 0, on obtient : 
al23 1/3123 = a\ 23-0123, 
al234 1/31234 = a/31234-1234, 
et de meme pour un determinant gauche et symetrique 
borde quelconque, suivant que l’ordre du determinant est 
pair ou impair.” 
To this there is added the suggestive commentary : — 
“ Je remarque a propos de cela, que dans le cas d’un 
determinant d’ordre pair, le terme aft est multiplie par un 
mineur premier lequel (comme determinant gauche et 
symetrique d’ordre impair) se reduit a zero ; le determinant 
YOL. XXIII. 0 
