1899-1900.] Dr Muir on the Theory of Shew Determinants. 187 
minant forme par un systeme de quantities X rs qui satisfont 
aux conditions 
K.s = - Kr (r s) . 
J’appelle aussi un tel systeme, systeme gauche .” 
So far as can be ascertained, the English equivalent ‘ shew,’ although 
it probably was the first of the two in order of thought, did not 
appear in print until a few years later. 
As has been pointed out elsewhere, the title of the paper is 
quite misleading, the real subject being the construction of a linear 
substitution for the transformation of aq 2 4- xf + xf+ .... into 
£i 2 + £ 2 2 + £ 3 2 + .... All. that can be said in defence of the 
inaccuracy is that skew determinants are made use of in obtaining 
the desired substitution. The proper place for giving an account 
of the contents of the paper is thus under the heading of £ orthogo- 
nantsf if we may so name the determinants of an orthogonal substi- 
tution. 
CAYLEY (1847). 
[Sur les determinants gauches. Crelle’s Journ., xxxviii. pp. 
93-96; or Collected Math. Payers , i. pp. 410-413.] 
Here the title and contents agree. At the outset the former 
definition is repeated, and then for a particular kind of skew deter- 
minant, viz., those in which the condition 
K> = ~ Kr (1) 
is to hold even in the case where s and r are equal, “ ou pour les- 
quels on a 
K.. = ~Kr (r.*«), \ r ,r — 0” , .... (2) 
the name c skew symmetric ’ (gauche et symetrique ”) is set apart. 
The reason for this is evident on the statement of the first theorem, 
which is to the effect that any skew determinant is expressible in 
terms of skew symmetric determinants and those elements of the 
original determinant which are not included in the latter. “ En 
effet,” he explains, 
