1899-1900.] Dr Muir on the Theory of Alternants. 
95 
sion, it would scarcely be possible for him not to anticipate Cauchy 
and Schweins in the discovery of the elementary properties of 
alternants. Thus, to take again the case of four variables, say the 
equations 
x + y + 2 + w = p 
ax + by + cz + dw = q 
a 2 x + b 2 y + c 2 z + dhu = r 
a 3 x + b 3 y + c 3 z + dhv = s 
Laplace’s process would have given the value of x in the form 
\W-cH 3 \p - \bW\q + 1 bWd'^ r - \ bWd 2 \s 
|6 W| - \b°c 2 d 3 \a + \b*c l d 3 \a 2 ~ \Wc l d 2 \a 3 * 
and Prony obtaining it in the form 
bed . p - (be + bd + cd)q + (b + c + d)r - s 
bed . a 0 - (be + bd + cd)a + (b + c + d)a 2 - a 3 
could not have failed to know in their general forms the theorems 
|6W 8 | -r \bWd 2 \ = bed , 
\b°c 2 d?\ ~ |6W| — be + bd + cd , 
|6W 3 | 4- \bW\ = b + c + d, 
and 
\a°b l c 2 d 3 \ ~ \b^d 2 \ = (d - a)(c - a)(b - a ) , 
and . • . | aPb i c 2 d 3 \ = (d - a)(c - a)(b - a)(c - b)(e — a)(b - a) . 
CAUCHY (1812). 
[Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs 
egales et de signes contraires par suite des transpositions 
operees entre les variables qu’elles renferment, Journ. de 
CEc. Polyt., x. pp. 29-51, 51-112.] 
By reason of the fact that Cauchy viewed determinants as a 
class of alternating functions, it has already been necessary to give 
an account* of a considerable portion of the first part (pp. 29-51) 
of this memoir : in fact, only five pages (pp. 45-51) remain to be 
dealt with if the portion referred to be borne in mind. 
* See Proc. Roy. Soc. Ediiib ., xiv. pp. 499-502. 
