1899 -im] Dr Muir on the Theory of Alternants. 
119 
the rule of signs is here fully expressed. Thus, if <f>(a,b,c, . . ., 1) 
were ab 2 c% we should have 
waW'\ aW aW 
P / (b - a)(c - a)(c - b) (c - a)(b - a)(b - c) 
& W , &W 
-P -f* 
(a - 6)(c - b)(c -a) (c- b)(a - b)(a - c ) 
c 1 ^ 2 /; 4 c^a 4 
(a - c)(& - c){b - a) (b - cj(a - c)(a - 6)’ 
aW - a 1 ^ 4 - b l a 2 rA + W + c ] a 2 & 4 - c 1 W 
(6 - a)(c - a)(c - £) ’ 
and . P 2,( ~ ) = aW - - We 4 + JlcW + cl< * 264 " clft2 “ 4 > 
which is an alternating function written by Cauchy in the form 
S( ± a l b 2 c^), and which, being a determinant, was written by 
Jacobi himself also in the form 2 ±al& 2 c 4 
It is pointed out that any term of which remains unchanged 
by the interchange of two of the variables may be left out of 
account; but the question raised by Cauchy regarding possible 
and impossible forms of <f> is not touched upon. As a corollary, it 
is stated that if 
Vi* • • • -, <*n) = % 0 a^ a®". 
the indices a 0 , cq, . . ., a n must be all different if the] alternating 
function is not to vanish. 
He then recalls- the known fact that, when the indices 
a 0 , a v . . ., a n are integral, the alternating function 
• • . aCln 
2 ± . . . a““ or P£ ' p - 
is divisible by P, the difference-produet of a 0 , and 
puts to himself the problem of finding the generating function of 
the quotient 
2 
a n 
n 
In the course of this quest his first proposition is — 
