1899-1900.] Dr Muir on the Theory of Alternants. 121 
and therefore could be simplified by having t x - t Q struck out of 
both numerator and denominator. This means that when m>n 
the second factor, like the first, can have only positive integral 
powers of the variables. As for the third and fourth factors, their 
product is the difference of the two fractions 
1 and 1 
Co - a)(t 2 - «)(<„ - d)...(t m -p) (<j - a)(t 2 - e)(t s -<?).. ,.{t m -p)' 
the former of which yields no negative powers of £ 1} and the 
latter no negative powers of t 0 . The proposition is thus 
•established. 
To prove the next proposition he utilizes the theorem that 
If F be any rational integral function of a number of variables , 
■the coefficient of x - x y - l z ~ 1 . . . . in the expansion of 
F(x,y,z, . . .) 
(x - a)(y - b)(z -c) ... . 
■ according to descending powers of x, y, z, . . . . is 
T(a,b,c, ....). 
This is spoken of as being well-known, and no proof of it is given. 
It is readily seen, however, that as the expansion referred to is 
got by performing the multiplications indicated in 
F (x,y,z, . . .) . {x~ l + ax~ 2 + a 2 x~ 3 + . . . .} 
{y- 1 + by~ 2 + b 2 y~ 3 + . . . .} 
{z -1 + cz~ 2 + f ,2 z~ 3 + . . . .} 
any term in F, say the term A xpyPtft . , would require to be 
multiplied by x~ a ~ 1 , y~P~ l , z~y~\ . . . in order to produce a 
term in x~ 1 y~ 1 z~ l . . . ., and that these multipliers being only 
found associated with the coefficients a a , bP, cv, . . . the term 
so produced would have for its coefficient A a a b&cy .... The 
full coefficient of x~^y~ 1 z~ x .... would thus be Y(a,b,c, . . .). 
He also uses an identity regarding difference-products which it 
may be as well to state separately, viz., that 
!n(rto,C£i, • • •> Mn) . n (a n 
= (-l)»tt+l)n(a 0 , > # On-m-i) .f(a n - m )f\a n -m+ 1 ) /{an ) 
where f{a r ) stands for the product of the n factors got by sub- 
tracting from a r each of the quantities a 0 , oq, . . ., a n except a r . 
