1905-6.] Dr Muir on the Theory of Alternants. 
379 
or, say, 
[1,2, . . . ,rc+l]. 
On multiplying each row of J by the product of all the 
denominators occurring in the row there is obtained a determinant 
Y whose r th row consists of elements which are expressible as 
polynomials arranged according to descending powers of u r , the 
index of the highest power of u r being 2?z — 2 in all the places 
except the last where it is 2 n. Y, which is equal to 
J • A;A 2 ... A 2 
12 n 
if we put 
A s = (fl- s + te 1 )(a s + M 2 ) • • • K + u n+1 ) , 
can thus be partitioned into (2ra - l) n (2?2+ 1) determinants, each 
expressible in the form 
a 
U a 1 
U a< * . . 
U an 
u an + 1 
1 
1 
1 
1 
U ai 
u 7 • • 
. u a * 
u“ n + 
2 
u ai 
u a<1 . . 
. u an 
U an + 1 
n+1 
n+1 
n+1 
where a is an integral function of the As- Further, Y in this way 
is seen to be not of higher order with respect to the As than the 
determinant 
u n x 1 
u i 
u n x +1 . . 
mJ"- 2 - 
uf 
'+ 1 
u” 
u T x • ■ 
u 2n-2 
uf 
<;! 
n+1 
J . . 
n+1 
U M - % 
n+1 
U ln 
n+l 
that is to say, its order-number cannot exceed ^n(?>n+ 1) ; and as 
it is exactly divisible by the difference-product of the u J s, which 
is of the order \n(n + 1), it follows that 
Y = A (u lt u 2 , . . ., u n+l ) • Y 1 
where Y x is a function whose order-number is not greater than 
n 2 . Noting now that the other form of the resultant, namely 
[1,2,. . . , n+ 1] , can by addition be transformed into 
U 
A]A 2 . . . A ?l 
where U cannot contain any of the differences of the A s , and in 
