The theory of determinants in the historical order of development, by Sir Thomas Muir.

178 HISTORY OF THE THEORY OF DETERMINANTS

or, say,
[1, 25...,1+]
On multiplying each row of J by the product of all the
denominators occurring in the row there is obtained a determinant
V whose rvt row consists of elements which are expressible as
polynomials arranged according to descending powers of 'a., the
index of the highest power of u, being 2n - 2 in all the places
except the last where it is 2'n. V, which is equal to
2~2 2 22
J.. A
if we put
As = (a,+'ab1)(8~'a2)... +,tb,,,,+
can thus be partitioned into (2n - 1)"(2n + 1) determinants, each
expressible in the form
al a2 aU ana+1
1 I 1 1
a1 a2 alE ant+i
2 2 2 2
al a2 tball Cn +
n+i Un+1. 1 'j 7+
where a is an integral function of the a's. Further, V in this
way is seen to be not of higher order with respect to the 'a's than
the determinant
n2~-i 1z z1+1 2n.-3 in-i2 2
'a U'a 'a,... a 'U, 'a
n-i n n-I-i 2n - 3 U n -2 2zz
'a2 U'2 Ua2. 2 '2
Fn-i 11 n-I-1 2in-i3 2in-2 2zz
'an'+i 'ani 'an+i. n+i 1 U+i Uni +1
that is to say, its order-number cannot exceed c n(3n + 1); and as,
it is exactly divisible by the difference-product of the 'a's, which
is of the order I n(ii I+ ), it follows that
V = A('al, 'a2),..,U+l V
where V1. is a function whose order-number is not greater than
'a2. Noting now that the other formi of the resultant, namely
[1, 2,..., +1, can by addition be transformned into
U
A1A2... AA