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

314 HISTORY OF THE THEORY OF DETERMINANTS

where a, = sk = a-+a- +... +a and the case where a, = ak+.a constant.
The next section (~ 3, pp. 7-11) is occupied with the discussion
of the case where
a7, = (x+k-al)(x+k-a2)... (x + k- c ),
and where therefore the constituents of the /ut'" difference-series are
each equal to! The value of the determinant is thus seen to be
(-l)(~-{ ( -1l)!}" or 0,
according as A = or <n-l, that is to say, is in both cases independent of the a's. By giving the a's the values
0, -1, -2..., — +l
in the first of these two results, and performing on the determinant
the operations
coll —col2, col2-col,..,
it is possible to remove the factor (n-l)1)- from both sides, then by
similar means the factor (n-2)"-2, and so on, the final outcome
being
P'(x+n-1, x+n,..., x+3n-4) = (n —l)l(n-2)2... (2)n-2ln-,
a result easily proved otherwise.
The fourth section is devoted to the case where
(I —q)(1 —q,,+l)... (1 —q'+-1)
a, = (1 —iq)(l —q-l).. (tq^ -1)'
Here, on the removal of the requisite factors, the P' form is readily
seen to be
/l-q lq —a+l 1__ a+2n1-+
pI q- 1_- '. ' 1 + ) or, say, F,(a, 7)
and on utilizing the column of l's to reduce the order it is found
that Fn(a, y) and Fni_(a(+l, y+2) are connected by a factor.
Repeated use of this result is then all that is wanted for the purpose
in view; but the connecting factor being
)-. q-(n-1)-2). (qy- qa)-l. A