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

198 HISTORY OF THE THEORY OF DETERMINANTS

the writer's predecessors Reiss and D'Ovidio are lost sight of, it
remains a valuable monograph.
In the first twelve pages there is little that is fresh. We may
note merely that formal proof is given (pp. 205-206) of the proposition that the mtt compound of a,, I is not altered in substance
if to each of its elements be prefixed + or - as determined by the signrule associated with Laplace's expansion-theorem; that (Cor. p. 208)
the mt" compound is equal to the (n-m+l)tl" compound: and that
(Cor. p. 211) to every minor of the order (n —1)1, in the mt` compound
there is an equivalent minor in the complementary compound. In
connection with the second of these we may remark that the lst
and ntll compound are not only equal but are identical.
The first thing noteworthy is a preliminary theorem, which,
although formally proved, rests manifestly on the simple fact that
the mtj7 compound of a minor of I a,, I is a minor of the m"t compound
of I a11,. It is to the effect that any m-line minor of a1,, is
expressible as the (m-1),"t root of a minor of the (m-t)t" compound of
| a,,. This is followed by the related result (p. 213) that if M,, be
an h-line minor of a,,, M, ',_l being its complementary, and there
be formed all the minors of the (h+k)th order which contain neither all
the rows nor all the columns of Ml,, then the determinant whose elements
are these minors is equal to
(Mt ought to be noted, although Picquet views the matter from a
lIt ought to be noted, although Picquet views the matter from a
different standpoint, that we have here again an instance of a minor
of the mntl compound being expressible as a product of a power of the
generating determinant by a power of one of its minors.
What is next reached is the interesting theorem (no. 8), which we
have already spoken of under Reiss as the so-called 'Picquet's
theorem.' As now stated, it is to the effect that if every set of q
colunms of I a,,, be replaced by every set of q columns of I b,,, 1, the
determinant, V say, of the square array of determinants thus obtained
is equal to
I al l("-q. I b", l("-I)q -.
A mode of proof is at once suggested if it be observed that the
first factor of the result is equal to the (n-q)tl compound of I a, I
and the second to the qth compound of i b6l, I. If in the formation