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

COMPOUND DETERMINANTS (PICQUET, 1878)

199

of the elements of this 'mixed nt' compound,' if we may so call it,
the two basic determinants were to change places we should have
another determinant V', which would consequently be equal to
I b -. I 1, la n -1;
so that there would result
vv' = {1, ba n
as an analogue to Cauchy's theorem regarding the product of
complementary compounds. Of course the elements of V and V'
might be so arranged that any element of the one would contain
those columns of I a I and I br, I which are not contained in the
corresponding element of the other, and be thus, in an extended
sense, complementary. Picquet might also have noted that if I b I
were the conjugate of I al, 1, his mixed compound would then be
equal to
Towards the close of the memoir (pp. 238-241) an important related
result is given, namely, that the ratio of any h-line minor of V to
the complementary of the corresponding minor of V is equal to
r l I bll 'h!la (n-1)- - | bill -1 )
Two modes of proof are given, the nature of which will be guessed
when it is recognized that the theorem is a generalization of Franke's
of 1862, into which, as a matter of fact, it degenerates when I b,,
is such that b,.,. = 1, b s = 0. In this again Picquet was forestalled
by Reiss.
The ninth theorem (p. 216) is professedly Sylvester's of March
1851 (Hist., ii. pp. 193-197): the tenth is an alternative mode of
stating the ninth: and the eleventh is essentially the same as one
of D'Ovidio's of 1876 (~ iii. of the 'Nota '). This last, for the sake
of uniformity, we may enunciate in our own way, namely, If Mh be
an h-line minor of I a, 1, M',_-h being its complementary, and there
be formed all the minors of the (n —h-k)t" order such that neither all
their rows nor all their columns belong to M',_ h, the determinant
whose elements are these minors is equal to
(Mh)(n-h-l)k. Ia I('-l)lh+k —(" —l —)k.