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

DETERMINANTS IN GENERAL (SCHLAFLI, 1851)

53

and it has to be shown that
3 2 3
a, (ia 3,aa2 3a2
a2bl 2ala2cb + a6b2 2aa2b2 + a2b1 2b2 1 6.
1 b~b~a, 1 1,2 72 2 = (/C2
albi 2blb2a1 + b1a2 2blb2a2 + b2a01 a2b2
3 86^2 3bib2 b3
b ~ Z3bb22 3blb2 b
To do this Schlafli takes the conjugate of the adjugate of alb2,
namely,
A1 B1
A, B2;
forms with its elements the like determinant
*A 3 3A2B1 3A1B2 B3
A1A2 2AlB1A 2+AB2 '2AIBB2 + BA2 BB2
A1A 2A2B2A1 + A2B1 2A2B2B1 + B2A1 B1B
A3 3A B2 3A2B2 B;
and then in row-by-column fashion multiplies the former fourline determinant by the latter, obtaining for the product
~ 3
alb2 3 '. alb2 3
~~* '* alb2 13
whence the desired result follows.
It is the penultimate step of the demonstration, namely, the
obtaining of the final form of the elements of the product, that
is attended with difficulty: and the difficulty is not lessened to
Schlafli by his taking n and r in all their generality. As soon,
however, as the determinant under investigation is known in
this or any other way to be a power of A, the index of the
power is readily found to be
r P
by noting that the determinant is of the order Cn+,._-,r and each
of its elements of the rth degree.