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

302 HISTORY OF THE THEORY OF DETERMINANTS

where rrp means the product of the ath and /th rows of
a h g
h b f
g f C.
By performing on the 3-by-9 array the operation
row3 - (a +b+c)row2 + (ab+bc+ca-f2 -g2-h2) row1
there is obtained... 1...
a h g h bf g f c
A H G H BF GF C
which is readily shown to be equal to Kummer's sum of squares.
It is a little curious that Borchardt nowhere draws attention
to the fact that the determinant of the coefficients in the righthand members of his mth set of equations is the mth power of the
determinant of the corresponding coefficients of the original set.
JACOBI, C. G. J. (1845, August).
[Ueber ein leichtes Verfahren die in der Theorie der Sacularst6rungen vorkommenden Gleichungen numerisch aufzulosen.
Crelle's Journ., xxx. pp. 51-94; or Gesamqnelte Werke, i.
pp. 227-270; or Nouv. Annales de Math., x. pp. 258-265.]
This long memoir being intended for astronomical mathematicians and computers, there is little of it that concerns us except
two of the introductory sections (~~ 2, 3, pp. 52-56); and even
these need not detain us, as they are in effect but a wellconstructed abstract of Cauchy's paper of 1829, the starting-point
being the set of n +1 equations
(al -0)x1 + a12x2 +.. + alx,, = O
a211 + (a/22 - 0)x2 + + a2 + = ~
aX1 + acn2x2+. * + (ann - )x = 0
1 + 2 x+.. + X2 =