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

AXISYMMETRIC DETERMINANTS (BRIOSCHI, 1854) 97

determinants which we may formulate for ourselves thus: If the
square of any determinant formed by row-multiplication be D, and
the square formed by column-multiplication be D', then the sum of the
m-line coaxial minors in D is equal to the corresponding sum in D'.
D'ARREST, H. (1857).
[Beobachtungen des Cometen iii. 1857. Astron. Nachrichten, xlvii.
col. 17-19.]
From consideration of three special cases d'Arrest ventures on
the generalisation that if in the application of the Method of Least
Squares the so-called 'normal' equations be
(aa)x+(ab)y-+(ac)z+.. = 0
(ab)x+(bb)y+(bc)z+... 0
(ac)x+(bc)y+(cc)z+... =0
then the weights of x, y, z,... are
A A A
[aa]' [bb]' [cc]' *
where A is the determinant of the set of equations and [aa] is the
cofactor of (aa) in A. In the same serial in 1866 (vol. Ixvii. pp.
174-175) a proof of the proposition is given but unaccompanied
by any author's name.
It is perhaps worth adding that the array of coefficients in the
'normal' set of equations is got by multiplying the array of the
original set columnwise by itself.
FERRERS, N. M. (1861).
[AN ELEMENTARY TREATISE ON TRILINEAR CO-ORDINATES (chap. iii.
pp. 58-71). xii+154 pp. London.}
One of the results, unfortunately inaccurate, given at the close
of the short chapter on determinants is. 1 1 1....
1. a+b a+c....
1 b+a b. b+c.b.... b..- 1 rb+ * * *
1 c+a c+b....
M.D. III. G