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

370 HISTORY OF THE THEORY OF DETERMINANTS
But the determinant on the right is resolvable into
1 1 a +o a3 a1, + a2112 + a013 ao + l1$2 + a2O2 + a3O23
1 /32 a&31 + oCt12 + a23a1 4 +OO 33 S2 + a,1/32+ a2323+ a,3f24
that is, into
1 1 'k(/ 9i) (132)
and therefore into
1 12
313 132
We thus have
S = b_,(131)>/02)'
and, consequently, if a,, a2, a3 be the roots of the first given
equation,
S = b23 a32(f3 - a,1)(31 - a2) (03 - a.)
(/2 - a1)(12 - a2)(02 - a3),
which is what was to be shown, the cofactor of b 3a32 being
Euler's product of differences.
ZEIPEL, V. v. (1859, April).
[Understikningar i h~igre algebran, jemte nagra deraf beroende
theoremer i determinant-theorien. K. Svenska Vet.-A kad.
Ha'nd[. (Stockholm), iii. No. 4, 32 pp.]
This simply written and fully detailed memoir is an extension
of the author's paper of the preceding year. It consists of five
chapters, of which the fifth contains the freshest matter.
The first, and much the longest (pp. 3-16), is occupied in ~
(pp. 3-11) with the application of the Sturmian division-process to
a xm + a xm'l + a xm-2 +... + a,,, or F(X),
b0Xm-l + bpe'n-2 +... + or f(),
with the object of obtaining Cayley's expressions of August 1848
for the remainders (PR). In this he is successful, and he adds