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

BIGRADIENTS (ISE, 1873: JANNI, 1874)

353

he, in effect, multiplies the columns in reverse order by yO, y1, y2 y3, y4
respectively, and then divides the rows in order by y4, y3, y2, y, yO
respectively, thus obtaining
a aly-1 a2Y-2 a3y-3
a aly-1 a2y-2 a3y-3
boy2 bly b2
I. boy2 by b2
In this equivalent form the elements of the first two rows are all now
of the degree 0 in y, and those of the last three rows are all of the
degree 2, whence comes at once the desired result.
It should be noted that the procedure shows each term to be of the
(mn)th degree in y; in other words, that the eliminant is homogeneous. Also, dispensing in the end with y, we may deduce the
isobarism of the eliminant, its weight being inn.
BJORLING, C. F. E. (1873): ZEUTHEN, H. G. (1874):
GARBIERI, G. (1874): MADSEN, V. H. 0. (1875).
[Sur les relations qui doivent exister entre les coefficients d'un
polynome F(x) pour qu'il contienne un facteur de la forme
x' —a. Archiv d. Math. u. Phys., lv. pp. 429-440.]
[En Bemaerking om Beviserne for Hovedscetningen om Elimination
mellem to algebraiske Ligninger. Tidsskrift for Math. (3), iv.
pp. 165-171.]
[DETERMINANTI, con.... xiii-267 pp. Bologna.]
[En Bemeerking om Sylvesters dialytiske Eliminationsmethode.
Tidsskriftfor Math. (3), v. pp. 144-145.]
All these deal with the subject of common roots, Bj6rling incidentally (~~ 3-6, pp. 431-437) and the others of set purpose.
Zeuthen repeats Salmon's mode of 1859 (Hist., ii. pp. 373-374) of
using Euler's treatment of two integral equations in x which have
more than one common root: he is, however, more detailed, and
takes the number of roots to be p.
Garbieri also repeats Salmon, taking indeed the same example,
but he is careful to add a proof (pp. 120-121), that of the six conditions there arrived at only two are independent.
M.D. III. Z