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

BIGRADIENTS (MANSION, 1878)

357

we have
(mn-5)ao + (rnn4)al + (mnn3)a2 + (nn2)a3 + (mnl)a4 + (mn0)a5 = 0
(mn6)a, + (mn5)a1 + (mn4)a2 + (mn3)a3 + (mn2)a4 + (mnl)a, = 0
(mit4)bo + (inn3)b1 + (mn2)b2 + (mnl)b, + (mn0)b4 = 0
(mn5)bo + (mn4)bl + (m~n3)b2 + (mn2)b3 + (mnl)b4 = 0
(mn6)bo + (mn5)b1 + (m74)b2 + (mn3)b, + (mn2)b4 = 0.
Here, however, by taking any two of the numbers 0, 1, 2, 3, 4, 5, 6
as values for m, n, two of the alternants will disappear, and we shall
be able to eliminate the five others, the final and complete result
thus being in Cayley's notation,
ao a, a2 a. a4 a.
ao a, a2 a. a4 a.
bo b, b2 b3 b 0=o.
bo b, b2 b3 b4
bo b, b2 b3 b4
For example, putting m, n equal to 0, 1, then equal to 0, 2, and finally
equal to 1, 2, we should have the particular three results, which, in
accordance with our usage under Trudi, we might write
1a ~..., a5)2 0.
(b0,... I, 4)
All this, however, is considerably modified from Mansion's
exposition.
MALET, J. C. (1878).
[On a problem in algebra. Annai di Mat. (2), ix. pp. 306-313.]
Malet first ascertains the conditions to be satisfied by the coefficients in order that two equations of the jtih degree in x shall be
such that to every root r in the one there is a root c/r in the other.
He then proceeds to show that the equation of the (mn)th degree,
whose every root is the product of a root of the equation
xm -aixm-1 +a2X2Xm-2.. = 0,
and a root of the equation
Xn - blxn-1 + b2X -2- = 0,