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

BIGRADIENTS (MALET, 1878)

359

Of these results it is the last that our subject requires us to be
most interested in. Malet himself notes it as involving a simpler
determinant than that usually given. It is necessary to point
out, however, that the said determinant is easily derivable from
Sylvester's bigradient
1 -a1 a2 -a3
1 — a a2 -a3
1. 1 - aI a2 - a. 1 -b1 b2 -b
1 -b1 b2 -b3
1 -b1 b2 -b3.
For, performing on the latter the operations
rows-row2, row6- row -(al -bl) row2,
we obtain
1 - al a2 - C3
1 -bi b2 - b,
al — bl b2 - a2 a3 - b3
b2-a2+ al2-albl a3-b3 —a1a2a2+ab a3a-a3b1.,
and the further operation
row4- a, row3 +a2 row2- b2 rowl
gives us substantially Malet's form.
GUNTHER, S. (1879): PAIGE, C. LE (1880).
[Eine Relation zwischen Potenzen und Determinanten. Zeitschrift
f. Math. u. Phys., xxiv. pp. 244-248.]
[Question 566. Nouv. Corresp. Math., vi. p. 333: Solution by
C. Le Paige, pp. 382-383.]
The subject here, proposed by H. Brocard, is simply the evaluation
of the bigradient which is the discriminant of (xm+2-l)/(x-1),
the result being (m + 2)m.
For example, when m is 2,
1 1 1 1..1111.I I I 1
123.. = 42.
1 2 3. 123