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

BIGRADIENTS (ZEIPEL, 1859)

371

Brioschi's of 1854 for comparison. In ~2 (pp. 12-15) he works
out for himself a third form, which, in the case of n = 4, gives
R1= a b, R= a0. bo, R3...
1 b b0.. a a b1 b.
a2 b2 b -1. - a6 b2 b2 bo
a% b8 b2 x -1 a %b a b2 b
3 &3 2 X -1 3 2 3 2 1
a4 b *b. a4 a3 b3 b2 -1. t4. b3 x
-expressions which take a more familiar appearance when
developed and arranged according to descending powers of x.
He then passes to the case where f(x)=F'(x), his second
chapter being occupied with a short but very complete historical
sketch of Sturm's functions.
In the third chapter the allied subject of the common roots
of two equations is taken up, and to illustrate the advantage
of his own procedure over Lagrange's and Brioschi's he takes
the equations
CoX3 + alx2 + a2 + 3 = 0,
box3 + b1x2 + b2x + b3 = 0,
and (1) supposing them to have one common root gives
0. a b
al.a bo bo a, a b b
1 o0 b1 bo0 + ao b bo O
X = 0
c2 a b2 b1 a2 a1 b2 bl
a3 2 b3 b2. a3.b3
as the equation for determining it; (2) gives the vanishing of
the two determinants in this equation as the conditions requisite
for the existence of two common roots; and (3) gives
a b o bo + ao bo 0
al b 2 as b as b
1 1a2 b2 a3 b3
as the equation for determining the said two. In this connection
Sylvester's paper of 1840 might well have been referred to
(see History, i. pp. 236-238).
The fourth chapter is headed "Relation between any three