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

AXISYMMETRIC DETERMINANTS (WILLIAMSON, 1872) 107
By dividing both sides by a, it is seen that in the case of the binary
quadric the conditions are
a, > 0, I ab2 > o.
In the next case it is equally evident that they are a1 > 0 and the
like conditions for the quadric in y, z: and as by the previous case
the latter are
I ab2 1> 0, I ab2 I I alb3 >
I alcs I Iac I1
we obtain in all for the ternary quadric
a, > 0, a |b2 > 0, I acb2c > 0.
Similarly for the quaternary quadric we must have a, a b2,
a1b2c3 i, I alb3d4 I all positive: and so on, the last determinant
in each case being the discriminant of the quadric. It is casually
added that if the 1st, 3rd, 5th,.... of the series be negative and
the others positive, the quadric will be negative for all real values
of the variables.
As might be expected from the connection with Lagrange, it is
also pointed out that, calling these determinants A1, A2, An,..,
we can by repeated applications of the fundamental transformation
change the n-ary quadric into the form
AU2 + U2 ~U+U + 1, UI +... + Ul,
A1,B2 ni A2 A I2
that is to say, into an aggregate of multiples of squares of linear
functions of the variables: for example,
ax2 + by2 + cz2 + 2dyz 2ezx -+2fxy
( f e \2 ab -f ad- abc+ 2def-...
-3f z2.
a +) +a a ab -ff2 z
Finally, as the quadric is invariant to a variety of sets of interchanges, there must be a corresponding variety of sets of conditions:
and, as these latter sets must be all coexistent, there follows an
interesting theorem which we may formulate for ourselves thus:
If the axisymmetric determinant I alb2c3d4 1 and its coaxial minors
I a1b2c3 1, I ab2 [, a1 be positive, then all the other coaxial minors are
positive also.