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

AXISYMMETRIC DETERMINANTS (BOOLE, 1862)

99

Following this lemma and made in part dependent on it comes
the rather notable proposition that if an axisymmetric determinant.
have all its elements of the form
Xa +b +vc.....
and the coefficients X,,, v,... in all the diagonal elements be positive,
and generally be such that all those joined to any one of the variables in
any row be iri order proportional to the coeficients of the same variable
in any other r'ow, then the final development of the determinant will
contain only positive terms. The proof is disappointingly lengthy,
occupying very nearly three pages (pp. 226-238).
The other proposition, which is a deduction from this, is to the effect
that if F be a rational integralfunction of n variables xl,, x, x, xn,
having no powers of them above the first and having all its terms
positive, then the final development of
F x1JF x9F2.... xnF11
xlF1 xlF1 x1x2F12.... xlX,,Fln
x2F2 x2xlFl1 x2F2..... X.XnF,2n
XnFnl XnxlFll xnXFn2.... XnF
where F1, Fs,, stand for EF/axr, D2F/axrax respectively, will contain
nothing but positive terms. For the case of two variables the result is
axy +bx+cy d axy+bx axy+ cy
axy + bx axy + bx axy = abcx2y2 + abdx2y + acdxy2
axy + cy axy axy + cy + bcdxy;
and for the development in the case of three variables we are referred
to a similar memoir * of the year 1857, where we find the expression
(d + h + e +f) (abc + acg +abg-+ bcg)
-+ (a-+b+c-+g) (dhe +-dhf +def +hef)
+ (ac+bg) (df+ dh+ ef +eh)
+ (ag+ bc) (df +-eh +de+fh)
+ (ab +-cg) (de+ dh +fe +fh)
+ 4agfe + 4bcdh,
* BoOLE, G., " On the Application of the Theory of Probabilities to the Question
of the Combination of Testimonies or Judgments," Transac. R. Soc. Edin., xxi.
pp. 597-652 (see p. 648).