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

ORTHIOGONANTS (SYLVESTER, 1852) 311
that Q1, Q2,... can be shown to be sums of squares; that
consequently the values of x2 in the equation
(X 2)f - Q,(X2)n-I +Q2(X2)'2 -... = 0
are all positive; and therefore, finally, that the values of x
in the equation
all - X a12 a a,2,
Ct21 a22-x.... a2, =
an1 a.2... C,, - x
are all real.The remainder of the paper deals with the " Law of Inertia for
Quadratic Forms," this law being "that by whatever linear
substitutions, orthogonal or otherwise, a given polynomial is
reduced to the form 7A1A512, the number of positive and negative
coefficients is invariable."
LAMP, G. (1852).
[LE9ONS SUR LA TH.IORIE MATH19MATIQUE DE L'ELASTICIT~~ DES
CoIRs SOLIDES. xvi+336 pp., Paris.]
While discussing (Q 18-22) the axes of the ellipsoid of
elasticity Lame gives in substance the theorem that if I a, 02 731
be an orthogonant, and the ordina'ry multiplication-theoremt
produee the identity
ai 3, y a f e a, a2 a3 P1 Q3 Q2
a2 0272 f b d 13 I32 /3 3= Q3 P2 Q1
a3/)337Y3 e d c 71 7273 Q2 Q1 P3
then
P,+2+P3 = a+b c,
P1 Q3 + P2 Q1 P1 Q2 -a f + b c + a el
Q3 p+Q1 3 +Q2 P3 f b de e c,
* This proof, for the case where n = 3, is given free of determinants by Grunert
in the Archiv d. Math. it. Phys., xxix. (1857), pp. 442-446.