1909-10.] Dr Muir on the Theory of Orthogonants. 
279 
where 
q rs = (a rX a r „ . . . a rn $ a ls a 2s . . . a ns ) , 
and where therefore 
I *Zll 9-22 ' * ’ 9nn | = | a u ®22 • * ' a nn | 2 • 
It is then pointed out that the lc*st determinant multiplied by ( — l) n is 
expressible in the form 
(x 2 ) n - Qi(a‘T -1 + Q 2 («T -2 - ; 
that Q x , Q 2 , . . . can be shown to be sums of squares ; that consequently the 
values of ce 2 in the equation 
(x 2 ) n - + Q 2 (a 2 r~ 2 - . . . = 0 
are all positive ; and therefore, finally, that the values of x in the equation 
— X $12 . • • $i n 
$2i $22 X • • • $2 n Q 
a n i 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 SA-^ 2 , the number 
of positive and negative coefficients is invariable.” 
Lame, G. (1852). 
[LEgoxs sue, la Theoeie Mathematique de l’Elasticite des 
Coeps Solides. xvi + 336 pp., Paris.] 
While discussing (§§ 18-22) the axes of the ellipsoid of elasticity Lame 
gives in substance the theorem that if | cq /3 2 y 3 1 be an orthogonant, and the 
ordinary multiplication-theorem produce the identity 
a i P 1 7i 
a / e 
a l a 2 a 3 
Pi Q 3 Q 2 
a 2 @2 1/2 
f b d 
CO 
02. 
(M 
02. 
= 
Q 3 f*2 Ql 
a 3 As 73 
e d c 
7i 72 7s 
Q 2 Qi ^3 
then 
P1 + P2 + P3 — fl + 5 + Cj 
?! 
Q 3 
t 
?2 
Qi 
+ 
?! 
q 2 
<* / 
1 
b d 
+ 
a e 
Q3 
?2 
T 
Qi 
?3 
Q2 
?3 
/ b 
T 
d c 
e c 
* This proof, for the case where n = 3, is given free of determinants by Grnnert in the 
Arcliiv d. Math. u. Phys ., xxix. (1857), pp. 442-446. 
VOL. XXX. 
19 
