136 
Proceedings of Royal Society of Edinburgh. [sess. 
of forces whose moduli are the n 2 constants which specify the 
original strain. Therefore that vibrating system has n real 
positive squares of periods. 
To pass to the case of systems of unequal masses, we have to 
extend to ^-dimensional space Tait’s process for the determination 
of non-rotated lines. The equality of action and reaction being 
postulated, any two constants, each of which is situated at the 
optical image of the position of the other relatively to the right- 
handedly downward diagonal of the determinant, bear to each 
other the inverse ratio of the corresponding masses. It follows 
at once that Tait’s condition (1), § 5, applies to every cubic minor 
situated on the right-handedly downward diagonal ; and that, 
in every square minor, and its image minor in the diagonal, the 
product of the right-hand diagonal terms of the one into the 
left-hand diagonal terms of the other is equal to the product of 
the left-hand diagonal terms of the one into the right-hand 
diagonal terms of the other. Thus, in the scheme below, 
a 2 c i^3 = a 3 c 2^i> ^4 C 2^3 = ^3 C 4^2> = d^e^f^g^. 
. a 2 
• 
C 1 C 2 
. d 9 
d, d* 
J 4 f 5 
y 4 
We may dispense with the special statement regarding cubic 
minors. If, for brevity, we use the term “image minors” with 
reference to any square minor and the minor situated at the image 
of its position with reference to the right-handedly downward 
diagonal, and if we use the term “cross-product” with reference 
to the product of the right-hand diagonal constituents of one 
minor into the left-hand diagonal constituents of the other, we 
can make the general statement that — 
The roots of an n - ic are real when the two cross-products of 
each pair of image square minors are equal. 
In the case of a diagonal cubic minor, one constant, e.g. c 3 in 
the above scheme, is common to each cross-product, so that we 
