137 
1905 - 6 .] Dr W. Peddie an Vibrating Systems. 
get Tait’s condition. A diagonal square minor, e.g. a l ,a 2 ,b 1 ,b 2 , is 
its own image, and the cross-products are identical. 
10. The ..(n - l)(n — 2)/2 square minors which are not self 
images give (n - l)(n - 2)/2 relations amongst the n 2 constants 
and leave a (n 2 + 3n-2)/2 fold infinity of examples subject to 
restriction by the inequalities and the conditions that the periods 
shall be real. The n - 1 square minors which are their own 
images give, by the terms not lying on the main diagonal, the 
n — 1 ratios of the masses in each example. 
If we postulate observance of the Boltzmann-Maxwell condition, 
and so change the inequalities into equalities, the number of 
possible infinities of examples is much lessened. 
11. In the preceding discussion, no conditions have been 
postulated for the purpose of ensuring that the centre of inertia 
of the n masses shall coincide with the origin. We shall see that, 
except when the periods are all coincident, the centre of inertia 
cannot lie at the origin. Hence we must presume the existence, 
at the origin, of a mass which is very large in comparison with 
the sum of the n masses. In any question regarding the total 
partitioning of energy in a system so constituted, the motion of 
this central mass must be considered. But the question of the 
partitioning of energy amongst the n satellites is not affected by 
the presence of the central mass. 
To see that we cannot dispense with this large mass at the 
origin if all the periods are not to be identical, we have to 
consider the various equations of the type 
£*A = [l,p]A x sin {n Y t + cq) + . . . + [n,p] A n sin (n n t + a n ), 
where 
A = 
1 X 
and [r, s] is the minor, of order n- 1, of the rth term in the 
sth column of A. Taking now the centre of inertia condition 
n 
2 m p £ p = 0, we get the necessary equations 
