1905 - 6 .] Dr W. Peddie on Vibrating Systems. 
135 
The values of the As being distinct, the values of the /x s are in 
general distinct. Also the periods, as given by — n 2 — a Y + b 2 \ + 
c 3 fx, are in general distinct, and can be made real by suitable choice 
of a r Reserving a 1 for this purpose, and reserving b ± to secure 
the first inequality, the remaining inequalities can be secured by 
reservation of d 2 and c v which do not enter into the expressions 
for the g s. Similarly, distinctness of periods can be provided for 
by reservation of b 2 and c 3 . The values of d 3 and a 2 are available 
to give each of /x 2 and fx 1 an infinity of values. 
But the reservations leave at disposal an infinity of values of 
each of a v b v c v b 2 , d 2 , and c 3 . On the other hand, the assumed 
values of the As impose three relations amongst the constants. 
There is therefore a five-fold infinity of solutions possible, in eacli 
of which equi-partitioning of energy between any pair of masses is 
impossible. The values of the As being at disposal, there is really 
an eight-fold infinity possible in the general case of three 
masses. 
This may be regarded as the tri-dimensional case of a single 
mass under the action of forces which are non-isotropic with 
reference to the co-ordinates. 
If the inequalities become equalities, choice of one A fixes in 
general the remaining As and the /ms and the values of b v d 2 , c v 
The fixing of the /xs gives three relations amongst constants, and 
the cubic for A gives three more. Thus, in general, there is only a 
single infinity of cases in which there is complete equi-partition of 
energy amongst three co-ordinates. 
9. The extension to n dimensions of the usual process for 
finding the necessary relations which subsist amongst the nine 
constants of a homogeneous tri-dimensional strain, in order that 
the strain shall be pure, leads to the result that the determinant 
of the nth. order, whose roots determine the directions of non- 
rotated lines, must be axisymmetrical. The roots are, in this 
case, necessarily real. 
If we now apply the process of “ flyping ” to the medium thus 
strained, we get a homogeneous, pure, w-dimensional strain, the 
roots of whose determinant for non-rotated lines are proportional 
to the squares of the fundamental frequencies of vibration of a 
system of n equal masses, which are acted upon by linear systems 
