1906-7.] Dr W. Peddie on Vibrating Systems. 
83 
Remembering now the condition k lql a rq = k lr a qlr , and noting that the quantity 
in the bracket in the last two double sums reverses its sign on interchange 
of q' and r, we see that the terms in these sums cancel in pairs, so that 
we finally obtain 
m m 
k lr {X rq X r ? -f-i)(l + k\q'X q i q \ q r t q+ifttyr — 0 . 
2 2 
Hence we have one or other (or both) of the conditions 
m m 
(1 + k lq r\q'q\ q '' ff+1 ) = 0 ) ^lr(Vg — V, q+l) a ir ~ 0 j 
2 2 
that is, one or other (or both) of the conditions 
(1 ki q’Xq’gXq,' 3+1 ) = 0 ; n q — n qJr x =0 . . . (2) 
2 
Each of the m— 1 possible values of q contributes a similar pair of 
alternative or simultaneous conditions. If, in any case, two frequencies 
specified by n q and n q+1 are unequal, the left-hand alternative must be 
satisfied.* 
It is necessary now to investigate the bearing of these conditions on the 
question of “ normality ” above alluded to. 
5. We have 
. m mm, \ 
2 < 4 2 \ =2 a>2 pr n r 2 -^r 2 + 2/S ) a ' pr a ' ps n r n s^r-^s C0S (M + a r) C0S n £ + OL s ) > 
where { } indicates an average taken over a time which is long in 
comparison with the longest of the fundamental periods, which we 
presume to be distinct. It is also implied that the average time between 
successive collisions is similarly large. In accordance with our postulate, 
every term in the double sum vanishes. If E be the total kinetic energy 
in the system in the interval between two successive collisions, its value is 
given by 
m , v mm 
M p a' 2 pr n^A^ 
2 2 1 M *4 2 \ = 22 
i 1 ' ii 
22 Mp2, a ' pr a ' ps n r v s-^r-^-s { C0S ( n r^ + a r) C0S ( n s^ + a s) {“ • 
111 ‘ ' 
The triple sum would vanish at every instant if we had 
m 
0 • 
(3) 
* Similar formulae hold in the m - 2 cases in which we take q + 2 instead of q, and so on. 
Altogether there are m(m - 1) equations of type (2). 
