187 
1906-7.] Dr W. Peddie on Vibrating Systems. 
the roots being 1 and 4 (triple) as above. The values of the quantities a' rs 
are as follows : — 
AV 2 n = 0, AV 2 12 =18 2 , AV 2 13 = 9 2 , aV 2 14 = 9 2 ; 
A 2 a' 2 21 = 0 , A 2 a' 2 22 = 6 2 , A V 2 23 = 3 2 , A V 2 24 = 3 2 ; 
AV 2 31 = 0, AV 2 32 = 0 , AV 2 33 = 0 , AV 2 34 = 0 ; 
A 2 a' 2 41 = 0 , A V 2 42 = 1. 2 2 , A 2 a' 2 43 = 6 2 , A V 2 44 = 6 2 . 
Hence, remembering the values of the ks in terms of the masses which 
give M 2 = 3M 4 /2 , M s = M x , M 4 = 3M 4 , we get 
A 2 (M 1 a' 2 n - M 2 a 2 21 ) = 0 ; A 2 (M 2 a' 2 21 - M 3 a' 2 31 ) = 0 
A 2 (M 1 a' 2 12 - M 2 « 2 ' 22 ) = A 2 M 1 (18 2 - f 6 2 ) ; A 2 (M 2 a' 2 22 - M 3 a' 2 32 ) = A 2 M 1 (|6 2 ) 
A 2 (M 1 a' 2 13 - M 2 a' 2 23 ) = A 2 M X ( 9 2 - 13 2 ) ; A 2 (M 2 a' 2 23 - M 3 a 2 33 ) = A 2 M 4 (f 3 2 ) 
A 2 (M 1 a 2 14 - M 2 a’ 2 24 ) = A 2 M X ( 9 2 - 13 2 ) ; A 2 (M 2 a 2 24 - M 3 a' 2 34 ) = A^f 3 2 ) 
A 2 (M 1 a' 2 11 - M 4 a' 2 41 ) = 0 , 
A 2 (M 1 a' 2 ]( , - M 4 a' 2 42 ) = A 2 M 1 (18 2 - 3(12)2) ? 
A 2 (M 1 a' 2 13 - M 4 a' 2 43 ) = A 2 M,( 9 2 - 3( 6) 2 ), 
A 2 (M 1 a' 2 14 -M 4 a' 2 44 ) = A 2 M 1 ( 9 2 -3( 6) 2 ), 
which show that the average energies of the masses are never equal, the 
energies of M 4 , M x , M 2 , M 3 being in descending order of magnitude. 
By change of the value of <x u a single infinity of such examples] is 
readily specified. 
10. The general values of a ' n , a 21 , a ' zl , are given by 
^22 ^32 * * 
■ • • Kn2 
\ 
v 32 ‘ " ‘ 
II 
'e 
<1 
; a «' 2 i = 
Vm 
Xjnm 
1 
Aa 31 — 
A 42 ..... A m2 1 A 22 
^oim 1 ^2 m 
and the expressions for other as can be written down by symmetry. 
Hence, selecting the m — 1 equations involving X 21 , . . . A ml out of the 
m(m— 1) equations (2) when all periods are distinct, we find 
