242 Prof. J. J. Thomson on the 
To solve equations A and B we notice that if w be any 
root of the equation x”=1, 2. e. if w be one of the nth roots 
of unity, equations A will be satisfied by 
P2=©P1, Ps=OP2, Py=OPs... G2=Oh1, P3=Oh2, Ps=OO; .. 
provided ° 
pi(A—mq’+@A,+o°A,+.. w”1A,_1) 
+ $,a(—2umag+oB,+o°B,+o"—'B,_1:)=0; (1) 
while equations B will be satisfied by the same values 
provided 
pi(2tmeq — ©B, —w*B,—"—!B,_1) 
+ d,a(C — mg? —w0,—@?0,—w"—'C,_1) =0. . (2) 
Hence, if both sets of equations are satisfied by these values; 
we have, eliminating p; and ¢, from (1) and (2), 
((A— mq?) +@A,+?A,4+.. 0" 'A,_3) 
(C—mg?— aC, —w2C, — oT Ona) 
— =—(—2uneg + oB, +B, +..@"-1B,)?, . (1) 
a biquadratic equation to determine g the frequency of the 
oscillations of the system. Now a is of the form 
2kar : Zim 
COS 7 te SL 
n n 
where & is an integer between 0 and n—1. Substituting 
this value for w, we find 
9 ‘ 
wA, + w’As + oA, aa ism  ( : ob u ) 


T nh 
. sin sin? — 
| Akar 1 ihe ee 1 1 
aba n rekon ak ear oe Bi. 
Sig == Sinai oa el 
We shall denote this by Ly; it will be noticed re Ly con- 
tains no imaginary terms. We find also that 


e? 2khar cos Tv. tae 
@ C, ae w’C.+ w°Cs ae w?=1C, 2) = ( COs (cot rs + 3 tan =) 

4q? p 
a "-- sin? 
cos a 
i Aor n 
+ COs —- (cot = ot — ae sta in) 
sm 
We shall denote this by Nz. ce ) 
