496 MR. 8. S. HOUGH ON THE OSCILLATIONS OF A 
values given in (37). If there be any external attracting system we must add to the 
right-hand members the couples due to this system, estimated as though the system 
were rigid throughout. In dealing with the “free” oscillations, however, we may 
omit these terms. 
Introducing the values of p, q, 7 from (39) and omitting small quantities of the 
second order . 
AG, — (B — C) a6, — of, (A + B— C0) =L, 
BO, + (C — A) a, + of (A + B— C)=M, 
C6; =. 
Hence, on replacing L, M, N by their values (37), putting 6, = @',e™, &, and 
omitting the time factor, we obtain 
6, {AN + (B—C) 0%} + ro’, (A+ B—C) —» {e+ Bel a 0 | 
0°, {BN + (A — C) w?} — cho’, (A + B—C) — ps {ar, s 2m oh \ . (40) 
0,{0N%} —(n—r)B, =0 
from (32) 
B= ae 6, 
Hence the freedom defined by the coordinate @; is neutral, and a disturbance, 
which causes the coordinate 0, to vary, will not give rise to an oscillatory motion. 
From (38) the values of B,/z, B,/7 are given by 
Bs (o,¢ — gy — 208 Bh 
T hr 
2nt B, 
T ws 
° 9 9.9 7 a) 
(p? — ¢?) —NC? W, = 0 | 
Weg (A), 
(pf? — 2?) + (2 = B) OL = oy 
Eliminating 6’,, 6’, B,/7, B,/7 from (40), (41) by means of a determinant, the 
periods of free oscillation are given by 


AN+(B—C—») a%, Ne AL =O), 0, = 
SF No (A-EB =O) BE (AE=C-= 1) aie emo 0 
0, —d2c?, 2p?—c”, — 2 (ppc) =e 
8 (e&—B%), 0, “2 (p—c%), 2p Pc? 
