(r)dr,
where 6 is a proper fraction.
The term involving S plays an important part in the theory
of capillary action; but for our present purpose we may
neglect it in comparison with the other term, if we make the
assumption usual in capillary theory that the smallest value
118 Mr. W. Sutherland on the
which it is permissible to take for L is negligible in com-
parison with the dimensions of experimental bodies.
Noticing that nm is the mass of the whole body, which we
assume to be unity, we get for the potential energy of the n
molecules,
fL
2 77-/91 r2(f>(r)dry
*) a
and for the virial of their forces,
irpi r3f(r)dr.
Now if (as Laplace does in his expression for the attraction
of a fluid on a column) we were to replace a and L in the two
integrals by 0 and ~~r, ^r, ©r vanish except <3>0, tyQ, ®0,
2)(0) reduces to the diagonal row simply, say 2/(0) . Let
c1; c2 be the roots of
, dhv ,T, dw ^ ,.
(39)
then
= 0.
(40)
SV (0) = A ( 1 — cos TTCi) — B sin irc^
= A (1 — cos 7rc2) — B sin irc2
so that the equation for c may be written
® (0), 1 — COS 7TC, Sill 7TC,
®'(0), 1— cos7tc1? sin7rc!,
2)'(0)? 1— COS7TC2, sin7rc2,
In this equation ~~~~0, c^ . . .
remain finite), 2)(c) is an even function of c, and the co-
efficient B vanishes in (38). In this case we have simply
1 — cos ire _S)(0)
l-coswV0o ""^(O)'
exactly as when ~~*i, _], 2, ^-2 • . • vanish,
Vibrations by Forces of Double Frequency. 153
Reverting to (24), we have as the approximate particular
solution, when there is no dissipation,
e(c--2)it ecit e(c + 2)it
w= (C_2)2-0O + W, +Xe + 2f-®0' ' ' (41)
If c be real, the solution may be completed by the addition
of a second, found from (41) by changing the sign of c. Each
of these solutions is affected with an arbitrary constant mul-
tiplier. The realized general solution may be written
_ R cos (c — 2)t + S sin (c-2)t
(c-2f-®3
_i_ R cos ct + S sin ct Rcos'(c + 2)£ + Ssin(c + 2)£ {.^
+ ~ ~@T '+ (c + 2)a-0o ' ' +**>
from which the last term may usually be omitted, in conse-
quence of the relative magnitude of its denominator. In this
solution c is determined by (26).
When c2 is imaginary, we take
4S2 = ©12-((H)0-1)2; (43)
so that
c2 = l + 2is, c=l+is, c— 2=— 1+is.
The particular solution may be written
w=e-st{®ie-it + (l-®0-2is)eit\ ; . . . (44)
or, in virtue of (43),
w = e-8t\(l—®0 + ®1)cost + 2ssmt\; . . (45)
or, again,
^=e-^{N/(e1+i-e0).cos^+v/(e1-i+@0).siiu}. . (46)
The general solution is
w ==]&-**{ (I— ©0 + e1)cosi + 2ssin$}l
+ Ses; {(1— ®0 + ©i) cos t — 2s sin*} J'
R, S being arbitrary multipliers.
One or two particular cases may be noticed. If ®0 = 1,
25 = ®!, and
ic=zWe-st{cost+ sin a )
V (48)
+ SV {cos*- sin*} J
Again, suppose that
012 = (0„-l)2, (49)
so that s vanishes, giving the transition between the real and
154 Lord Rayleigh on the Maintenance of
imaginary values of c. Of the two terms in (46), one or
other preponderates indefinitely in the two alternatives.
Thus, if ®1 = 1— ©0, the solution reduces to cost; but if
©i = — 1 + ©0, it reduces to sin t. The apparent loss of gene-
rality by the merging of the two solutions may be repaired in
the usual way by supposing s infinitely small.
When there are dissipative forces, we are to replace c by
(c—ik), and © by (SQ—k2); but when k is small the latter
substitution may be neglected. Thus, from (26).
c=i+^+i s{{&t-ij*-.e1*). . . . (50)
Interest here attaches principally to the case where the radical
is imaginary ; otherwise the motion necessarily dies down.
If, as before,
4s2 = ©12-(©0-1)2, (51)
c = l + ik + is, c— 2=— 1 + iJc + is, . . (52)
and
or
or
e(c-2)it ecit
W= (c-ik-2f-(&0 + ©7'
w=e-*