(20) [rho] = M/2(a + l), the mass per unit length of the chain,
(21) [kappa] = K/2(a + l), the gyrostatic angular momentum per unit
length,
(22) [alpha] = A/2(a + l), the transverse moment of inertia per unit
length,
(23) 1/2c = (a + l)n/U,
equation (19) can be written
(24) {sin (a + l)n/U}^2
[rho] Ta + [kappa]nl - [alpha]n^2l
= (a + l)^2n^2 ----- -------------------------------------------------------------,
T Ta + [kappa]n(a + l) - [alpha]n^2(a + l) + [rho]n^2a^2(a + l)
/ (a + l)n \^2
(25) ( --------------- )
\ sin (a + l)n/U /
T T + ([kappa]n - [alpha]n^2) (1 + l/a) + [rho]n^2a(a + l)
= ----- . --------------------------------------------------------.
[rho] T + ([kappa]n - an^2)l/a
In a continuous chain of such gyrostatic links, with a and l
infinitesimal,
T / [kappa]n - [alpha]n^2 \
(26) U^2 = ----- ( 1 + ------------------------------- )
[rho] \ T + ([kappa]n - [alpha]n^2 l/a) /
for the vibration of helical nature like circular polarization.
Changing the sign of n for circular polarization in the opposite
direction
T / [kappa]n + [alpha]n^2 \
(27) U'^2 = ----- ( 1 - ------------------------------- ).
[rho] \ T - ([kappa]n + [alpha]n^2 l/a) /
In this way a mechanical model is obtained of the action of a
magnetized medium on polarized light, [kappa] representing the
equivalent of the magnetic field, while [alpha] may be ignored as
insensible (J. Larmor, _Proc. Lond. Math. Soc._, 1890; _Aether and
Matter_, Appendix E).
We notice that U^2 in (26) can be positive, and the gyrostatic chain
stable, even when T is negative, and the chain is supporting a thrust,
provided [kappa]n is large enough, and the thrust does not exceed
(28) ([kappa]n - an^2)(1 + l/a);
while U'^2 in (27) will not be positive and the straight chain will be
unstable unless the tension exceeds
(29) ([kappa]n + [alpha]n^2)(1 + l/a).
15. _Gyrostat suspended by a Thread._--In the discussion of the small
vibration of a single gyrostat fly-wheel about the vertical position
when suspended by a single thread of length 2l = b, the suffix k can
be omitted in the preceding equations of S 14, and we can write
(1) A :[~omega] - K .[~omega]i + Ta[~omega] - Ta[sigma] = 0,
(2) M :w + T[sigma] = 0, with T = gM,
(3) w - a[~omega] - b[sigma] = 0.