180-183] 
Rotating Systems 
199 
Secular Instability. 
182. Let us next suppose that F lt F z ,... do not vanish, but represent 
dissipative forces such as viscosity. The special characteristic of these forces 
is that they always act against the velocity in such a way as to compel the 
system to work against them; in brief, they resist the motion. Thus, what 
ever the algebraic signs of 0 U 0 Z ,... may be in equation (181’1), the signs of 
the terms F 1 0 1 , F z 0 z , ... on the right will always be negative unless they 
are zero. Thus the right-hand member of equation (181*1) will be negative 
except when the system is relatively at rest, so that T R + W— %ao‘ 2 I must 
decrease indefinitely. If W— %co*I is an absolute minimum in the position of 
equilibrium, this condition can only be satisfied by T R being reduced to zero, 
and the system coming to rest in its position of equilibrium. But if IT— ^o> 2 / 
is not an absolute minimum in the configuration of equilibrium, there will be 
a possible motion in which W — ^w 2 / continually decreases while T R remains 
small at first, but may increase beyond limit when W — |g> 2 / is sufficiently 
decreased. The system is now in a restricted sense unstable. 
Instability of the kind just discussed is called “secular instability.” The 
conception of “secular instability” was first introduced by Thomson and 
Tait*. It has reference only to rotating systems or systems in a state of 
steady motion; for systems at rest secular stability becomes identical with 
ordinary stability. 
So long as W — |<w 2 / remains an absolute minimum the system is stable 
both secularly and ordinarily. At the moment at which W — \w 2 I ceases to 
be an absolute minimum, secular instability sets in, although the system may 
remain ordinarily stable. It follows that a system which is ordinarily stable 
may or may not be secularly stable, but a system which is ordinarily unstable 
is necessarily secularly unstable. 
Further, we see that as the physical conditions of a system gradually 
change, secular instability necessarily sets in before ordinary stability. Thus, 
for problems of cosmogony it is secular instability alone which is of interest. 
A system never attains to a configuration in which ordinary instability 
comes into operation, since secular instability must always have previously 
intervened. 
Mass rotating freely in space. 
183. We have so far considered only the case of a mass constrained to 
rotate with a constant angular velocity <w. Schwarzschildf has shewn that 
the conditions of secular stability assume a somewhat different form for a 
mass rotating freely in space. In this case the rate of rotation is not constant, 
* Nat. Phil., 2nd ed., vol. i, p. 391. 
t See Schwarzschild: “Die Poincarö’sche Theorie des Gleichgewichts.” Neue Annalen d. 
Sternwarte München, m. (1897), p. 275, or Inaugural Dissertation, München (1896).