197 
178-iso] Rotating Systems 
These equations only differ from the simpler ones for a system at rest in 
that W has become replaced by W- |o> 2 /. The configurations of relative 
equilibrium may accordingly be found just as though the system were at rest 
under a potential W — £&> 2 /, and these configurations will fall into linear series 
as before. 
Small Oscillations. 
180. To discuss the small oscillations of such a system, we have to return 
to the equations of motion (177-2). Suppose we are considering the oscilla 
tions of a configuration which is one of equilibrium under no applied forces, say 
6 1 = ©x, etc. 
Let the co-ordinates be replaced by Q x — etc., so that the new values 
of 0 U 6 2 , ... all vanish in the configuration of equilibrium. The values of 
W— I» 2 / and of T R for any small displacement may now be expressed in the 
forms . . 
2T B = o 11 0 a 2 + 2 o a 0 1 0 2 + (1801), 
2 ( W- £a> 2 /) = Mi 2 + 2MA + (180-2), 
the preliminary condition that equations (179'2) shall be satisfied in the 
configuration of equilibrium requiring the omission of terms of first degree in 
0 a , 0 2 ,.... By a linear transformation, T R and W — ^eo 2 / may be simultaneously 
reduced to a sum of squares, so that we may legitimately assume the still 
simpler forms 
2T B = M 1 2 +a 2 0 2 2 + (180-3), 
2 ( W- ¿a, 2 /) = Mi 2 + M 2 2 + (180-4), 
With these values for T R and W —|&> 2 /, the equation of motion (178*2) 
assumes the form 
a s 6 8 + b s 0 s to (fisidi 4* $* 2^2 + •••) = etc (180*5). 
Had the system been at rest, these equations would have reduced to 
a s 6 s + b 8 0 s = Fg, etc (180-6), 
a system in which each equation depends on only one co-ordinate. From the 
form of these equations it follows that the different co-ordinates 0 1} 0 ^,... 
execute entirely independent vibrations; the changes in any one co-ordinate 
6 X does not influence, and is not influenced by, whatever changes may ,be 
occurring in the other co-ordinates 0 2 , These are of course the well- 
known properties of “principal co-ordinates.” 
But a glance at equations (180"5) and (180*6) will shew that these 
properties no longer persist when the system is in rotation. A disturbance 
in which 6 X at first exists alone will soon set up oscillations in which 0 2 , 0 3 ,... 
have finite values, and the co-ordinates 0 1} 6 2 ,... no longer correspond to 
independent vibrations. 
Since these equations are linear with constant coefficients, it is clear that 
there will be systems of separate free vibrations. These may be found by