25-27] Rotating Systems 29 



W i&> 2 / and of T R for any small displacement may now be expressed in the 

 forms 



2 (W - i&) 2 /) = 6 11 6> 1 2 4- 26 18 ft ft, + ... 



the condition that equations (32) shall be satisfied in the configuration of 

 equilibrium requiring the omission of terms of first degree in ft, 2 , .... By 

 a linear transformation, T R and W ^ 2 / may be simultaneously reduced 

 further to a sum of squares, so that we may assume the still simpler forms 



2T E = a 1 1 2 4-a 2 2 2 + ........................ (35), 



>2(W - X 7 ) = Mi 2 + M 2 2 + ........................ (36). 



The equations of motion (32) now reduce to 



ft* Mi + a>(&808 + /Ms +...) = F, .................. (37), 



0, + M.+ fi>(&i ft + ft,+ ...) = ^ etc (38). 



Had the system been at rest, these equations would have reduced to 



a, 8 d s +b s s = F s , etc. 



and all the properties of " principal coordinates " would have been immediately 

 deducible. But a glance at equations (37) and (38) will shew that these 

 properties no longer persist when the system is in rotation. A disturbance 

 in which 0j exists alone at first will soon set up oscillations in which 2 , 3 ... 

 have finite values, and the coordinates ft, 2 , ... no longer correspond to 

 independent vibrations. 



Since equations (37) and (38) are linear with constant coefficients, it is 

 clear that there will be a system of separate free vibrations. These may be 

 found by putting F 1 = F 2 ... = 0, and assuming ft, 2 , ... each proportional 

 to the same time-factor e Kt . The equations reduce to 



2 + b,) O l + &>X/3 12 2 -I- a>\ 13 ft + . . . = 0. 



+ *&2 ft, + . . + KX 2 + b 8 ) e s + . . . = 0, etc (39). 



Eliminating the 0's, we find as an equation for X, 



0. 



Since /8 rs = /3 sr , it appears that this equation is unchanged when the 

 sign of X is changed. Thus the equation is an equation in X 2 , just as when 

 the system is at rest. But the roots in X 2 are no longer all real as they are 



