195-197] GENERAL EQUATIONS. 325 



where the coefficients may be treated as constants. The terms of 

 the first degree in V T Q have been omitted, on account of the 

 stationary property. 



In order to simplify the equations as much as possible, we will 

 further suppose that, by a linear transformation, each of these 

 expressions is reduced, as in Art. 165, to a sum of squares; viz. 



22T = a 1 1 2 + a 2 g 2 2 + (3), 



2(F-r o )= c iqi *+c 2 q*+ (4). 



The quantities q lt q 2) ... may be called the principal coordinates 

 of the system, but we must be on our guard against assuming that 

 the same simplicity of properties attaches to them as in the case 

 of no rotation. The coefficients a 1? a 2j ... and c lf c 2 ,... may be 

 called the principal coefficients of inertia and of stability, respec 

 tively. The latter coefficients are the same as if we were to 

 ignore the rotation, and to introduce fictitious centrifugal forces 

 (mn z x, mn z y, 0) acting on each particle in the direction outwards 

 from the axis. 



If we further write, for convenience, /3 rs in place of [r, s], then, 

 in terms of the new coordinates, the equation (7) of the preceding 

 Art. gives, in the case of infinitely small motions, 



If we multiply these equations by q lt q 2 , ... in order, and add, 

 then taking account of the relation 



Prs = Psr (6), 



wefmd j t (^+V-T,) = q i q l + Q4, + (7). 



This might have been obtained without approximation from the 

 exact equations (7) of Art. 195. It may also be deduced directly 

 from first principles. 



197. To investigate the free motions of the system, we put 

 Qi = &amp;gt; Q 2 = 0, ... in (5), and assume, in accordance with the usual 

 method of treating linear equations, 



ft = 41^, q* = AteV,..., (8). 



