197-198] FREE AND FORCED OSCILLATIONS. 329 



harmonic motion of period 2-Tr/cr ; the directions 



f/P=V<2=?/-R> and f/P' = ,/Q'=^' (17), 



being those of two semi-conjugate diameters of the elliptic orbit, 



of lengths (P 2 + Q 2 + R 2 ) h . K, and (P' 2 + Q' 2 + R'rf. K, respectively. 

 The positions and forms and relative dimensions of the elliptic 

 orbits, as well as the relative phases of the particles in them, 

 are in each natural mode determinate, the absolute dimensions 

 and epochs being alone arbitrary. 



198. The symbolical expressions for the forced oscillations 

 due to a periodic disturbing force can easily be written down. If 

 we assume that Q lf Q 2 , ... all vary as e ifft , where a- is prescribed, 

 the equations (5) give, omitting the time-factors, 



.(18). 



The most important point of contrast with the theory of the 

 ' normal modes ' in the case of no rotation is that the displacement 

 of any one type is no longer affected solely by the disturbing force 

 of that type. As a consequence, the motions of the individual 

 particles are, as is easily seen from (15), now in general elliptic- 

 harmonic. 



As in Art. 165, the displacement becomes very great when 

 A (ia) is very small, i. e. whenever the ' speed ' er of the disturbing 

 force approximates to that of one of the natural modes of free 

 oscillation. 



When the period of the disturbing forces is infinitely long, the 

 displacements tend to the ' equilibrium-values ' 



2i = Qi/Ci, &=&/<*,..., (19), 



as is found by putting <r=0 in (18), or more simply from the 

 fundamental equations (5). This conclusion must be modified, 

 however, when any one or more of the coefficients of stability 

 Cj, C 2 , ... is zero. If, for example, Cj=0, the first row and 

 column of the determinant A (X) are both divisible by X, so 



