138 Lord Rayleigh on the 



equations may be reduced to the form 



a i'qi + c l q 1 +/3i2^2 + /3 13 t} 3 4-. . . = Qi, 1 



«2 ^2 + ^2 ^2 + ^21^1 + /3 2 3?3+- • - = Q 2 , [ 



(1) 



in which 



&* = -&, (2) 



From these we may fall back upon the case of small oscil- 

 lations about stable equilibrium by omitting the terms in ft ; 

 but in general tidal theory these terms are to be retained. 

 If the oscillations are free, the quantities Q, representing 

 impressed forces, are to be omitted. 



If the coefficients ft are small, an approximate theory of 

 the free vibrations may be developed on the lines of ' Theory 

 of Sound/ § 102, where there are supposed to be small dissi- 

 pative (but no rotatory) terms. For example, the frequencies 

 are unaltered if we neglect the squares of the ft's. Further, 

 the next approximation shows that the frequency of the 

 slowest vibration is diminished by the operation of the ft's ; 

 or more generally that the effect of the ft } s is to cause the 

 values of the various frequencies to repel one another. 



To investigate forced vibrations of given period we are to 

 assume that all the variables are proportional to e i<7 \ where a 

 is real. If the period is very long, a is correspondingly 

 small, and the terms in q and q diminish generally in impor- 

 tance relatively to the terms in q. In the limit the latter 

 terms alone survive, and we get 



?l = Ql/Cl, q2 = Q,2/C2, &c (3) 



which are the " equilibrium values. " But, as Prof. Lamb 

 has shown, exceptions may arise when one or more of the 

 c's vanish. This state of things implies the possibility of 

 steady motions of disturbance in the absence of impressed 

 forces. For example, if c 2 — 0, we have as a solution, 

 q 2 = constant, with 



qi = — /3i2 sbM, q$ = — A$2 £2/^, &c. 



In illustration Prof. Lamb considers the case of two 

 degrees of freedom, for which the general equations are 



(h qi + ^1 ?i + /3? 2 = Qi- ") , 4 x 



a 2 q-2 + c 2 q 2 — ft qi = Q 2 ', J 



