154 Lord Rayleigh on the Maintenance of 



imaginary values of c. Of the two terms in (46), one or 

 other preponderates indefinitely in the two alternatives. 

 Thus, if ® 1 = 1— © , the solution reduces to cost; but if 

 ©i = — 1 + © , it reduces to sin t. The apparent loss of gene- 

 rality by the merging of the two solutions may be repaired in 

 the usual way by supposing s infinitely small. 



When there are dissipative forces, we are to replace c by 

 (c—ik), and © by (S Q —k 2 ); but when k is small the latter 

 substitution may be neglected. Thus, from (26). 



c =i+^ + i s{{&t-ij*-.e 1 *). . . . (50) 



Interest here attaches principally to the case where the radical 

 is imaginary ; otherwise the motion necessarily dies down. 

 If, as before, 



4s 2 = © 1 2 -(© -1) 2 , (51) 



c = l + ik + is, c— 2=— 1 + iJc + is, . . (52) 



and 



or 



or 



e (c-2)it e cit 



W= (c-ik-2f-(& + ©7' 

 w=e-< k + 8 »{® 1 e- it + (1 -® -2is)e u \ , 



w. 



■(k+s)t 



{(l-© H-© 1 )cos^+2ssin4. • • (53) 



This solution corresponds to a motion which dies away. 

 The second solution (found by changing the sign of s) is 



w; = ^ s - / ^{(l-©o + ©i)cos^ — 2s sin 4- • • (54) 



The motion dies aw T ay or increases without limit according as 

 s is less or greater than k. 



The only case in which the motion is periodic is when s = k, 

 or 



4# = e 1 2 -(0 o -l) 2 ; (55) 



and then 



w = (l — ©o— ©i)cosf— 2ksmt. . . . (56) 



These results, under a different notation, were given in my 

 former paper*. 



If ©o=l, we have by (51), 2s=© ; and from (53), (54), 

 w—'Re-( k+8 »{G08t+ sin^} +Se- ( *- s) '{cos£-sin t}.. . (57) 



* In consequence of an error of sign, the result for a second approxima- 

 tion there stated is incorrect. 



