Vibrations by Forces of Double Frequency. 149 



The value of □ (0) is calculated for the purposes of the Lunar 

 Theory to a high order of approximation. It will here suffice 

 to give the part which depends upon the squares of ® u ® 2 , & c « 

 Thus 



TrcotftTrveo) !-©, 2 <s) 2 2 e 2 -, 

 D(0) - 1+ — v®o Ll^e; + 43@ o + 9^0 o + ---J- (17) 



Another determinant, V(0), is employed by Mr. Hill, the 

 relation of which to Q (0) is expressed by 



V(0) =2 sin 2 (i7rV© ). □(()); . . . (18) 



so that the general solution for c may be written 



cos(ttc) = 1-V(0) (19) 



Mr. Hill observes that the reality of c requires that 1 — V (0) 

 should lie between —1 and +1. In the Lunar Theory this 

 condition is satisfied; but in the application to Acoustics the 

 case of an imaginary c is the one of greater interest, for the 

 vibrations then tend to increase indefinitely. 



Cos (ire) being itself always real, let us suppose that ttc is 

 complex, so that 



c = a + i/3, 



where a and are real. Thus 



COS 7TC = COS ITU COS ilT0 — SHI ITU sin VTT$\ 



and the reality of cos ire requires either (1) that /3 = 0, or (2) 

 that oc = n, n being an integer. In the first case c is real. In 

 the second 



COS7TC= + COS 177-/3=1 — V(0), . . . (20) 



which gives but one (real) value of j3. If 1 — V (0) be positive, 

 e=±i0 + 2n; (21) 



but if 1 — V (0) be negative, 



COS 7TC= —COS 17T/9, 



whence 



c=±ij3 + 2n + l (22) 



The latter is the case with which we have to do when ® , and 

 therefore c, is nearly equal to unity ; and the conclusion that 

 when c is complex, the real part is independent of ® lf © 2 , &c. 

 is of importance. The complete value of w may then be 

 written 



w=eP t $b n e it ( 1 + 2n ) + e-P t Zb' n e it v+ 2n \ . . . (23) 



the ratios of b n and also of b n f being determined by (9). After 

 the lapse of a sufficient time, the second set of terms in e~P* 

 become insignificant. 



