w- 



Vibrations by Forces of Double Frequency. 159 



In conclusion, it may be worth while to point out the ap- 

 plication to such a problem as the stationary vibrations of a 

 string of variable density fixed at two points. A distribution 

 of density, 



2lTX 4:7T,V ,__. 



Po + pxcos—^ \-p 2 cos—j- + (75) 



is symmetrical with respect to the points x = and % = \l, 

 and between those limits is arbitrary. It is therefore possible 

 for a string of this density to vibrate with the points in ques- 

 tion undisturbed, and the law of displacement will be 



f . . 2irx . . A:irx . . 6wx ") .-^ 



cospM AiSin —j- +A 2 sm— -, — hA 3 sm— = h. . . Y . (76) 



When, therefore, the problem is attacked by the method of 

 Mr. Hill, the value of c obtained by the solution of (69) must 

 be equal to 2. By (15) this requires 



q(0)=0 (77) 



This equation gives a relation between the quantities © , © 1? 

 © 2 , . . . ; and this again, by (70), determines p, or the fre- 

 quency (p/2tt) of vibration. 



Since © = 4 nearly, the most important term in (17) is 

 that involving © 2 2 . The first approximation to (77) gives 



© = 4 + © 2 ; 

 whence, by (70), 



(|r)' = *Vn-*ft) (78) 



To this order of approximation the solution may be obtained 

 with far greater readiness by the method given in my work 

 on Sound*; but it is probable that, if the solution were 

 required in a case where the variation of density is very con- 

 siderable, advantage might be taken of Mr. Hill's determinant 

 □ (0) . There are doubtless other physical problems to which 

 a similar remark would be applicable. 



Terling Place, Witham, 

 June 19, 1887. 



* ' Theory of Sound,' vol. i. § 140. In comparing the results, it must 

 be borne in mind that the length of the string in (78) is denoted by £ /. 



