150 Lord Kayleigh on the Maintenance of 



In the application of greatest acoustical interest © (and c) 

 are nearly equal to unity; so that the free vibrations are per- 

 formed with a frequency about the half of that introduced 

 by © x . In this case the leading equations in (9) are those 

 which involve the small quantities [0] and [ — 1]; but for 

 the sake of symmetry, it is advisable to retain also the equa- 

 tion containing [1]. If we now neglect © 2 , as we ^ as the ^' s 

 whose suffix is numerically greater than unity, we find 



and 



[0][l][-l]-© 1 2 {[l] + [-l]}=0. . (25) 



For the sake of distinctness it will be well to repeat here 

 that 



[0]=c 2 -<h)o, [-l] = (c-2) 2 -© , [l] = (c + 2) 2 -e°. 



Substituting these values in (25), Mr. Hill obtains 



( C 2 -©o){(c 2 + 4-© ) 2 -16c 2 }-2@ 1 2 { c 2 + 4-(h)o} = 0, 



and neglecting the cube of (c 2 — © ), as well as its product with 



(S) 2 



( C 2 -©„) 2 + 2(0 o -l)( C 2 -@ o )+0 1 2 = O; 

 and from this again 



^l-fv/U^o-l) 2 -©/ 2 } (26) 



It appears, therefore, that c is real or imaginary according 

 as (© — l) 2 is greater or less than ©! 2 . In the problem of 

 the Moon's apse, treated by Mr. Hill, 



O =1-1588439, ©!= -0-0570440; 



and in the corresponding problem of the node, investigated by 

 Prof. Adams, 



© = 1-17804,44973,149, 



©, = 0-01261,68354,6. 



In both these cases the value of c is real, though of course 

 not to be accurately determined by (26). 



Mr. Hill's results are not immediately applicable to the 

 acoustical problem embodied in (1), in consequence of the 

 omission of k, representing the dissipation to which all actual 

 vibrations are subject. The inclusion of this term leads, 

 however, merely to the substitution for (c + 2«) 2 — © in (8) of 



{c + 2n) 2 -2ik(c + 2n)-® () ; 



so that the whole operation of k is represented if we write 



