of Young s Modulus for Glass. 423 



where S is constant, and SS/S is supposed small everywhere 

 along the tube. 



Let T denote the kinetic, V the potential energy, then, as 

 in Lord Rayleigh's * Theory of Sound,' 



= 1 tyvP&dz, 



Thence from (1) and (2) 



T = ipA-KS sin- kt CJcos 2 (irz/l)(l + 8S/S )dz, 



V = iA-E(7r//) 2 cos 2 ^ ^ sin 2 (irz/l) (1 + SS/S )«fe. 



But T + V is constant throughout the motion, and so the 

 coefficients of sin 2 kt and cos 2 kt are equal. Equating them, 

 we find 



fW(wfl)(l +»/£«& 



lS-= B/p (w/0 s -4 > 



,\ co S 2 (7r;7)(H-8S/S y- 



o r l+^r'sin 2 ( TO /Q(8S/S )<fo 



lr = (E/p)(n/iy- jfa — . . (3) 



1+"' 



Tcos s M/0(SS/S„>fe 



As 5S 8 is by hypothesis everywhere small, we may 

 neglect squares of either integral as compared to unity, and 

 so deduce 



Now suppose that the rod has the same period as if it 

 ss ed a uniform section but were of slightly different 

 length 1+&L Then 



/2 _E J E^/ 2SZ1 



p (7 + S/) 2 -> P\ O " ' * W 



Comparing the identities (4) and (5) we deduce at once 



