336 Dr. C. Chree on Longitudinal 



Owing to the omission of i in the expression on p. 120 it 



looks as if zr vanished for the same values of z as the normal 



stress zz. In reality, as I showed in (A) p. 295, zr does not 

 vanish over a terminal free section of radius a, but is of the 

 order r(r 2 — a 2 )/l 3 , where r is the perpendicular on the axis. 



We are quite justified in neglecting zr when terms of order 

 (a/1) 3 are negligible, but strictly speaking the solution is so 

 far only an approximate one when the ends are free. 



A New Method. 



§ 5. In the Camb. Phil. Soc. Trans, vol. xv. pp. 313-337 

 I showed how the mean values of the strains and stresses 

 might be obtained in any elastic solid problem independently 

 of a complete solution. For isotropic materials I obtained 

 (/. c. p. 318) three formulas of the type 



E \\\ d -ldxdydz= N[ {I*-ri£Kw'+Yy))dx dy dz 



■i;(Fff + (fy)}dS, . (4) 



jj> 



where a, fi, y are components of displacement, X, Y, Z of 

 bodily forces, and F, G, H of surface forces. The volume 

 integrals extend throughout the entire volume, and the surface 

 integrals over the entire surface of the solid. As was explicitly 

 stated in proving the results (I. c. p. 315), X, Y, Z may include 

 ' reversed effective forces ' 



dPa d 2 /3 d 2 ry 



p dt 2 ' p dF' - p W 



where p is the density. 



In the present application there are no real bodily forces ; 

 we may also leave surface forces out of account, if we suppose 

 that when one end of a rod is held, that end lies in the plane 

 2 = 0. F 



Supposing the rod to vibrate with frequency &/27T, we have 



so that we replace (4) by 



"^JJJ ^ € *^^ €fe= ^^jHU ^^ — ^C«a?+ifiEy)l^»^«fc, . (5) 



and similarly with the two other equations of the same type. 

 I he only other result required is one established in my 



