130 Transverse Vibrations of Bars of Uniform Cross- Section, 



equation for transverse vibrations of rods, if this is supple- 

 mented in the manner explained in my previous paper, 

 where the correction to the frequency p was given in the 

 form of a multiplying term of amount * 



[\ Itt 2 /; 2 /-, E\"l 



In this expression, 7r/L is equivalent to the a of the 

 present paper, and P = c 2 /3. X is a constant relating the 

 shearing force with the angle of shear at the section con- 

 sidered : its value for a rectangular section (if we make use 

 •of the experimental results of L. N. G. Filonf) is 8/9. The 

 ratio E/C must be replaced, for comparison with our present 



problem of plane strain, by the quantity -^ — -~ (in the 



notation of the present paper). We then have from (17), 

 in our present notation, as the correcting factor required in 

 the expression for the square of the frequency, 



[-K-'t 1 *!^)]- 



With o- = 025, or X — fi, the correction to p 2 , given by 

 {17), is thus 



[l-f(«c) 2 ], ........ (18) 



whilst (16) gives 



Li-iiW*]. ...... (19) 



§ 5. This close agreement gives us some confidence in 

 applying the approximate solution of my earlier paper 

 to other shapes of cross-section. We take, for example, 

 the case of a circular cross-section J : the exact solution 

 may be written in the form 



Tr ire /hi r ttV/7 E fl\\ 



(20) 



where c denotes the radius of the cross-section ; whilst the 

 methods of my earlier paper would give 



where k is the " shear constant 5 ' previously denoted by X. 



* >Cf. equation (13) of the paper referred to. 

 t A. E. H. Love, op. cit. §245. 



t See Pochhammer, Journal f. d. reine u. angw. Math. Bd. lxxxi. 

 p. 335 (1876). 



