Vibrations of Bars of Uniform Cross- Section. 129 



Replacing \ by the quantity 



V-^j. ...... (12) 



we should obtain the corresponding approximate solution 

 for the case of plane stress, and it is easily verified that 

 this is equivalent to the solution of equation (1). 



Proceeding to a further approximation in the case of 

 plane strain, we observe that the quantity d in (6) is small 

 in comparison with a and b, and that the solution of this 

 equation can therefore be written as follows : 



0<=-f(l + f4) d3) 



The second term in the bracket on the right-hand side 



is a small correction. In calculating it, we may take for ^ 



the first approximation (8), and in the expression for a\b 

 we may retain the terms of the order h 2 and f 2 only. We 

 then have 



£• ! = *(«)■ a*) 



The first term on the right of (13) must be calculated 

 more exactly. Retaining in the expression for b terms 

 of the order f* and A 4 , and substituting for V the first 

 approximation (10), we have 



Substituting from (14) and (15) in (13), and neglecting 

 small quantities of higher order, we obtain 



or 



_ «V 4KX + A*) n _ i . « /3X+V , 12\"1 

 P - 3 (X+2/») L 3 ^ c) U + 2 /t + W 



. . . (16) 



§ 4. The square brackets in (16) contain the required 

 corrections to (11). These correspond principally to the 

 effects of "rotatory inertia " and of shearing force, 'and 

 could have been obtained with sufficient accuracy from the 



Phil. Mag. S. 6. Vol. 43. No. 253. Jan. 1922. K 



