Uniform and Varying Sectional Area. 125 



.-. _y= 199 J 63 M^. (-06192 / 6 --15936 Px 4- '14517 l l x 2 



J 29 EA 2 / 4 



-•06206 JV 4- -01678 ZV--0027 loo" + -0003 ^ 6 ) 

 -^ = 23445^ 



, XT 1-5312 AU 

 and N=^ p (11) 



In this case the period is independent of the breadth of the 

 bar, and the influence of the depth is exhibited by means of 

 the constant A. 



9. Kirchhoff' s Investigation. 



The solution for some cases of non-uniform cross-section 

 was considered by Kirchhoff in 1879 *. He remarked that 

 the general equation 



d 2 C^-rd 2 }/ \ d' 2 u „ , 



d 



can be solved in general terms when the depth and breadth 

 of the cross-section are of the form 



d = ^Mx\ *-*/•&), 



% being any variable quantity. 



The only cases he investigates fully are 



(a) m = l. n = 0, % = constant. 



(b) m=l } n=l, % = constant. 



The former corresponds to that in the last section and the 

 numerical result obtained by Kirchhoff may be written 



N _ 1-534 AU 



2tt / ' 



Agreeing well with equation (11) above. 



University College, Bristol, 

 January 1905. 



* Berliner Monatsberichte, 1879, p. 815. 



t Rayleigh's ' Theory of Sound,' vol. i. art. 187 



MG*0 



