Vibrations in Solid and Hollow Cylinders, 339 



by which to judge of the limitations of our results. If the 

 correction is large even for the fundamental note, it is pretty 

 safe to conclude that the section is not one adapted for the 

 ordinary type of longitudinal vibrations. If a section, for 

 instance, were of an acutely stellate character, with a lot of 

 rays absent and the centroid external to the material, I for 

 one should be extremely chary of applying to it the ordinary 

 formula. 



§ 7. For deiiniteness let us consider the fixed-free vibra- 

 tions, taking the origin of coordinates at the centroid of the 

 fixed end. Our terminal conditions are 



7 = when z = 0, 

 S=0 „ z = l; 

 the latter condition being the same thing as 



_¥ =0 when z = L 

 dz 



These conditions give at once 



6=0, p=(2* + l)w/2Z, 



and hence 



I pz sin {pz — e) dz = 1 cos (pz — e) dz. 



J Jo 



Take the axes of x and y along the principal axes of the 

 cross section cr, so that 



\\ xy dx dy = 0, 



ij x 2 dx dy = k. 2 2 (t, Jj y 2 dx dy = k?<t. 



Then, substituting from (6) in (5), we obtain at once 



(Ep-Fp/ / >)(C + Crf + C 2 / V + ...) 



= -^(A 1 « 2 HB 1 V+...) . • (?) 



The section is supposed small, L e. terms ^ in k{ 2 , k 2 2 are 

 small compared to O , though large compared to the terms 

 of orders /q 4 , &c, which are omitted in the above equation. 

 Thus as a first approximation the coefficient of C must vanish, 

 or 



*=?VE/p, (1) 



which is simply the ordinary frequency equation. 



■* Treating the other two equations of the type (5) similarly, 



