260 Miss Lorna M. Swain on the Period of 



and the depth, measured parallel to oy, decrenses uniformly, 

 remaining finite at the free end. The taper is defined by 

 taking the length to be I and the additional length, which 

 would have to be added to make it taper to a knife-edge, to 

 be c. 



The case of a uniform rod is then obtained by making c 

 tend to go and that of a rod tapering to a knife-edge by 

 putting c=0*. 



The problem is to find the period of the gravest mode for 

 vibrations in the plane yoz. 



Let a be the area of any cross- section of the rod, perpen- 

 dicular to oz, at distance z from o, 



let I be the moment of inertia of the section about its 

 mean line perpendicular to the plane yoz, 



let <r , lo De ^ ne va ^ ues °f °"> I respectively for z = 0. 



mi l + C-Z T T (I + C-ZV 



The rotation of the rod causes a tension in it and we 

 proceed to evaluate this tension in the steady motion. 



Let E be Young's modulus of elasticity, 

 F be the shear, ] 



M be the bending moment, 

 T be the tension, assumed j> at distance z from o, 



to act along the axis of | 



the rod, : 



p be the density. 



These quantities are measured in gravitational in.-lb.-sec. 

 units. 



Resolving along the axis of the rod, 



dT pjo* 



dz g 



dT pco 2 l + c-z 



dz g I -\- c 



Integrating this equation and expressing the condition 

 that T = 0, when z=l, we find 



* For this method of treatment the author is indebted to Mr. H. A. 

 Webb. 



