4 MEMOIRS NATIONAL ACADEMY OF SCIENCES, VOL. X, NO. 1. 



It is evident, however, that any nonhomogeneity or figure error which is symmetrical with 

 respect to the center of the ring can not be detected by observations on the period; hence the 

 necessity of comparing results with a number of rings. With this in mind, five forgings for 

 rings were obtained, of which one was broken during construction; but on account of the great 

 labor involved in making a finished ring, only two have been completed and used. 



What has been said as to the simplicity of the theory is also true of the measurements 

 involved; the difficulty of the problem lies in the construction of the ring. But before taking 

 up this, it will be well for the sake of reference to consider the following: 



1. The vibration of a perfect ring, that is, one having no irregularities of figure or density. 



•2. The effect of "flaws" in general. 



3. Special cases of nonhomogeneity, and error in figure: 

 (a) Nonparallelism of faces. 



(f>) A particular case of irregular density. 



(c) One face conical. 



(d) Inner edge conical. 



4. Effect of error in adjusting the plane of the ring at right angles to the knife edge. 



1. PERFECT RING. 



[After Kimball. 1 



Let: 



Then: 



Fig. 1. 



R=external radius. 

 r= internal radius. 

 M = mass of ring. 

 T= period of vibration around P. 

 p, f, =density and thickness of ring, which may lie assumed unity. 



I = ^(R*— >•*) = moment of inertia around O. 



I p = I +7t(R 2 — r)r 2 = nioment of inertia around P. 

 = |(R 2 -r) (R 2 + 3r 2 ) 



= 1JVI (R 2 +.V). 





