8 MEMOIRS NATIONAL ACADEMY OF SCIENCES, VOL. N, NO. 1. 



Now let there be a number of flaws Jni x . Jm.,. dm 3 , .... whose position is determined by 

 the radii /, p" and the angles t,. e a , which these radii make with a diam- 

 eter fixed in the ring, which, in turn, makes an angle a with the vertical line through the knife- 

 edge. We should then have, if 



t=24T 



is the resultant change in period produced by all the flaws: 



- _/,//,/(•<>* (r,— «) + J,//(/'cos ((',, — a) + . . .) 



or. if- 



A=2J l /nr'cos e, 

 B=2-J 1 //// , 'sin e l 



-T^tan' 1 x 



We have- 



4 = A \24mr' 2 -2 2A m - l n/A 2 +B s cos(«- r)~] 



The effect of the flaws is then in general to produce a certain change in T which is independent 

 of the position of vibration (i. e., of the angle «) and in addition a change dependent on this 

 position. As is otherwise evident, for any S3'mmetrical distribution of flaws, A = B = and the 

 change in T will be independent of a. If at the same time — 



\2Jmr ri = 22Jw 

 r 



the constant part of the change, t, will also be O; and this determines the radius of the equivalent 

 "simple ring" previously deduced from other considerations. 



3. SPECIAL CASKS OF NONHOMOGENEITY AND ERRORS OF FIGURE. 



In order to the better interpret the behavior of the actual rings, it will he well to consider 

 a few special cases of departure from a perfect ring which, on account of the method of making 

 the ring, are especially to be guarded against. The first of these is the matter of nonparallelism 

 of the side faces, which is equivalent to the addition of a wedge-shaped ring to a perfect ring. 

 Exactly the same effect would be produced by a special case of nonuniform density — i. e.,that in 

 which the density varied as the distance from a plane tangent to the outer circumference of the 

 ring. The method of procedure has been to determine the period of the " ring and wedge," for 

 three positions, thick edge up, thick edge down, and one of the symmetrical positions with thick 

 edge to right or left. By comparing these periods with that of the perfect ring, the effect of a 

 given wedge as to both the " constant " and " variable" changes in the period can be determined. 



