10 



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



At . . 



If we assume as a special case , =0.001, that is, for the actual ring At=\bn, approxi 



mately. the above expressions become — 



TV- = 0.999952 T 



TV= 1.000039 T 



From which 



TV -TV = 0.0000S6 T 



^(T'p + P'p) = D.999995 T. 



Also for either symmetrical position in which the thick edge is to right or left, we have, 

 neglecting terms of second and higher orders in 



MR (1+ap 



^1> 



=T. 



T in all these expressions being the period of a perfect ring having the same outer and inner 

 radii as the imperfect ring under discussion. The above conclusions will be of interest in 

 connection with the experimental results later discussed. They will, of course, apply equally 

 well to the special case of nonuniform density before mentioned, where the increment of 

 density at a point ■/■', 0, is given by 



Ap'=^(R-7-' cos 9) 



Ap being the maximum increment in density (at the bottom of the ring in this case); in the final 



Ap Af 



result — would appear in the place of - above. 



Another error of figure which must be especially guarded against is that shown in figure 6, 

 one face being symmetrically conical; here, if Am is the increment of mass 

 and Al the increase in moment of inertia about Pdue to the added conical ring, 

 we find 



4 



and 



Am .At 

 =0.54— 



in t 



f=o.*f 



1+0.6 



At 



from which 



If 



T'=T' 



l-|-0.54- 



At 



At 



Fig. 6. 



= 0.0001 



(At=1.5M) 



T = 1. 000006 T. 



Even a systematic error in thickness as small as this could be detected and corrected in the 

 course of construction. 



