16 



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



The measurements made with this comparator may be grouped as follows: 



1. Measurement of various diameters to test the 



24 



28 



22 



21 



20 



0.9 



28.849 cm 



roundness" of the ring. No variations 

 as great as 0.001 mm. could with cer- 

 tainty be detected, and the rings were 

 therefore considered round. 



2. A series of measurements of a 

 single diameter of the rings at several 

 temperatures to determine the tem- 

 perature coefficient. It was not pos- 

 sible to work over a very large range 

 of temperatures, but the agreement of 

 the results is quite satisfactory, as can 

 be seen from figure 9. From figure 9 

 the uncorrected values of D, for the 

 standard temperature (23° C.) have 

 obtained, and in a similar way for D r 



3. Measurements of diameters at 

 various points between the two faces. 

 Differences were here found in each 

 ring ranging over about 0.01 mm., due 

 undoubtedly to errors in the vertical 

 guideways of the milling machine. 

 The following figure 10 shows the dis- 

 tribution of these variations, the quan- 

 tities plotted being differences in diam- 

 eter and the corresponding position 



on the ring. A sufficiently exact allowance for these irregularities has been made, as follows: 



Consider the actual ring as made up of a perfect 

 ring with a certain small mass, the irregular rim, 

 added: 



Let d)n=this, small additional mass. 

 M = mass of actual ring. 



H = minimum external radius of actual ring. 



K = R+JR = radius of gyration of Am about the center 



of ring. 



J'R = an increment of R such that 



R+ J'R = radius of a perfect ring equal as to mass 



and moment of inertia to the real ring. 



Then: 



moment of inertia of the actual ring=s(M— Am)R. 2 -\-AmK." 



= 3 MR2 



-|z////R 2 



= |mR 8 +^»iR 2 



-zb»R 2 -M-////RJR 



[Neglecting terms of second order in Jm and JR. 



