VALUE OF ACCELERATION OF GRAVITY— MENDENHALL. 



13 



About ii parallel axis through the inner circumference, we have: 



Ip = I„+ //(/•'■ 



anl 



unci, if we put /' ; -KK'- 



: f| RS + 3 ^+|(|- RE - 



, /R 2 + 3r 2 +^-R 2 -r 2 Ysin 



t«=«wv w — 



/r 2 , f-iR 1 . t 

 T«=T ^/l3 



[Since R 2 = 3r 2 J 



4 • , 

 - sin" « 



12 



*~± sin 2 « + 

 24 



:T(l 



) 



Iii the actual rings K=0.0113 [about J so that: 



T« = T (1-0.166 hr + a ) 



From which expression the following table of changes in the period (4T) produced by various 

 errors in azimuth has been calculated: 



Evidently if « can be made less than 8' in practice, the azimuth correction will be quite 

 negligible. 



CONSTRUCTION OF THE RINGS. 



The first question in the matter of construction is that of choice of material. Obviously the 

 material should be of a hardness comparable with that of agate, of which the knife-edge is best 

 made: homogeneous, free from slow changes in shape, having a definite temperature coefficient, 



i i ..• Young's Modulus 



and a large ratio: - — g en3ity 



Probably the ideal material would be fused quartz, and next, properly annealed glass; but 

 the difficulty and expense of construction with either of these would be much greater than with 

 the tool steel actually used. Troubles with permanent magnetization which might perhaps be 

 anticipated with a hardened steel ring have not been found, and of course the finished ring can 

 I >e demagnetized if necessary. Uncertainties as to the degree of homogeneity and permanence 

 of form which could be obtained with this material could only be settled by trial. 



Owing to the mechanical difficulties of the work and the necessarily exacting conditions, no 

 manufacturing mechanician could be found who was willing to undertake the making of these 

 rings, and it was therefore taken up and completed at the -University of Wisconsin. On this 

 account it will, perhaps, be desirable to devote somewhat more space to the matter of construction 

 than would otherwise be done. The finishing of the glass-hard ring must, of course, be clone by 

 grinding, and during this process the ring must be so held as not to distort its figure and, at the 

 same time, in such a way that the inner and outer cylindrical surfaces and one face can be 



