Elastic Solids to Metrology, 605 



The first equation will apply when b < a, or the end of the 

 unaffected portion lies between the centre of the bar and 

 the support ; similarly the second equation applies when 

 ■c>b>a, and the third when b>c. In all three cases we 

 have supposed c > a. Under certain circumstances it might 

 be necessary to have W negative, i. e. to have not a weight 

 but an upward pull from a belt. 



Suppose, as a particular case, that the auxiliary weight is 

 .applied at the end, and that a 2 = l' 2 /3, so that the bar when 

 unweighted has its whole length unaffected by bending. The 

 relations then are : — 

 for b>a{>lx 0-5774), 



W = w(l-b) 3 +{2l 2 -3(l-by\ ; .... (93) 

 for b<a, 



W = w\{2 \''3-3)F-b 2 } + {{6-2\/3)l\. . (94) 

 In both these cases the value of W is always positive. 



As a second example suppose that the bar is supported at 

 its centre so that a = Q. We then find from (91) and (92) 



W=-±w{l 3 -(l-b) 3 \ + {c 2 -(c-b) 2 \ . . (95) 



or w=-i^ 3 -(Z-/>) 3 Kc 2 , (96) 



.according as c is greater or less than b. 



In both instances W is negative and so represents an 

 upward pull. If this pull is applied at the extreme end of 

 the bar we have 



~W=>Z{i-(i-6/O 3 }^U-(i~&/0 2 }; • (97) 



so that the pull ( — W) increases from (wl/3) to (wI/2),sls 

 the portion of the bar to have its length unaffected is reduced 

 from being the whole length to being an infinitesimal length 

 at the centre. 



Numerical Applications to Standards of Lenytli. 

 § 28. To illustrate the numerical application of the results 

 in §§19 to 27 we shall consider the types of bars shown in 

 fig. 1. Particulars as to the dimensions &c. of the bars are 

 given in Table I. The data as to the types B and C are 

 from the papers by Broch &, Benoit already referred to. 

 The dimensions of A, D, and E are from measurements of 

 bars in the National Physical Laboratory. In the case 

 of these bars, and to some extent in the bars B and C, the 

 values ascribed to Young's modulus and the density are 

 derived not from actual experiments on particular standards 

 but from records of mean values for the materials of which 

 the standards are composed. The letters have the meanings 

 assigned in § 21. 



