Elastic Solids to Metrology. 599 



■so that (dyjdx) 12 is neglected as compared to unity whenever 

 the use of this theory is warranted. 



On the theory, the length of the neutral fibres (for which 

 h = 0) measured along the arc is unaffected by stretching. In 

 standards of length, however, it is the horizontal projection 

 of the graduated surface that usually concerns us. We thus 

 require to know how much shorter x is than s, the latter 

 denoting the length of a neutral fibre measured from the 

 centre of the bar along the arc. 



We have in any plane curve 



s-x= V \(l+(dyldw) 9 f-l\dx, . . . (72) 



As (dy/dxf is supposed neglected compared to unity, we 

 can consistently retain only the lowest power of dyjdx in 

 such a formula as (72), and so get 



= [\(dyjdxfdx 



(720 



For instance, we have from (69) and (70) for the difference 

 between arc and chord for the entire length o£ the bar, 



2(s t -l)= (ic/Eco* 2 ) 2 \l(-a + il)x + irx s \ 2 dx 



+ (uiE(OK*f{-ila 2 + ±P-±(l-x) 3 \ 2 dx 



-«*^{sri©' + sCi) 4 -5fi) ,+ ifi) , }w 



An equivalent formula is given by Broch (I. c. formula (16) 

 p. B. 67). The right-hand side represents the quantity by 

 which the distance between the extreme ends of the bar is 

 shortened through the bending which it experiences under 

 gravity, when supported symmetrically at two points at a 

 distance 2a apart. 



It will be observed that <x r is of the order July/dx, or 

 h(wl* l*Ea>fc 2 ) , and s — ,r is of the order xlivfijEcotc 2 ) 2 ; while by 

 the hypothesis made as to the smallness of dyjdx we must 

 suppose wP/Ecdk 2 a small quantity. Thus unless x/h be 

 very big, t. e. unless we have a very long bar, or are consi- 

 dering fibres very close to the neutral plane, the difference 

 between chord and arc is small compared to the direct 

 stretching or shortening of the fibre. 



§ 23. Owing to the symmetry, we need consider only the 

 half of the bar for which x is positive. In all cases dyjdx 



2R2 



