Elastic Solids to Metrology. 611 



§ 34. In the metre prototypes B or B' the divisions occur 

 on the neutral surface, and on the Bernoulii-Euler theory 

 their distance apart is unaffected by stretching of the material. 

 In the other types, stretching is a much more important 

 factor than the difference between the arc and its horizontal 

 projection. As shown in § 22, the stretching in any sym- 

 metrical central portion of a " fibre " in the upper half of 

 the cross section is 2hdy/dx, where h is the height of the 

 u fibre " considered above the C.Gr. of the section, and dyjdx 

 is the tangent of slope at one end of the portion considered. 

 When ajl is less than 05 the stretching is greater for the 

 whole length of the bar than for any shorter symmetrical 

 part ; when, however, ajl exceeds 0'5 the stretching is less for 

 the whole than for the central length 2#j, for which it is a 

 maximum. 



Table VII. gives the stretching in the divided upper 

 surface for the whole length 2/, as well as for the central 

 portion 2<z , 1 . The sign is + or — according as the upper 

 surface is lengthened or contracted. 



Table VII. 



Stretching of upper (divided) surface (unity = 1// = 0*001 mm.). 



Bar. 



Lsngth 

 considered. 



a/l=0. 



•2. 4. -5. 



•5. 



Vl/3. 



■6. 



•8. 



10 





(21 



+20-11 

 + 10-20 



+ 1770 +1046 +503 

 00 



+ 9 03 + 5-34 +2-57 

 -<X) 



+ 1-49 

 -1-49 

 +0-70 

 -0-76 



•00 

 -2-45 



•00 

 — 1-25 



-161 



-360 

 -0-82 

 -1-84 



-18-50 

 - 18-69 



- 9-44 



- 9 54 



-40-22 

 -4022 

 -20-53 

 -20-53 



A. 



U*i 



f 21 



C. 



lar a 



The values of x x will be found in Table VI. 



As an illustration, take the case of bar A when a/Z=0'6. 



Between the centre and the point ./: = <r 1 ( = 23*37 cm.) the 

 upper surface is all contracted, the total shortening in this 

 portion amounting to \ of 3*60 or 1'8/a. Between x = x x and 

 x = l, i. e. in the 2&H\) cm. nearest the end, the upper surface 

 is all extended, the total lengthening in this portion amounting 

 to VOfi. Thus in the entire half length of hl"h) cm. there is 

 a shortening of 0*8m in the upper surface. 



When a/l= V^l/3, the contraction of the upper surface 

 between <r = and x = x i is exactly balanced by the extension 

 between x = x^ and x = l; the fact, however, remains that 



