Elastic Solids to Metrology, 601 



the centre, that particular length whose total extension is a 

 maximum. If dyjdx vanishes for two values of m, then the 

 portion of any longitudinal fibre included between the two 

 corresponding transverse sections has its total length un- 

 changed by stretching. These remarks will, it is hoped, suffice 

 to explain how the conclusions in the next paragraph are 

 reached. 



§ 24. The bending of a symmetrically supported bar under 

 its own weight may be conveniently dealt with under several 

 cases as follows : — 



Case (i.) Fig. 2,a/l<0'5. 



The upper surface is everywhere convex (stretched). The 

 slope vanishes only at the centre, and is greatest at the end. 



Case (ii.) Fig. 3, 0-5505 >a/l>0'5. 



The upper surface is concave (contracted) between x=x 

 and as=x l (where Xi<a), and convex (stretched) between 

 x=x x and the end. The " summit," where the slope vanishes, 

 occurs at x = x 2 (where x 2 = &x 1 ) between the centre and the 

 support. 



Between the centre and the summit the steepest slope 

 occurs at x = x x ; beyond the summit the slope is steepest at 

 the end, and the slope at the end is greater than at a? = <r x . 



Case (iii.) Fig. 4, 0-5654 >a/Z> 0*5 505 



This agrees with Case (ii.) except in the following 

 respects : — 



The summit occurs at x = x- 6 between the support and the 

 end. 



The slope is greater at x = x l than at the end when a\l> 0*5. 



Case (iv.) Fig. 5, 0'57735>a/Z>0'5b , 54. 



The upper surface is concave (contracted) between x = 

 and x = x l (where x 1 <a), and convex (stretched) between 

 #= a?j and the end. 



The summit occurs at x = x% between the support and 

 the end. 



The slope is greatest at x = x Y between the centre and the 

 support, and is greater at the support than at the end or at 

 any intermediate point. 



Case (v.) Fig. 6, «//>0'57735. 



This agrees with Case (iv.) except that the slope vanishes 

 only at the centre. 



What is said above of the "upper surface" applies to all 

 longitudinal " fibres " above the neutral plane. For absolute 

 accuracy the decimals 0*5505, 0'5654, and 0*57735 are to 



