376 Prof. Harrison and Mr. S. K. Chakravarti on the 



not even approximately hold in the neighbourhood of the 

 maximum, and any series of graphs at different temperatures 

 above and below the maximum must necessarily bear to one 

 another some such relation as that shown diagram matically 

 in fig. 2. 



Ffc. 2. 



Load 



For, from the series of curves in that figure, in which 

 #b ^25 #3? • • • & c «> represent temperatures in ascending order 

 of magnitude, it is clearly possible to derive the two types 

 of elasticity temperature curves shown in fig. 1. 



First using loads smaller than c, as the temperature rises 

 successively from 1 to # 7 , the slopes of the load elongation 

 graphs at first decrease and then increase. Now Young's 

 modulus is inversely as the slope of a load elongation graph ; 

 hence the modulus reaches a maximum about 6±. Next 

 consider the case where loads greater than c are applied. 

 It is evident that the only way to obtain curve B (fig. 1) is 

 for the slopes of the elongation graphs above load c to 

 increase steadily as the temperature rises. In order that 

 this may be the case, and at the same time that a curve at 

 any one temperature may be a continuous function of the 

 variables, it is necessary for some, at least, of the graphs to 

 be bent lines. In the example given, for loads below c 

 curve 0i has a larger slope than 7 ; while for loads above c 

 6 1 has a smaller slope than 6 7 . Consequently, one at least 

 of the two graphs must be non-linear ; that is to say, for one 

 or both of these two temperatures Hooke's law fails throughout 

 the range of stress shown. 



