236 Mr. C. Chree on some Applications of 



certain limits. The existence of these limits, it must be 

 admitted, is seldom recognized, and experimental results are 

 not infrequently referred to which are inconsistent with the 

 view taken here, viz. that rj must lie between and *5. If, 

 how r ever, 77 were negative in any material a circular bar of 

 this material, when subjected to uniform longitudinal tension, 

 would increase in diameter ; while if rj were greater than '5, 

 the bar, when fixed at one" end and subjected to a torsional 

 couple at the other, would twist in the opposite direction to 

 the applied force. Until these phenomena are shown to 

 present themselves in isotropic materials — and the experi- 

 mental verification ought to be easy — it seems legitimate to 

 suppose that when experimentalists deduce values for rj which 

 he outside of these limits, their experiments refer to bodies 

 whose constitution is different from what is assumed in their 

 mathematical calculations. 



The properties attributed to an isotropic elastic solid by the 

 ordinary mathematical theory are as follows : — 



(A) The strain must be elastic, L e. it must disappear on 

 the removal of the stress. 



(B) The ratio of stress to strain must be independent of 

 the magnitude of the stress, or, in Professor Pearson's words, 

 the stress-strain relation must be linear. 



(C) The strains must be small. 



(D) The values of Young's modulus and Poisson's ratio in 

 a bar of the material must be independent of the direction in 

 which the axis of the bar is taken. 



The last property alone distinguishes isotropic from seolo- 

 tropic elastic solids. 



(A) answers to MaxwelFs definition, but (B) and (C) are 

 not assumed by Maxwell. In other words, a solid may be 

 perfectly elastic without showing a linear stress-strain rela- 

 tion, or possibly even after the strains have become large. 

 Thus, for the sake of clearness, I shall call Maxwell's limit of 

 perfect elasticity the Physical limit, and the limits supplied 

 by (B) and (0) the first and second Mathematical limits 

 respectively. 



It is not infrequently taken for granted that the physical 

 and the first mathematical limit are necessarily identical, L e. 

 that the elasticity is certainly not perfect when the stress- 

 strain relation ceases to be linear. According, however, to 

 some experimentalists cast iron is as perfectly elastic as any 

 other metal in the sense of MaxwelFs definition, but the stress- 

 strain relation for even small strains is sensibly not linear*. 



* See Todhunter and Pearson's ' History of Elasticity/ vol. i. art. [1411] 

 and pp. 891-3. 



