130 SOME APPLICATIONS OF 
The last property alone distinguishes isotropic from wolotropic elastic 
solids. 
(A) answers to Maxwell’s 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 relation, or possibly even after the strains 
have become large. Thus, for the sake of clearness, I shall call Max- 
well’s limit of perfect elasticity the physical limit, and the limits sup- 
plied by (B) and (C) the first and second mathematical limits respec- 
tively. 
It is not infrequently taken for granted that the physical and the 
first mathematical limit are necessarily identical, ¢. ¢., 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 Maxwell’s definition, 
but the stress-strain relation for even small strains is sensibly not 
linear.* This is, of course, a question for experimentalists to decide, 
but in any case where their final verdict is, that the stress-strain rela- 
tion is sensibly not linear, the employment of the ordinary mathemati- 
cal theory is unjustifiable. It must be admitted that the principle (C) 
is a very vague one, leading to no exact limit, and that it seldom re- 
ceives any very formal acknowledgment. It is, however, clearly ree- 
ognized, and a reason for it assigned in the following statement, due to 
Thomson and Tait:+ “The mathematical theory of elastic solids imposes 
no restrictions on the magnitudes of the stresses except in so far as that 
mathematical necessity requires the strains to be small enough to admit of 
the principle of super-position.” The italics are mine. ‘The meaning is 
that the strains must be small fractions whose squares are negligible 
compared to themselves. If this principle be neglected and the mathe- 
matical equations be supposed to apply when the strains are large, the 
difficulty of giving them a consistent physical interpretation is very 
great if not wholly unsurmountable. 
In most materials having any claim to be regarded as elastic solids 
the stress-strain relation for most ordinary stress systems certainly 
ceases to be linear while the strains are still small. We shall thus in 
the meantime leave the condition (C) out of account, though we shall 
have to return to it in treating of the so-called “theories of rupture.” 
The existence of the properties (A), (B), (D), presupposed by the 
mathematical theory, is determined not solely by the chemical constitu- 
tion of the body, but also by the treatment to which it has been sub- 
jected. Thus a freshly annealed copper wire may, when loaded for the 
first time, be far from satisfying conditions (A) or (B), and yet by the 
process of loading and unloading itmay be brought into a state of ease, 
wherein these two conditions are very approximately, if not exactly 
*See Todhunter and Pearson’s History of Elasticity, vol. 1, art. [1411] and pp. 
891-893. 
t Nat. Phil., vol. 1, part 0, p. 422. 
