Physics and Mathematics to Geology. 237 



This is of course a question for experimentalists to decide, 

 but in any case where their final verdict is that the stress- 

 strain relation is sensibly not linear the employment of the 

 ordinary mathematical 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 receives any very formal 

 acknowledgment. It is, however, clearly recognized, 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 mathematical 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 unsur- 

 mountable. 



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 constitution of the body, but also by the treatment 

 to which it has been subjected. 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 it may be brought into a state of ease, 

 wherein these two conditions are very approximately, if not 

 exactly fulfilled, so long as the stress does not exceed a certain 

 limit. Again, the fact that a large mass of metal is sensibly 

 isotropic is no sufficient reason for attributing isotropy to the 

 same metal when rolled into thin plates or drawn into thin 

 wires. 



It is quite possible that the three conditions (A), (B), and 



(D) represent an ideal state which is never actually reached, 

 and that a divergence may always be shown by the use of 

 very delicate apparatus. If this be true, then the results 

 obtained by the mathematical theory cannot claim absolute 

 correctness. It seems, however, to be satisfactorily established 



* Nat. Phil. vol. i. Part ii. p. 422. 



