Maiuhestcr Memoirs, Vol. Iviii. (1914), No. 5- 5 



some length tlie mathematical forms necessarily imposed 

 on solutions of the statical stress equations. In the 

 present Part, I have found it possible to proceed without 

 much reference to the earlier Parts. 



In this Part I propose to deal also with elastic stresses 

 for which the mathematical forms of expression are based 

 on the hypothesis of Hooke's Law. On this basis it is 

 fundamental that the stresses, which are of considerable 

 and even of great magnitude, should appear as large 

 multiples of strains which are themselves extremely 

 minute and certainly not measurable by any direct obser- 

 vation. The large multipliers we call elastic constants. 

 To form a mathematical theory it is presumed that the 

 material is absolutely uniform in character, and that the 

 strains are everywhere continuous. In fact, an ideal 

 mathematical clastic material is presumed, and it must 

 always be borne in mind that the elastic theory applies 

 with accuracy only to such an ideal substance. When 

 the results are taken to apply to ordinary material, we 

 have to remember that such material is irregular and 

 coarse-grained and differs in many ways from the ideal 

 substance to which the theory actually applies. 



It appeals to be thought by some writers on the sub- 

 ject, that by proceeding to a higher degree of accuracy' in 

 the mathematical expressions for the strains, a closer 

 representation of the actual state of a natural body under 

 stress might be obtained. The notion seems to me 

 fundamentally wrong and fallacious if it is supposed that 

 by introducing still more minute terms in the expression 

 for the strains in the ideal substance, any closer approxi- 

 mation can be made to the conditions in the natural 

 substances. A further refinement in the mathematical 

 expressions including terms of a higher order should 

 be deferred until the terms of the first order have been 



