346 G. F. Becker — Finite Elastic Stress-Strain Function, 



and tan x. In a case of this kind, however, dh would not be a 

 function of Q/M — v + x excepting for infinitesimal strain; 

 the exponent then taking the form of a series of terms 

 A m (v m 4- x m ) instead of Ajy + x) m . Finally it is conceivable 

 that the expanded function should contain in the higher terms 

 moduluses not appearing in the first variable term ; but this 

 would be inconsistent with continuity. In short I have been 

 unable to devise any change in the functions which does not 

 conflict with the postulate of isotropy as defined or with 

 some kinematical condition. 



Abbreviation of proof . — In the foregoing the attempt has 

 been made to take a broad view of the subject in hand lest 

 some important relation might escape attention Merely to 

 reach the equations (5) only the following steps seem to be 

 essential. Exactly one-third of the external initial stress in a 

 simple traction is employed in dilation, and of the remainder 

 one-half is employed in each of the two shears. An ideal iso- 

 tropic homogeneous body is postulated as a material present- 

 ing equal resistance to strain in all directions, the two 

 resistances to deformation and dilation being independent of 

 one another ; the strains moreover are to be of the same order 

 as the loads, and continuous functions of them. In such a 

 mass the simplest conceivable strains, shear and dilation, can 

 each involve only a single unit of resistance or modulus. The 

 principle of superposition is applicable to a simple traction 

 applied axially to the unit cube however great the strain. It 

 follows that the length of the strained unit cube is a function 

 of Q/M. 



Together these propositions and assumptions give (1) and 

 without further assumptions the final equations sought (5) fol- 

 low as a logical consequence. 



Data from experiment. — No molecular theory of matter is 

 essential to the mechanical definition of an isotropic substance. 

 An isotropic homogeneous body is one a sphere of which 

 behaves to external forces of given intensity and direction in 

 the same way however the sphere may be turned about its 

 center. There may be no real absolutely isotropic substance, 

 and if there were such a material we could not ascertain the 

 fact, because observations are always to some extent erroneous. 

 It is substantially certain, however, that there are bodies which 

 approach complete symmetry so closely that the divergence is 

 insensible or uncertain. Experience therefore justifies the 

 assumption of an isotropic substance as an approximation 

 closely representing real matter. 



All the more recent careful experiments, such as those of 

 Amagat and of Yoigt, indicate that Cauchy's hypothesis, lead- 

 ing for isotropic subtances to the relation 3h = 5n, is very far 



