338 G. F Becker — Finite Elastic Stress-Strain Function. 



In the first part of this paper finite stress and finite strain 

 will be examined from a purely kinematical point of view ; 

 then the notion of an ideal isotropic solid will be introduced 

 and the attempt will be made to show that there is but one 

 function which will satisfy the kinematical conditions con- 

 sistently with the definition. This definition will then be com- 

 pared with the results of experiment and substantially justified. 



In the second part of the paper the vibrations of sonorous 

 bodies will be treated as finite and it will be shown that the 

 hypothesis of perfect isochronism, or perfect constancy of pitch, 

 leads to the same law as before, while Hooke's law would in- 

 volve sensible changes of pitch during the subsidence of the 

 amplitude of vibrations. 



Analysis of shearing stress. — Let % ft and X be the resul- 

 tant normal and tangential stresses at any point. Then if N x , 

 N 9 and N 8 are the so-called principal stresses and A, /i, v the 

 direction cosines of a plane, there are two stress quadrics 

 established by Cauchy which may be written 



r = n;a 2 + n> 2 + n 3 v, 



ft - N^ a + N ay u 2 + ^y. 

 Since also V = W - ft 2 , 



X 2 = (N 1 - N a ) 2 AV + (N 1 - N 3 ) 2 AV 2 + (N, - N 3 )>V a ; 



and these formulas include the case of finite stresses as well as 

 of infinitesimal ones. 



In the special case of a plane stress in the xy plane, 1ST, = 

 and v = 0, and the formulas become 



at- = n;a 2 + n 2 v, 



ft = N X A 2 + N 2 //, 

 V = (N, - NJ 2 A 2 /A 



In the particular case of a shear (or a pure shear) there are 

 two sets of planes on which the stresses are purely tangential, 

 for otherwise there could be no planes of zero distortion. 

 On these planes ft = 0, and if the corresponding value of 

 A/ti is a, 



— Njtf = N 9 /or. 



If this particular quantity is called Q/3, one may write the 

 equations of stress in a shear for any plane in the form 





