G. F. Becker — Finite Elastic Stress-Strain Function, 353 



This result may also be expressed geometrically. The 

 quantity Q/6n is simply the area swept by the radius vector of 

 the point x Q = 1, y Q = 1. This area is also the integral of 

 ydx from x = 1/a to x = 1, or the integral of xdy from y = a 

 to y = 1. Thus (p represents any one of three distinct areas. 

 In terms of hyperbolic functions, a = Sin <p + Cos <p and the 

 amount of shear is 2Sin <p. 



It appears then that isochronous vibrations imply that in 

 pure shear the area swept by the radius vector of the corner 

 of the cube, or l?ia, is simply proportional to the load. The 

 law proposed by Hooke implies that the length a — 1 is pro- 

 portional to the same load. The law commonly accepted as 

 Hooke's makes a — 1 proportional to the final stress, or (a— l)/a 

 proportional to the load. 



Value of h. — Knowing the value of «, the value of h can 

 be found without resort to the extreme case n = oo . In the 

 case of pure elongation, unattended by lateral contraction, 

 h = a and 97c = 6)i. If a l and h x are the ratios for this case, 



If three such elongations in the direction of the three axes 

 are superimposed, the volume becomes 



and this represents a case of pure dilation without distortion. 

 Here however a 1 = h x and therefore the case of no distortion, 

 irrespective of the value of n, is given by 



The values of a and h derived from the hypothesis of iso- 

 chronous vibrations when combined evidently give the same 

 value of dh which was obtained from kinematical considera- 

 tions and the definition of isotropy in equation (5). 



Law of elastic force. — Let s be the distance of a particle on 

 the upper surface of a vibrating cube from its original position 

 or 



Then the elastic force per unit volume is minus Q, or 



d~s , ,_^ , __ Ms 2 



P^--- Q=-Mln{s + l)=-Ms+ — - .... 



When the excursions of the particle from the position of no 

 strain are very small, this becomes 



d's ,_ 



p^-r, = — Ms 

 ' df 



a familiar equation leading to simple harmonic motion. 



