604 Mr. G. F. Becker on the Gudermannian 



vibration is not harmonic; it becomes harmonic, however, for 

 infinitesimal amplitudes *. 



Thus the angles A and B are of interest in the dynamics 

 of a vibrating elastic mass. 



Interesting also are the planes of maximum tangential 

 strain or maximum slide. In the general ellipsoid there are 

 four sets of such planes ; two of these are symmetrically 

 placed with reference to the greatest axis, make equal acute 

 angles with it, and are perpendicular to the plane of the 

 greatest and least axes. The other two are symmetrical with 

 reference to the intermediate axis with which they make 

 acute angles, and their directrices lie in the plane of this and 

 the least axis. The strain along the first pair of planes is 

 inaximax and along the second pair minimax. 



The angles which these planes make with the axes are 

 dependent wholly on pure deformation, and are independent 

 of dilatational stresses or pure rotation. They are thus re- 

 ducible to terms of pure shear. Any number of pure axial 



* The finite elastic load-strain function, whatever it is, must fulfil 

 the reciprocal conditions set forth above. It is manifest that they would 

 be satisfied by the hypothesis 



« = eQ/6fc A = eQ/9*, « 2 /i = eQ/^ 



ivhere n is the modulus of rigidity, k the modulus of cubical dilatation, 

 and M Young's modulus. To the best of my knowledge, I long since 

 proved that only this hypothesis will satisfy the conditions (Airier. Journ. 

 Sci. vol. xlvi. 1898, p. 337). Finite strain, however, seems to excite 

 almost no interest, and, so far as I know, the only authority who has 

 discussed my conclusions is Ostwald, who approved them (Zeitsch. Phys. 

 Chemie, vol. xiii. 1894, p. 13f>). These functions can, of course, be ex- 

 pressed in terms of the gudermannian complement, or, in other words, u 

 may be regarded as a load expressed in terms of an appropriate modulus. 

 The hypothesis leads to the inference that Poisson's ratio 



dy 



-dx I dy 



* I ~y 



is constant, irrespective of the state of elastic strain. Here x/x =h/ct 

 arid y/y ==ct 2 h. By integration it follows that the equation of continuity 

 is xy** = x y Q a , which is unquestionably true for the three cases o-=l/2, 

 a-=0, and cr=— 1. Of these the first is nearly realized by indiarubher 

 and the second by cork. The case of an infinitely rigid but compressible 

 mass is given by cr= — 1 or, what amounts to the same thing, the case of a 

 real mass subjected only to hydrostatic pressure. 



Only on this hypothesis will the frequency of vibrations be independent 

 of their amplitude. 



For small strains let a~h—l=f, then 



/=Q/M+ .... 

 which is Ilooke's law, and my results do not conflict in any way with 

 those of the classical investigations of elasticity, which, however, they 

 tend to simplify 



