344 G. F. Becker — Finite Elastic Stress Strain Function. 



compressibility or to pure distortion.* Now the functions are 

 related by the equation 



f(r)cp(H) = i/j{v + x) (3) 



and if v and x are alternately equated to zero 



f(v) — ip{r) and <p (x) = ip (x). 

 Hence the three functions are identical in formf or (3) becomes 



,/M/(«) =/(*+*); W 



Developing the second member by Taylor's theorem and 

 dividing by fix) gives a value for f {y\ viz : 



J(x)dx 



Since the two variables are algebraically independent, this 

 equation must answer to McLaurin's Theorem, which implies 

 that the expression containing x is constant, its value being 

 say b. Then 



%& = Mn 



Hence since y(0)=l 



fix) = e h " 



and since all three functions have the same form 

 f(v+x) = e b ( v + K ) = 1+bQ/M + . . . 



* Compare Thomson and Tait Nat. Phil., section 179. 



f This proposition is vital to the whole demonstration. Another way of ex- 

 pressing it is as follows: — If the functions are continuous, 



a * h=]+A (l + i) + B (l + t) 



9k/ 



where A, B, etc ; are constant coefficients. Then since n and k are algebraically 

 independent, or since the principle of superposition is applicable, the develop- 

 ment of a 2 is found by making h= 1 and /c=oo . Thus 



'♦I? < 



\:W 



A, B, etc., retaining the same values as before. Consequently a 2 is the same 

 function of Q/'&n that a 2 h is of Q/M. By equating a to unity and n to infinity, 

 it appears that h also is of the same form as a 2 h. 



There is the closest connection between this method of dealing with the three 

 functions and the principle, that, when an elastic mass is in equilibrium, any por- 

 tion of it may be supposed to become infinitely rigid and incompressible without 

 disturbing the equilibrium. For to suppose that in the development of a 2 h, k=co 

 is equivalent to supposing a system of external forces equilibrating the forces 

 Q/dk. This again is simply equivalent to assorting the applicability of the prin- 

 ciple of superposition to the case of traction. 



In pure elongation, unaccompanied by lateral contraction, it is easy to see that 

 h=a and that a varies as Q/6n. In this case also 6n=9k because Poisson's 

 ratio is zero. Hence without resorting to the extreme cases of infinite n or k, it 

 appears that h is the same function of 9k that a is of 6n. This accords with the 

 result reached in (5) without sufficing to prove that result. 



