of varying Elasticity. 261 



For this end the fundamental equations of my previous 

 paper will suffice, and a reference to them will be denoted by 

 the letter a attached to the number of the equation. The 

 notation of that paper is also followed in the present. 



Let us, then, suppose a straight bar, of length I and uniform 

 cross section, to be composed of two different materials ; the 

 elastic constants of the inner material being m, n, and those of 

 the outer m x , n^ For greater generality we may suppose the 

 bar to be hollow, its inner surface being the cylinder r = b ; 

 the surface of separation and the outer surface are coaxial 

 cylinders whose radii are % and a respectively. 



From (4a), (12a), (13a), and (14a) we see that suitable 

 solutions are, in the case of longitudinal traction or pressure, 

 for the inner material, 



a) 



and for the outer, 



S 1 = 2A 1 + B, 



Ml =A^+^> (2) 



iv x =Bz ; 



where B, A, A', A l7 A/ are constants to be determined from 

 the surface-conditions. We shall suppose the bar fixed at the 

 end -; = so that w and iv x both must vanish with z. It is 

 then obvious that if the materials are to stick together through- 

 out, B must be the same in the expressions for w and for i^. 

 At each of the three surfaces r = b, r = a 1} r = a we have to 

 satisfy the equations (10 a), putting X = 1 and /jl = v = 0. Now 

 the only stresses corresponding to the solution (1) are 



? = ( m -n)S + 2n C ^=(m-n)B + 2mA-^A / , . (3) 



R, = ( m _ w )a + 2w|^=(m + n)B + 2(m-n)A; . (4) 

 and to the solution (2) in like manner, 



P 1 =(m 1 -n 1 )B + 2m 1 A 1 -?jA 1 ', (5) 



R 1 =(m 1 + %)B + 2(m 1 -/i 1 )A 1 (6) 



Thus the equations (10 a) give, neglecting any surface 

 normal forces such as atmospheric pressure, which can, how- 



