Wires of varying Elasticity. 87 



E 2 ^ 2 + D 2 = E 3 e 2 -fD 3 , I (33) 



E a., +D,_,=E^,_, + D„, 1 



where Gr denotes the couple of torsion about the axis applied 

 at the second end of the cylinder. 



We thence get for a typical layer of suffix s, 



2G 1 



E s - 



iraT n s 



irar ln l n s n s ) 



and the solution, 



v =mh. + ... + h.^]. . . (34) 



Thus, at the second end the displacement is 



y= ^ 2: =;A) (35) 



These results are also obviously true when we pass to the case 

 when the material of the cylinder varies continuously after the 

 fashion already considered ; for this case, at a distance z from 

 the fixed end, 



V =^{ n- •■■•■■• (36) 



a. 



while the stress is given by 



a 2Gr 



S= ^ (37) 



ira 



The vibrations of an elastic cylinder of the kind whose 

 equilibrium we have already considered may be treated in the 

 same method. 



As regards the torsional vibrations, the ordinarily accepted 

 equations are perfectly satisfactory, as they satisfy the surface- 

 conditions without making any assumption as to the cross 

 section of the bar being small compared to the length ; thus, 

 in applying them to a bar formed by successive layers, no 

 error can possibly be introduced by regarding the layers as 

 thin. Thus, the results afterwards obtained for. the torsional 

 vibrations are more to be relied on than those obtained for the 

 longitudinal, which, however, as being best known, we shall 

 first consider. 



For the longitudinal vibrations of a uniform bar, we shall 



