of varying Elasticity. 267 



values of m, n, A, and A' are substituted in (3) , 



2A [m 5 (l+pa) + ca(m Q + ±n )] = -B[m -n 



+ (m p—n q)a]. . . . (32) 

 The first approximation gives 



A =-<r B. 

 Writing this in (31), we get 



v a - y; g ' (33) 



Substituting this value of c in (32), we get, after reduction, 

 for a second approximation, 



A 0= -Bk + ffi7 )a J . • • • (34) 

 L ° 3m (m + 7i )J 



Introducing these values of A , c, A! inj(l), we get, after 

 reduction, 



«=-<, Br[l+| ? ^rf( V +m r)] ; . . (35) 



while throughout, 



w = Bz. 



It should be noticed that correct values of P and R are to be 

 obtained only from (3) and (4), substituting therein the above 

 values of A, A', m, and n. 



If, as previously, F denote the total traction over the 

 terminal section, we get 



F= f a 27rrRdr, 



where R has the value (4) , when m, n, and A are regarded as 

 variables given by (28), (29), (33), and (34). 

 This equation easily leads to 



F=^M B[l + |a ? +|^- ? )3^-], . (36) 



where M is the value of Young's modulus for the material at 

 the axis. 



When — is constant, p = q ; and the above value of B 



obviously agrees with that derived from (25), noticing that 

 then M = M o (l + 0r). 



The torsion of a cylinder, hollow or solid, formed of differ- 

 ent materials or of one continuously varying material, as in 

 the cases just considered, presents no difficulty. Regarding 



T2 



