84 Mr. 0. Chree on Bars and 



whence 



»-!» A =-w F 19 > 



Thus this is the solution for a uniform normal tension or 

 pressure. M is Young's modulus, and o- the ratio of lateral 

 contraction to longitudinal expansion. 



Suppose, now, that the cylinder consists of successive por- 

 tions, or layers, of different isotropic materials, all of the same 

 circular cross section when undisturbed, the planes of separa- 

 tion being all perpendicular to the axis of the cylinder. Let 

 one end e=0 be fixed, and a uniform tension or pressure be 

 applied over the other. 



Then obviously the internal equations of equilibrium for 

 any one layer, whose elastic constants are m and n, are given 

 by (12), (13), and (14), and thus will be satisfied by a solu- 

 tion of the form (15). Also, in order that the cylindrical 

 surface be free from forces, a relation of the form (17) must 

 exist between the A and the B of each layer. 



Now suppose that, starting from the fixed end, the suffixes 

 1, 2, 3, . . . i refer to the successive layers, and that z l9 z 2 . . . 

 denote the undisturbed distances from that end of the succes- 

 sive surfaces of separation. It is clear that at a surface of 

 separation the values of w and of R, cf. (10) , for the adjacent 

 layers must be equal. The latter consideration gives 



M 1 B ] = M 2 B 2 = .., = M 2 -B i =F; . ... . (20) 

 the former, 



B^x + Oj = 6^ + 02, 



B 2 02 + C 2 = B 3 ^2 . /.o 



+ ^2 A 

 + C 8 ,f 



i- a. ) 



Further, if we suppose the initial end of the first layer held, 

 wi=0 when z=0, .*. Ci = 0. Thus if s denote the suffix of a 

 typical layer, we get 



B *=w s > ( 22 > 



C «= F Hi ~w) +z >(% - w) +•••+*«- (sh ~w)] 



Or, if l J} k, • • • denote the lengths, when undisturbed, of the 

 successive layers, 



h 1 . h 



a 



= F lh-- + wrw] ( 23 > 



