264 Mr. C. Chree on Bars and Wires 



Thus the solutions (1) and (2) coincide, both reducing to 



u= — a-Br, 



w= Bz 



?} (19) 



where B is simply determined by (18). 



According to many foreign elasticians, following Poisson, 

 m = 2n for any elastic solid ; if this were correct, then neces- 

 sarily cr=o- 1? and the simple solution (19) would be true for 

 any combination of two elastic materials forming a cylinder, 

 whether hollow or not. This theory, however, seems to be 

 contradicted by experiment ; still if, as in the case of wire, 

 both materials are of the same metal but have been exposed 

 to different treatment, it seems by no means unlikely that the 

 variation of the elastic constants m and n will follow the same 

 law, in which case a = a i obviously. In actual w T ire, of course, 

 there is no strict surface of demarcation answering to the 

 cylinder r = a Y of the above problem ; but in many cases the 

 transition is very rapid, and the absolute amount of change 

 considerable, and in such cases the previous solutions should 

 give results comparing favourably with those obtained by 

 neglecting the change altogether, as is usual. 



The same method will apply to any number of materials in 

 contact, the surfaces of separation being coaxial cylinders. 

 Thus, suppose there to be i + 1 materials whose elastic con- 

 stants, proceeding outwards, are in order (m, n), (m v w,), . . . 

 (mi, iii). Let the surfaces of separation be in order r=a 1 . . . 

 r = ai, the outmost surface of all being r = a, and the inmost, 

 if the cylinder be hollow, r = b. Employing also suffixes to 

 distinguish the constants for the several materials, we may 

 take as the solution for the (s + l)th material, following (1), 



S S =2A S + B, 



Us= A s r+^,} (20) 



r 



w. 



= Bz; 



where, as previously, the B is the same for all the media. 

 At the common surface r — a s of this and the -sth medium we 

 have, precisely as in (9) and (10), 



(m s ^-n s ^)B-{-2m s _ l A s . 1 -^A / s . l = (m s ^n s )B 



+ 2mA- J* A/, . (21) 

 and A s _ 1 « t +^i = A,a s +^ (22) 



