266 Mr. C. Chree on Bars and Wires 



r determined by the equations 



r2 dA + dA^ =Q (27) 



dr dr 



Being given the law of variation of m and n, we can from 

 these equations get a differential equation in A or in A' as is 

 desired. To determine the constants of the solution, we have 



A' 

 2mA + (m— n)B — 2n— =0 ; 



when r = a, and also when r = b if the cylinder be hollow ; if 

 it be solid, instead of the latter equation we have A' = when 

 r = 0. In obtaining these equations we have, it seems to me, 

 tacitly assumed that the force at any point is the same as if all 

 the neighbouring material, at least on one side of a plane 

 through the point, within the distance at which molecular 

 forces are sensible, were the same as at the point considered. 

 Thus, if the variation in the material were very rapid, the 

 validity of deductions from these equations might be questioned. 

 As an example of the use of (26) and (27), let us consider 

 the case of a solid cylinder of which the material has elastic 

 constants given by 



m=zm (l-\-pr)j 



n = n (l + qr) 



'.} (28) 



where m , n , p, q are absolute constants ; while pa : 1 and 

 qa : 1 are so small that terms containing their squares or pro- 

 ducts may be neglected. The form of (26) and (27), then, 

 suggests 



A = Ao(l + (»•), (29) 



where A and c are constants, the latter being of the same 

 order of quantities as p and q. Then from (27) we get 



A^-Ao-g-, (30) 



no constant being required as A! vanishes when r—0. Sub- 

 stituting (29) and (30) in (26) and retaining only the principal 

 terms, we get 



A „ B{m p-n q)+2m pA 



AoC= 2(m + rc ) ' * * ' ( 31 ) 



The surface-condition P = when r=a gives, when the above 



