260 Mr. C. Chree on Bars and Wires 



ordinarily accepted equations seems to me liable at least to 

 criticism. 



Since, however, the range of molecular action, and thus the 

 distance between these two imaginary surfaces, must be ex- 

 tremely small, the elastic stresses over these surfaces must 

 form equilibrating systems ; and these systems of stresses are, 

 without any assumption, each expressible by the ordinary 

 formulae. It would thus appear that the ordinarily accepted 

 equations between the stresses at the common surface of two 

 media are, on the whole, as trustworthy as the three equations 

 usually given for the stresses at the surface of a single medium. 



In the case of media of finite thickness, the equations I have 

 applied would thus appear as reliable as the ordinary equations 

 for a single medium ; but their extension to a continuously 

 varying medium is open to objection. If, however, as in the 

 cases worked out in my previous paper and in the present, the 

 change be so gradual as to be inappreciable at distances of the 

 same order as the range of molecular action, the results ob- 

 tained should be, at least for practical purposes, comparatively 

 satisfactory. 



If, as in the present paper, the surfaces of separation be not 

 plane, but cylindrical, no fresh difficulty is introduced, as the 

 range of molecular action may be regarded as vanishing- 

 compared to the radii of curvature. 



There are a great variety of structures whose elastic pro- 

 perties vary with the distance from a central axis, though the 

 same at corresponding points in all cross sections. Such a 

 condition of matters is sometimes intentionally produced ; as 

 when a bar, solid or hollow, is protected from the action of 

 certain fluids or gases by a coating of some other material. 

 The effect may also be due to the gradual operation of natural 

 agents, as when girders are exposed to the action of air or 

 water. Sometimes the process of manufacture involves such 

 variation in the material. Thus the process of drawing in- 

 creases the density and tenacity of the surface portions of a 

 wire, which form a sort of rind to the inner and softer por- 

 tions. Again, in metal casts of large cross section, it is well 

 known that the surface-portions differ very markedly from 

 the interior, and that in particular the strength does not by 

 any means increase as fast as the cross section. 



It thus seems desirable to find a solution for the equilibrium 

 of a cylindrical bar exposed to longitudinal traction or pres- 

 sure, or to torsion, the bar being composed of two or more 

 materials, or of one continuously varying material, such that 

 surfaces of equal elasticity are cylinders coaxial with the outer 

 and inner surfaces of the bar. 



