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XXXI. Bars and Wires of varying Elasticity. 

 By C. Chree, B.A., Fellow of King's College, Cambridge* . 



IN a previous paper f I considered some cases of bars and 

 - wires whose elastic properties, though constant over a 

 cross section, varied from point to point of their length. In 

 the present paper a similar treatment is applied to certain 

 cases where the elastic properties vary with the distance from 

 the centre of the circular cross section, but are independent 

 of the distance of the cross section from the ends. Before 

 entering on this question, however, I wish to make some 

 remarks on the trustworthiness of the method employed in my 

 last and in the present paper. 



The mathematical determination of stress as a function of 

 strain is based upon a consideration of the molecular forces 

 operating at points in the interior of a solid. A difference 

 of opinion as to the nature of these forces has separated 

 elasticians into two schools, who differ as to the number of 

 independent elastic constants. Both schools, however, appear 

 to agree in assuming the same relation between stress and 

 strain to hold at the surface of a solid body as in its interior. 

 On this assumption, generally tacitly made, are based the 

 ordinary surface-equations whose validity seems to have been 

 seldom questioned. 



The phenomena existing at the common surface of two 

 distinct elastic solid media have not been considered by many 

 mathematicians. Green, however, in his celebrated papers on 

 light, and others have supposed the same relations between 

 stress and strain to exist in the surface- layers of each medium 

 as in its interior, and this was assumed without comment in 

 my previous paper. 



It is, however, apparent, that there will be in, say the first 

 medium, a surface-layer of molecules within the range of 

 molecular action of the molecules of the second medium. 

 Thus, when one of these molecules in the first medium is dis- 

 placed, the change in the molecular force acting on it will be 

 due partly to the action of the molecules of that medium, and 

 partly to the action of those of the second medium. It is 

 thus by no means obvious that the resultant change of mole- 

 cular force (i. e. the stress) will depend only on the elastic 

 constants of the first medium. We may in fact imagine two 

 surfaces constructed, the one in the first medium and the 

 other in the second, between which the molecules are sub- 

 jected to forces which have their seat in both media ■ and to 

 this, extremely limited, region the strict applicability of the 



* Communicated by the Author, 

 t Phil. Mag. Feb. 1886, p. 81. 



