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XX. An Analytical Discrimination of Elastic Stresses in an 

 Isotropic Body. By R. F. Gwyther, M.A* 



Sir G. B. Airy obtained from mechanical considerations 

 (British Association Reports, Cambridge, 1862) a solu- 

 tion in Cartesian coordinates of the mechanical stress- 

 equations, but he ignored all elastic requirements. In one 

 sense this paper may be regarded as an extension of Airy's 

 scheme, though it has nothing in common with that scheme 

 either in general plan or detail. 



The method is purely analytical, depending upon general 

 solutions of the mechanical stress-equations and upon the 

 development of a scheme for the selection of those stress- 

 systems which satisfy the stress-strain relations, briefly 

 called Hooke's law, from the general mechanical stress- 

 systems. 



It is shown that the elements of a mechanical stress 

 depend upon an arbitrary primary stress-system, and, to 

 form a connexion with the stress-strain relations, I introduce 

 a subsidiary, but allied, stress-system which is such that the 

 vector system naturally deduced from it possesses inherent 

 qualities distinctive of the displacement corresponding to an 

 elastic stress-system. 



The main body of the paper consists in discussing the 

 requirements necessary to ensure that the stress-system 

 should be an elastic stress-system. 



The displacement, according to this method, becomes 

 somewhat incidental, however necessary, and the elements 

 of stress are given prominence. There are no displacement 

 equations. 



In the first instance I deal with a body under tractions 

 only, and extend the scope of the results later. 



I. Introduction. 



1. By first treating certain ancillary matters as lemmas, 

 the steps in the final stages can progress more 

 steadily. 



The mechanical equations of stress in a body under 

 * Communicated by the Author. 



