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April 24, 1856. 

 The LORD WROTTESLEY, President, in the Chair. 



The following communications were read : 



I. " Elements of a Mathematical Theory of Elasticity." By 

 Professor WILLIAM THOMSON, F.R.S. Received April 16, 

 1856. 



(Abstract.) 



This paper consists of two parts : Part I. on Stresses and Strains ; 

 Part II. on the Mechanical Conditions of Relation between Stresses 

 and Strains experienced by an Elastic Solid. 



Part I. The terms Stress and Strain are used in accordance with 

 the valuable definitions by which they were first distinctively intro- 

 duced into the Theory of Elasticity by Mr. Rankine*; with only 

 this deviation ; that instead of defining a stress as the reactive force 

 exerted by an elastic body when in a condition of strain, the author 

 of the present paper defines stress as " a definite external application 

 of force to a body." 



Various well-known theorems regarding the geometrical relations of 

 the displacements among the parts of a body in a state of strain, and 

 the geometrical representation of stresses and strains are enunciated, 

 and briefly demonstrated, for the sake of convenience. A mode of 

 expressing in absolute measure the magnitude of a stress or a strain, 

 which the author believes to be new, is laid down nearly in the follow- 

 ing terms. The amount of work done by a stress applied to a body 

 of unit volume, while acquiring a strain of the same type as the stress, 

 is measured by the product of the magnitude of the stress into the 

 magnitude of the strain. 



* " On Axes of Elasticity and Crystalline Forms," Proceedings, June 21, 1855. 

 VOL. VIII. K 



