198 



the Coefficients of Elasticity of the body, for the system of orthogonal 

 axes chosen. 



Twenty-one equations express the relations between the systems 

 of coefficients of elasticity in a given body for any two different sy- 

 stems of orthogonal axes. 



When a body possesses a system of orthogonal axes of elasticity, 

 its coefficients of elasticity, when referred to these axes, are reduced 

 to nine. 



A body isotropic with respect to elasticity has but three coefficients 

 of elasticity, which are the same for all sets of orthogonal axes, and 

 are connected with each other by an equation. 



If the Potential Energy of Elastic Forces be expressed as a homo- 

 geneous quadratic function of the six elastic pressures, its coefficients 

 constitute the coefficients of compressibility and extensibility, and of 

 pliability. They have relations to the coefficients of elasticity which 

 are consequences of the properties of determinants. 



The Second Section of the paper relates to the problem of the ge- 

 neral integration of the equations of the internal equilibrium of an 

 Elastic Solid, especially when it is not isotropic. The method of so- 

 lution consists of the following eight processes : 



I. The centre of gravity of the body being (in general) taken for 

 the origin of co-ordinates, the forces applied to the surface of the 

 body are subdivided into nine systems of " REDUCED EXTERNAL 

 PRESSURES, " which are of such a nature, that for any integration of 

 the external forces as originally expressed over a portion of the sur- 

 face of the body, may be substituted the sum of three integrations of 

 certain of the reduced external pressures over the three projections 

 of that portion of the surface upon-the co-ordinate planes. 



By such integrations, extended to the whole of the body, are 

 found the mean values of the nine reduced external pressures, which 

 are connected by simple equations with the mean values, or constant 

 terms, of the six internal elastic pressures. 



The deviations of the reduced external pressures above and below 

 their mean values, constitute nine systems of variable parts of those 

 pressures. 



II. The eighteen coefficients of the three co-ordinates in the linear 

 terms of the six internal elastic pressures are determined by means 

 of eighteen equations ; viz. three equations of internal equilibrium 



