302 Royal Society. 



Twenty-one equations express the relations between the systems 

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

 systems 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 sir 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 " Rehuced 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 

 between certain of these coefficients and the force of gravity, and 

 fifteen equations formed by means of the conditions of equilibrium 

 of portions of the body cut off by the co-ordinate planes, and planes 

 parallel to them. 



III. The six constant terms, and the eighteen linear terms, of the 

 three dilatations or compressions and the three distortions, are com- 

 puted from the corresponding terms of the internal pressures by 

 elimination, or by means of the coefficients of extensibility and com- 

 pressibility, and of pliability. The coefficients of the co-ordinates in 

 those twenty-four terms bear linear relations to the coefficients in 

 the linear and quadratic terms of the three projections of the mole- 

 cular displacement. 



IV. The parts of the nine reduced external pressures correspond- 

 ing to the constant and linear terms of the internal pressures having 

 been determined for each element of the body's surface and sub- 



