[ 301 j 

 XXXIX. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from p. 237.] 



June 21, 1855. — The Lord Wrottesley, President, in the Chair. 

 n'^HE following communication was read : — 



-*- " On Axes of Elasticity and Crystalline Forms." By W. J. 

 Macquorn Rankine. C.E., F.R.SS.L. & E. &c. 



An Axis of Elasticity is any direction with respect to which 

 any kind of elastic force is symmetrical. 



In this paper the deviation of a molecule of a solid from that con- 

 dition as to volume and figure which it preserves when free from the 

 action of external forces, is denoted by the word " Strain," and the 

 corresponding effort of the molecule to recover its free volume and 

 figure by the word " Stress." 



In devising a nomenclature for quantities relating to the theory 

 of elasticity, strain is denoted in composition by d\i\pis, and stress 

 by Tciais. 



Everj' possible strain of a molecule, when referred to rectangular 

 axes, may be resolved into six elementary strains ; three elongations 

 or linear compressions, and three distortions. Every possible stress, 

 when referred to rectangular axes, may be resolved into six elemen- 

 tary stresses ; three normal pressures, and three tangential pressures, 

 which tend to diminish the corresponding elementary strains. 



The elementary strains being in fact approximately linear func- 

 tions of the elementary stresses, are treated in this investigation as 

 exactly so. 



The sum of the six integrals of the elementary stresses, each taken 

 with respect to the corresponding elementary strain from its actUcil 

 amount to zero, is the Potential Energy of Elasticity, and is a homo- 

 geneous function of the elementary strains of the second order, and 

 of twenty-one terms, whose constant coefficients are here called the 

 Tasinomic Coefficients, or coefficients of Elasticity. 



The principles of the Calculus of Forms, and in particular the 

 Umbral Notation of Mr. Sylvester, are applied to the Orthogonal 

 Transformations of the Tasinomic Coefficients. 



Several functions of these coefficients are determined, called Tasi- 

 nomic Invariants, which are equal for all systems of orthogonal axes 

 in the same solid. 



Certain functions of the Tasinomic Coefficients constitute the 

 coefficients of two Tasinomic Ellipsoids, designated respectively as 

 the Orthotatic and Hetcrotatic Ellipsoids, whose axes have the fol- 

 lowing properties. 



Orthotatic Axes. 

 At each point of an elastic solid there is one position in which a 

 cubical molecule may be ait out, such, that a uniform dilatation or 

 condensation of that molecule by equal elongations or compressioris of 

 its three axes, tvill produce no tangential stress at the faces of the 

 molecule. 



