495 



of the first, but with sign changed. The modulus is given for every 

 O'Ol from O'OO to 0'80. The corresponding deviations are given in 

 degrees and minutes. For each modulus there is also given the 

 mean of all the deviations in the semicircumference, for that modu- 

 lus ; by use of which, in comparison with the mean in any given 

 instance, the modulus in that instance is determined. 



XVIII. " On Axes of Elasticity and Crystalline Forms." By 

 W. J. MACQUORN RANKINE, C.E., F.R.SS. L. & E. &c. 

 Received June 14, 1855. 



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 BXtyis, and stress 

 by rdffts. 



Every 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 actual 

 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. 



