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XII. On Axes of Elasticity and Crystalline Forms. 
By William John Macquorn Ranking, C.E., F.R.SS. Lond. and Edin. 
Received June 15, — Read June 21, 1855. 
1. General Definition of Axes of Elasticity. 
As originally understood, the term ^‘Axes of Elasticity” was applied to the inter- 
sections of three orthogonal planes at a given point of an elastic medium, with respect 
to each of which planes the molecular actions causing elasticity were conceived to be 
symmetrical. 
If the elasticity of solids arose either wholly from the mutual attractions and repul- 
sions of centres of force, such attractions and repulsions being functions of the mutual 
distances of those centres, or partly from such mutual actions, and partly from an 
elasticity like that of a fluid, resisting change of volume only, it is easy to prove that 
there would be three such orthogonal planes of symmetry of molecular action in every 
homogeneous solid. 
But there is now no doubt that the elastic forces in solid bodies are not such as 
can be analysed into fluid elasticity and mutual attractions between centres simply; 
and though there are, as will presently be shown, orthogonal planes of symmetry for 
certain kinds of elastic forces, those planes are not necessarily the same for all kinds 
of elastic forces in a given solid. 
The term “Axes of Elasticity f therefore, may now be taken in a more extended 
sense, to signify all directions, with respect to which certain kinds of elastic forces are 
symmetrical ; or speaking algebraically, directions for which certain functions of the 
coefficients of elasticity are null or infinite. 
The theory of Axes and Coefficients of Elasticity is specially connected with 
that branch of the Calculus of Forms which relates to linear transformations, 
and which has recently been so greatly advanced by the researches of Mr. Syl- 
vester, Mr. Cayley, and Mr. Boole. In such applications of that Calculus as 
occur in this paper, the nomenclature of Mr. Sylvester is followed*; and by the 
adoption of the “ Umbral Notation" of that author, immense advantages are gained 
in conciseness and simplicity 
* See Cambridge and Dublin Mathematical Journal, vol. vii.; and Philosophical Transactions, 1853, 
f See the Note at the end of the paper. 
2 M 
MDCCCLVI. 
