262 
MR. MACQUORN RANKINE ON AXES OF 
2. Strains, Stresses, Potential Energy, and Coefficients of Elasticity. 
In this paper, the word “Strain" will be used to denote the change of volume and 
figure constituting the deviation of a molecule of a solid from that condition which 
it preserves when free from the action of external forces ; and the word “Stress" will 
be used to denote the force, or combination of forces, which such a molecule exerts 
in tending to recover its free condition, and which, for a state of equilibrium, is equal 
and opposite to the combination of external forces applied to it. 
In framing a nomenclature for quantities connected with the theory of elasticity, 
OXixpic is adopted to denote strain, and raaic to denote stress. 
It is well known that the condition of strain at any given point in the interior of a 
molecule may be completely expressed by means of the following six elementary 
strains, in which ri, ^ are the components of the molecular displacement parallel to 
three rectangular axes x, y, %. 
Elongations . . . ^=a; ^=/3i ^=y; 
Distortions 
dy'dz ’ dz'dx dx'dy 
It is also well known that the condition of stress at a given point may be com- 
pletely expressed, relatively to the three rectangular coordinate planes, by means of 
six elementary stresses, viz. — 
Normal Pressures . . . P,, Pg, P3, 
Tangential Pressures . . Q,, Qa, Q3; 
these quantities being estimated in units of force per unit of surface. 
Let each elementary stress be integrated with respect to the elementary strain 
which it tends directly to diminish, yrom the actual amount of that strain, to the con- 
dition of freedom ; the sum of the integrals is the Potential Energy of Elasticity of 
the molecule dxdydz, expressed in units of work per unit of volume ; viz. — 
^0 /*0 ^0 
U=l P^r/jS-fi Vfy 
«y a ^ ^ %) y 
To ^0 
-f-j Q3C?r. 
4//A •Jv 
(!•) 
I'he condition that the function U shall have the same value, in what order soever 
the variations of the different elementary strains take place, amounts to supposing, 
that no transformation of energy of the kind well distinguished by Professor Thomson 
as frictional or irreversible takes place during such variations ; in other words, that 
the substance is perfectly elastic. 
Each of the elementary stresses being sensibly a linear function of the six elementary 
strains, the Potential Energy of Elasticity is, as Mr. Green first showed, a function 
