496 
PROFESSOR THOMSON’S ELEMENTS OF 
The concurrences of the stress-components used in interpreting the differential 
equation of energy with the types of the strain-coordinates in terms of which the 
potential of elasticity is expressed, being perfect when these constitute an orthogonal 
system, each of the quantities denoted above by a, b, c,f, g, h, is unity when the six 
principal strain-types are chosen for the coordinates. The equations of equilibrium 
of an elastic solid may therefore be expressetl as follows; — 
P=A.r, Q=B?/, R=:C%, 
S=F|, T = Gn, U = H^, 
where x, y, z, rj, ^ denote strains belonging to the six Principal Types, and P, Q, 
K, S, T, U the components according to tlie same types, of the stress required to 
hold the body in equilibrium when in the condition of having those strains. The 
amount of work that must be spent upon it per unit of its volume, to bring it to 
this state from an unconstrained condition, is given by the equation 
w; = 1 -h -f W -h H . 
Def. The coefficients A, B, C, F, G, H are called the six principal Elasticities of 
the body. 
T!ie equations of equilibrium express the following propositions; — 
Prop. If a body be strained according to any one of its six Principal Types, the 
stress required to hold it so is directly concurrent with the strain. 
Examples inserted September 16, 1856. 
4 ) If a solid be cubically isotropic in its elastic properties, as crystals of the cubical class pro- 
bably are, any portion of it will, when subjected to a uniform positive or negative normal pressure 
all round its surface, experience a uniform condensation or dilatation in all directions. Hence a 
uniform condensation is one of its six Principal Strains. Three plane distortions with axes bisect- 
ing the angles between the edges of the cube of symmetry are clearly also principal strains, and 
since the three corresponding principal elasticities are ecpial to one another, any strain whatever 
compounded of these three is a principal strain. Lastly, a plane distortion whose axes coincide 
with any two edges of the cube, being clearly a principal distortion, and the principal elasticities 
corresponding to the three distortions of this kind being equal to one another, any distortion com- 
pounded of them is also a principal distortion. 
Hence the system of orthogonal types treated of in Examples (3) Art. IX., and (7) Art. X., or 
any system in which, for (11.), (Ill,), and (IV,), any three orthogonal strains compounded of them 
are substituted, constitutes a system of six Pi'incipal Strains in a solid cubically isotropic. There 
are only three distinct Principal Elasticities for such a body, and these are (A) its cubic com- 
pressibility, (B) its rigidity against diagonal distortion in any of its principal planes (three equal 
elasticities), and (C) its rigidity against rectangular distortions of a cube of symmetry (two equal 
elasticities). 
(2) In a perfectly isotropic solid, the rigidity against all distortions is equal. Hence the 
rigidity (B) against diagonal distortion must be equal to the rigidity (C) against rectangular distor- 
tion, in a cube; and it is easily seen that if this condition is fulfilled for one set of three rectangular 
planes for which a substance is isotropic, the isotropy must be complete. The conditions of perfect 
