A MATHEMATICAL THEORY OF ELASTICITY, 
495 
Now in general there will be 21 terms, with independent coefficients, in this function ; 
but by a choice of types of reference, that is, by a linear transformation of the inde- 
pendent variables, we may, in an infinite variety of ways, reduce it to the form 
m;= C^^H- Ff -f 0;?"+ 
The equations of equilibrium then become 
the simplest possible form under which they can be presented. The interpretation 
is expressed as follows. 
Prop. An infinite number of systems of six types of strains or stresses exist in any 
given elastic solid such that, if a strain of anyone of those types be impressed on the 
body, the elastic reaction is balanced by a stress orthogonal to the five others of the 
same system. 
Article XV. — On the Six Principal Strains of an Elastic Solid. 
To reduce the twenty-one coefficients of the quadratic terms in the expression for 
the potential energy to six by a linear transformation, we have only fifteen equations 
to satisfy ; while we have thirty disposable transforming coefficients, there being five 
independent elements to specify a type, and six types to be changed. Any further 
condition expressible by just fifteen independent equations may be satisfied and makes 
the transformation determinate. Now the condition that six strains may be mutually 
orthogonal, is expressible by just as many equations as there are of different pairs of 
six things ; that is fifteen. The well-known algebraic theory of the linear trans- 
formation of quadratic functions shows for the case of six variables, (1) that the six 
coefficients in the reduced form are the roots of a “determinant” of the sixth degree 
necessarily real; (2) that this multiplicity of roots leads determinately to one, and 
only one system of six types fulfilling the prescribed conditions unless two or more of 
the roots are equal to one another, when there will be an infinite number of solutions 
and definite degrees of isotropy among them ; and (3) that there is no equality 
between any of the six roots of the determinant in general, when there are twenty- 
one independent coefficients in the given quadratic. 
Prop. Hence a single system of six mutually orthogonal types may be determined 
for any homogeneous elastic solid, so that its potential energy when homogeneously 
strained in any way is expressed by the sum of the products of the squares of the 
components of the strain, according to those types, respectively multiplied by six 
determinate coefficients. 
Def. The six strain-types thus determined are called the Six Principal Strain types 
of the body. 
