A MATHEMATICAL THEORY OF ELASTICITY. 
497 
or spherical isotropy are therefore expressed in terms of the conditions referred to in the preceding 
example, with the farther condition, B = C. 
A uniform condensation in all directions, and any system whatever of five orthogonal distortions, 
constitute a system of six Principal Strains in a spherically isotropic solid. Its Principal Elasticities 
are simply its Cubical Compressibility and its Rigidity. 
Prop. Unless some of the six Principal Elasticities be equal to one another, the 
stress required to keep the body strained otherwise than according to one or other of 
six distinct types is oblique to the strain. 
Prop. The stress required to maintain a given amount of strain is a maximum or 
minimum if of one of the six Principal Types. 
Cor. If A be the greatest and H the least of the six quantities A, B, C, F, G, H, the 
principal type to which the first corresponds is that of a strain reijulring a greater stress 
to maintain itAhixei any other strain of equal amount ; and the principal type to which 
the last corresponds is that of a strain which is maintained by a less stress than any 
other strain of equal amount in the same body. The stresses corresponding to the 
four other principal strain-types have each the double, maximum and minimum, 
property in a determinate way. 
Prop. If a body be strained in a direction of which the concurrences with the 
principal strain-types are /, w, n, X, g, v, and to an amount equal to r, the stress 
required to maintain it in this state will be equal to fir, where 
fl = -P B W + C V -f 4- Gy + 
and will be of a type of which the concurrences with the principal types are respectively 
Al B/n Qn Fx Gpu Hv 
n’ IT’ xF’ H’ IT' 
Prop. A homogeneously strained elastic solid, crystalline or non-crystalline, subject 
to magnetic force or free from magnetic force, has neither any right-handed or left- 
handed, nor any dipolar, properties dependent on elastic forces simply proportional 
to strains. 
Cor. 1. The elastic forces concerned in the luminiferous vibrations of a solid or 
fluid mediunj possessing the “right- or left-handed isotropic axial property,” or the 
completely “ isotropic rotatory property,” (such as quartz crystal, right- or left-handed 
tartaric acid, solution of sugar,) or the dipolar axial rotatory property discovered by 
Faraday in his heavy glass and other transparent bodies, solid and fluid, in the 
magnetic field, either depend on the heterogeneousness or on the magnitude of the 
strains experienced. 
Hence as they do not depend on the magnitude of the strain, they do depend on 
its heterogeneousness through the portion of the medium containing a wave. 
Cor. 2. There cannot possibly be any characteristic of elastic forces simply propor- 
tional to the strains in a homogeneous body, corresponding to certain peculiarities 
of crystalline form which have been observed ; for instance corresponding to the 
