ELASTICITY AND CRYSTALLINE FORMS. 
267 
The properties of ihQ Heterotatic Axes are expressed by the following equations : — 
(p) — (a>i) = 0; ({'X) — (/3^) = 0 ; — (yv) = 0 ; . . . . (14.) 
or by the following 
Theorem as to Heterotatic Axes. 
At each point of an elastic solid, there is one position in which a cubical molecule map 
be cut out, such, that if there be a distortion of that molecule round x (x being any one 
of its three axes) and an equal distortion round y (y being either of its other two axes), 
the normal stress on the faces normal to x arising from the distortion round x, shall be 
equal to the tangential stress round z arising from the distortion round y. 
The six coefficients of the Heterotatic Ellipsoid may be called the Heterotatic Dif- 
ferences. For a solid whose elasticity is wholly due to the mutual attractions and 
repulsions of physical points, each of those differenees is necessarily null ; therefore 
they represent a part of the elasticity which is necessarily irreducible to sueh attrac- 
tions and repulsions. There is reason to believe that part at least of the elasticity of 
every substance is of this hind. 
If this part of the elastieity of a solid be, as suggested in a series of papers in the 
Cambridge and Dublin Mathematical Journal for 1851-52, a species of fluid elasti- 
city, resisting change of volume only, the solid may be said to he Heterotatically Iso- 
tropic. The equations (14.) will be fulfilled for all directions of axes, and also the 
following equations : — 
— = — = — (15.) 
that is to say, the excess of the Platytatic above the Goniotatic Coefficient will be 
the same in every plane. 
In a substance Orthotatically Isotropic, the equations (13.) are fulfilled for all direc- 
tions, and also the following : — 
(a') + («(3)-i-(ya) = ((3")-f (/3y)-i~(a(3) = (/)-j-(ya) + ((3y), . . . (16.) 
that is to say, a uniform compression in all directions produces a uniform normal 
stress in all directions, and no tangential stress. 
The equations (16.) may be reduced to the following form : — 
(«^) - (^y) = ((3^) - (y«) = (/) - (u(3) (17.) 
In a substance which is at once Orthotatically and Heterotatically isotropic, there 
may still be eleven independent quantities amongst the tasinomic coefficients, viz. — 
Three Euthytatie Coefficients, 
The isotropic excess . . . . 
The isotropie exeess . . . . 
Six Plagiotatic Coefficients 
(«^), (^^), (/), 1 
(«^)-(/3y), ! 
m-if), f 
(/3X), (yX), {yfj, {af), {av), (/3r)J 
Such a substance may therefore be far from being completely isotropic with respect 
to elasticity. 
