ELASTICITY AND CRYSTALLINE FORMS. 
271 
Symmetry of this kind may be called Cijhotatic, from its analogy to that of crystals 
of the Tessiilar System. 
12. Of Pantatic Isotropy. 
When a body fulfils the conditions of Cybotatic Symmetry, and at the same time 
those of Metatatic Isotropy, it is completely isotropic with respect to Elasticity, or 
Pantatically Isotropic. It has but three tasinomic coefficients, viz. tlie Euthytatic, 
Platytatic, and Goniotatic coefficients, which are equal for all sets of axes, and are 
connected by the following equation, expressing the condition of Metatatic Isotropy: 
■ («^) = (^y) + 2(?.^) (26.) 
The properties of such bodies have been fully investigated by various authors. 
13. Of Thlipsinomic Coefficients. 
If the six elementary strains a, &c. at a given point in an elastic solid, be expressed 
as linear functions of the six elementary stresses Pj, &c., these expressions will con- 
tain twenty-one coefficients of compressibility, extensibility, and pliability, which are 
the second differential coefficients of the potential energy of elasticity with respect to 
the six elementary stresses ; that energy being represented as follows : — 
u=K)f+w5+(c*)-|2+(/‘')-f+wf+('''')f 
-}- {hc)V^V^-{- (ca)P3Pi-l-(«&)PiP2+ (w.w)Q 2Q3-1- (n/)Q3Qi-I- (/m)Q,Q2 
-\-{{al) Pi-l-(^/) P2+(c/) PsiQi 
-h { (am) P, -f ( 6 m) P2-I- ( cm) Pg } Q2 
+ {(««) Pi+(M ^2-\-{cn) PsiQa (27.) 
The twenty-one coefficients in the above equation may be comprehended under the 
general term Thlipsinomic, and classified as follows : — 
Designations of Coefficients. 
Orthothliptic^ 
Euthythliptic 
Platythliptic 
Goniothliptic 
Plagiothliptic 
Properties expressed by them. 
Longitudinal Extensibilities 
Lateral Extensibilities . . 
Pliabilities 
Unsymmetrical Pliabilities . 
Symbols. 
{a?), {P), 
ic% 
{he). 
(ca). 
{ah). 
{1% 
(m"). 
{n^). 
{mn), &c. &c. 
14. Of Thlipsinomic Transformations, Umhrce, Surfaces, and Invariants. 
The equations of transformation of the Thlipsinomic Coefficients are easily deduced 
from the principle, that the operations 
d d d d d d 
are respectively covariant with 
d?^ dV^ dV^ dQ^ dQ^ dQ.^ 
l\, P3, 2Q., 2Q.„ 2Q3, 
and these with 
y\ 
s’, "^yz, 2zx, '2xy. 
2 N 2 
