ELASTICITY AND CRYSTALLINE FORMS. 
265 
The following Table of Covariants is readily deduced from the principles stated at 
the end of § 2 : — 
("Squares of Strains . 
Covariant< 
[Tasinomic Coefficients (cd), 
Mf), 
4f) ; 
[Products of Strains . 
Covariant<! 
ya, 
a/3, 
^v, 
vk. 
X^, 
[Tasinomic Coefficients (f3y), 
( 7 ^), 
(u(3), 
4(p), 
A{vk), 
4(Xia-), 
ock, ccfJij, av, /3X, 
/3^, 
j3u. 
7 X, 
7(^, 
yp. 
2 (a?i), 2 (a^), 2{av), 2((3X), 2(/3^), 2(^v), 2 (yX), 2 ( 7 ^), 2{yv). ^ 
Each Tasinomic Function being equated to a constant, forms the equation of a 
Tasinomic Surface-, and on the geometrical properties of such surfaces depend many 
of the laws of coefficients and Axes of Elasticity. 
A convenient and expeditious mode of forming Tasinomic Functions is obtained 
by the aid of an Umhral Notation analogous to that introduced by Mr. Sylvester 
in the Calculus of Forms. 
Let each Tasinomic Coefficient be regarded as compounded of two Tasinomic 
Umbras, those umbrae being expressed by the following notation : 
(«), (( 3 ), W), if), (f, (v ) ; 
then the following equation, deduced from that of the Thlipsimetric Surface (3), by 
substituting umbrae for elementary strains according to the following Table of 
Covariance, 
Strains .. a, (3, 7 , X, v, 
Umbrae . . (a), (/3), ( 7 ), 2 (X), 2{f), 2{v), 
is the equation of the Tasinomic Umhral Ellipsoid, from which, by elimination, mul- 
tiplication, involution, addition, subtraction, and differentiation, various Tasinomic 
Functions may be deduced, 
{u)x^f{^)y^-\-{y)z^-^2{X)yz-{-2{f)zx-\-2{v)xy={(p) = \ (8 a .) 
5. Tasinomic Invariants and Spheres. 
Tasinomic Invariants are constant Isotropic functions of the Tasinomic Coefficients, 
which are deduced, either by substitution from Thlipsimetric Invariants, or directly 
from the Umhral Matrix, 
{dc) (v) (im)' 
(*') m f) > 
(f (X) ( 7 ) 
(9.) 
The following invariant is umbral of the first order: — 
( ^2 ^2 ^ 2 \ 
• ('P) = (“) + ((3) + (y) = (<’.) 
(9n.) 
