ELASTICITV AND CRYSTALLINE FORMS. 
263 
of those strains of the second degree, having twenty-one constant coefficients, which 
are the coefficients of elasticity of the body, and will in this paper be called the 
Tasmomic Coefficients-, that is to say, adopting Mr. Green’s notation for such 
coefficients, — 
+ (/37)/3y+(7«)7“ +(a/3)a/3 
-}-(aX)a7 -i-{(5[M)(3iJj + {yv)yv 
+ (7^)7^ ( 2 .) 
From a theorem of Mr. Sylvester it follows, that every such function as U is 
reducible by linear transformations to the sum of six positive squares, each multiplied 
by a coefficient. The nature and meaning of this reduction have been discussed by 
Professor William Thomson. 
The following classification of the Tasinomic Coefficients will be used in the 
sequel : — 
Designation of Coefficients. 
Orthotatic 
Euthytatic. 
Platytatic . 
Goniotatic . 
Elasticities. 
D irect or Longitudinal . 
Lateral 
Rigidities 
Symbols. 
(«^) m (/) 
((3y) (ya) (a(3) 
(X^) (f.^) 
Plagiotatic 
Unsymmetrical . . . {^v), &c. &c. 
The twenty-one equations of transformation by which the values of these coefficients, 
being known for any one set of orthogonal axes, are found for any other, are founded 
on the following principles. 
It is well known, that for rectangular transformations, the operations 
d d d 
dx dy dz 
are respectively covariant with 
X, y, %, 
from which it is easily deduced, that because the displacements 
1, n, 
are respectively covariant with 
X, y, 
therefore the elementary strains. 
a, 
7: 
the operations. 
d d 
d 
0 ^ 
0 ^ , 
To^ If 
d<y 
Zi—i 
d\ 
^diX 
2 M 2 
I 
