ELASTICITY AND CRYSTALLINE FORMS. 
273 
each coefficient or function belonging to one system contravariant to the correspond- 
ing coefficient or function belonging to the other system. 
The values of the coefficients in either of those matrices are expressed in terms of 
those in the other matrix, in Mr. Sylvester’s umbral notation, by twenty-one equa- 
tions, of which the following are examples : — 
(«^) = 
{ah) = 
O), 
(i3), 
(/3), 
(a)j 
(y). 
(y)> 
(y)j 
(y)^ 
({^)> 
(/3), 
( 7 ), 
w, 
(f^)> w 
• 
(a), 
(/3), 
(y)> 
w, 
(0 
9 
(“)j 
0), 
( 7 ), 
w, 
({^)> w 
r 
((3), 
(y)j 
w. 
(f^), (*') 
9 
(32.) 
16. Of Thlipsinomic Axes. 
If, under given conditions, any symmetrical system or function of the constituents 
of one of the above matrices be null, then under the same conditions will the contra- 
variant system or function of the constituents of the inverse matrix be null or infinite. 
Therefore Systems of Thlipsinomic Axes coincide with the corresponding systems of 
Tasinomic Axes. 
17 . Platythliptic Coefficients are negative. 
It may be observed as a matter of fact, that in consequence of the largeness of 
the Euthytatic Coefficients (a^), iffi), (ffi), as compared with the other Tasinomic 
Coefficients, the Platythliptic Coefficients (^c), (ca), {ah') are generally, if not always, 
negative. 
To illustrate this, the case of Pantatic Isotropy may be taken, for which the two 
matrices have the following forms : — 
(a^) 
'(M 
(M 
0 
0 
0 
(a^) 
(he) 
{he) 
0 
0 
0 ■ 
(M 
(«^) 
0 
0 
0 
(he) 
{a^) 
{he) 
0 
0 
0 
(M 
(M 
(«^) 
0 
0 
0 
{he) 
{he) 
{a^) 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
(n 
0 
0 
0 
0 
0 
0 
(x^) 
0 
0 
0 
0 
0 
in 
0 
0 
0 
0 
0 
0 
(X^) 
0 
0 
0 
0 
0 
in . 
from which it is easily seen 
value : 
that the sole Platythliptic coefficient has the following 
— (/3r) 
{he): 
•(aT+(«")(/3y)-2(/3y)^ 
(33a.) 
The denominator of this fraction is always positive so long as {od) exceeds (j3y) ; a 
condition invariably fulfilled by solid bodies, and, in fact, necessary to their exist- 
ence. 
