ELASTICITY AND CRYSTALLINE FORMS. 
275 
then the coordinates are the following linear functions of the contraordinates : — 
Also, 
x-=. h{u — 
y= — \ L . . 
z= —k.2U—h{v-\-h^w. 
r^z=zx^-\-y^-\-z^-\-2c^yz-\-2c2Zx-{-2c^xy . . 
= A, — 2Ti{vw — 2h^wu — 2k ^uv. 
(36.) 
( 37 .) 
(37a.) 
Differentiations with respect to the contraordinates are obviously covariant with 
the coordinates, and vice versd ; that is to say. 
the operations 
are respectively co- 
variant with 
d 
d 
d 
d 
d 
d 
dx’ 
dy 
dz^ 
du’ 
dv 
dw 
u. 
V, 
w, 
X, 
y. 
z. 
(38.) 
By making substitutions according to the above law of covariance in the equations 
(34.), ( 37 .), (37a.), three equivalent symbols of operation are obtained, which, being 
applied to isotropic functions of the second degree, produce invariants of the first 
degree. 
19. Of Molecular Displacements and Strains as referred to Oblique Axes. 
If the displacement of a particle from its free position be resolved into three com- 
ponents, I, 7i, parallel respectively to three oblique axes, Oj?, Oy, O2, those com- 
ponents are evidently covariant respectively with the coordinates x, y, z. 
It is now necessary to find a method of expressing the strain at any particle in an 
elastic solid by a system of six elementary strains, which shall be covariant respect- 
ively with the squares and doubled-products of these oblique coordinates. This con- 
dition is fulfilled by considering the elementary strains as being constituted by the 
variations of the components of the molecular displacement with respect to the 
distances of the strained particle from three planes passing through the origin, and 
normal respectively to the three axes ; that is to say, with respect to the contraordi- 
nates of the particle, as expressed in the following equations : — 
Elongations . . . 
Quasi-Distortions . 
dv'dw' dw ' du' ^ du'dv. 
(39.) 
The six elementary strains, as above defined, are obviously covariant with the 
squares and doubled-products of the coordinates, according to the following table : — 
a, 7, X, f/., V, ] 
X*, y\ z\ 2yz, 2zx, 2xy.\ 
( 40 .) 
