ELASTICITY AND CRYSTALLINE FORMS. 
269 
Let the suffix 1 designate coordinates and coefficients referred to these axes. Let 
Oy, be any new pair of orthogonal axes in the plane y^z^. Then since ((3X) — ('yX) 
is covariant with {y'^ — z^)yz, it follows that 
(3\)-(yX)={2(P7).+4OT,-((3’),-(/).}.5i^^ .... (20.) 
(where (y= <y,Oy), 
a quantity which is =0 for all values of which are multiples of 45°. There are of 
course similar equations for the other metatatic planes. Hence it appears that in each 
of the three Metatatic Planes there is a pair of Diagonal Metatatic Axes, bisecting the 
right angles formed by the Principal Metatatic Axes. 
Each pair of diagonal axes is metatatic for that plane only in which it is situated. 
Thus there are in all nine metatatic axes, three orthogonal axes, and three pairs of 
diagonal axes. The diagonal axes are normal to the faces of a regular rhombic dode- 
cahedron. 
Let Oy, Oz be a pair of rectangular axes in any plane whatsoever ; Oy’ , 0;s' any 
other pair of rectangular axes in the same plane ; and let 
<yOy’=cJ ; 
then 
Cl p AfjJ 
(/3?.)'-(yXy=:{2(/3y)-h4(X^)-(/3^)-(/)}^-+{(/3X)-(yX)}cos4^', . (21.) 
a quantity which is null for eight values of cJ, differing from each other by multiples 
of 45°. Hence, in each plane in an elastic solid, there is a system of two pairs of axes 
metatatic for that plane and forming with each other eight equal angles of 45°. 
In equation (21.), make 
(3X)'-(yX)'=(3^),-(y>.).=0; 
then from equations (20.) and (21.), it is easily seen that 
2(f3y) + 4(X") — (^") — (y") = {2(^7).-f4(X")i— (/3"),— (/),}.cos4^. . . (22.) 
The trigonometrical factor cos 4iy is -f-l for all values of a; which are even mul- 
tiples of 45°, —1 for all odd multiples of 45°, and =0 for all odd multiples of 22^°. 
Hence, in every plane in an elastic solid, the quantity (22.), which may be called the 
Metatatic Difference, is a maximum for one of the two pairs of Metatatic Axes, a 
minimum of equal amount and negative sign for the other, and null for the eight 
intermediate directions. 
9. Of Metatatic Isotropy. 
A solid is Metatatically Isotropic, when if a cubical molecule, cut out in any position 
whatsoever, undergo simultaneously an elongation along one axis, and an equal and 
opposite linear compression along another axis, no tangential stress will result on the 
faces of that molecule. 
2 N 
MDCCCLVI. 
