ELASTICITY AND CRYSTALLINE FORMS. 
277 
as in equation ( 27 .), with twenty-one thlipsinomic coefficients, forming a system con- 
tragredient to that of the tasinomic coefficients. 
21. Of Tasinomic and Thlipsinomic Umhrce for Oblique Axes. 
The tasinomic coefficients for oblique axes may be regarded as compounded of 
Umbrse 
(«). (j')’ 
contravariant respectively to the elementary strains 
Pi 7} 
and consequently covariant with the squares and products of the contraordinates 
uf, vw, wu, uv ; 
and the thlipsinomic coefficients for Oblique Axes may be regarded as compounded of 
Umbrse 
{a), (&), (c), {1), (m), (n), 
contravariant respectively to the stresses 
Pi, P 2 , P 3 , 2 Q,, 2 Q 2 , 2 Q 3 , 
and consequently covariant with the squares and products of the coordinates 
x^, z^, 2yz, 2zx, 2xy. 
22 . Of the Biquadratic Surface, and of Principal Euthytatic Axes. 
For oblique as well as for rectangular axes of coordinates, the characteristic function 
of the Biquadratic Tasinomic Surface is represented by equation (19.) ; and the fifteen 
Homotatic Coefficients are covariant respectively with suitable multiples of the fifteen 
biquadratic powers and products of the contraordinates. 
If by linear transformations a system of three axes, oblique or rectangular, be found 
which reduce the characteristic function of the Biquadratic Surface to the canonical 
form, consisting of not more than nine terms, viz. — 
(^)^=(o.^)^H(W+(/)^^ 
+ 2 { (/3y)-l-2(X^) } 3 /V-|- 2 { (ya)-f 2 (/^") 2 { (a/3) -]- 2 (i/^)}^y 
+4{2(f/.v)-\-{a'k)}x^yz-\-4{2{vX)-\-{^lx)}xy^z+4{2(X[A)-{-('yv)}xyz^=l ; . (42.) 
then for that system of axes, the following six Plagiotatic Coefficients are null, 
(|3X) = 0; (yX)=0; (y^)= 0 ; ^=0; («v) = 0; (/30=0; . . (43.) 
and each of those axes is Euthytatic, according to the definition in § 7, that is to 
say, is a direction of maximum or minimum direct Elasticity (absolute or relative), 
and also a direction in which a direct elongation or compression produces a simply 
normal stress. 
There are necessarily three Euthytatic Axes at least in every solid, viz. the three 
Principal Euthytatic Axes as above described, which are normal to the faces of a 
MDCCCLVI. 2 o 
