268 
MR. MACQUORN RANKINE ON AXES OF 
7. Biquadratic Tasinomic Surface. Homotatic Coefficients. Euthytatic Axes defined. 
If the equation (8a.) of the Urnbral Ellipsoid be squared, there is obtained the fol- 
lowing equation of a Biquadratic Tasinomic Surface : — 
+ 2 { (/3y) -f 2 + 2 { (ya) + 2 V-f 2 { (a/3) -f 2 (v") } xffi 
-{-4{2((Mi>)-{-(aX)}x^yz-{-4{2(vX)-i-((3[J.)}xy^z-jr4{2{'k[/.)-\-{yv)}xyz'^ 
-\-4{(5X)y^z-y4('yX)yz^-\-4{'y(jj)z^x-y4{a(j!j)zx^-y4{av)x^y-y4{(3v)xy^=]. . (19.) 
The fifteen coefficients of this surface (which will be called the Homotatic Coeffi- 
cients) are covariant respectively with the fifteen biquadratic powers and products 
of the coordinates, with proper numerical factors. 
It is obvious, that when the fifteen Homotatic Coefficients, and the six Heterotatic 
Differences, are known for any set of Orthogonal Axes, the twenty-one tasinomic 
coefficients are completely determined. 
Mr. Haughton, in the paper previously referred to, discovered the biquadratic sur- 
face for a solid constituted of centres of force. It is here shown to exist for all solids, 
independently of hypotheses. 
Those diameters of the Biquadratic Surface which are normal to that surface, are 
axes of maximum and minimum direct elasticity, and have also this property, that 
a direct elongation along one of them produces, on a plane perpendicular to it, a 
normal stress, and no tangential stress ; so that they may be called Euthytatic Axes. 
Though such axes sometimes form Orthogonal Systems, their complete investiga- 
tion requires the use of oblique coordinates, and is therefore deferred till after the 
eighteenth section of this paper, which relates to such coordinates. 
8 . Orthogonal Axes of the Biquadratic Surface. Metatatic Axes, Orthogonal 
and Diagonal. 
By rectangular linear transformations, it is always possible to make three of the 
terms with odd exponents, or three functions of such terms, vanish from the equation 
of the Biquadratic Surface. Thus are ascertained sets of Orthogonal Axes having 
special properties. 
To exemplify this, let the rectangular transformation be such as to make the fol- 
lowing functions vanish : — 
{ (PO — W ) cy — ; { {yf) — (“^) } {z^ — x^)zx ; { (av) - {(5i >) } {x^—y’^)xy. 
A cubical molecule having its faces normal to the axes fulfilling this condition has 
the following property: — if there he a linear elongation along y, and an equal linear 
compression along z (or vice versd), no tangential stress will result round x on planes 
normal to y and z ; and similarly of other pairs of axes. 
This set of axes may be called the Orthogonal or Principal Metatatic Axes, and 
their planes, Metatatic Planes. 
