278 
MR. MACQUORN RANKINE ON AXES OF 
Hexahedron, right or oblique as the case may be ; but in special cases of symmetry 
there are additional or secondary euthytatic axes, of which examples will now be given. 
23. Of Rhombic and Hexagonal Symmetry. 
When a solid has three oblique principal euthytatic axes making equal angles with 
each other round an axis of symmetry, and having equal systems of Homotatic 
Coefficients corresponding to them, viz. — 
(«=) = (/3’) = (/); ((3>)+2(X») = (y«)+2(f<,=) = («(3)+2Wl 
Y . . . (43 A.) 
2 igv) + (aX) = 2 (j/X) + ((3g) = 2 (X^) + (yv) J 
it may be said to possess Rhombic Symmetry, because the three oblique axes are 
normal to the faces of one Rhombohedron, and to the edges of another belonging to 
the same series, crystallographically speaking. It is evident in this case, that the 
Axis of Symmetry must be a fourth Euthytatic Axis. 
In the limiting case, when the three oblique axes make with each other equal 
angles of 120°, they lie in the same plane, normal to the axis of symmetry, and are 
normal to the faces of one hexagonal prism, and the edges of another. 
Let 0?/i denote the longitudinal axis of symmetry of the prism ; O^i any one of 
the three transverse axes perpendicular to Oy^. The equation of a section of the 
Biquadratic surface by the Plane of Hexagonal Symmetry y^Zi, is as follows : — 
0’).y;+(/).^:+2{((3y).+2(x’),}^;*;=i (44.) 
The equation of the same section, referred to any other pair of orthogonal axes 
Oy, 0^2, in the plane of y^Zi, is as follows : — 
(/3')./+(y').a'+2{(/3y)+2(X")}3/V+4{(^X)3/"+(yX)2:"}3/2: = l. . . (44 a.) 
From considerations of symmetry, it is evident that the coefficient ((3i^) must be 
null for every direction of the axis in the plane of yiZ ^ ; consequently, every 
direction Oy in that plane, for which ((3X)=0, is an Euthytatic Axis. 
To ascertain whether, and under what conditions, there are other Euthytatic Axes 
in the planes of hexagonal symmetry besides the longitudinal and transverse axes, it 
is to be considered, that for rectangular coordinates (/3x) is covariant with y^z ; 
hence, let 
Zy,Oy=^, 
then = [{2(|3y). + 4(X*), -(/),} cos 2»-(/3'). + (/),] . (45.) 
The first factor of the above expression is null for the longitudinal and transverse 
axes only. The conditions of there being additional euthytatic axes in the plane y,z, 
is, that the second factor shall vanish ; that is to say, that 
(/B^)4-(y^)i 
COS 2iy 
• (46.) 
