270 
MR. MACQUORN RANKINE ON AXES OF 
For such a substance, the metatatic differences must be null for all sets of axes, viz. — 
2(i3y)+4(X^)-(/3^)-(/)=0;- 
2(ya)+4(^") — (y")-(a")=0; > (23.) 
2(a/3)+4(i/") —(«")—(, S") = 0. 
In a paper in the Cambridge and Dublin Mathematical Journal, vol. vi., this theo- 
rem was alleged of all Homogeneous solids, it having been, in fact, tacitly taken for 
granted, that Homogeneity involves Metatatic Isotropy, as above defined. 
10. Of Orthotatic Symmetry. 
If it be taken for granted that symmetrical action with respect to a certain set of 
axes, between the parts of a body under one kind of strain, involves symmetrical 
action with respect to the same axes under all kinds of strains, then one and the 
same set of orthogonal axes will be at once Orthotatic, Heterotatic, Metatatic, and 
Euthytatic, and for them the whole twelve plagiotatic coefficients will vanish at onee, 
and the independent tasinomic coefficients be reduced to the nine Orthotatic Coeffi- 
cients enumerated in Article 2. As long as the rigidity of solid bodies was ascribed 
wholly to mutual attractions and repulsions between centres of force, it is difficult to 
see how, with respect to homogeneous substances, the above assumption could be 
avoided. It is probable that there exist substances for which it is true. Such sub- 
stances may be said to be OrtJiotatically Symmetrical. 
Orthotatic Symmetry requires that the equation (19.) of the Biquadratic surface 
should be reducible by rectangular transformations to its first six terms, and that the 
axes so found should also be those of tlie Heterotatic Ellipsoid. The conditions 
which must be fulfilled in order that a Biquadratic function of three variables may be 
reducible by rectangular transformations to its first six terms, have been investigated 
by Mr. Boole*. 
1 1 . Of Cyhotatic Symmetry. 
Let a substance be conceived which is not only Orthotatically Symmetrical, but 
for which the three kinds of Orthotatic Coefficients are equal for the three orthotatic 
axes, viz. — 
(«') = (/3') = (r') ; (l3y) = (ya) = (af3) ; = — . . . (24.) 
Then for such a substance the Metatatic Difference may be expressed by 
2(/37)-1-4(X^)-2(«^); (25.) 
and if the body be not Metatatically Isotropic, this difference will have equal maxima 
and minima for the three Orthogonal Axes, normal to the faces of a cube, and con- 
versely, equal minima or maxima for the six diagonal axes, normal to the faces of 
a regular rhombic dodecahedron. 
• Cambridge and Dublin Mathematical Journal, vol. vi. 
