THE ELECTRIC AND LUMINIFEROUS MEDIUM. 
741 
30 . But these conditions are not sufficient to insure that the elasticity shall 
be purely rotational, and in no wise distortional. For example, as may be seen 
from the above, the elasticities of Lord Kelvin’s labile elastic-solid rether and of 
Green’s incompressible aether satisfy them. What is required is that for any dis¬ 
placement of a given portion of the medium, the total work done by both the bodily 
forcive and the surface tractions shall be expressible in terms of the rotations of its 
elementary parts alone. In the particular case in which the medium is in internal 
equilibrium in a state of strain, the part of this work which is due to bodily forcive is 
of course null ; so that the surface-tractions are then all-important. 
31 . Now let us examine a form of Wg, the dispersional part of the energy, which 
has been put forward by MacCullagh solely in order to explain the fact that the 
character of the crystalline wave-surface is not altered by the dispersional energy. 
He assumes that Wg is a function of {f, g, h) and of its vorticity or curl, and of 
the curl of that curl, say its curl squared, and so on; and he observes that if this 
quadratic function only involve squares and products of the respective components of 
odd powers of the curl, Fresnel’s wave-surface is unaltered, while if even powers 
come in, the surface is modified in a simple and definite manner it will be clear on 
consideration that if an odd power of the operator is combined with an even power, 
in any term, rotational quality of the medium must be introduced. It will be 
sufficient for practical applications to attend to the dispersional terms of lowest order. 
Since in an incompressible medium (ciirl)^ = — V“, these terms yield two possible 
forms for the dispersional part of the energy, 
and 
(V3^)2-f- (V^ if- 
or in a crystalline medium we might take the corresponding forms 
and 
« “ (V3 (v^ gf + y 3 (V^ if ; 
or we could have more generally the lineo-linear function of {f, g, U) and V“ [f, g, h) 
and the general quadratic function of (f, r), i), respectively, which would not be 
symmetrical with respect to the principal optical axes of the medium. 
The first of these forms, the intermediate case being taken for brevity, yields a 
bodily forcive 
^2 A/ 7 -A (/ dy-li djB“ g dor f\ 
\ dy dz dz djo ’ dx dy j ' 
* MacCullagh, ‘‘ On the dispersion of the Optic Axes and of the Axes of Elasticity in Biaxal 
Crystals,” ‘Phil. Mag.,’ October, 1842, ‘Collected Works,’ pp. 221-226; “On the law of Double 
Refraction,” ‘Phil. Mag.,’ 1842, ‘Collected Works,’ pp. 227-229. 
