734 
MK. J. LAKAIOR ON A DYNAMICAL THEORY OF 
reclacible to MacCullagh’s by changing (^, y), Cj (®i/) ^'9> making the 
corresponding change for (/", g, li), and taking (a^, /3^, y~) = p {a~^, c~'~); Avhile 
the surface conditions are easily seen by taking [I, m. n) = (1, 0, 0) to be continuity 
of tangential elastic-solid tractions, and continuity of tangential displacement; both 
these results might of course have been foreseen from the formulee for the tractions 
in an elastic solid, without special analysis. The surface condition involving normal 
disj^lacement can be adjusted by the lability of the medium as regards simple 
elongation ; and the continuity of its coefficient, that is, of the normal forcive as 
determined by the lateral contraction, is already secured by the other surface con¬ 
ditions, provided the elasticity is continuous. The mode in which lability thus 
affects the surface-conditions in the method of variations, is the chief point that 
required illustration; the addition to the energy of § 18 of terms which form a perfect 
differential is seen to be immaterial, provided they show no discontinuity at the inter¬ 
face. 
20. It is of interest to observe that a geometrical transformation, specified by tbe 
equations"^^ 
/ x' y' \ 
{x, y, z) = pqr \^— y, , and (^, y, Q = p)qr {p^', qy , rC) , 
leads to 
dr = (//, and (/, g, h) = ixqr ^ 
and so leaves the elastic quality of a purely rotational medium unaltered. 
Also, the variational equation of MaoCullagh 
= 0 
may be expressed, so far as regards vibrations of period 27^/?^, in the form 
S ^dt 1 Jpn" + 7^2+ r) dr - 1 + hY + cdd) dr 
= 0, 
in which the distinction between co-ordinates and velocities, between irotential and 
kinetic energy, has been obliterated, if we regard n as simply a numerical coefficient. 
If in the above transformation, [p, q, r) is taken equal to {a, h, c), this variational 
equation of MaoCullagh is changed into the one appropriate to an aether of isotropic 
rotational elasticity and molotropic effective density, as discussed above ; and the 
wave-surface is changed into its polar reciprocal, which is also a Fkesnel’s surface in 
which a, h, c, are replaced by their reciprocals ; and the geometrical relations between 
the two schemes may be correlated on this basis. This mode of transformation does 
not however extend to surface integral terms, and so cannot be applied to the problem 
of reffexion. 
^ Cf. ‘ Proc. Loud. Math. Soc.,’ 1893, p. 278. 
