ON DOUBLE REFRACTION. 263 
which as a whole should possess this number of arbitrary constants, could be 
built up of isotropic matter. 
Green supposes, in the first instance, that the medium is symmetrical with 
respect to planes in three rectangular directions, which simplifies the investi- 
gation and reduces the twenty-seven or twenty-one arbitrary constants to 
twelve (entering the partial differential equations of motion in such a manner 
as to be there equivalent to only eleven) or nine. It may be useful to give a 
Table of the constants employed by Green, with their equivalents in the theo- 
ries of Cauchy and Neumann, the density of the medium at rest being taken 
equal to unity for the sake of simplicity. The Table is as follows :— 
Green ABC GHI LMN PQR 
Cauchy GHI LMN P QR PQR 
Neumann 000 DCB AAA, AAA; 
so that Green’s equations are reduced to Cauchy’s by making = 
L=P, M=Q, Neko aogier «0 a 
For a plane wave propagated in any given direction there are three velocities 
ef propagation, and three corresponding directions of vibration, which are 
determined by the directions of the principal axes of a certain ellipsoid U=1, 
which he proposes to call the ellipsoid of elasticity, the semiaxes at the same 
time representing in magnitude the squared reciprocals of the corresponding 
velocities of propagation; and Green has shown that U may be at once 
obtained from the function —2% by taking that part only which is of the 
second order in u,v, w, and replacing u,v, w by 2, y, 2, and the symbols of 
Lod ; é 
differentiation > ala by the cosines of the angles which the wave-normal 
makes with the axes. This applies whether the medium be symmetrical or 
not with respect to the coordinate planes. Green then examines the conse- 
quences of supposing that for two of the three waves the vibrations are strictly 
in the front of the wave, as was supposed by Fresnel, and consequently that 
the vibrations belonging to the third wave are strictly normal. This hypothesis 
leads to five relations between the twelve constants, namely 
G=H=I=yp suppose, P=p—2L, Q=p—2M, R=p-—2N; . (7) 
and gives for the form of the fundamental function 
du du dw 
—2 @=2A—+2B —+2C —_ 
Sheu aa dy e dz 
+41 (ce) +(@5) +(az) | + | Ga) +a) +) | 
+l) Ge) +e) } ror) 
+N{ (74g) 4% dt ee a ee eee) 
from which the equations of motion, the expressions for the internal pressures, 
and the equation of the ellipsoid of elasticity may be at once written down. 
- The simpler case in which the medium in its natural state is supposed free 
