ON OPTICAL THEORIES. 175 
F’, F’, F’” being symbolic functions obtained by substituting integral 
functions of Nas d 4 sy the periodic coefficients of F, and similarly for 
dx’ dy’ dz 
Er". 
The second memoir ! is devoted to the consideration of the problem on 
the supposition that the ether in a crystal is isotropic as regards its 
elasticity, and that the variations in density are all which we have to 
consider. Again following Cauchy, and treating the ether asa system of 
attracting and repelling points, Sarrau arrives at the equations 
du 
HBV ut Hv) Th. 1 2 . (10) 
etc., where E and F are certain connected functions depending on the law 
of force, and 6 the dilatation. 
For free space— 
E(v’) =ev’, 
LA Sa Soe 
eand f being constants. 
For the ether in a crystal, omitting the consideration of dispersion, it 
is shown that it is probable that E and F have the same forms, only 
now e and f are periodic functions of the co-ordinates. 
If we denote d/dw, d/dy, d/dz, d/dt by a, (3, y, o, respectively, then 
the equations in the crystal become, in conformity with the general rule, 
eu 7*(Fyu + Foy + F3w) + (fia + fo2 + fay), 
etc., where F, G, H, etc., f, g, h, etc., denote now symbolic functions of © 
a, B, y. 
’ “These general equations are simplified by the consideration of the 
various kinds of symmetry possible, and it is shown that in the case of 
ordinary biaxial crystals they reduce to 
i ee . dO. 
dt, =v U+ Ji da| 
Pirin cers dd | 
gitan a OF ae te ie mg a 
dw 
dé 
Greiy el hh a} 
It is further assumed that f+f,=g+g9,=h+h=0. This, of 
course, is the condition that the velocity of the normal wave should be 
zero. 
These equations are solved by putting u=Pel(/e+my+nz—wi), ete, and 
lead to E 
Poe, Ge i 
| ae ad (P1+ Qm+Rn), 
w?—f w*—g w—h 
whence 
"(2 m* n? 
ee ee ao : d 2 0GE2) 
) Liouville’s Journal, Ser. ii. t. xiii. p. 59. 
