ON OPTICAL THEORIES. 161 
According to Green’s theory of double refraction, founded simply on 
the supposition that the displacements are in the wave front in a crystal, 
1 = i aay 
P=p—2L 
Q=,1—2M ; ; - (4 
R=p—2N 
The equations of motion are given by 
day de} p(T eS) ri) — a9} Onda Soke 
di? dt? dt? 
In treating of the problem of reflexion this integral is applied to the 
whole of the two media, and is transformed by partial integration into a 
volume integral, which may be written 
2 2, 2 
|\|@edy a { (Ge _ x) ou + (os -¥) de + (Gr 2 Jew, 
and a surface integral, which we may write 
[Jay dz (Kew — X, bm) 
+ dzdu (¥ év — Y, 6v,) 
+ da dy (Lew — Z, dw,). 
These two integrals must vanish separately. Green’s work as to the 
former, on which the propagation of light depends, has been considered 
by Professor Stokes. It leads to the three equations— 
d?u d fdu . dv , dw 7 
Se CA = By? 
aia idari st ake a nes 
d?y d {du , dv , dw 
panes CANE etl et toe By? : ‘ 
P ae ( ») dy (Etzt a) ke Ss (6) 
daw __ d (du , dv , dw 9 
pov=(A—B) 4 atta) Bye | 
which form the basis of the whole theory of isotropic elastic solids. 
§ 2. The latter integral equated to zero gives us the surface condi- 
tions; for over the surface, according to Green, who treats the ether in 
the two media as two separate elastic solids always in contact with each 
other, we must have 
U=U,, V=V}, W=W), ; d ; gS) 
and hence eines) ‘prrolivt ue 
eK ¥ SV; BA : ; - (8) 
These six equations determine the motion completely. 
Using Green’s notation, and considering only the case of two homo- 
geneous media, let us take the plane z=0 as the separating surface. 
Then the surface conditions become 
U=U), V=V1, W=V, 
1885. M 
