470 = oS. W. Gibbs—Double Refraction and Circular 
The axes of the displacement-ellipse coincide in direction with 
those of a central section of the ellipsoid (24) by a wave-plane. 
we write U,, U, for the reciprocals of the semi-axes of 
the central section of the ellipsoid (24) by a wave-plane, 
being the reciprocal ist the one to which the displacement a, 
A, 7: 18 parallel, we 
aa +bB Ye +ey, ee We (a+ f,°+7”,’), (25) 
as is at once evident, if we substitute the codrdinates of an 
extremity of the axis, for the proportional quantities a, (3, 7. 
So also 
G0" + O82 + ey. + PoVotfY 22+ J Af.=U,* (ay + B+ 2"). (26) 
If we write V for the velocity of Poe of the system 
of progressive waves corresponding to the system of stationary 
waves which we have been considering, we shall have 
; : 
Vv=-. 
< (27) 
By most? (22), (25), and (26), equations (18) and (20) are 
reduced to the f 
Usp Lop.=V'p!, Us" po ‘+ Fpp=V'p:, (28) 
where we are to read + or — according as the ane has 
the character of a rae handed or a left-hande 
progressive system of waves, when the sonic bneatich of displace- 
ments has the character of a right-handed screw, the rotations 
will be such as appear clock-wise to the observer, who looks in 
the direction opposite to be tek propagation of light. We 
shall call such a ray right-ha 
We may here observe Hes in aH y=0 the solution of these 
a is very simple. We have oe either p.=0 
=U;, or pj=0 and V*=U,". In this case, the light is 
linearly polarized, and the directions of veritas and the 
velocities of propagation are given by Fresnel’s law. Experi- 
ment has shown that this is the usual case. We wish, however, 
to sins See the case in which ¢ does not vanish. Since the 
term containing a arises from the consideration of those 
ciuantities which was allowable to neglect in the first 
approximation, we may, assume that ¢ is always very small in 
comparison with V*, or U,*. 
17. Equations (28) may ie written 
2 Pp Pr Oy YP 
ViUSty 5 V-US=2e (29) 
Pr 
Py 
