158 
PROFESSOR POWELL’S SUPPLEMENT TO REMARKS ON THE 
With the equations in this form we easily pursue the investigation of the dispersion 
formula •, and this condition is, in fact, assumed by M. Cauchy, by Mr. Kelland 
and others, though without explicit reference to the axes of elasticity. 
In the case of elli ptically polarized light we are obliged to adopt a peculiar method. 
We have here to consider not, as in the common case, a rectilinear displacement g>, 
and its resolved parts f, q, £, but a curvilinear displacement, which is the result of two 
virtual rectilinear displacements acting at right angles to each other, and in a plane 
to which the ray is perpendicular ; and one of which is always retarded behind the 
other by an interval b, which we consider as an arc less than r. In this case, there- 
fore, we must proceed (as in my paper) by making one of the coordinate axes, as x, 
coincide with the ray, (or, more correctly and generally, the normal to the wave 
surface,) whence | = 0, A | = 0, &c. The first of the three equations (2.) disappears, 
and ri, £ coincide with the components which give the elliptic motion and are of the 
forms 
71 = 2 a sin (n t — k x) 
£ = 2 (3 sin (n t — k x -f- b ) 
In pursuing the investigation on the principles now referred to, Mr. Lubbock shows 
that in this case we have always 
2 {a ^ r A z Ay sin (k A x)} = 0 "I 
f * * . „ / k A 
>.... 
. . . (4.) 
< a ip r A % A y sin 2 y n J ( 
[=01 
Thus upon the whole we have (in my notation) 
2 {a q sin 2 0} = O' 
2 {a q 2 sin 2 0} = 0 
2{?A^ =0' 
2{<?A?j} =0 
( 5 .) 
( 6 .) 
S = 2{ P A,}' 
, /; <8 
A?}] 
From the equation to the ellipse obtained from equations (3.), and since, in elliptic 
polarization, all the ellipses included in the summation must be equal and similar 
with their axes parallel, the semiaxes being respectively a sin b, and (3 sin b, where 
a = A sin i, and (3 = B sin i, it follows that we have a, (3, and b , constant for all the 
ellipses. Hence, the factors involving those terms become common multipliers to 
the several sums. 
