462 Mr. MacCuttacu on the Laws of 
Now, from the nature of equations (1.) and (2.), the vibrations must be elliptical. 
In fact, if we put 
b=p cos} Zist-2b, n= sin} ot-2)t, (4.) 
where p, 9, s, J, are constant quantities, the differential equations will be satisfied by 
assigning proper values to s and to the ratio 1. For, after substituting in equations (1.) 
and (2.) the values of the partial differential coefficients obtained by differentiating 
formule (4.), we shall find that every term of each equation will have the same sine or 
cosine for a factor ; omitting, therefore, the common factors, and making ies we 
shall get the two following equations of condition : ; 
F=A— ae Ck, (5 ) 
2 27, C 
s=B-—"y-. (6.) 
Subtracting these, we have 
27C / 1 
i Bt (= ~k) =0, (7.) 
which, by formule (3.), becomes 
U : 
ke — al a*—0* ) sin’ 4. ha (8.) 
Let us now interpret these results. It is obvious, from formule (4.), that s is the 
velocity of propagation for a wave whose length is /, and that each vibrating molecule 
describes a little ellipse whose semiaxes p and q are parallel to the directions of w and y. 
But the number k, expressing the ratio of the semiaxes, has two values, one of which 
is the negative reciprocal of the other, as appears by equation (8.); and each value of 
Kk has a corresponding value of s determined by equation (5.) or (6.) Hence there 
will be two waves elliptically polarized, and moving with different velocities, the ratio 
of the axes being the same in both ellipses ; but the greater axis of the one will coin- 
cide with the less axis of the other. The difference of sign in the two values of &, 
shows that if the vibration be from left to right in one wave, it will be from right to 
left in the other. These laws were discovered by Mr. Airy. 
The law by which the ellipticity of the vibrations depends on the inclination ¢, 
and on the colour of the light, is contained in equation (8.). The value of the 
constant C will be determined presently. In the mean time we may observe, that 
C denotes a line, whose length is very small, compared with the length of a wave. 
a 
