ON OPTICAL THEORIES. 225 
while the ratio of the amplitudes is given by 
Fane fgiintols esl -(ngeadei? 
2 2 2) 
suena ae 
J { ie a te 
We can give asort of physical meaning to the constants in these formulee 
as follows: \, is the wave length of the natural vibrations of the matter, 
freed from any action of the ether; Ay is their wave length on the suppo- 
sition that the action between the ether and matter is proportional to 
the displacement, while the ether remains fixed; while v, and v9 are the 
frequencies of these vibrations. B yanishes when there is no matter 
present, and since the expression shows that B/m is a number, it is 
probable that B will be proportional to the matter density ; while K is a 
number on which the strength of the frictional retardation depends. 
The quantity 4,, the wave length of the free vibrations (i.e. the dis- 
tance the light-wave travels in a natural free matter period) is immensely 
great compared with X, so that A is small compared with YI, except in the 
cases in which \ does not differ greatly from \o. 
It will be seen at once that the formula for F, on which, when the 
absorption is small, the refractive index depends, in terms of the wave 
length is very complicated. Iam not aware that any attempts have 
been made to compare it carefully with theory. 
In the cases in which K is small (i.e., for transparent media) Ao will 
be an approximate lower limit to the wave length of the light trans- 
mitted. 
If we integrate the equation given by Lommel’s hypothesis, modified 
so as to agree with the principle of action and reaction, we find 
(34) 
B” 2 (x q 
2 2 = -) = (35) 
a2 2 2 B/ 
St 7 paces, 
where B’ is a constant related to the 8? of Lommel’s equations in the 
same manner as B is to GB? above. If, however, we take Lommel’s ex- 
pression strictly, to which he still adheres,! the sign of the fractional 
expression must be changed. 
If we retain the negative sign the formula (35) fails to represent the 
facts. Neglecting for a moment the effect of absorption, and supposing the 
ether to be of the same rigidity and density as in free space, the square of 
the refractive index will be rather less than unity for the longest waves; it 
will then decrease to a minimum value, which will be positive, and then 
rise rapidly through the absorption band, for which A=Ag, reaching a 
maximum a little above the band, from which it will again fall. Absorp- 
tion effects will only slightly modify these conclusions. Thus the 
spectrum above the band ought to be more refracted than that below, and 
except just near the band the refractive index should decrease as the 
wave length decreases. This is fatal to the theory in this form. In its 
* This becomes the expression given by Lommel on substituting B//u—K = &, 
4, = A,, B‘ =m(K — e), and interchanging m and u. 
? Lommel, ‘Zur Theorie des Lichtes,’ Wied. Ann. t. xix. p. 908. 
1885. Q 
