: 
ON OPTICAL THEORIES. 197 
with the solutions for the two media, 
Lo Le taxtbytel) 4. 2M" ¢ (—actby +e) 
C= Le larrry+} } (63) 
where 
a = = cos 6, b=" sin 6, o = RY, 
6 being the angle of incidence. If we put y? = n/D, y?; =”/D,, we get 
from the differential equations— 
Sea et tag 2 A 
raaiar peng hae == p?, say. ‘ ‘ (64) 
From this we get sin 6’= Tain 6, and hence p is the quantity which 
we have denoted by Re’. 
Ci 7 kh a 
Hence R2e ae € _ NG ‘ ; ; (65) 
Thus R? cos 2a is positive, and R? sin 2a is negative, so that 2a lies 
between 0 and — 47 and tan 2a =h/D,v. Again, in the expression for 
the refracted wave we have a, = pa when @ is zero, and hence we find 
that the real part of is positive, the imaginary part negative, so that 
finally a lies between 0 and —iz. This result is contradicted by Hisen- 
lohr’s value for silver, in accordance with which a = 83°, from which it 
follows that the real part of »? is negative, and this Lord Rayleigh says 
is tantamount to assuming the medium to be unstable. MHisenlohr ' has 
replied to this that the objection is really one to the form of equation 
assumed by Lord Rayleigh, and that according to other theories (ey. 
Helmholtz on anomalous dispersion ”) real negative values of py? are con- 
templated. With this reply we may ina sense agree. Lord Rayleigh’s 
objection is a valid one, however, against the supposition that the 
peculiar effects of metallic reflexion may be explained by the introduction 
q2n+ 1¢ 
df2r+1 
ether, and forms an insuperable argument against the attempt to account 
for the effects on a purely elastic solid theory. When, however, we come 
to consider the theories depending on the mutual reaction of the ether 
and matter, we shall see that under certain circumstances the relation 
between the periods of the ether and matter molecules may be such as to 
= a negative value to m?, and thus render possible Hisenlohr’s value 
or a. 
The general value for a, for any angle of incidence may be shown to 
be given by 
ai= == Re { cos (w+a)+e sin (w+ a) } r j (66) 
of terms such as in the differential equations of an elastic solid 
1 Hisenlobr, ‘ On the Reflexion of Light from Metals,’ Wied. Ann. t. i. 
2 See p. 220. 
