246 REPORT—1885. 
These are simplified if we put— 
2 2 242 | Vis 
h? = {(v2 +1) 82 + 20%} [e+ tan 0 a): =tanf 
7 Li te 
(96) 
Se Sees. 
and the displacements in the transparent medium are then, for the incident 
wave, 
_ 2a 
» 
and for the reflected, 
PE S sin (— ae + ly + wt —f). 
S sin (aw + by +t +f), 
in this case the rigidities in the two media are supposed to be equal. 
Sir William has also worked out the problem in the case in which the 
rigidities are not equal, in the hopes that by this assumption combined 
with variations in the density—or rather effective density—the variations 
from Green’s formule in the case of light polarised at right angles to the 
plane of incidence may be accounted for. He finds, however, that 
any difference of rigidity which might, combined with a difference of 
density, be sufficient to reconcile Green’s theory with experiment would 
cause the proportion of light reflected at normal incidence to be greater 
than {(#—1)/(#+4+1)}?, and this value, given by Green’s theory, 
agrees closely with Rood’s experiments. Weare thus driven back to 
Lord Rayleigh’s case of equal rigidities in the two media. For metals, 
then, we are to have the rigidities equal, and the value of »? decreasing 
from — o when r=x, to zero when 7 =«,/N, N being some large nume- 
rical quantity, and then again augmenting from zero to unity as 7 
decreases from «,/N to 0. 
The dynamics give no foundation to a theory such as Canchy’s, in 
which p? is a complex quantity. For light polarised in the plane of inci- 
dence we have, if x and m’ be the rigidities, and 
r=n' /n, ) 
(97) 
and tan e=r{v? sec? + tan26}!) 
= 4${1+7°(v?sec? 6 + tan?6) 
f= RB. cos (az + by + wt —e) . F - (98) 
eed ene Wiieeede- boyh i 
for the incident and reflected wave ; and for the refracted wave, 
2nn - 2 cay et 
Gey tO) cos(bytot) . « « (99F 
According to these formule the reflexion is total from a metal surface at 
all angles of incidence. Sir John Conroy has recently shown that the 
loss is exceedingly small. If light be polarised in any plane, then the 
vibration in the plane of incidence is retarded relatively to that at right 
angles to that plane by the amount 2f+2e—7. If we suppose v and rv 
to be both very large numerics, this retardation becomes— 
2| tan} (+ tan 0) = tan (SS) \ 
Z 
Tv 
nt Math 
