188 ' REPORT—1885. 
(1) Light polarised in the plane of incidence— 
sin (« — 3) . 
aha a i : 4 
R=<A shies BED cos kt + tan A sin ae (28) 
where 
° B > 
tan A = el (cos? Ptanz — sin? cot z \ ede P (29) 
(2) Light polarised at right angles to the plane of incidence— 
Gz2GR ran OT ht Ae ] 
R’=—A ia cos kt + tan A’sin kt - (30) 
where 
te ctl __Sin 2a sin 2 T sin 2¢ sin 26 7] dé ois (31) 
sin? 2a — sin? 23),| sin26 sin 2a | dz 
Now = is always small, hence A is small; but for sin 2a = sin 2), 
or tan a = p, tan A’ is infinite. 
Jamin’s results as to positive and negative reflexion are shown to 
follow, and if it be assumed that the density is approximately proportional 
to »?—1, the thickness of the variable sheet can be estimated, and is found 
to lie between 4', and +3, of the wave length. 
In criticising this theory, Lord Rayleigh, in a paper we shall shortly 
consider, has pointed out that Fresnel’s tangent formula does not express 
the result of sudden transition, and that Green’s formula, which does, 
deviates from the truth on the other side. On the electro-magnetic 
theory, however, the tangent formula is strictly true, and Lorenz’s 
investigations regain their interest. 
Another objection which Lord Rayleigh has made to the supposition 
of gradual transition, however, may be a serious one. It is that there 
should be some indication of colour in the light reflected near the polaris- 
ing angle, since it is to all intents and purposes a case of interference 
produced by a thin plate. It may, however, happen that the thickness of 
the plate is comparable with that of the black spot in Newton’s rings, 
and so, though big enough to modify the quantity of light reflected, is too 
small to show colour. According to Newton, the thickness of the black 
spot in a soap film is about 4, of a wave length, while Reinold and 
Riicker have recently determined it as 5, and these fall within the 
limits required by Lorenz to explain the variations from Fresnel’s tangent 
formula. 
In another paper! the problem of reflexion at a surface across which 
the density varies gradually has been more fully considered by Lorenz, 
and the surface conditions on either side of the variable layer are deduced 
according to a strict elastic solid theory, and lead to similar conclusions. 
§ 4. Cauchy gave the results of his theory of reflexion and refraction 
without the calculations which were supplied by Briot ? in France, and 
Beer? and EHisenlohr ‘4 in Germany. 
L. Lorenz, Pogg. Ann. t. cxiv. p. 238. 
Briot, Liouville’s Journal, t. xi. p. 305; t. xii. p. 185. 
Beer, Pogg. Ann. t. xci. and xcii. 
Eisenlohr, Pogg. Ann. t. civ. p. 346. 
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