ON OPTICAL THEORIES. 195 
and Case II.— 
pt, BF m>m'* — 2mm’ cos x 
~ 1 + mm? + 2mm’ cos x 
‘Nae (52) 
an on” — _ 2mm sin x 
r mm’? — 1 
§ 2. Cauchy ' has also given equations founded on his principle of con- 
tinuity and the assumption of a peculiar form for the refracted ray which 
agree closely with those just established. His complete theory was never 
published by himself, and was first given by Hisenlohr. It has been 
further developed and criticised in some important points by Lord 
Rayleigh. LHisenlohr? takes for the displacement in a metal at a dis- 
tance r from a source of light the expression e *’ "where X! is @ com- 
plex quantity connected with A, the wave length in air, by the equation 
A=D Re. 
Hence, using 6 and 6’ to denote the angles of incidence and refraction, 
we have 
sin 0 = Re sin 0’ . : ; : (53) 
The surface conditions of the continuity of the displacement and of 
the stresses become, as we have seen, identical with Cauchy’s conditions 
of continuity of motion in the case in which the rigidity of the ether is 
the same in the two media, and the expressions for the intensity and 
change of phase for light polarised in the plane of incidence are most 
easily obtained by transforming Fresnel’s sine formula, which is strictly 
true. 
To effect the transformation put 
cos 2a sin? 6 
CAICUR 226 le a ee renee 
R?2 
ha ate (98) 
2 gin Qy = Sin 2a sin 7) 
c? sin ze 
Then the intensity in the reflected wave is 
Bo. P=tan(f—ir) . ‘ ; : (54) 
cot f = cos (% + a) sin 2tan—1 (): 
c 
while d, the change of phase, is given by 
tan d = sin (a + w) tan 2tan-! () agphtc6 aaa 
These values agree with those given by MacCullagh if we put 
R=m a=-—y, 
* Cauchy, C. RB. t. ii. p. 427; t. viii. pp. 553, 658; t. ix. p. 727; t. xxvi. p. 86. 
Liowvilte’s Journal, t. vii. p. 338. 
* Hisenlohr, Pogg. Ann. t. civ. p. 368. 
02 
