190 REPORT—1885. 
Then ¥ refers to the light wave and ® to the pressural wave; let ’ 
refer to the incident wave, ¥” to the reflected, ¥, to the refracted, so that 
yy — Wr! euiar + by + ct) + wp! ei(-ax +by + vg 
etc. Then the surface conditions become in general, if we put 
w+" =X, y — w= Y, 
i(® + ®,) SS — OX 
® {m(a'? + b?) — 2nb?} + 2nabY 
(35) 
=©, {m'(a,’? + b?) — 2n'b*} + 2n'a,b¥, . . (36) 
n{b¥,— aX + ib(aY — a,¥,)} 
= n' {(b?X — a,?¥, + ib(aY — a,¥,)} ; » G4) 
MacCullagh, in his original work, neglects the pressural waves en- 
tirely, and puts 6 = ©, = 0, deriving his result (Fresnel’s sine formula) 
from equation (35), These results are inconsistent with (36) and (37), 
and therefore wrong. To obtain the correct solution we must remember 
that m is infinitely great, while a’? + b? is vanishingly small, and m(a’? + b?) 
=Dc?. This is what has been done by Green, and applied by Lorenz 
and Lord Rayleigh to MacCullagh’s theory. 
[Cauchy puts a’? + b> = —k?. We shall consider the consequences 
of this shortly. | 
Hence (36) becomes 
Do — D/6, = Pen = nl) {™ ————e . (38) 
Cast I, n= ~’ (Green). 
Then i 
, _ p tan (¢ — ¢') (1 + M’ tan? (¢ + ¢’)}? 
B= B= eaters! ee 
R’ and R being the amplitudes of the reflected and refracted waves, and 
= ; ; 
M equal to “ ry : while the difference of phase between the incident 
? 
and refracted waves is e where 
cot e = 2 cot (¢—¢') . ; : . (40) 
while between the reflected and refracted waves it is e’, where 
cot e = s cot (¢ +’) . § : oe (aly 
Casr II. D =D’ (MacCullagh’s corrected theory). 
The equations are very complicated and lead, when the difference in 
the rigidities is very small, to two polarising angles of 224° and 674° 
respectively, results which are thus utterly at variance with experiments. 
Cauchy’s theory leads to results the same in form as Green’s, if we 
substitute — ¢ sin @ for M, « being a certain small constant. 
The solution is contained in the above equations if we talve 
a? += — PB, a? +0 = — b,?, 
