858 MR. G, A. SCHOTT OH THPJ REFLECTTOH AHD REFRACTIOH OF LIGHT. 
These expressions are true as far as order provided -^Xj^ird > greatest 
v'alue of p. 
To get some idea of the limiting thicknesses of the film, let us compare them with 
soap-films ; Reinold and PlUCKEr estimate the thickness of a black soap-film at 
aoout HI 7 X lO”-^ centim., that of a film showing red of the 1st order at about 
2'84 X 10“® centim. Hence for 
black soap-film Xl'Iird is, for line A 10, D 8, H 6 
red of 1st order ..|, ^ 
Since the refractive indices of transparent substances lie between 1 and 3, it 
follows that a transition layer to which the above analysis is to be applicable must 
certainly be less than that necessary to show even a I’ed of the 1st order. 
§ 7. Comparison of Tlieoiy ivith Experiment—Elastic Solid Theory. 
The expression found for the change of phase is by (XII.)— 
tan {p 1. — p \ \) = 
cot (« + /3) -t- D cos ify 
cot-(a + /3) (1 -I- lJ-cos-?o) -1- D cos io cot (a + ^) + D-cos^'io 
where tan a = M tan (tQ -pfi), tan /S = M tan (7 q — ij), M ^ 
and D is 
a disposable constant. 
'I'he denominator of tan (p _L — p 11) may be written cos" f [f + cot” (« + /3)] 
-f [cot (a + y8) -p D cos ?|,]^ and this cannot vanish even to order D” unless 
a -j- /3 = ^77. 
Now, a-\- /3= ^TT gives cot (^ 0 + g) cot (4 — fj) = IVH, or 
1 — siii^ L— sin- i 
siii- ?Q — sin- 
0 HRHH ^ whence 
h) = 4a-b 
we should obtain Brewster’s angle it is necessary that M should be only a small 
instead of Brewster’s angle f = tan ^In order that 
fraction e of > which would give sin^ L ~ 
/^i" ■+ 1 + 
H'i)' 
1 
/ip + 
This, as 
is well knoivn, was pointed out by Haughton, who thought it possible that a smallei 
effective refractive index for the pressural-wave Avould lead to such a value of M, but 
the rigid theory developed above, which includes the most general theory possible, 
according to VoiGT, ivithout absorption, shows that any alteration in the refractive 
index for the pressural-waves consistent wdth keeping their velocity of propagation 
large could only produce a very sliglit change in the value of M—and that an 
increase—except at very small angles of incidence. It is clear then that a rigid 
Elastic Solid Theory cannot explain the change of phase at reflection. 
