MR. G. A. SCHOTT ON THE REFLECTION AND REFRACTION OF LIGHT. 839 
12 — , 17 ~ > and 20 f . If for a we take 1 ’5 they are 7 ^ , 9-^, H’rJ if we take 
X A, A AAA 
ix = 1‘334, the value for water, the ratios are 6 -— , 8 — , 10 . 
Reinold and Rucker found for the thickness of a black soap-film *7 X 10~® to 
1'4 X 10“® cms., for red of tlie first order 2‘8 X 10“^, blue of second order 3’5 X 10“®; 
these are in wave-lengths of yellow light ^ ^ ^, ''y • It follows that for the 
Ol> 4:0 5 
series to converge at all, the thickness of the film must be less than that of a soap-film 
giving the red of the first order. 
§ 5. Equations Determining the Constants r, r, . . . 
Vibrations Peiyendicular to the Plane of Incidence. 
The equations (V'.) (p. 837) give, on substituting for iVq . . . 
— (1 + '?’) {1 — (F — + s (1 + 8 ^ (E — ^ cos ^0 (1 “ 
— (l -)- ?’) [1 fi- (E — i^~)s "k {1 — “ 3 ^')] ~ cos ii. s, 
or 
r. {1 - t 8 /xo cos 4 - 8 ^ (F - 1^2)} - ,. (1 + §2 (E _ a^ 2 )| 
= — (1 + I 8 /ro COS Iq — 8 ^ . (F — 
— r. {1 -f 8 ^ (K—^V^)} + s.{1 — lS}x^ cose — 8 ‘^ (D — ^v~)] = 1 + 8 h(E — ^v-), 
whence 
. [(1 — t Sfjt.Q cos e — 8 ^ (F — ^w)} {1 — t Vi cos e — 8 ^ (D — 
- {1 + 8 ME - 
= — [{1 -fi 18/^0 cos Iq — 8 ^ (F — ^v^)] 11 — t 8 ej^ cos e — 8 ^ ^ ^ I 
- {1 + 8 ME~^.^)}^], 
or 
r [gQ cos e + gi cos e — I 8 (A. + gQg^ cos e cos e — 
— 8 ^. {go cos Iq (D—+ gi cos e (F— 
= + [go cos e — gi cos 7^ + t 8 . (A — gogi cos 7o cos 7^ — C) 
— 8 ^. {gocos7o (D — — gi cos (F — since D + 2 E + F = A (p. 836). 
Similarly 
s. [go cos 7o + gi cos 7i — 7 8 (A + go^i cos i.Q cos 7^ — v^) 
— 8 ^. {go cos 7o (D — -f gi cos 7^ (F — ^v-)] = 2go co 5 7o. {1 -f 8 ^. (E — ^v^)}. 
Since r = Re‘'‘, s = Se“^, we get, by changing t into — t, multiplying and dividing 
W . [(go cos 7o + gi cos 7j)2 — 2 8 w {(xq cos 7o + cos 7^) {go cos 7o (D — 
+ gi cos 7i (F — -g-v^)] + 8 ^ (A + gogi cos io cos 7^ — v‘^f'] 
= (go cos 7o — gi cos 7i)^ — 28 ^. (go cos f — g^ cos ?q) {go cos 7o (D — 
— gi cos iy (F — f r^)] + 8 '^. (A — gogi cos 7o cos 7i — v"f 
