76 ON THE PKOPAGATION OF WAVES [J
thence calculate in order from the pairs of simple equations Hm^1} Kn ffm-2, J£m-z: etc., until H^ and K-^ are reached; and then determine ratio B-ijA^ The procedure would entail no difficulty in any special c numerically given ; but the algebraic expression of Hl and Kl in terms Hm and Km soon becomes complicated, unless further simplifying conditi are introduced. Such simplification may be of two kinds. In the first il supposed that the total thickness between the initial and final media small relatively to the wave-lengths, so that the phase-changes occurr within the layer are of subordinate importance. In the second kind simplification the thicknesses are left arbitrary, but the changes in character of the medium, which occur at each transition, are supposed sins
The problem of a thin transitional layer has been treated by seve authors, L. Lorenz*, Van Bynf, DrudeJ, Schott§, and Maclaurin||. A j account will be found in Theory of Light by the last named. It \ therefore not be necessary to treat the subject in detail here; but it may worth while to indicate how the results may be derived from our equatic For this purpose it is convenient to revert to the original notation so far to retain a and <r. Thus in place of (17), etc., we write
Os (c, j6Ta + Sj#s) == a9ffa, o-2 (s^Ez + On-STa) = ^3^-3, etc. . . .(33;
In virtue of the supposition that all the layers are thin, the c's are nea equal to unity and the s's are small. Thus, for a first approximation, identify c with 1 and neglect s altogether, so obtaining
fti-Si = a>2^2 = • • . = amHm, ^JKi = <r2K2 = ... = <rmKm. . . .(34
The relation of H1} K^ to Hm, Km is the same as if the transition betwc the extreme values took place without intermediate layers, and the law reflection is not disturbed by the presence of these layers, as was to expected.
For the second approximation we may still identify the c's with uni while the 's's are retained as quantities of the first order. Adding togetl the column of equations constituting the first members of (32), (33), etc., find
Oi Hl + a2sj^ + ass2K3 + . . . + am_x sm__2 Km-i = amHm ; ...... (35"
and in like manner, with substitution of a- for a and interchange of K and
r2+ ... + o-m_1Sm_2^Tm_1 = crmjarm .......... (36)
* Fogg. Ann. 1860, Vol. cxi. p. 460. t Wied. Ann. 1883, Vol. xx. p. 22. J Wied. Ann. 1891, Vol. XLIII. p. 126. § Phil. Trans. 1894, Vol. CLXXXV. p. 823. || Boy. Soc. Proe. A, 1905, Vol. LXXVI. p. 49:ransmitted about one-third of the incident light. In carrying out the exposures suitable stops, cemented to the negative, must be provided to guide the lantern-plates into position, and thus to ensure their subsequent exact superposition by simple mechanical means.