830 MR. G. A. SCHOTT ON THE REFLECTION AND REFRACTION OF LIGHT. 
this latter case we shall neglect 1/m^ in the small terms, which are themselves only 
corrections due to the finite, though small, thickness of the transition layer. 
Vibrations 'per'pendicular to the plane of incidence {all theories). 
The equation to be solved is (IV.), p. 829, viz. :— d^wjd^ + w — 0, 
with the conditions that when ^ ■= 0, w — iOq, when 1, ty = iv^. 
Put w = 'Up hhd + . . . , where dhv^jd^^ = 0, d^ifijd^^ + (jjd — v^) = 0, . . . 
and when ^ = 0, uP = iVq, tv^ = . . . = 0, when ^ = I, = iv-^, iv^ = . . . = 0. 
These give 
'W^=:tVQ{l — w^= — f (ijd — v‘^)w^ .{^ — r))dy] [ {iM^ — vflV^.{l—'r})d'r),. . . 
io Jo 
whence 
dip fd^ =—IVq dw^ld^= —[ {ix~ — v^)'W^.d7j-]-i {fjd — v^).iv'^.{l—y])dr), . . . 
Jo Jo 
Let a bar - written over a quantity denote its greatest numerical value between 
^ = 0 and ^=1, e.g., p the maximum refractive index, 
is given by dw^^yd^ = 0, or by 
[ {fjd — vf . ^h(l — 7]) dr) = f {[jp — v^) ^idr). 
Jo Jo 
Hence = (p,®— v^) r) dg where ^ lies between 0 and 1, and therefore 
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^yG) < — j,2)_ Now = [Xq^ sin^ 2 q < Pq^< jjp, and therefore (p.- — w)</a“, 
_ _ _ _ _ _ q/P 
and ^y("J < |/x^. ^yG-l). It follows that iy < ^y°. {1 +1^ S'^.+ . . . ] < - -r^ 2 > 
l~2 cl 1 
which is finite as long as 8 < or We shall neglect powers of 8 
above the second, so that we may write iv = + 8®. ui. 
Then dvPjd^ — iv-^ — iVq. 
= ^yo.[ (p,2 — r/2)(i _ rjfdr) + [ {)r — vfr){l — g) dr) 
Jo Jo 
= iyo|j^/x2 (1 — r)fdr) — y| + ty^ (1 - r))dr) — y|. 
= — (1 - r)) dr) “ y} ” — y} * 
