834 ME. G. A. SCHOTT OH THE EEFLECTIOH AHD REFRACTION OF LIGHT. 
wO) < L -j- IlO) < i/yO) -j- _ y^{>i 1)^ 
^(0 < I yn(»-2) + 1 (P - v^) . yO-2), 
and since v < . 
u < Vq + y Sn + S . (l/?rd) .FI, y < y® + ^ (/u.^ — 2 ^'). . v, 
n < yy + S . (/x^ — v~). n, 
or if 0 < e < ], 0 < e' < 1, 
u. (1 — 8^. (/x^ — V")] — yS.y. {] + (l/m^)} = eu^, 
— 8®. "U (ju,^ — + y . {1 — . 8^} = e'y°, 
whence 
e«o.(l - ISV) + {1 + (IM) 
7 / ' - ■ - -- - — -:rr- - - ■ 5 
{1 - S', (/x' - v^)] {1 - iS^. /x'j - ^v\ sc (1 + (I/m^)} (^2 _ ^ 2 ) 
and this is finite, and so also are v, IT, as long as the denominator does not vanish. 
Writing this denominator in the form (1 — |-a8^) (l — |/38^), we have 
a -j- ^ = 2 (/X" — y") + /x^,. a/3 = ^ (/x' — y') {/x' — 
Now m is large, at least 10, and jx^ — y‘' is at least /xq" cos^ hence is + or 
very small negative, in which latter case /x^ — y^ is small; thus, a, the larger of the 
two, < 3/xC Hence u, v, IT are finite as long as 8 < \/ or ^ —= • 
, o V \ 7‘bo X fi 
We shall, as before, neglect 8\ . . ., but it will be necessary to go to order 8^ 
in ^ in order that the result should be correct to 8^: we shall also neglect — when 
° m~ 
multiplied by 8^, since is about 100 ('B. A. Hep.,’ 1885, p. 192). 
We have 
z=i 
(w')j ^- 4 m {vq + y^) -t- IV (y^ - Vq) “ Hq J 
IT' 
= i u" being multiplied by 8^ 
(n°)j =1 — iIq — Lv (I’l vq) 
(n')i = i = Uo- 
