MR. G. A. SCHOTT OH THE REFLECTION AND REFRACTION OF LIGHT. 833 
with the conditions that when ^ = 0, u = Uq, v = Vq, IT = and when 1, 
V = v^. 
Vntu-u^ + hu + 8ht" + .., v = Vq + hv + 8V'+.., n = no + 8n' + S-u" + .., 
where 
clu^ 
clu' 
dAt" 
= 0 , 
no 
+ + ^ = 0, 
n' 
rff + 
d^v 
dW f 1 ^ 
- 
^ no + -Z.2)^0 = 0, 
dno dv^ 
f/y -r — 0, 
dn' 
d^ 
dv' 
d| + ^ ~ ~ 
dU" dv" 
-i- ^ ~ = 0, 
and when ^ = 0, = u^, = Vq, n® 
nQ, u' — v' = li' = . . =0, when 1, 
-yO = i\, V = . . = 0. 
We have at once 
u 
0 — 
uq, v^ = Vq{i — ^) + no = no - iz/. {v^ - Vq) t 
- rmo 
u = 
fl f? TTO 
— Lv\^^v^ .drj — \^^—d'q-= — lvVq^ (1 — \^) —'\iv . 
V = 0. 
n' = 
U — 
V = 
[ {jjd — v‘^) vP drj = Uq . I [ dr) — 
Jo I Jo 
— LV \ V dy] — \ —^ = — — . dr). 
Jo U m- Jo«n 
lo “ i) n° . (^ — r)) dr) — (y — V-) . 'yO . (f - ^q) dq 
“ 1^(1 - ^-^^‘no. (1 — r))dr) + “* v-)v^.{l - r))dr). 
— Lvv” + [ (ju.^ — v^) U dq. 
Jo 
n" = 
In the same way as before denoting maximum value of vhj v . . . we have, 
MDCCCXCIV.-A. 5 O 
