MR. POWER ON THE ABSORPTION OF THE SOLAR RAYS, ETC. 
21 
being - perpendicular to the plane of incidence and therefore parallel to each other, we 
have not two equations, as in the last case, but only one, namely, 
k=k'-\-k t ; 
but 0, being already determined, we have sufficient data for determining k' and k r 
Putting the equation of vis viva under the form 
{k-kY, 
we see that it is satisfied by making either k—k' = 0, 
a ^\k+k') = ^(k-k>). 
or 
The first solution gives k'=k and k ( — 0, so that the ray is totally reflected ; the vis 
viva of the reflected ray being equal to that of the incident ray. 
The second solution gives in conjunction with 
or 
whence we obtain 
-5 sin 0 ,=—i sin 6 . 
OL / 
sin 6 cos 6 . (k+k 1 ) — sin 0 , cos 0 l {k—K) i 
(sin 20+ sin 20,)&' = (sin 20,— sin 20 ) k, 
ft' _ _ ft sin (20) — sin (23 ,) 
" sin (29) + sin (29,) 
— _ft tan 
’ tan (9+9,) 
k,=k-K=k. Un (»+V+ tan ( 9 -+ 
tan (9 + 9,) 
or 
k.=k.- 
1 c 
2 sin 29 
sin 29 + sin 29, 
The vis viva of the reflected ray is therefore 
2 ^ 2 „ , oA tan 2 (9 -9,) 
— 3 - avoi cos 0. — 5+ — 
« tan 2 (9 + 9,)’ 
and its relative brightness, compared with that of the incident wave, namely, 
27T 2 /t 2 „ . tan 2 (9 -9,) 
am, cost, is 
The vis viva of the refracted wave is 
2 tt 2 A 2 4 sin 2 29 
o ay co cos u i~? * r%/\ . • o a \ o 5 
a? i 1 (sin 20 + sin 2 0 y ) 2 
and its relative brightness is obtained by first multiplying it by 
dividing it by 
m cos 9, 
and then 
/ru/.i nr\ c 
/ . 
2tt 2 ^ 2 
