Encyclopaedia Britannica > Volume 15, NIC-PAR

Oftlic
|i Rainbow.
'ait II. q p -p
angle of reflection : consequently, the fliflerence bet ween
^ the angles of incidence is equal to the difference between
the angles of reflection, or b e l—d c l~g e l~fe /, or
b e d—g ef Since therefore either the lines eg, e f or
the lines e b, e d, are equally inclined both to one an¬
other and to the surface of the drop ; the rays will be
refracted in the same manner, whether they return in
the lines e b, e d, or are reflected in the lines eg ef
But if they return in the lines c b, e d, the refraction!
when they emerge at b and (/, Would make them parallel.
Therefore, if they are reflected from one and the same
point c in the lines <? g, e f the refraction, when they
emerge at g- and/; will likewise make them parallel.
But though such rays as are reflected from the same
point in the hinder part of a drop of rain, are parallel
to one another when they emerge, and so have one
condition that is requisite towards making them effec¬
tual, yet there is another condition necessary; for rays
that are effectual must he contiguous as well as paral¬
lel. And though rays, which enter the drop in different
places, may be parallel when they emerge, those only
will be contiguous which enter it nearly at the same place?.
Bet be a drop of rain, a g the axis or diameter
of the drop, and s a a ray of light that enters the drop
at a. This ray s a, being perpendicular to both the sur¬
faces, will pass through the drop in the line ag h without
being refracted; but any collateral rays, such as those
that fall about s b^ will be made to converge to the
axis, and passing out at n will meet the axis at h : Rays
which fall farther from the axis than s b, such as those
which fall about scy will likewise he made to converge ;
but their focus will be nearer to the drop than^h.
Suppose therefore i to be the focus of the rays that fall
about 5 c, any ray s c, when it has described the line
c o within the drop, and is tending to the focus i, will
pass out of the drop at the point o. The rays that fall
upon the drop about ,y d, will converge to a'focus still
neaier than i, as at Jc. I hese rays therefore go out of
the drop at p. The rays, that fail about ^ c, will con¬
verge to a focus nearer than k, as suppose at l; and the
vav s e, when it has described the line e o within the
drop, and is tending to /, will pass out at the point o.
Ibe rays that fall still more remote from the axis will
converge to a focus still nearer Thus the ray .?/will
after refraction converge to a focus at m, which is
nearer than /y and having described the line f n with¬
in the drop, it will pass out to the point n. Now we
may here observe, that as any rays.? b or s c, fall farther
above the axis .? o, the points or o, where they pass out
behind the drop, will be farther above g; or that, as
the incident ray rises from the axis s a, the arc g n o
me'eases, till we come to some ray s d, which passes
oiii of the drop at p ; and this is the highest point where
any ray that falls upon the quadrant or quarter ax can
pass out: for any rays / e, or sf that fall higher than
^ d, will not pass out on any point above p, but at the
points o, or which are below it. Consequently,
though the arc g no p increases, whilst the distance of
the incident ray from the axis s a increases, till we come
to the ray s d; yet afterwards, the higher the ray falls
above the axis 5 a, this arc pong will decrease.
Ve have hitherto spoken of the points on the pos¬
terior part of the drop, where the rays pass out of it ;
ut this was for the sake of determining the points
trom which those rays are reflected, which do not pass
Vql. XV. Part I. 1
I C S.
out behind the drop. For, in explaining tire rainbow,
we have no further reason to consider those rays which R
go through the drop; since they can never come to the
eye of a spectator placed anywhere in the lines r vox a t
with his face towards the drop. Now, as there are
many rays which pass out of the drop between and /,
so some rays will be thence reflected : and consequent!?
the several points between g and p, which are the points
where some of the rays pass out of the drop, are like-
wise the points of reflection for the rest which do not
pass out. 1 herefore m respect of those rays which are
reflected, We may call ^ p the arc of reflection ; and
may say, that this arc of reflection increases, as the dis¬
tance of the incident ray from the axis s a increases, til!
we come to the ray .9 d; the arc of reflection is g- „ for
the ray .9 b it w g.0 for the ray s c, and gp for the rav
s d. But after this, as the distance of thelncident rav
trom the axis sa increases, the arc of reflection de¬
creases ; for og less than/ig- is the arc of reflection for
the ray x e, and ng ,s the -arc of reflection for the ray .9 f
Hence it is obvious, that some ray, which falls abov?
s d, may be reflected from the same point with some
other ray which falls below 5 d. Thus, for instance, the
ray * 6 will be reflected from the point n, and the ravs
4/ will be reflected from the same point; and conse¬
quently, When the reflected rays n r,nq, are refracted
as they pass out of the. drop at r and q, they will be pa-
raHei. But since the intermediate rays, which enter
the drop between s/and 5 b, are not reflected from the
same point a, these two rays alone will be parallel
to one another when they come out of the drop, and the
intermediate rays will not be parallel to them. And
consequently these rays r v, q t, though they are parallel
after they emerge at r and q, will not he contiguous,
and for that reason \vill not be effectual ; the ray s din
reflected from p, which lias been shown to be the limit
of the arc of reflection ; such rays as fall just above * d
and just below s d, will be reflected from nearly the’
same point//, as appears from what has been already
shown. These rays therefore will be parallel, because
they are reflected from the same point /;; and they will
likewise be contiguous, because they all of them enter
the drop at the same place very near to d. Conse¬
quently such rays as enter the drop at d, and are re-
ileeted from p the limit of the arc of reflection, will
be effectual ; since, when they emerge at the part of
the drop between a and p, they will be hath parallel
and contiguous.
If it can he shown that the rainbow is produced by
the rays of the sun which are thus reflected from drops
of lam as they fall while the run shines upon them, this
proposition may serve to show us, that this appearance
is not produced by atip rays that fall upon am/ part,
and are reflected from any part of those drops*: since
this appealance cannot he produced by any rays but
those which are effectual; and effectual rays must al¬
ways enter each drop at one certain place in the ante-
nor part of it, and must likewise be reflected from one
certain place in the posterior surface.
Prop. IV.
When rays that are effectual emerge from a drop
of lain after one reflection and two refrac¬
tions, those which are most refrangible will,
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