Whatever may be the cause of the inflection of light passing through
a small aperture, the light nearest its centre must be the least
diverted, and the nearest to its sides the most: another portion of
light falling very obliquely on the margin of the aperture, will be
copiously reflected in various directions; some of which will either
perfectly or very nearly coincide in direction with the unreflected
light, and, having taken a circuitous route, will so interfere with it,
as to cause an appearance of colours. The length of the two tracks will
differ the less, as the direction of the reflected light has been less
changed by its reflection, that is, in the light passing nearest to the
margin; so that the blues will appear in the light nearest the shadow.
The effect will be increased and modified, when the reflected light
falls within the influence of the opposite edge, so as to interfere
with the light simply inflected by that also.
But, in order to examine the consequences more minutely, it will be
convenient to suppose the inflection caused by an ethereal atmosphere,
of a density varying as a given power of the distance from a centre,
as in the eighth proposition of the last Bakerian Lecture. (Phil.
Trans. for 1801, p. 83.) Putting r = 3, and x = 1/2, I have constructed
a diagram, (Fig. 4.) which shows, by the two pairs of curves, the
relative position of the reflected and unreflected portions of any
one undulation at two successive times, and also, by shaded lines
drawn across, the parts where the intervals of retardation are in
arithmetical progression, and where similar colours will be exhibited
at different distances from the inflecting substance. The result fully
agrees with the observations of NEWTON’s third book, and with those
of later writers. But I do not consider it as quite certain, until
further experiments have been made on the inflecting power of different
substances, that Dr. HOOKE’s explanation of inflection, by the tendency
of light to diverge, may not have some pretensions to truth. I am sorry
to be obliged to recall here the assent which, at first sight, I was
induced to give to a supposed improvement of a late author. (Phil.
Trans. for 1800, p. 128.)
_Scholium_. In the construction of the diagram, it becomes necessary
to find the time spent by each ray in its passage. Since the velocity
was denoted by x^{–1/r}, on the supposition of a projectile, it
will be as x^{1/r} on the contrary supposition, (Phil. Trans. for
1801, p. 27. _Schol_. 2. Prop. I.) and the fluxion of the distance
described being ẋ/sqrt(1–yy), that of the time will be x^{(–1/r)ẋ}/sqrt(1–yy) or
rs/(1–r)· ẏ/(yy·sqrt(1–yy)), of which the fluent is r/(1–r)·s/y·sqrt(1–yy).
Therefore, with the radius x^{1–(1/r)}, describe a circle concentric with
the surfaces of the inflecting atmosphere, then the angle described
by the ray during its passage through the atmosphere, will always be
to the angle subtended by the line cut off by this circle from the
incident ray produced, in the ratio of r to r–1; and the time spent in
this passage, will be in the same ratio to the time that would have
been spent in describing this intercepted portion with the initial
velocity. For y, being equal to sx^{(1/r)–1} is the sine of the
inclination of the incident ray to the radius, where it meets this
circle; therefore, by the proposition quoted, the angle described is in
a given ratio to the angle at the centre, which is the difference of
the inclinations. Making x^{1–(1/r)} or s/y radius, the sine, instead
of y, becomes s, and the cosine sqrt((ss/yy)–ss) or s/y sqrt(1–yy),
and, when y = ss, sqrt(1–ss); therefore the line intercepted is to the
difference of the fluents as r to r–1. (See also YOUNG’s Syllabus, Art.
372.)