kinds of coloured rays, that this period _d_, or the length of the
undulation, varies greatly, according to the [p127] colour of the
light, and that for the extreme red rays, for example, it is [more
than] half as great again as for the violet rays situated at the
other extremity of the spectrum.
It may easily be imagined that the number of different undulations is
not limited to the seven principal ones which are indicated in the
table, and that there must be a multitude of intermediate magnitudes,
and others beyond the red and the violet rays: for the ponderable
particles, of which the oscillations give rise to them, must be
subjected to forces that are infinitely varied, in the combustion
or the incandescence of the bodies which excite the motions of the
ether: and it is on the energy of these forces that the duration
of each oscillation depends, and consequently the length of the
undulation produced by it. It is found that all the undulations
comprehended [in the air] between the lengths .0000167 E.I. and
.0000244, are visible; that is, are capable of exciting vibrations in
the optic nerve: the rest are only sensible by their heat, or by the
chemical effects which they produce.
It has been remarked, that when two systems of waves differ half
an undulation in their progress, two of the semiundulations must
escape from interference; that six must be exempt when the difference
amounts to three semiundulations; and that, in general, the number
of undulations exempt from interference is equal to the number of
lengths of a semiundulation separating the corresponding points of
the two systems. While this number is very small in proportion to
that of the waves contained in each system, the motion must be nearly
destroyed, as in the case of the exemption of a single undulation.
But it may be imagined that, as we increase the difference of
the progress of the two pencils, the undulations exempted from
interference may become a material portion of each group, and that
it may finally become so great as to separate the groups entirely
from each other; and in this case the phenomena of interference would
no longer be observable. If, for example, the groups of undulations
consisted but of a thousand each, a difference of one-twentieth of an
inch in their routes would be much more than sufficient to prevent
the interference of the rays of all kinds. [p128]
But there is another much more powerful reason which prevents our
perceiving the effects of the mutual influence of the systems of
waves when the difference of their routes is considerable; which is
the impossibility of rendering the light sufficiently homogeneous:
for the most simple light that we can obtain consists still of an
infinity of heterogeneous rays, which have not exactly the same
length of undulation; and however slight the difference may be, when
it is repeated a great number of times, it produces of necessity, as
we have already seen, an opposition between the modes of interference
of the various rays, which then compensates for the weakening of some
by the strengthening of others; [while the shades of colour are not
sufficiently distinct to allow the eye to remark the difference.]
This is without doubt the principal reason why the effects of the
mutual interference of the rays of light become insensible when the
difference of the routes is very considerable, so as to amount to 50
or 60 times the length of an undulation.
It has already been laid down as one of the conditions necessary for
the appearance of the phenomena of interference, that the rays which
are combined should have issued at first from a common source: and it
is easy to account for the necessity of this condition by the theory
which has now been explained.
Every system of waves, which meets another, always exercises on
it the same influence when their relative positions are the same,
whether it originates from the same source or from different sources;
for it is clear that the reasons, by which their mutual influence
has been explained, would be equally applicable to either case. But
it is not sufficient that this influence should exist, in order that
it may become sensible to our eyes: and for this purpose the effect
must have a certain degree of permanence. Now this cannot happen when
the two systems of waves which interfere are derived from separate
sources. For it is obvious that the particles of luminous bodies,
of which the vibrations agitate the ether, and produce light, must
be liable to very frequent disturbances in their oscillations, in
consequence of the rapid changes which are taking place around them,
which may [p129] nevertheless be perfectly reconciled, as we have
seen, with the regular continuance of a great number of oscillations
in each of the series separated by these perturbations. This being
admitted, it is impossible to suppose that these perturbations should
take place simultaneously and in the same manner in the vibrations
of separate and independent particles; so that it will happen, for
example, that the motions of the one will be retarded by an entire
semioscillation, while those of the other will be continued without
interruption, or will be retarded by a complete oscillation, a change
which will completely invert the whole effects of the interference of
the two systems of undulations which originate from them; since if
they had agreed on the first supposition, they would totally disagree
on the second. Now these opposite effects, succeeding each other with
extreme rapidity, will produce in the eye a continuous sensation
only, which will be a mean between the more or less lively sensations
that they excite, and will remain constant, whatever may be the
difference of the routes described.
But the case is different when the two luminous pencils originate
from a common source: for then the two systems of waves, having
originated from the same centre of vibration, undergoing these
perturbations in the same manner and at the same instant, undergo no
changes in their relative positions: so that if they disagreed in the
first instance at any given point, they would continue to disagree at
all other times; and if their motions cooperated at first, they would
continue to agree as long as the centre of vibration continued to be
luminous: so that in this case, the effects must remain constant,
and must therefore be sensible to the eye. This is therefore a
general principle, applicable to all the effects produced by luminous
undulations; that in order to become sensible, they must be permanent.
We have hitherto supposed that the two systems of waves were moving
exactly in the same direction, and that consequently their elementary
motions, to be combined with each other, were precisely limited to
one single line: this is the simplest case of interference, and the
only one in which the one motion can be completely destroyed by the
other: [p130] for in order that this effect may be produced, not only
the two forces must be equal and in contrary directions, but they
must also act in the same right line, or be directly opposed to each
other.
The phenomenon of coloured rings, and that of the colours developed
by polarised light in crystallised plates, present a particular case
of interference, in which the undulations are exactly parallel. But
in the phenomena of diffraction, or in the experiment with the two
mirrors, which has been already described, the rays which interfere
always form sensible though very small angles with each other. In
these cases the impulses to be combined with each other at the same
points, as belonging to the two systems of undulations, will also act
in directions forming sensible angles with each other: but on account
of the smallness of these angles, the result of the two impulses
is almost exactly equal to their sum, when the impulses act in the
same direction, and to their difference, when they are in contrary
directions. Thus, in the points of agreement or disagreement, the
intensity of the light will be the same as if the directions agreed
more perfectly; at least the nicest eye will not be able to discover
any difference in them. But although, with respect to the intensity
of the light, this case of interference resembles that which has
already been considered, there are other differences which modify the
phenomenon very greatly, both with respect to its general form, and
to the circumstances necessary for producing it.
We may take, as a convenient example, the case of diverging rays
originating from the same luminous point, and reflected by two
mirrors slightly inclined to each other, so as to produce two pencils
meeting each other in a sensible angle: the two systems of waves
will then meet each other with a slight inclination; and it follows
from this obliquity, that if a semiundulation of the first system
coincides perfectly in one point with a semiundulation of the second,
urging the fluid in the same direction, it must separate from it to
the right and left of the point of intersection, and must coincide,
a little further off, on one side with the preceding semiundulation
which is in a contrary direction, [p131] and on the other side with
the following semiundulation, and then be separated from this again,
and at a distance twice as great as the first, must coincide with
the second semiundulation before and behind it, of which the actions
will coincide with its own: whence there will arise, on the surface
of this undulation, a series of lines, at equal distances from each
other, in which the motion is destroyed and doubled alternately by
the action of the second series. Thus if we receive this luminous
undulation on a white card, we shall observe on it a series of
dark and bright stripes, if the light employed is homogeneous; or
coloured fringes of different tints, if we employ white light for the
experiment.
[Illustration: untitled, diagram of two mirrors and reflected
undulations]
This will be more easily understood by the inspection of a figure,
which represents a section of the two mirrors and of the reflected
undulations, formed by a plane drawn from the luminous point
perpendicularly to the mirrors represented by DE and DF. The luminous
point is supposed to be S, and A and B are the geometrical positions
of its two images, which are determined by the perpendiculars SA
and SB falling from S on the mirrors, taking in them PA = SP [p132]
and QB = SQ. The points A and B, thus found, are the centres of
divergence of the rays reflected from the respective mirrors,
according to the well known law of reflection. Thus, in order to
have the direction of the ray reflected at any point G of the mirror
DF, for example, it is sufficient to draw a right line through B and
G, which will be the direction of the reflected ray. Now it must be
remarked, that, according to the construction by which the position
of B is found, the distances BG and SG will be equal, and thus the
whole route of the ray coming from S and arriving at _b_, is the
same as if it had come from B. This geometrical truth being equally
applicable to all the rays reflected by the same mirror, it is
obvious that they will arrive at the same instant at all the points
of the circumference _n′bm_, described on the point B as a centre,
with a radius equal to B_b_; consequently this surface will represent
the surface of the reflected undulation when it arrives at _b_, or,
more correctly speaking, its intersection with the plane of the
figure: the surface of the undulation being understood as relating
to the points which are similarly agitated at the same instant: the
points being all, at the commencement of the whole oscillation, for
example, or at the middle or the end, completely at rest; and in the
middle of each semioscillation, possessed of the maximum of velocity.
In order to represent the two systems of reflected undulations, there
are drawn, with the points A and B for their centres, two different
series of equidistant arcs, separated from each other by an interval
which is supposed equal to the length of a semiundulation. In order
to distinguish the motions in opposite directions, the arcs on which
the motions of the ethereal particles are supposed to be direct, are
represented by full lines, and the maximum of the retrograde motions
are indicated by dotted lines. It follows that the intersections
of the dotted lines with the full lines are points of complete
discordance, and of course show the middle of the dark stripes; and,
on the contrary, the intersections of similar arcs show the points
of perfect agreement, or the middle of the bright stripes. The
intersections of the arcs of the same kind are joined by the dotted
lines _b′p′_, _br_, _b′p′_, and those of arcs of [p133] different
kinds by the full lines _n′o′_, _no_, _no_, _n′o′_: these latter
representing the successive positions or the trajectories of the
middle points of the dark stripes, and the former the trajectories of
the bright bands.
It has been necessary to magnify very greatly in this figure the real
length of the luminous undulations, and to exaggerate the mutual
inclination of the two mirrors, so that we must not expect an exact
representation of the phenomenon, but merely a mode of illustrating
the distribution of the interferences, in undulations which cross
each other with a slight inclination.
It is easy to deduce from geometrical considerations, that the length
of these fringes is in the inverse ratio of the magnitude of the
angle made by the two pencils which interfere, and that the interval,
comprehended between the middle points of two consecutive dark or
bright bands, is as much greater than the length of the undulation,
as the radius is greater than the sine of the angle of intersection.
In fact the triangle _bni_, formed by the right line _bi_, and the
two circular arcs _ni_ and _nb_, may be considered as rectilinear
and isosceles, on account of the smallness of the arcs; and the
sine of the angle _bni_, considered as very small, may be called
_ib_/_bn_: so that _bn_ being the radius, _ib_ will represent the
sine of the angle _bni_, which has its legs perpendicular to those
of the angle A_b_B: consequently, these angles being equal, one of
them may be substituted for the other; and representing by _i_ the
angle A_b_B, formed by the reflected rays, we have _bn_ = _ib_/sin
_i_; consequently _nn_, which is twice _bn_, will be equal to
2_ib_/sin _i_. But _nn_ is the distance between the middle points
of two consecutive dark stripes, and is the distance which has
been called the breadth of a fringe; and _ib_ being the breadth
of a semiundulation, according to the construction of the figure,
2_ib_ will be that of a whole undulation; consequently the breadth
of a fringe may be said to be equal to the length of an undulation
divided by the [numerical] sine of the angle made by the reflected
rays [p134] with each other, which is also the angle under which the
interval AB would appear to an eye placed at _b_. We find another
equivalent formula, by remarking that the two triangles, _bni_ and
A_b_B, are similar, whence we have the proportion _bn_: _bi_ = A_b_
: AB, and _bn_ = (_bi_ × A_b_)/AB, or 2_bn_ = (2_bi_ × A_b_)/AB:
which implies that we may find the numerical breadth of a fringe
by multiplying the length of an undulation by the distance of the
images A and B from the plane on which the fringes are measured, and
dividing the product by the distance of the two images.
It is sufficient to inspect the figure, in order to be convinced of
the necessity of having the two mirrors nearly in the same plane, if
we wish to obtain fringes of tolerably large dimensions; for in the
little triangle _bni_, the side _bi_, which represents the length of
a semiundulation, being little more than the hundred thousandth of an
inch for the yellow rays, for example, the side _bn_, which measures
the half breadth of a fringe, can only become sensible when _bn_
is very little inclined to _in_, so that their intersection may be
remote from _ib_; and the inclination of _bn_ to _in_ depends on the
distance AB, which is the measure of the inclination of the mirrors.
If A and B, instead of being the images of the luminous point, were
the projections of two very fine slits cut in a screen RN, through
which the rays of light were admitted from a luminous point placed
behind the screen in the continuation of the line _b_DC, the two
paths described between the point and the slits A and B being equal,
it would be sufficient to compute the paths described by the rays,
beginning from A and B, in order to have the differences of their
lengths; and it is obvious in this case, that the calculations which
we have been making of the breadth of the fringes, produced by the
two mirrors, would remain equally applicable, at least as long as
each slit remained narrow enough to be considered as a single centre
of undulation, relatively to the inflected rays which it transmits.
It may therefore be said that the breadth of the fringes, produced
by two very fine slits, is equal to the length of an undulation
supposed [p135] to be multiplied by the interval between the two
slits, and divided by the distance of the screen from the wires of
the micrometer employed for measuring the fringes.
This formula is also applicable to the dark and bright stripes which
are observed in the shadow of a narrow substance, substituting the
breadth of this substance for the interval which separates the two
slits, as long as the stripes are far enough from the edges of the
shadow: for when they approach very near to the edges, it is shown,
both by theory and by experiment, that this calculation does not
represent the facts with sufficient accuracy; and it is not perfectly
correct in all cases, either for the fringes within the shadow, or
for those of the two slits, but only for the fringes produced by the
mirrors, which exhibit the simplest case of the interference of rays
slightly inclined to each other. In order to obtain from the theory,
a rigorous determination of the situation of the dark and light
stripes in the two former cases, it is not sufficient to calculate
the effect of two systems of undulations, but those of an infinite
number of similar groups must be combined, according to a principle
which will shortly be explained, in treating of the general theory of
diffraction.
ii. _Rule for the Correction of a_ LUNAR OBSERVATION. _By_ Mr.
WILLIAM WISEMAN, _of Hull_.
RULE.
Add together the reserved logarithm (found as directed, page 111 and
112 of the Appendix to the third edition of the Requisite Tables) the
log. sines of half the sum, and half the difference of the apparent
distance, and difference of apparent altitudes, and 0.3010300, the
log. of 2. Then, to the natural number corresponding to the sum of
these four logarithms, add the natural verse sine of the difference
of true altitudes, and the sum will be the natural verse sine of the
true distance.
Or, having obtained the natural number, as directed above, subtract
it from the natural cosine of the difference of the true altitudes,
and the remainder will be the natural cosine of the true distance.
[p136]
EXAMPLE.
(_From page 112, Appendix to Requisite Tables_.)
Reserved log. from Tables (Req.) 9th and 11th 9.9938860
Log. sin. 43° 23′ 5″ = 1/2 sum of app. dist. and diff.
app. altitudes 9.8368895
Log. sin. 6° 45′ 36″ = 1/2 diff. ditto ditto 9.0708157
Log. of 2 0.3010300
---------
Nat. num. to sum of 4 logarithms .1594488 9.2026212
Nat. vers. 37° 13′ 12″ = diff. true altitudes .2036812
-------
Nat. vers. 50° 26′ 28″ = true distance .3631300
Or, Nat. cos. 37° 13′ 12″ = diff. true altitudes .7963188
Nat. number found above .1594488
-------
Nat. cosin. 50° 26′ 28″ = true distance .6368700
DEMONSTRATION OF THE RULE.
Let M′, S′, D′, d′ and M, S, D, d, respectively denote the true and
apparent altitudes, distances, and differences of true and apparent
altitudes of the moon and sun (or a star); then will the theorem
answering to the above rule be expressed by
vers. D′ = ((2 cos M′ cos S′)/(cos M cos S)) sin 1/2(D + d) ×
sin 1/2(D−d) + vers. d′.
By Bonnycastle’s Trig. p. 175, the cosine of the angle contained
by the co-altitudes is
(cos D − sin M sin S)/(cos M cos S) =
(cos D′ − sin M′ sin S′)/(cos M′ cos S′);
consequently the verse sine of the same angle
= 1−(cos D − sin M sin S)/(cos M cos S) =
1−(cos D′ − sin M′ sin S)/(cos M′ cos S′); that is,
(cos M cos S + sin M sin S − cos D)/(cos M cos S) =
(cos M′ cos S′ + sin M′ sin S′ − cos D′)/(cos M′ cos S′).
Substituting cos d and cos d′ for cos M cos S + sin M sin S and
cos M′ cos S′ + sin M′ sin S′. (Bon. Trig. p. 282), we have
(cos d−cos D)/(cos M cos S) =
(cos d′−cos D′)/(cos M′ cos S′); whence
cos D′ = cos d′−((cos M′ cos S′)/(cos M cos S)) (cos d−cos D);
or, which is the same,
cos D′ = cos d′−((cos M′ cos S′)/(cos M cos S)) (vers D−vers d);
or, (Bon. Trig. p. 286.) [p137]
cos D′ = cos d′−((cos M′ cos S′)/(cos M cos S))·
(2 sin^2 ((1/2)D) − 2 sin^2 ((1/2)d)); that is,
cos D′ = cos d′−((2 cos M′ cos S′)/(cos M cos S)) sin ((1/2)(D + d))·
sin 1/2(D−d); whence
also vers D′ = vers d′ + ((2 cos M′ cos S′)/(cos M cos S)) ·
sin ((1/2)(D + d)) sin ((1/2)(D−d)).
It may be observed, that Requisite Tables 9–11, answer
logarithmically to (cos M′ cos S′)/(cos M cos S); and the verse
sines, and the cosines can be very readily taken out of the tables in
the Appendix. Also no ambiguity can arise from the application of the
rule before given: for all the arcs concerned in the operation will
always be (each of them) less than a quadrant, except the resulting
true distance, which cannot cause any ambiguity; and the verse sines
are given in the Appendix, to 126°.
EXAMPLE.
(_Example 2nd, p. 39, Requisite Tables_.)
Reserved log. from Tables 9 and 10 9.995307
Log. sin 62° 45′ 56″ = 1/2 sum app. dis. and diff. app. alts. 9.948971
Log. sin 40 43 31 = 1/2 diff. ditto ditto 9.814536
Log. 2 0.301030
--------
Nat. num. corres. 1.147741 0.059844
Nat. vers. 22° 48′ 16″ = diff. true alts. 0.078167 --------
--------
Nat. vers. 103 3 23 = true distance 1.225908
--------
_De l’Influence des Agens Physiques sur la Vie_. Par W. F. Edwards,
D.M., Membre associé de l’Académie royale de Médicine de Paris,
Membre de la Société Philomatique, de la Société de Médicine de
Dublin, &c.
The researches of science among the phenomena of the physical world
have long obtained a high degree of estimation and interest in
general society; but it is of late years only that their application
to living functions has attracted much of the attention of the
literary world.
The laws which govern the action of animal organs (the proper
department of Physiology) have usually been investigated by the
medical profession, to which they especially [p138] refer. Now
we find the public take some pains, and with reason, to inform
themselves upon subjects connected with physiological knowledge. A
well-educated person, disposed to philosophical inquiries, is not
merely contented with the consciousness of living, and the common
information he derives of its means by experience, but he seeks also
to comprehend the relations subsisting between his own organisation
and the matters with which he is surrounded, and which at once
furnish him with nutrition, life, and support, and assail him with
disease and annihilation. His own instincts and observation, joined
to the more learned experience of his medical advisers, help him
through the precarious stages of life, and these may perhaps be
sufficient for all its purposes; and under this impression many will
seek to know no more of the secrets of nature.
But we live in an inquiring and scrutinising age, when the demand
for scientific principles is very generally urgent. All, therefore,
relating to organisation seems of equal interest with that
appertaining to what is termed the physical creation or inert matter.
Under this impression we have perused the book before us with great
satisfaction, and propose to present our readers with an analysis of
the valuable materials which it contains. We have some knowledge of
Dr. Edwards, a countryman domiciliated in France, and long resident
in Paris. We have confidence in his reports, and highly estimate
his philosophical skill, extensive acquirements, and accuracy of
observation, ranking him among the first physiologists of the age.