CONCLUDING CHAPTER.
Comprehensiveness of the Nebular Theory—Illustration—Huxley and the
Origin of Species—Rudimentary Organs—The Apteryx—Its Evanescent
Wings—The Skeleton—An Historical Explanation—Application of the Same
Method to the Nebular Theory—The Internal Heat of the Earth—The Lady
Psyche.
IT is not difficult to show that the nebular theory occupies a unique
position among other speculations of the human intellect. It is so
comprehensive that almost every conceivable topic will bear some
relation to it. Perhaps I may venture to give a rather curious
illustration of this fact, which was told me many years ago by one who
attended a course of lectures by an eminent Professor in the medical
faculty at, let us say, Vienna. The subject of the course was the no
doubt highly important, but possibly not generally interesting, subject
of “inflammation.” I think I am right in saying that the course had to
last for six months, because the subject was to be treated with
characteristic breadth and profundity. At all events, I distinctly
remember that the learned Professor commenced his long series of
professional discourses with an account of the nebular theory, and from
that starting point he gradually evolved the sequence of events which
ultimately culminated in—inflammation!
It may be remembered that in the year 1880, Professor Huxley delivered
at the Royal Institution a famous lecture which he termed “The Coming of
Age of the Origin of Species.” Among the many remarkable and forcible
illustrations which this lecture contained, I recall one which brought
before the audience, in the most convincing manner, the truth of the
great Darwinian Theory of Evolution. Huxley pointed out how the
discoveries in Biology, during the twenty-one years which immediately
succeeded the publication of the “Origin of Species,” had been so
numerous and so important, and had a bearing so remarkable on the great
evolutionary theory, that even if the Darwinian Theory had not been
formed to explain the facts of Nature, as they were known at the time
when Darwin published his immortal book, the same theory would have had
to be formed, were it only to explain the additional facts which had
come to light since the great theory itself had been first given to the
world.
I believe we may use similar language with regard to the nebular theory
and its great founders, Kant, Laplace, and Herschel. If the facts which
were known to these philosophers led them to adopt in one form or
another that view of the Origin of the Universe which the nebular theory
suggests, how stands the theory now in the light of the additional facts
that have been since disclosed? If we merely took the discoveries which
have been made since the last of the three great philosophers passed
away, it might well be maintained that a nebular theory would be
demanded to account for the facts brought to light, in the interval.
The argument on which the nebular theory of the solar system is founded
has other parallels with that wonderful doctrine of Natural Selection by
which Darwin revealed the history of life on our globe. It not
unfrequently happens that an animal has in its organisation some
rudiments of a structure which is obviously of no use to the animal in
his present mode of life, and would be unintelligible if we supposed the
animal to have been created as he is. A curious instance of a
rudimentary structure is furnished in the apteryx, the famous wingless
bird which still lives in New Zealand.
The arrival of civilisation in New Zealand seems likely to be
accompanied with fatal results, so far as the unfortunate apteryx is
concerned. Weasels and other fierce enemies have been introduced, with
which this quaint bird of antiquity is unable to cope. The apteryx is
defenceless against such foes. Nature had not endowed it with weapons
wherewith to fight, for it had, apparently, no serious adversaries until
these importations appeared in its island home. Unlike the ostrich, the
apteryx has neither strength to fight his enemies, nor speed to run away
from them, though, like the ostrich, it has no wings for flight; indeed,
the apteryx has no wings at all. As its name signifies the apteryx is
the wingless bird. Living specimens are still to be seen in the
Zoological Gardens. The special point to notice is that, though he has
no wings whatever, still there are small rudimentary wing-bones which
can be easily seen. You need not be afraid to put your hand on the
apteryx, and feel the puny little remnants of wings (Fig. 55).
If, having seen the bird in the Zoological Gardens, you go to the
Natural History Museum, you will there find a skeleton of the apteryx
(Fig. 56). Look near the ribs in the photograph, and there you will see
those poor little wing-bones—wing-bones where there never was a wing.
From our present point of view these wings are, however, more
interesting and instructive than the most perfect wings of an eagle or a
carrier-pigeon. Those wings in the apteryx may be incapable of flight,
but they are full of instruction to the lover of Nature. As it is
certain that they are absolutely of no use whatever to the bird, we may
well ask, why are they there? They are not there to give assistance to
the bird in his struggle for life; they cannot help him to escape from
his enemies or to procure his food; they cannot help him to tend and
nurture the young one which is hatched from the egg; they can help him
in no way. The explanation of those ineffectual wings is historical.
Those bones are present in the apteryx simply because that bird has come
down by a long line of descent from birds which were endowed with
genuine wings, with wings which enabled them to fly like rooks or
partridges.
[Illustration: Fig. 55.—THE APTERYX: A WINGLESS BIRD OF NEW ZEALAND.]
[Illustration: Fig. 56.—SKELETON OF THE APTERYX, SHOWING RUDIMENTARY
WINGS.]
But if this be the explanation, how has it come to pass that the wings
have dwindled to useless little bones? We cannot of course feel certain
of the reason, but it seems possible to make surmises. In early times
winged birds flew over the sea into New Zealand, and found it a country
of abundance, as many other immigrants have done in later times. It may
have been that the food in New Zealand was so plentiful that the wants
of the birds could be readily supplied, without the necessity for
ranging over large tracts. It may have been that the newly arrived birds
found that they had few or no enemies in New Zealand, from which flight
would be necessary as a means of escape. It may possibly have been both
causes together, and doubtless there must have been other causes as
well. The fact is, however, certain, that in the course of long
generations this bird gradually lost the power of flight. Natural
selection decrees that an organ which has ceased to serve a useful
purpose shall deteriorate in the course of generations. If the wings had
become needless in the search for food, unnecessary for escape from
enemies, and useless for protection of its young, they would certainly
tend towards disappearance. The organism finds it uneconomical to
maintain the nutrition of a structure which discharges no useful end.
The wings, in such circumstances, would be an encumbrance rather than an
aid, and so we may readily conjecture that, in accordance with this
well-known principle, the wings gradually declined, until they ceased to
be useful organs, so that now merely a few rudimentary bones remain to
show that the bird’s ancestors had once been as other birds. Whatever
may have been the cause, it seems certain that in the course of
thousands of years, or it may be in scores of thousands of years, these
birds lost the power of flight; thus they gradually ceased to have
wings, and these little bones are all that now remain to render it
almost certain that, if we could learn what this bird’s ancestry has
been, we should find that it was descended from a bird which had useful
wings and vigorous flight. Whenever we find an organ which is obviously
rudimentary, or of no use to its possessor in its present form, Darwin
has taught us to look for an historical explanation. Let us see if we
cannot apply this principle to the illustration of the nebular theory.
[Illustration: SPIRALS IN OTHER DEPARTMENTS OF NATURE.
Fig. 57.—FORAMINIFER.
Fig. 58.—NAUTILUS.]
We liken the internal heat of the earth to the rudimentary wing-bones of
the apteryx. In each case we find a survival devoid of much
significance, unless in regard to its historical interpretation. But
that historical significance can hardly be over-estimated. Unimportant
as the wing-bones may be, they admit of explanation only on the
supposition that the apteryx was descended from a winged ancestor.
Unimportant as the internal heat, still lingering in our globe, may
seem, it admits of explanation only on the supposition that the earth
has had the origin which the nebular theory suggests.
That the earth’s beginning has been substantially in accordance with the
great Nebular Theory is, I believe, now very generally admitted. But the
only authority I shall cite in illustration of this final statement is
the Lady Psyche, who commences her exquisite address to her “patient
range of pupils” with the words:—
“This world was once a fluid haze of light,
Till toward the centre set the starry tides,
And eddied into suns, that wheeling, cast
The planets;”
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APPENDICES.
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I.—ON THE HEAT GIVEN OUT IN THE CONTRACTION OF THE NEBULA.
§ 1. FUNDAMENTAL THEOREMS IN THE ATTRACTION OF GRAVITATION.
The first theorem to be proved is as follows:—
_The attraction of a thin homogeneous spherical shell on any point in
its interior vanishes._
[Illustration: Fig. 59.]
Take any point P within the sphere. Let this be the vertex of a cone
produced both ways, but with a very small vertical angle, so that the
small areas S and S´, in which the two parts of the cone cut the sphere,
may be regarded as planes. Draw the tangent planes at S and S´. Let the
plane of the paper pass through P and be perpendicular to both these
tangent planes. Let O P O´ be one of the generators of the cone, and let
fall P Q perpendicular to the tangent plane at O, and P Q´ perpendicular
to the tangent plane at O´. The volume of the cone with the vertex at P
and the base S is ⅓ P Q × S, and the other part of the cone has the
volume ⅓ P Q´ × S´.
As the vertical angles of the cones are small, their volumes will, in
the limit, be in the ratio of O P^3 to O´ P^3, and accordingly ⅓ P Q · S
÷ ⅓ P Q´ · S´ = P O^3 ÷ O´ P^3. But from the figure P Q ÷ P Q´ = P O ÷ P
O´, and hence S ÷ O P^2 = S´ ÷ O´ P^2.
As the shell is uniform, the masses of the parts cut out by the cones
are respectively proportional to S and S´. Hence we see that the
attractions of S and S´ on P will neutralise. The same must be true for
every such cone through P, and accordingly the total attraction of the
shell on a particle inside is zero.
The second fundamental theorem is as follows:—
_A thin spherical homogeneous shell produces the same attraction at an
external point as if its entire mass were concentrated at the centre of
the sphere._
This is another famous theorem due to Newton. He gives a beautiful
geometrical proof in Section XII. of the first book of the “Principia.”
We shall here take it for granted, and we shall consequently assume
that—
_The attraction by the law of gravitation of a homogeneous sphere on an
external point is the same as if the entire mass of the sphere were
concentrated at its centre._
§ 2. ON THE ENERGY BETWEEN TWO ATTRACTING MASSES.
Let _m_ and _m´_ be two attracting bodies supposed to be small in
comparison with their distance _x_. Let the force between them be ε _m
m´_ ÷ _x_^2 when ε is the force between two unit masses at unit
distance. It is required to find the energy necessary to separate them
to infinity, it being supposed that they start from an initial distance
_a_. The energy required is obtained by integrating between the limits
infinity and _a_, and is consequently ε _m_ _m´_ ÷ _a_.
§ 3. ON THE ENERGY GIVEN OUT IN THE CONTRACTION OF THE NEBULA.
We assume that the nebula is contracting symmetrically, so that at any
moment it is a homogeneous sphere. We shall consider the shell which
lies between the two spheres of radii, _r_ + _dr_ and _r_ respectively.
Let M´ be the mass of the nebula contained within the sphere of radius
_r_, and let _d_M´ be the mass of the shell just defined. Then it
follows from § 1 that the condensation of the shell will have been
effected by the attraction of the mass M´ solely. The exterior parts of
the nebula can have had no effect, for the outer part has always been in
symmetrical spherical shells exterior to _d_M´, and the attraction of
these is zero. We see from § 2 that the contraction of _d_M´ from
infinity, until it forms a shell with radius _r_, represents a quantity
of energy,
(ε M´_d_M´)/_r_ ;
for it is obvious that the energy involved in the contraction of the
whole shell is the sum of the energies corresponding to its several
parts.
If M be the total mass and _a_ the radius of the nebula always supposed
homogeneous
M´ = M (_r_^3/_a_^3),
and therefore
_d_M´ = 3 M (_r_^2/_a_^3) _dr_.
Hence the work done in the contraction is
(ε/_r_) M (_r_^3/_a_^3) · 3 M (_r_^2/_a_^3) _dr_ = (3 ε/_a_^6) M^2 _r_^4
_dr_.
Integrating, therefore, the total work of contraction is
⅗ (ε M^2/_a_)
At the present moment a mass of 1 lb. at the surface of the sun would
weigh 27 lbs. if tested by a spring balance. Hence
ε M/_a_^2 = 27.
With this substitution we find the expression for the foot-pounds of
work corresponding to the contraction of the nebula from infinity to a
sphere of radius _a_ to be,
⅗ · 27 _a_ M = 16 _a_ M very nearly.
Hence we have the following fundamental theorem due to Helmholtz, which
is the basis of the theory of sun heat.
_If the sun he regarded as a homogeneous sphere of mass_ M _pounds and
radius a feet, then the foot-pounds of energy rendered available for sun
heat by the contraction of the solar material from, an infinite distance
is_ 16 a M.
§ 4. EVALUATION OF THE SUN HEAT GIVEN OUT IN CONTRACTION.
The number of foot-pounds of work given out in the contraction from
infinity is 16 _a_ M. As 772 foot-pounds are equal to one unit of heat,
_i.e._ to the quantity of heat necessary to raise 1 lb. of water 1°
Fahrenheit, we see that 772 M is the work required to raise a mass of
water equal to the mass of the sun through 1° Fahrenheit. Hence the
number of globes of water, each equal to the sun in mass, which would be
raised 1° Fahrenheit by the total heat arising from the contraction, is
(16 _a_)/772,
but _a_, the radius of the sun in feet, is 2,280,000,000, and hence we
have the following theorem:—
_The energy liberated in the contraction of the sun from infinity to its
present dimensions would, if turned into heat, suffice to raise
47,000,000 globes of water, each having the same mass as the sun,
through 1° Fahr._
It is found by experiment that 1 lb. of good coal may develop 14,000
units of heat, and is therefore equivalent to 14,000 × 772 foot-pounds
of work. A mass of coal equal to the sun would therefore (granted oxygen
enough) be equivalent to 14,000 × 772 × M foot-pounds of work. But we
have
(16 _a_ M) 16 × 2,280,000,000
–––––––––––––– = ––––––––––––––– = 3,400.
14,000 × 772 × M 14,000 × 772
Hence we see that
_The energy liberated in the contraction of the sun from infinity to its
present dimensions, is as great as could be produced by the combustion
of 3,400 globes of coal, each as heavy as the sun._
We may speak of 3,400 in this case as the coal equivalent.
§ 5. ON THE FURTHER CONTRACTION OF THE SUN AND THE HEAT THAT MAY THUS BE
GIVEN OUT.
Let us suppose the sun contracts to the radius _r_, and then, as already
proved, § 3, the energy it gives out is
⅗ (ε M^2)/_r_,
but we have
ε M/_a_^2 = 27,
whence on contraction to the radius _r_ the total energy given out from
the commencement is
16 M (_a_^2/_r_)
The average density of the sun at present is 1.4. Let us suppose it
condenses until it has a density ρ.
_r_^3 ÷ _a_^3 = 1.4 ÷ ρ,
whence the energy becomes
14 _a_ M · ∛ρ;
but the coal equivalent of 16 _a_ M has been found in § 4 to be 3,400,
and hence the coal equivalent in this case is
3,000 ∛ρ.
If we take ρ to be the density of platinum (21.5), we get a coal
equivalent 8,300. This, therefore, seems to represent a major limit to
the quantity of heat which can be obtained from the condensation of the
nebula from infinity into a sun of the utmost density.
§ 6. ON THE PRESENT EMISSION OF SUN HEAT.
According to Scheiner, “Strahlung und Temperatur der Sonne, Leipzig,
1899,” the value of the solar constant, _i.e._ the number of cubic
centimetres of water which would be raised 1° Centigrade by the quantity
of sun heat which, if there were no atmospheric absorption, would fall
perpendicularly on a square centimetre, at the earth’s mean distance
from the sun, is between 3.5 and 4.0. If we take the mean value, we have
(translated into British units), the following statement:—
_If at a point in space, distant from the sun by the earth’s mean
distance, one square foot was exposed perpendicularly to the solar rays,
then the sun heat that would fall upon it in one minute would raise one
pound of water 14° Fahr._
This shows that the solar energy emitted daily amounts to
700,000,000,000 × 4 π _a_^2 foot-pounds.
§ 7. ON THE DAILY CONTRACTION OF THE SUN NECESSARY TO SUPPLY THE PRESENT
EXPENDITURE OF HEAT.
We have seen that at the radius _r_ the energy is
16 M (_a_^2/_r_).
Hence for a change _dr_ it is
–16 M (_a_^2/_r_^2) _dr_.
At its present size, accordingly, the energy given out by a shrinkage
_dr_ is
16 M _dr_.
One cubic foot of the sun averages 87 pounds, so that
M = 4/3 π _a_^3 × 87
16 M _dr_ = 464 × 4 π _a_^3 _dr_.
We have to equate this to the expression in the last article, and we get
_dr_ = 700,000,000,000/(464 _a_) = .65.
This is the shrinkage of the sun’s radius expressed in feet. Hence the
daily reduction of the sun’s _diameter_ is 16 inches.
One coal equivalent possesses energy represented by M × 14,000 × 772.
Hence we can calculate that one coal equivalent would supply the solar
radiation at its present rate for about 2,800 years.
II.—THE CONSERVATION OF MOMENT OF MOMENTUM.
We give here an elementary investigation of the fundamental dynamical
principle which has been of such importance throughout this volume.
§ 8. CASE WHERE THERE ARE NO FORCES.
Newton’s first law of motion tells us that a particle in motion if
unacted upon by force, will move continuously in a straight line without
change of velocity.
Let A_{0}, Fig. 60, be the position of the particle at any moment. Let
A_{1} be its position after the time _t_; A_{2} be the position at the
time 2_t_; A_{3} be the position at the time 3_t_, and so on.
Then the first law of motion tells us that the distances A_{0} A_{1},
A_{1} A_{2}, A_{2} A_{3}, A_{3} A_{4}, must form parts of the same
straight line and must be all equal.
If lines O A_{0}, O A_{1}, O A_{2}, etc., be drawn from any fixed point
0, then the areas of the triangles O A_{0} A_{1}, O A_{1} A_{2}, O A_{2}
A_{3}, 0 A_{3} A_{4}, will be all equal. For each area is one-half the
product of the base of the triangle into the perpendicular O T from O on
A_{0} A_{1}, and, as the bases of all the triangles are equal, it
follows that their areas are equal.
Thus we learn that a particle moving without the action of force will
describe around any fixed point O equal areas in equal times.
[Illustration: Fig. 60.—FIRST LAW OF MOTION EXEMPLIFIES CONSTANT MOMENT
OF MOMENTUM.]
The product of the mass of the particle and its velocity is termed the
momentum. If the momentum be multiplied by O T the product is termed the
moment of momentum around O. We have in this case the simplest example
of the important principle known as the conservation of moment of
momentum.
The moment of momentum of a system of particles moving in a plane is
defined to be the excess of the sum of the moments of momentum of those
particles which tend round O in one direction, over the sum of the
moments of momentum of those particles which tend round O in the
opposite direction.
If we deem those moments in one direction round O as positive, and those
in the other direction as negative, then we may say that the moment of
momentum of a system of particles moving in a plane is the algebraical
sum of the several moments of momentum of each of the particles.
§ 9. A GEOMETRICAL PROPOSITION.
The following theorem in elementary geometry will be required:—
[Illustration: Fig. 61.—A USEFUL GEOMETRICAL PROPOSITION.]
Let A B and A C be adjacent sides of a parallelogram, Fig. 61, of which
A D is the diagonal, and let O be any point in its plane. Then the area
O A C is the difference of the areas O A D and O A B.
Draw D Q and C P parallel to O A. Then O A D = O A Q, whence O A D – O A
B = O B Q = O A P = O A C.
§ 10. RELATION BETWEEN THE CHANGE OF MOMENT OF MOMENTUM AND THE FORCE
ACTING ON THE PARTICLE.
[Illustration: Fig. 62.—ACCELERATION OF MOMENT OF MOMENTUM EQUALS MOMENT
OF FORCE.]
Let A_{1} and A_{2}, Fig. 62, be two adjacent points on the path of the
particle, and let A_{1} Q and A_{2} R be the tangents at those points.
Let S Q represent the velocity of the particle at A_{1}, and SR the
velocity of the particle at A_{2}. Then Q R represents both in magnitude
and direction the change in velocity due to the force F, which we
suppose constant both in magnitude and direction, while the particle
moves from A_{1} to A_{2} in the small time _t_; we have also Q R = F
_t_ ÷ _m_.
Complete the parallelogram S Q R U, and let fall O P_{1}, O P_{2}, O T
perpendiculars from O on S Q, S R, S U respectively. Since S Q is the
velocity of the particle when at A_{1} the moment of momentum is _m_ O
P_{1} × S Q; when the particle is at A_{2} the moment of momentum is _m_
O P_{2} × S R. Whence the difference of the moments of momentum at A_{1}
and A_{2} is _m_ (O P_{2} × S R - O P_{1} × S Q) = 2 _m_ (O S R - O S Q)
= 2 _m_ O S U = _m_ O T × S U = _m_ O T. Q R = F _t_ × O T. But in the
limit S coincides with A_{1} and A_{2}, and we see that the gain in
moment of momentum is _t_ times the moment of the force around O. Hence
we deduce the following fundamental theorem, in which, by the expression
acceleration of moment of momentum, we mean the rate at which the moment
of momentum increases:—
_If a particle under the action of force describes a plane orbit, then
the acceleration of the moment of momentum around any point in the plane
is equal to the moment of the force around the point._
If the force is constantly directed to a fixed point, then the moment of
the force about this point is always zero. Hence the acceleration of the
moment of momentum around this point is zero, and the moment of momentum
is constant. Thus we have Kepler’s law of the description of equal areas
in equal times, and we learn that the velocity is inversely proportional
to the perpendicular on the tangent.
§ 11. IF TWO OR MORE FORCES ACT ON A POINT, THEN THE ACCELERATION OF THE
MOMENT OF MOMENTUM, DUE TO THE RESULTANT OF THESE FORCES, IS EQUAL TO
THE ALGEBRAIC SUM OF THE MOMENTS OF MOMENTUM DUE TO THE ACTION OF THE
SEVERAL COMPONENTS.
Let A D, Fig. 61, be a force, and A C and A B its two components. Then,
since O A D = O A B + O A C, we see that the moment of A D around O is
equal to the sum of the moments of its components. Hence we easily infer
that if a force be resolved into several components the moment of that
force around a point is equal to the algebraical sum of the moments of
its several components.
The acceleration of the moment of momentum around O, due to the
resultant of a number of forces, is equal to the moment of that
resultant around O. But, as we have just shown, this is equal to the sum
of the moments of the separate forces, and hence the theorem is proved.
§ 12. IF ANY NUMBER OF PARTICLES BE MOVING IN A PLANE, AND IF THEY ARE
NOT SUBJECTED TO ANY FORCES SAVE THOSE WHICH ARISE FROM THEIR MUTUAL
ACTIONS, THEN THE ALGEBRAIC SUM OF THEIR MOMENTS OF MOMENTUM ROUND ANY
POINT IS CONSTANT.
This important theorem is deduced from the fact stated in the third law
of motion, that action and reaction are equal and opposite. Let us take
any two particles; then, the acceleration of the moment of momentum of
one of them, A, by the action of the other, B, will be the moment of the
force between them. The acceleration of the moment of momentum of B by
the action of A will be the same moment, but with an opposite sign.
Hence the total acceleration of the moment of momentum of the system by
the mutual action of A and B is zero. In like manner we dispose of every
other pair of actions, and thus, as the total acceleration of the moment
of momentum is zero, it follows that the moment of momentum of the
system itself must be constant.
This fundamental principle is also known as the doctrine of the
conservation of areas. It may be stated in the following manner:—
_If a system of particles are moving in a plane under the influence of
their mutual actions only, the algebraic sum of the areas swept out
around a point, each multiplied by the mass of the particle, is directly
proportional to the time._
§ 13. IF A PARTICLE OF MASS _m_, IS MOVING IN SPACE UNDER THE ACTION OF
ANY FORCE F, THEN THE PROJECTION OF THAT PARTICLE ON ANY FIXED PLANE
WILL MOVE AS IF IT WERE A PARTICLE OF MASS _m_ ACTED UPON BY THAT
COMPONENT OF F WHICH IS PARALLEL TO THE PLANE.
This is evident from the consideration that the acceleration of the
particle parallel to the plane must be proportional to this component of
F.
Let us now suppose a system of particles moving in space under their
mutual actions. The projections of these particles on a plane will move
as if they were the particles themselves subjected to the action of
forces which are the projections of the actual forces on the same plane,
and as the reactions between any two particles are equal and opposite,
the projections of those reactions on the plane are equal and opposite.
Hence the proof already given of the constancy of the moments of
momentum of a plane system, will apply equally to prove the constancy of
the moments of momentum of the projections of the particles on the
plane. Hence we have the following important theorem:—
_Let a system of particles be moving in space under the action of forces
internal to the system only. Let any plane be taken, and any point in
that plane, and let the momentum of each particle be projected into the
plane, then the algebraic sum of the moments of these projections around
the point is constant._
§ 14. ON THE PRINCIPAL PLANE OF A SYSTEM.
Let us suppose a system of particles moving under the influence of their
mutual actions. Let O be any point, and draw any plane L through O. Then
the moment of momentum of the system around the point O and projected
into the plane L is constant. Let us call it S. If another plane, L´,
had been drawn through O, the similar moment with regard to L´ is S´.
Thus for each plane through O there will be a corresponding value of S.
We have now to show that one plane can be drawn through O, such that the
value of S is greater than it is for any other plane. This is the
principal plane of the system.
If _v_ be the velocity of a particle, then in a small time _t_ it moves
over the distance _v t_. If _p_ be the perpendicular from O on the
tangent to the motion, then the area of the triangle swept round O in
the time _t_ is ½ _p v t_, and we see that the momentum is proportional
to the mass of the particle multiplied into the area swept over in the
time _t_. The quantity S will, therefore, be proportional to the sum of
the projections of the areas in L, swept over in the time _t_, each
increased in the proportion of the mass of the particle. It is easily
seen that the projection of an area in one plane on another is obtained
by multiplying the original area by the cosine of the angle between the
two planes. For if the area be divided into thin strips by lines
parallel to the line of intersection of the planes, then in the
projection of these strips the lengths are unchanged, while the breadths
are altered by being multiplied by the cosine of the angle between the
two planes. If, therefore, we mark off on the normal to a plane L a
length _h_ proportional to any area in that plane, then the projection
of this area on any other plane L´ may be measured by the projection of
_h_ on the normal to L´.
[Illustration: Fig. 63.—MOMENT OF MOMENTUM UNALTERED BY COLLISION.]
To determine the moment of momentum resolved in any plane we therefore
proceed as follows: Draw a plane through O, and the tangent to the path
of one of the particles, and mark off on the normal drawn through O to
this plane a length _l_ proportional to the moment of momentum. Repeat
the same process for each of the other particles with lengths _l´_,
_l″_, etc., on their several normals. Suppose that _l_, _l´_, _l″_
represent forces acting at O, and determine their resultant R. Then R,
resolved along any other direction, will give the component of moment of
momentum in the plane to which that direction is normal. In any plane
which passes through R the component of moment of momentum is zero. The
plane perpendicular to R contains the maximum projection of moment of
momentum. This is the principal plane of the system which we have seen
to be of such importance in connection with the nebular theory.
§ 15. COLLISIONS.
The conservation of moment of momentum remains true in a system, even
though there may have been actual collisions between the several parts.
This is included in the proof already given, for collisions are among
the mutual actions referred to. It may, however, be instructive to give
a direct proof of a particular case.
Let two particles collide when meeting in the directions A P and B P
(Fig. 63) respectively. Whether the particles be elastic or inelastic is
quite immaterial, for in both cases the action and reaction must be
equal and opposite, and take place along some line P Q. The action on
the particle moving along A P will give to it an acceleration of moment
of momentum which is equal to the moment of the action around O. The
acceleration of the moment of momentum coming along B P will be equal
and opposite. Thus the total acceleration of the moment of momentum is
zero. Hence the collision has no effect on the total moment of momentum.
§ 16. FRICTION AND TIDES.
We have shown that such actions as collisions cannot affect the moment
of momentum of the system, neither can it be affected by friction of one
body on another. Here, as in the former case, the actions and reactions
are equal and opposite, and consequently the accelerations of moment of
momentum are zero. Nor is it possible for any tidal action to affect the
total moment of momentum of the system. Every such action must be
composed of the effects of one particle in the system on another, and as
this must invariably produce an equal and opposite reaction the total
moment of momentum is unaltered.
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INDEX.
Acceleration of moment of momentum, 377
Aldebaran, 27, 28
Anderson, Dr., 356
Andromeda, Great Nebula in, 43, 204
Antinous, Cluster of stars in, 353
Apteryx, Rudimentary wing-bones of, 364, 366
——, Skeleton of, 364, 366
——, The, 363, 365
Arcturus, Spectrum of, 85
Argon, 265
Argus and surrounding stars, 103
Ariel, 338
Boring, The great, 123
Brooks’ comet, 89
Bunsen burner, The, 283
Butterfly and the oak-tree, The, 15
Calcium, 274
Capella, Spectrum of, 61–64
Carbon, 280
Ceres, 311
Change of moment of momentum, 375
Cluster, Nebulous region round a, 33
Clusters of stars, 53–60, 203
—— —— of 17th magnitude, 353
Coal-unit, 110
Collisions, 219, 380
——, Cause of formation of nebulæ, 356
Comets, 37
Comet, Brooks’, 89
—— of 1882, 119
——, Spectrum of, 290
Common, Dr. A. A., 44
Concord, The first, 294–307
——, The second, 308–323
——, The third, 324–336
Conglomerates, 159
Conservation of moment of momentum, 374
Corona of the sun, 117
Crab nebula, The, 19, 44
Crossley Reflector, The, 45, 46, 48, 49, 50, 67, 199
Cygnus, Nebula in, 329
Dark bodies in universe, 355
Darwin, Professor G. H., 153, 254, 332
Darwinian theory, 10, 268, 362
Dewar, Professor, 144, 272
Diurnal motion, The, 21
Dumb-bell nebula, 43, 44, 45, 46, 50, 74, 195
Dust from Krakatoa, 185
Earth, Heat in interior of, 134, 367
——, ——, Cause of, 153
——, History of, 122–157, 251
——, Rigidity of, 162
Earth-moon system, 253, 332
Earthquakes, 158–190
—— in England, 175
——, Routes of, 171
Emission of sun heat, 373
Energy between two attracting masses, 370
—— given out in contraction of nebula, 370
—— of a system, 216, 235
Equivalent of heat, 88
Eros, 312
Evaluation of sun heat given out in contraction, 371
Everett, Professor, 149
Fire-mist, The, 268
“Flash” spectrum, 70
Foot-pound, 91
Foraminifera, 367
Friction and tides, 381
Gas in rarefaction, 118
H and K lines, 70, 276
Heat, Cause of, 153
——, Equivalent of, 88
—— given out in contraction of nebula, 369
—— in interior of the earth, 134, 367
——, Unit of, 80, 89
Helium, 277
Helmholtz, 86, 96, 100
Hercules, Star-cluster in, 52, 53, 56, 57, 59
Herschel, Sir William, 4, 11, 72, 73, 74
Huggins, Sir W., 60, 61, 63, 65
Huxley, Professor, and Darwinian theory, 362
Huyssen, Captain, 127
Hydrogen in spectrum of Nova Persei, 358
“Inflammation” and nebular theory, 361
Joule’s equivalent of heat, 88
Jupiter, 23, 25, 26, 29, 208, 237, 310, 327
K and H lines, 70, 276
Kant, Immanuel, 4, 5, 72, 73, 74, 327
Keeler, Professor, 45–48, 67, 73, 199, 200, 202, 245
Kelvin, Lord, 153, 162
Krakatoa, 176–189
Langley, Professor, 78
Laplace, 4, 72, 73, 74, 206
Lassell, Mr., 338, 340
Lick Observatory, 41, 43, 44, 45
Lockyer, Sir Norman, 277
Lyra, Ring nebula in, 249
Mars, 25, 26, 27, 28, 29, 311, 341, 349
——, Satellites of, 341
_Mécanique Céleste_, 5, 8
Mercury, 23, 25, 26, 29, 208
Meteors, 37
Milky Way, 205, 206, 214, 220
Milne, Professor, 165
Moment of momentum, 222, 226, 240, 352
—— ——, Acceleration of, 377
—— ——, Change of, 375
—— ——, Conservation of, 374
Momentum, Moment of, 222, 220, 240, 352
Monoceros, Nebulous region round a cluster in, 33
Moon, Origin of, 254
——, Surface of, 255
Nautilus, The, 367
Nebula, Contraction of, Heat given out in, 369
——, ——, Energy given out in, 370
—— in Orion, The great, 40, 41, 42, 44, 46, 50, 74, 195, 242
—— ——, ——, Spectrum of, 63, 64, 65
——, The great spiral, 192, 193
Nebulæ, 40, 41, 43, 45, 47, 50, 57, 58, 66, 67, 71, 73, 105, 120, 157,
191–206, 242, 247, 249, 256, 257, 258, 259, 296, 329, 345, 348–360
——, Development of, 242
——, Discovery of, 11
——, Number of, 67, 200
Nebular anecdote, 362
—— Theory, The, 2, 3, 72, 74, 157, 205, 266, 292, 307, 323, 328, 331,
337–347, 362, 368
Nebulosity, Faint diffused, in Perseus, 17
Neptune, 37
——, Satellites of, 330, 340
Newcomb, Professor, 238
Norway, Conglomerates in, 159
Nova Persei, 358
—— ——, Spectrum of, 358, 359, 360
Oak-tree and the butterfly, The, 15
Oberon, 338
Orbits of the planets, 208
Orion, 22
Orion, Great nebula in, 40, 41, 42, 44, 46, 50, 74, 195, 242
——, Spectrum of, 63, 64, 65
Pegasus, Nebula in, 47, 345
Perseus, A faint diffused nebulosity in, 17
——, New star in, 356
Photosphere, The, 69
Pickering, Professor, 358
“Plane, Principal,” The, 225, 352, 379
Planetary system, The, 37, 208
Planets, 22, 26, 28
——, Movement of, 35, 311
——, Orbits of, 208, 298
——, Rotation of, on their axes, 325
Platinum, 263
Pleiades, 22
——, Nebulae in, 71
Potassium, 272
“Plane, Principal” The, 225, 352, 379
Probabilities, Theory of, 305
Radiation of sun’s heat, 82
Ramsay, Professor, 278
Ray nebulæ, 201, 211
Rigidity of the earth, 162
Ring nebula in Lyra, The, 249
Roberts, Dr. Isaac, 198
Rosse, Lord, 57, 196–201
Rowland, Prof. Henry, 273
Sagittarius, Nebula in, 105
Satellites, 37, 209
Saturn, 25, 26, 29, 220, 233
——, Dweller in, and the Sirian, 14
——, Ring of, 210, 220, 231, 232, 233, 234
Scheiner, Professor, 202, 204
Seismometer, The, 165
Sirian, The, and the dweller in Saturn, 14
Sirius, 215, 216
Smiths, The parable of the, 303
Solar system, 36, 207
——, Energy of, 350
——, Evolution of, 20, 246–260, 349
——, Origin of, 351
Solar system, Scale of, 29, 30, 31
Spectra, Continuous, 68, 203
——, Discontinuous, 68
Spectroscope, The, 60, 271
Spiral form in Nature, 256, 257
Spiral nebula, The great, 192, 193, 247
Spiral nebulæ, 191–206, 211, 212, 213, 220, 243, 247, 256, 257, 258,
259, 296, 345
Star-clusters, 53–60
Star, Spectrum of, 64
Stars distinguished from planets, 28, 29
Stoney, Dr. G. Johnstone, 279
Sun compared with the planets, 26, 29
——, Corona of, 117
——, Contraction of, 99, 373
——, Density of, 102, 115
——, Heat of, 75–94, 95–111, 371, 372, 373
——, History of, 112–121, 251
——, Nebulous part of, 121
——, Spectrum of, 61, 62, 69, 70, 85, 273
——, Surface of, 278
——, Velocity of, 354
——, Weight of, 101
Sun heat, given out in contraction, Evaluation of, 371, 372
——, Present emission of, 373
Sunsets, The Krakatoa, 189
_Système du Monde_, 8
Thermometer for testing the heat of the earth’s interior, 129
Thomson, Prof. J. J., 316
Tides and friction, 381
Titania, 338
Umbriel, 338
Unit of heat, 80, 89
Uranus, 37, 238, 239
——, Satellites of, 238, 338
Venus, 23, 25, 26, 29, 208, 325
Volcanoes, 158–190
Voltaire, Fable of, 14
Waves caused by Krakatoa earth quake, 179, 182, 183
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● Transcriber’s Notes:
○ The raised dot (·), used to indicate multiplication, was confused
with the decimal point in several numbers. The raised dot has been
changed to the decimal point in numbers and the raised dot used
only for multiplication.
○ In places where “×” was used for multiplication, it was left as
the author wrote it.
○ Missing or obscured punctuation was silently corrected.
○ Typographical errors were silently corrected.
○ Inconsistent spelling and hyphenation were made consistent only
when a predominant form was found in this book.
○ Superscripts are used to indicate numbers raised to a power. In
this plain text document, they are represented by characters like
this: “P^3” or “10^{18}”, i.e. P cubed or 10 to the 18th power.
○ Variables in formulæ sometimes use subscripts, which look like
this: “A_{0}”. This would be read “A sub 0”.
○ Text that was in italics is enclosed by underscores (_italics_).