CHAPTER XIX.. Concluding Chapter.

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.

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.

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.

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 dM´ 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 dM´, and the attraction of these is zero. We see from § 2 that the contraction of dM´ from infinity, until it forms a shell with radius r, represents a quantity of energy,

(ε M´dM´)/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

dM´ = 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 2t; A{3} be the position at the time 3t, 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.

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:—

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.

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´.

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_).