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M E C H A N I C A L Q U E S T I O N S BY O.E. WESTIN
The omission in olden times to formu- late clear and complete definitions of the first elements of mechanics, position, rest and motion, has for thousands of years been the cause of deep-going misconceptions of exceedingly important natural phenomena.
STOCKHOLM 1922
This little treatise contains some explanations concerning a few elements of mechanics which in the essential I have formulated during the first decade of my employment a teacher at the Royal Technical College of Stockholm, in all embracing the period 1877 – 1913. They have been published before sporadically and are now brought together, in some measure completed, and illustrated by a few examples. A particular reason for me to return to the subject is the animated discussion of late in several countries concerning mechanical questions, and I wish to give a contribution to the exchange of opinions on the matter, especially concerning some questions of which the published opinions differ. In later times some treatises have been published as mechanics, which certainly contains news which, however, may be considered to be of a dubious value. For example, concerning the question whether the dimensions of rigid bodies are changed by displacement, whether the space is crooked, whether time ought to be considered as imaginary, and so on. Such questions scarcely belong to mechanics or mathematics. Mechanic is a science several hundred years old, but a sufficient investigation of its very first elements has not been made. I now permit me to ask : Is it not high time to bring these details into an earnest discussion ? – I think the answer ought to be : certainly. – There are several reasons to suppose that there is more need of an investigation in this respect in our time than before. My contribution to the matter is given in the following discourse.
O.E.W.
Åmål Sweden May 1922
I N T R O D U C T I O N
1. The ancient idea of motion. Though every change of the objects is a consequence of motion, for which reason this word is of the greatest importance, its meaning has during long periods of time not been explained in a satisfactory manner. In ancient times everybody certainly saw that many things in the surrounding were in motion, but any deeper insight into the phenomena or their causes obviously was not connected therewith ; the matter was covered in deep obscurity.
2. Different views on this matter later on. – When I later time the stars and other objects in the heavens were carefully observed, examined and made subject for calculation, which of course characterised essential progress, the element in question was not subjected to an exact definition. Some speculations were certainly made concerning the construction of the universe, but the meaning of the word motion was not explained. Not only unlearned people but also men of scientific training had in that time an uncertain intuition concerning how the word motion was to be understood. That appeared especially by their attempts to make clear the grand phenomena in the heaves. A few recollections in this respect may be noted here. In the beginning the heavens were regarded as a spherical vault in rotation round the earth, upon which sphere the stars were supposed to be fixed. The earth was thought to be a round plate at absolute rest in the centre of this sphere. The grand Grecian philosophers THALES, PYTHAGORAS and others, who lived several hundred years before Christ, however, had discovered that the earth and the sun are spherical and free in space. Another discovery of theirs was, that the earth was rotating round an axis through its centre and round the sun. These discoveries were made two thousand years before COPERNIC appeared before the publicity with his world’s system. Copernic, who in our time usually is regarded as the discoverer of the fact that the earth is rotating not only around its axis through its centre but also around the sun, had probably got impulses hereto from the old scientists whose writings he may have studied. How incomplete his knowledge was about is to be understood by motion is shown by the fact that he did not perceive that nor only the earth but also the sun is in motion. According to his opinion the sun is at absolute rest. The philosopher ARISTARCHOS had, however, already more than two hundred years before Christ accentuated that with equal right it could be assumed that the sun is at rest and that the earth is describing a yearly path around it as the contrary. He understood that the sun is in motion relatively to the earth. His declaration is, though it is only told concerning two special objects, remarkable especially because it is probably the first establishment of the particularly important fact that real motions always exist in couples.
2. Incomplete definition of motion in the present time. In the latter part of the last century motion was usually defined quite simply thus : An object is in motion when it changes its position in space. So it is still defined in our time. This definition, however, is incomplete in no less than two respects, namely, first, because the meaning of the word ‘position’ is not explained, and further, because the same can be said about the word ‘space’. It is evident that that the authors have thought it to be superfluous to give these explanations; but explications of such details are not in the least superfluous; they are on the contrary of great importance. In our century some progress in the matter certainly has been made, bur there is more to observe in this respect.
4. Two great classes of objects. Though a multitude of essentially differing things are to be found in the universe, in mechanics or theory of motion they can be divided into only two kinds, namely, substantial and geometrical.
a. Substantial. Every such object consists of a great number of small parts, particles or material points. The material point has in every direction an infinitely small extension. The room which it occupies cannot at the same time be occupied by such another point. It can be pushed aside by other material points bat cannot be annihilated. It can further be combined with other such points and resolved into others. Its state of aggregation can be changed from one form into another. The material points attract and repel each other, have weight, mass and inertia. Some exert chemical affinity and magnetical and electrical influence. Some of them are luminous and visible; others reflect the light and are therefore also visible.
b. Geometrical. The geometrical point is a remarkable contrariety to the material point. It has no extension in any direction, it occupies, therefore, no room, an infinite number of such points can be placed upon one another without occupying any room, and through a group of such points material bodies can travel without any hindrance at the greatest possible velocity, even at a velocity of several light-years in the second ; the geometrical points have under all circumstances place in all parts of substantial bodies without having any influence on them. They can penetrate one another and material points without resistance. They are not influenced by changes of temperature and are not visible. Attractions, repulsions, chemical, magnetic, electrical and other influences do not exist between them. The geometrical points, lines surfaces, spaces and so on, are thus merely objects conceivable in thought but, notwithstanding this fact, realities of the greatest importance in mechanics and other sciences and are quite indispensable as thought for the thinkers. Geometrical points and lines are, for the natural reason that they are invisible marked upon paper and other things as more or less large spots and bands of ink, colour, chalk or incisions, so as to give people something visible to fix their thoughts on. The differences between material and geometrical objects are thus many and considerable, but they are not always properly noted.
P A R T I
Kinematic expansions
CHAPTER I Outlines
5 Systems of reference. While the principal purpose of mechanics is to treat problems concerning motions, and since such a treatment would be completely meaningless if nothing was to be found, to which the object regarded could be referred, therefore in every case considered a system of reference must be used. Hereto any materiel object can be applied, the configuration of which, is as invariable as the investigation requires.
Strictly rigid bodies are not to be found in the universe, but many bodies experience so small changes in their configuration by acting forces that they can be considered as rigid.. If they are thus supposed to be perfectly rigid this is of course an abstraction.
Sometimes not only solid but also liquid or gaseous bodies can be regarded as systems of reference, namely, when partial motions of theirs are of such inferior influence on the result of the problem treated that they can be neglected.
In some cases it might be sufficient to regard a few points or even a single point as the system of reference.
6, Positions pf material or geometrical points and bodies. The position of a material or geometrical point P is determined by a geometrical point p which, in order to answer the purpose must 1:o) coincide with P in the moment in question and 2:o) be thought fixed to determinative parts of the system of reference.
This universal determination is in its simplicity one of the most important in mechanics.
The position of a body is, of course, the comprehension of the positions of its different pointes.
As it is necessary for the determination of every position to use a system of reference, the positions determined will always be relative.
7, Spaces The comprehension of all possible positions relatively to the system of reference in question is a space belonging to this system, namely the space in which all motions relatively to that system take place. Thus a space is an aggregate, unlimited in all directions of geometrical points of essentially unchangeable mutual positions.
As there is an infinitude of things which can be used as such systems, the number of spaces is indefinite. All spaces are further relative.
It is to be borne in mind that space is not the same as air, steam, ether or other substantial things – see 4 b !
8. The dimensions of a space are never more than three As all positions in relation to a system of reference can be exactly and completely determined by the use of the coordinates x, y, and z each varying from -- ∞ to + ∞ in a coordinate system belonging to and fixed to that system of reference, the dimensions of a space are never more than three.
To be sure, other quantities, as time, temperature, density, attraction, repulsion, kinetic energy, and so on, also continually change their values, but that is not a sufficient reason for calling them dimension of space, Time and space for instance are quantities of different kinds ; time does not change into space by putting into the calculation minus instead of plus or a sign of equality between quantities belonging to different orders of number. Such tricks are condemnable ; they do not contribute to progress in science. The phenomena in nature are, in general, so complicated that the greatest care must be used in their investigation, if attainment of the necessary clearness shall be possible.
In this connection it may be remarked that the custom of calling numbers used in the solution of real problems ‘imaginary’ is not satisfactory and ought to be extended. The imaginary numbers have another unit, i=√-- I, than the ordinary, I, but in spite of this they exist, though they have obtained a misleading denomination. They are complementary numbers which together with the ordinary give the complex numbers.
Further it may be remarked, that the addition of the ordinary number a to the complementary number ib is an addition of another kind than a1+a2+a3+ … and also than ib1+ib2+ib3 … The latter are both straight—lined, the first along one line, the second along another. The sum of a and ib on the other hand is rectangular. A special addition—sign therefore ought to be used such as (+ with a v on top, not to be found in this computer) The sum of a and ib would then be written : a+v ib , that is a geoplus ib .
Unit – circles Elements of ordinary of complement-- of complex numbers tary numbers numbers
Number plane I II III
Coordinate OX, OY OiX, OiY OX, O/Y axes
Radius I i cos ∂ geoplus i sin ∂
Arc ∂ i∂ ∂
Absciss cos ∂ cos i∂ cos ∂
Ordinate sin ∂ sin i∂ i sin ∂
[In a little publication, Grafisk räkning med komplexa tal (Graphic computation with complex numbers) Stockholm 1918, the author has treated of this question and shown how the three systems of numbers belonging to the elementary algebra can be graphically expressed, how easy its six species of calculation are performed in this manner, and that in these systems of numbers the units and the elements belonging to them can be represented by their unit—circles. The following little table contains the elements belonging hereto : ]
9. Real rest Points and bodies are in every case at real rest when their positions relatively to the system of reference in question remain unchanged.
The real rest is thus always relative.
Every body is at rest in the space which is determined by the said body and the different parts of that space are att rest mutually, or, in other words, every body is always at rest in its own space.
10. Real motion Points and bodies are always in real motion when they change their positions relatively to the system of reference used.
All real motions are thus relative, and the expression ‘real motion’ therefore, have the same significance.
Real motion is neither dependent on the size of the moveable object nor on the size of the system of reference. The essence of the matter is, that the positions change, nothing else.
11. Graphical demonstration of real motion The real motion of a substantial object is characterized by the fact it can be graphically shown by the object itself. For this purpose the later is to be furnished in a suitable manner with pens and tools for drawing lines or cutting incisions upon the system of reference or on an external elevation upon it. These lines or incisions show that the pens or tools together with their fastenings have passed a series of positions relatively to the system of reference and that they thus have been in real motion in relation to it.
12. Couples of real motion All real motions appear in couples, that is : when one object A which so ever is in real motion relatively ton another one B which so ever, then B is simultaneously in real motion relatively to A.
This remarkable fact depends on the circumstances that when the distance from A to B is changed, then at the same time the distance from B to A changes. It is evidently impossible to change the one of these distances if the other remain constant ; variation of positions or distances, however is real motion.
As rest is an special case of motion – the velocity being equal to zero – thus holds without exception the thesis that when A is at rest relatively to B then B is at rest relatively to A. An if two bodies X and Y are at rest relatively to each other and X is in real motion relatively to a third object, Z, then Y and Z are mutually in real motion relatively to each other. These remarks perhaps seems to be quite superfluous, but they contain more than might be seen at the first view.
13. Wear as proof that couples of motion are realities All substantial, more or less rigid, bodies, finished as well as others, are in some degree rough on their surfaces, that is, they have elevations and depressions,. When two such bodies A and B touch each other, the elevations of A catch in depressions in B, and B thus relatively to A, a twofold wear always arises. Herein no exception is given, and that constitutes the most obvious proof that couples of motions are realities. A is in motion in the space in which B is at rest, and B is in motion in the space which belongs to A or in which A is at rest.
14. Every body always has an unlimited number of real motions In universe an unlimited number of bodies B, C D …. exist in real motion relatively to a body A which so ever and relatively to each other. Since such motions always take place I couples, A has one real motions relatively to B an other relatively to C, a third relatively to D and so on in the indefinite.
15. Every point of a space represents an unlimited number of positions The position of a material or geographical point P relatively to the bodies A, B, C, … may be denoted by pa, pb, pc … respectively. All these geometrical points which are in motion -- pa rel. to B, C, D, …, pb rel. to A, C, D … pc rel. to A, B, D, -- coincide with one another and are belonging to the spaces Sa, Sb, Sc … connected with A, B, C … respectively. Since that holds in general, every point of a space which so ever represents an unlimited number of positions.
16. Apparent rest An object is at apparent rest when from some point of view it appears to be at rest, though in reality it changes its position
If for instance a source of light moves to and from the observer undr such circumstances that he can note no change in its intensity, then the light is at apparent rest relatively to him.
17. Apparent motion An object is in apparent motion when appears to change its position though it is at real rest.
If for instance, the trees in a forest or the buildings of a town are observed from a railway train passing by, then the seem to change place relatively to one another, though that, of course is not the case. This phenomenon evidently depends on the fact that the observer sees the objects from different points of view, which results in changed impressions on the retina of his eyes.
Apparent motion, moreover, occurs not only when the objects are observed from different lines of sight but also when regarded from one and same line. If the observe is moving toward a rigid body or this to him, then it seems to grow. This is obviously due to the fact that the angle of sight is growing ; the phenomenon is only a consequence of a change of the impression in his eye.
In this connection it may be pointed out that the deflection of the light emitted from a luminous body at a great distance from the observer can cause him to believe that the light--source has another position that the actual one. The sun, for example is seen arising before it has arisen over the horizon. The image of the sun advances before the sun itself. Moreover from the earth we seldom see the bodies in the heavens passing in their actual positions and never of their right size; we only see diminutive images of stars and planets in positions which they are not passing at the same moment. The heavenly bodies are relatively to the earth – it must be borne in mind – in real motion combined with an apparent one, the latter being that of their image on the line of sight tangential to the deflected light ray at its terminal point in the eye. The luminant object itself follows after its little image.
The apparent motion cannot like the real one be shown by the object itself, because it is a fictitious phenomenon. Further it is to be observed, that apparent motions do not exist I couples as the real ones do.
However it is exceedingly common that apparent and real motions are confounded. Very often people says that the trees of a forest have an apparent motion relatively to a train passing by, but this motion is quite as real as that of the train on the track – see 22 ! Bear in mind that there is an essential difference between the motion of the trees relatively to one another and their motion relatively to the train passing. by !
18. Absolute rest and absolute motion These two conceptions are quite as old as mechanics itself. In olden times, obviously, the first idea that came in mind concerning rest and motion was to regard the earth as a body at absolute or complete rest and the motion upon it as absolute ; other possibilities were out of account. Not only in these early times, however, the opinion that absolute rest and absolute motion are realities had adherents, but during long periods of time afterwards and even in modern times efforts have been made to get such ideas accepted. The cause of this remarkable fact is undoubtedly to seek in the omission of formulating sufficiently complete definitions of the word motion. The circumstances which these ideas have appeared are many and varied in comparison. We refrain from going more deeply to the bottom of this detail. A characteristic instance, however, may be cited.
In at little publication under the title :”Über die Prinzipen der Galilei—Newtonschen Theorie” C. Neumann has vindicated the idea that the absolute motion is the only right one and that by the motion of a points not to be understood its change of place relatively to the earth or sun but its change of place relatively to a certain at all times in configuration and size completely unchangeable body which he calls Alpha. He formulates his idea thus : “Unter der Bewegung eines Punktes ist nicht etwa seine Ortsveränderung in Bezug auf Erde oder Sonne, sondern seine Ortsveränderung in Bezug auf jenen Körper Alpha zu verstehen,“ Neumann says that such a body certainly must exist and that is unavoidable, but he acknowledges that it is impossible to determine the same.
Thus according to Neumann’s opinion the motion of the earth round the sun, the sunrise and the sunset, the passage of a ship on the sea, the running of a train on a track, a man’s walk in one place or another, and so on, should not be realities, and the reason is, that they are relativities. Such an opinion we cannot endorse. On the contrary, we are of the opinion that the relative motions are the only ones that are to be acted on, the only real ones which exist. Moreover it may be remarked, that if a body B which so ever changes its position relatively to the supposed body Alpha, then the latter changes its position relatively B, and since there is an unlimited number of bodies C, D, E … in motion relative to Alpha, and this body has a motion relatively everyone of them, then Alpha must have an indefinite number of real, that is relative, motions. That means : There is in the universe no absolute motion. The supposition that this body Alpha exists leads to the conclusion contrary to all experience that the whole universe is dead.
The term ‘absolute motion’ is, however, used as an abbreviated expression concerning fundamental motion – see 38
CHAPTER II
Some well-acquainted and some overlooked motions 19. Conditions for visibility of motion The fact that real motions always exist in couples is not easy to comprehend. The difficulty is mostly due to the fact that people from their youth are accustomed only to think of motion relative to the earth. It is, furthermore, never possible for an observer to see at once both parts of such a couple ; but he can see either the one of said parts or the other whichever it may be, under the following conditions : 1:o) that he has opportunity of occupying a suitable position at rest upon the system of reference in question, and 2:o) that he has the necessary capability of objective intuition. Here a few examples will be given.
19. Rotations
That rotations appear in couples can be graphically demonstrated by an arrangement as shown in fig. 1. Upon the upper side of a fundament—plate A placed on the earth is fixed a spindle B enclosed in a tube C which at right angles to it supports a disk D. On a is fixed an arm E, and it holds a pen F resting on D. Another arm G is fixed on D and holds a pen H resting on A.
Should an observer which to see the motion of the disk relatively to the earth, then he has of course only to place himself on the ground near the disk and fix his eyes on the same. I order to see the other motion in the couple he has to take his place on the disk and view his surroundings. He must then remember that the earth is either at rest or in motion relatively to the disk. Since it now is not at rest in relation thereto, it must be in motion relatively to it.
In equally simple manner the two notions in the couple can in every comparable case be observed; but it is evident that in many cases it is difficult and even impossible to obtain and hols a suitable place for the observation.
With another very usual combination, figs. 3 and 4 the couple of rotation can be shown.
A shaft and its bearing in a machinery are both always worn. When they are new, the journal of the shaft fills the bush of the bearing, fig. 3, but later on both will have changed their form and size, as shown in fig. 4. The journal is rotating in the bush and wears the latter, and, inversely, the latter wears the former. If the bearing is fixed to an machine—foundation standing on the earth, then the bush and the earth all things upon it are rotating round the shaft. How wondrous this may seem at the first sight, never the less it is an irrefragable fat. 21, Foucault’s pendulum—experiment Though beyond the above mentioned, there are many other arrangements, which are suitable for showing that a rotary motions always appear in couple, none of them, however, equalling in importance to the pendulum--experiment performed by the Frenchman Leon Foucault. As is well known, he made use of a long pendulum, fig. 5, with a heavy ball so suspended, that it could swing freely to and from the vertical line through its point of support. On a horizontal graduated concentric ring below the turning of the swinging plane relatively to the earth could be read off, and at the same time the turning of the earth relatively to that plane. Upon the ring mentioned he had placed a little wall of sand of triangular section, fig. 6.
A pin in the lower part of the ball pushed the sand aside, so that the section was changed from the triangular into a four—sided one, se the figs.! The pins pushed off a little portion of sand when it was moving in the arc OA0, figs. 5 and 7, another one in the arc A1O, a third in AO2 and so on. These lines are in the said figs. marked in a plane parallel with that of the graduated ring. In this manner the reality of the turning of the pendulum—plane relatively to the earth was clearly shown.
When by such an experiment anybody wishes to see the real motion of the pendulum plane to the earth, he, of course has to take a place on the floor in the floor where the experiment is performed and look at the swingings. Should he on the contrary wish to see the rotation of the earth relatively o the said plane, then he has to take place himself on the pendulum—ball and to eye the surroundings. Certainly he then has to regard not only the slow rotation of the earth relatively to the plane but also the swingings in them same, but it obviously requires no great capability of observation to distinguish the essential one in the different parts of the phenomenon.
The motion of the pendulum—plane has, however, been called ‘apparent’, and usually it is said : “It is not the pendulum—plane but the earth that is turned ; the motion of the plane is only apparent, but it shows that the earth really is in motion”.
A statement thus formulated is not satisfactory. It is a matter of fact, as here has been shown, that Foucault has proved that the said motion of the pendulum plane is a reality. Under such circumstances it may be asked : Did not Foucault perceive that it was a reality ? – And if he did, why de he not use the correct denomination ? – He evidently knew that this motion was a reality; of course he could not escape from seeing the change from one form to another of the section of the sand--wall which he had arranged, but he had en important reason for using a a term which could involve a certain indefiniteness. He was born and lived in a catholic country; he was aware of the brutal treatment Galileo had suffered when he was bold enough to declare that the world was in motion. Foucault wanted to live and work in liberty; he therefore used the incorrect expression, though he knew that it was not the right one. He surely thought that the fact would once be clear, and ha was modest enough to feel satisfied with that.
The experiment of Foucault, moreover shows more than may be seen at first sight. By means of it is shown that the rotating of the earth round its central axis is varying with the sine of the latitude. And as the latitudes are many abd there are more than one meridian, the earth consequently has a great number of real motions round its central axis. At the poles one revolution lasts 24 hours, at Paris 32 at the equator there is no motion of this kind. Since these rotations are varying with the sine of the latitude, the rotation observed is only a component of the rotation at the pole, The other component is perpendicular to the one observed, according to a known thesis in mechanics concerning the compositions of rotations. Fig. 8 shows the connection between these rotations.
O denotes the centre of the earth, N the north pole, a the angle of latitude of the place A for the experiment, OB is perpendicular to OA; Oa, Ob and On denote the angular—velocities round the axis OA, OB, ON respectively, and r the radius of the earth. As known, the earth rolls around the sun, the centre of the former describing an ellipse, in one focus of which is situated the centre of the sun through which passes the axis of rotation of the latter. The space S, in which the earth is rolling, is determined by the plane through this ellipse and by positions on its both sides. As rolling is a motion composed of rotation and translation, the motion of the earth mentioned is a rotation around the polar axis in a space S1 and the translation S1 in S. S1 is determined by this axis and an arbitrary line fixed to it.
Speaking of the rotation of the earth it is customary only to think of such a motion around the polar axis. But the fact is that the earth has a real rotation around every of its perpendiculars, unlimited in number, at the different places of its surface. The proof of this remarkable fact Foucault has given through his pendulum—experiment. This result, however, has not before been sufficiently noted.
The triangle in fig. 8 shows the relation of two angular velocities, namely, that at the pole of the value w and that at the place A of the value w1 . The third side of the triangle signifies the angular velocity w2 of a space S2 rotating around the axis BO perpendicular to AO. For, the rotation of the earth in S 1, around the polar axis can be said to be composed of its rotation in the space S2 around AO and the rotation in the space S2 around BO in S1.
Through the pendulum—experiment, performed at A, firstly the velocity w1 is directly determined, and further, indirectly the velocity w at N. because between them, according to the experiment, there exists the relation w1=w sin a. Moreover, through the same experiment the velocity v in S1 for the place C on the earth at the latitude beta can be indirectly determined. For attaining this purpose there are two different methods, namely, partly through multiplication of the distance r cos b(eta) from C to the polar axis by the angular velocity w, which gives the result v = wr cos b(eta), and partly through calculation of the component velocities around the axes AO and BO. Thereby it sis to be observed, that the first component wr sin (a - b) is the velocity of C in S1 whereas the other one wr cos (a – b) is the velocity of a point C1 of S1 coinciding with C, thus not the velocity of C. Using the relations w1 = w sin a and w2 = w cos a , the sum of the two components, both of equal direction and perpendicular to the plane CON is = wr [sin a sin (a --- b) + cos a cos (a -- b)] .
According to a well known trigonometric relation this sum is = wr cos b, or the same result as in the first case.
The simple calculation is of interest, because it conforms the fact, that Foucault’s pendulum—experiment shows that the earth has a real rotation around every one of its perpendiculars, except those at the equator.
22. Translations When a railway-train is running upon a straight—lined track, the bodies of the wagons have, as known, a translatory motion relative to the earth. At the same time the earth has such a motion relatively to them. That can be shown I so evident manner, that every normally thinking person ought to understand it.
If on the train is fixed in a suitable manner a scythe A, fig. 9, the A can cut off the tops of the trees near the railway, it being thus clear that the train has been in motion on the track. That motion can for the rest, as everybody knows be observed from the ground in the neighbourhood
The other motion of the couple can be shown in just as drastic a way. For if on a scaffold by the track another scythe B is fixed and some plants are placed on a wagon of the train, the latter can be cut off in just the same manner as the trees in the forest. The plants cut off show, that B has passed through a series of positions relatively to the train. If anybody thinks, that both these two motions are not realities and that there is no danger in taking a seat upon the train where the scythe B is passing, yet he had better not make such an experiment. If however he should risk it, he will in a perceivable manner indeed be convinced of the reality in question, that is, if he should be fortunate enough to escape alive from the adventure.
It may lastly be remarked that it is not necessary to use such forcible means to show the reality of these motions. It is quite convincing to use pens fixed upon the objects in motion, which draw lines on band fixed on the two systems of reference. The essential part of the matter is, as before pointed out, to show how the positions do change, which is shown by the lines drawn by the pens.
As upon every railway generally several trains are running at great or small velocities, and the number of railways is considerable, it is evident that the earth has many translations with varying directions and velocities.
However numerous the motions may be, there are, moreover, many, many more of the same kind. When a vehicle, or an other object, changes its position on a highway or path and thus has the real motion relatively to the earth, the earth has at the same time a real motion relatively to the object. It will be easy to find out an arrangement by which it could be graphically shown, but it may seem superfluous to enter upon a further examination in this respect.
23. Rollings Also in this case it is very easy to show that real motions appear in couples. The two motions of every such combination are, however, not equal as in rotations and translations. The differences are considerable concerning both paths and changes of velocities.
If a wheel without gliding upon a rail on the earth,, every point of its periphery describes, relatively to the earth a cycloid. During the same time the rail rolls upon the wheel, and every point of it describes an evolute—curve relatively to the wheel. But not only that : the earth, which is fixed to the rail, also rolls around the wheel.
Ann arrangement suitable for the graphical demonstration of these motions is shown in figs. 10, 11 and 12.
On a railway there is running a wagon W, and at its sides are two vertical screens, placed parallel to the rail, one A, fastened in the ground at the other, B, applied to the wheel on the other side.
On the periphery of a wheel of a wheel at A is placed a pen P drawing the cycloid of A and on an arm C, fastened in the ground are placed pens Q and R on a level with the rail.
These two pens draw the evolute—lines mentioned on B. In this simple way the two motions in the rolling—couple are shown.
The velocity in the cycloidal path incessantly is changed continually between zero and twice the velocity of the centre of the wheel relatively to the earth. The velocity in the evolute path also changes continually, but not periodically,
namely from zero upwards for the point Q, fig. 12, of the rail which lies behind the tangential point T between the wheel and the rail, but downwards to zero for the point R of the rail in front of T.
There is moreover, another essential difference between the two motions in the couple: the rolling of the wheel on the earth is a fundamental motion, that is, a motion in which both the dynamical and the kinematical laws hold true, but the rolling of the earth on the wheel is a consequential motion, or a motion in which certainly the kinematical, but not the dynamical, laws are applicable – see 38 !
The rolling of the earth around the wheel is quite as irrefragable a fact ass the rolling of the wheel upon the earth. When the distance from Q to D increases, then at the same time the distance from D to Q increases, and when the distance from R to E is diminished, then at the same time the distance fro E to R is reduced. Further it should be observed that DQ is nor the same as QD and that ER is not identical with RE; the difference between the directions must not be neglected; to do so would be quite as great a cardinal error as to put the sign plus where it should be minus.
24. General remark Couples of real motions and other kinds than those related in the foregoing of course occur, but the examples mentioned above will suffice.
It certainly is remarkable that real motions always are to be found in such a way combined with each other, but there is in this circumstance nothing mysterious. And when nature itself gives us information in this respect through the always twofold wear which everybody recognizes as at fact when two solid bodies touching each other glide mutually, there is no good reason for doubting.
CHAPTER III
Composition of motions and velocities
25. Preparatory remark. The theory of composition, combining or compounding of motions and velocities is in all its simplicity not seldom misconceived, and the reason is, in general, an unclear comprehension of what is to be understood by real motions and velocities belonging to them. Not seldom it is said , when a point has two simultaneous velocities, that the resultant velocity is determined by the parallelogram—law; but that is an essential error, because these velocities never can be combined. If the preceding investigations be thoroughly considered, there will be no difficulty in gaining necessary insight into the part of mechanics now before us. But without knowing that everything in every moment of time has an infinite number of real motions it is almost fruitless to spend time upon this part of the matter.
26. Convenient indices To three, or more in general n, velocities of real motion can in certain cases be combined in accordance with the parallelogram--, triangle--, or polygon—law. Thereby, however, some details are to be remembered which rather often seem not to have been considered. We are going to call attention to them in the following..
The reality of the theory of composition will appear more clear, if at the signs of velocity indices are used, which indicate firstly the point in motion to which the velocities belong, and, secondly, the system of reference relatively to which the point is in motion. For example, if the point P changes position relatively to the system of reference A, then the velocity conveniently can be marked vPA. If a is the point in question and B the system of reference, the in accordance with what has been said the velocity ought to be signed vaB and so on. For the resultant of vPA and vaB it is convenient to use the sign VPB which of course signifies the velocity of P relatively to B. If the components are more than two, vPA, vaB, vbC … vwX, then the resultant in accordance herewith is to be signed VPX.
Inasmuch as to every system of reference A belongs a space in which the motions relatively to A take place, this space can be signed SA. The meaning of the expression : the motion of P in the space SA and the motion of P relatively to A is therefore one and the same.
27. Composition of two straight—lined and uniform motions It is well known that in this case the resultant motion also is straight—lined and uniform and the resultant velocity is geometrically determined either as the diagonal in the parallelogram of the two components combined, with their initial points coinciding and with unchanged magnitude and direction, or as the third side in the triangle in which the two other sides represent the said the said velocities so placed in succession that the terminal point of the first coincides with the initial point of the second.
Fig. 13 illustrates this case which, however simple it may be, is of fundamental importance.
The point P may pass the position a relatively to a system of reference A at the constant velocity vPA, while a at the same moment has the constant velocity vaB relatively to another system of reference B. The resultant velocity then is VPB.
It may at once be emphasized that the two components vPA and vaB always belong to two different points, P and a respectively, which certainly coincide at the moment in question but never the less are distinctly separated from each other, because the former, P, is the object whose resultant velocity VPB is to be determined while the latter, a, belongs to the object A relatively to which P has its primary velocity vPA. The two velocities vPA and VPB of P can never be composed, because of a resultant never being composed or compounded with its own component.
The differences now pointed out, however, are often overlooked, which is the cause of misconception of the whole construction. The first component vPA and the resultant, VPB, belonging to the same point P, are in fig. 13 represented by full drawn lines, the second component, vaB, by a broken one.
As one half of the parallelogram of the velocities is sufficient to determine the resultant, the triangle—law can be used instead of the parallelogram--law. Hoe the two components in such a case are compounded has been mentioned before and appears for the rest from fig. 14.
28. Composition of velocities in two curvy—linear and variable motions Here is to be remembered that, since by the velocity of a point in every case is to be understood the way which this point would traverse during the unit of time if the motion ceased to change, and consequently the variable motions are thought to be unchanged as well in intensity as in direction, the resultant velocity is to be determined in exactly the same manner as for the uniform and straight—lined motions. The velocities also are tangential to the three corved paths: two of them belonging to the component motions and the third belonging to the components are belonging to the component motions and the third belonging to the resultant motion. Also in this case it must be observed 1:o) that the two components are belonging to different but coinciding point and 2:o) that the two velocities – the first component and the resultant – which are belonging to one and the same point, can not be compounded.
29. Composition of n velocities in general For this purpose the polygon—law is to be used. This law is, as well known, a consequence of the triangle—law, being a repeated application of the same. When the components are vPA, vaB, vbC … vwX, fig. 15,
the triangle—law gives the resultant VPC and the third component vbC. In the same way we get the third resultant VPD through combination of VPC and the fourth component vcD, and so on. Under such circumstances it is evident, that by combination of n components the polygon will have n +1 sides and at in the same manner, the components will enter after one another in such a manner, that the initial point of every one from the second one to the n:th one.
The resultant, all represented in fig. 16 by full--drawn lines
can, together with the first component, be regarded as a bundle of rays proceeding from the initial point P and spread in a plane or a space.
30. Condition for composition of velocities Our experience has proved it useful to enter a little more fully into particulars on this matter and to remind of the fact the following conditions must be satisfied, viz., the velocities must
1:o) be simultaneous ; 2:o) be mutually connected ; 3:o) belong to different points.
Velocities at different instants of time can not be compounded. The reason is simply that in every case the law of composition is only applicable for the moment of time in question. It may perhaps seem as if the addition and dv to the velocity v at the end of time t in order to get the velocity v + dv at the end of the time t + dt would contradict that statement, but dv is a change of velocity or an acceleration and thus not a velocity.
Velocities which are not mutually connected cannot be compounded, because they do not belong to the problem.
Velocities belonging to only one point P can not be compounded, though this point, like every other one, at every moment of time has an unlimited number of real motions and velocities. It is to be remembered, that while the resultant velocity of P has n components vPA, vaB … vwX. only one of them, vPA, is the velocity of P. All the others are velocities of coinciding points, a, b … w, which at the moment in question determine the positions of P relatively to the systems of reference A, B … W, respectively which are used in the solution of the problem respectively. The velocities of P, vPa, VPB, VPC …vPX cannot be compounded, because such a composition would be contrary to the law.
Further it is to be observed, that the components fpr the composition must be taken in an order which is determined by a suitable but in some degree free choice of reference.
31. Guide in composition of velocities When a point P is in motion relatively to a system of reference A or in a space Sa , and A has a mutual motion relatively to a system B or in a space Sb, B has a mutual motion in the space Sc and so on, and, finally the system W has a motion in the space Sx, then it is useful to apply the following short guide :
P – X P – A A – B B - -- -- -- -- -- -- -- -- -- W W – X
[an impulse to this valuable guide was given by the late professor Hj. Holmgren at the Royal Technical College of Stockholm]
It will appear that the guide consists of two parts : a head for the resultant and a list of the components. In the head the first place Is occupied by the sign for the point for which the resultant is to be determined. The first place in the first line of the component—list is occupied by the same sign which has the first place in the head, and the last place in the last line is occupied by the same sign which has the second place n the head. For the rest, in every line the first place is occupied by the sign which has the last place in the preceding line. This guide can be regarded as closed memorandum--chain.
32. Remark concerning the order of the signs in the composition—guide It is well known that in a projection—sum the term can be displaced without changing the total of the sum. From this, however it may not be inferred that the order of the object—signs in the composition—guide also can be changed without inconvenience. Here follow for comparison two combinations of such signs, one correct I and one incorrect II:
I II III P – X P – X P – X P – A P – A P – A A – S S – A A – S S – X S – X Q – Z
II is wrong because the order is disarranged, and It cannot be used as a guide in the application. It may be observed that S – A signifies the contrary to A – S. The combination III is more erroneous than II, as it contains signs for velocity, Q – Z which do not belong to the problem. Such an error, remarkably enough, however, has been made and has said to prove that the triangle—law concerning composition of velocities were not correct, but such an application of this law has no influence upon the validity of the same.
33. Applications The reality of the matter will undoubtedly be made more lain if it is illustrated by concrete examples. We therefore will give a few of practical interest.
a. Shooting a flying object Every marksman knows that if he is to shoot a flying object, F, fig. 17. he must take sight ahead of it, if he is going to hit, and still more ahead of it as the distance between himself and the object is greater and as F has greater velocity relatively to himself.
The condition of hitting evidently is that the trajectory of the ball B relatively F will pass through F. When the velocity vFE relatively to the earth E is vFE, them it is to be observed that the earth has a real velocity to F, equal to the former but contrary to it. The motion of B relatively to F is compounded, and the guide is
B – F B – E E – F
The resultant velocity VBF has the components vBE and vEF. and this resultant is geometrically determined in fig. 17 which, moreover, shows the trajectory BF. The construction of the triangle of velocities can be carried out anywhere in the figure, at the gun G or at any other place quite optional, if only the magnitudes and the directions of the components are unchanged.
The two velocities of B, vBE and VBF cannot be compounded, because they belong to the same point B and are not both components.
The numerical calculation is easy when the components are known ; it is only to solve the triangle in question.
b. Danger of collision at sea Two ships A and B travelling with parallel—motions on the sea S, fig. 18, have on S the velocities vAS = 3 and vBS = 4 miles per hour eastwards and northwards respectively. With regard to danger of collision the velocity of B, VBA, relatively to A shall be determined.
Since A has the real velocity vAS eastwards relatively to S, S has a real velocity relatively to A, equal to the former but westwards. The guide is then :
B – A B – S S – A
The resultant—velocity therefore is
VBA = √v²BS + √v²SA = √4² + √ 3²= 5 miles per hour
The direction of VBA is a = 37 westerly, because
tan a = vSA/vBS = ¾
The two velocities vSA and VBS of B cannot be compounded because a resultant velocity cannot be compounded with its own component.
The stem F of B travels in the path FP, indicated by the broken line, to A and strikes it in the point P after a time of FP/5 hours. If FP = 1/12 mile, then B will strike A in one minute, if the motion remain unchanged for so long a time.
c. Three component velocities in lifting bridge A bridge, figs. 19 and 20, l m. long and b m. broad, has a machinery for lifting it to the height h m. In this machinery a mechanism may be used which among other things consists of a screw—nut, N, movable on a l m. long screw, S, parallel to the length of the bridge, which screw is movable sideways along the breadth of the bridge. N, S an B may during the lifting have motions which are rectangular to each other. The time for the lifting may be t seconds.
If the velocity of N relatively to the earth E is to be determined, we have the guide :
N – E N – S S – B B – E
The velocities are :
vNS = l/t of N relatively to S, vSB = b/t of S relatively to B, vBE = h/t of B relatively to E.
The resultant velocity thus will be :
VNE =√ v²NS + v²NS + v²BE =√ l² +²b +h²/t
Except the three motions just mentioned there is in this case a fourth real motion, namely, N:s relatively to S and S:s relatively B, fig. 21. The resultant velocity in the same will be :
VNB = √v²NS + v²SB =√ l² + b²/t m/s
The three velocities if N: vNS, VNB and VNE, in fig. 21 indicated by full-drawn lines, cannot be combined, though they are simultaneous and mutually connected. The reason is that they belong to one and the same object N.
d. Four component velocities in shooting
From a boat travelling on a river, R, a shot is to be fired at an automobile, A, on the shore, S. The centre, C, of the projectile may relatively to B have a velocity vBR = 500 m/s eastwards ; R has relatively to S a velocity vRS = 5 m/s south—eastwards, and A relatively to S a velocity vAS = 10 m/s westwards.
B, R, and A may have parallel—motions mutually.
To determine the initial velocity of C relatively to A we have the guide :
C – A C – B B – R R – S S – A
In fig. 22 the polygon of velocities is constructed. Therein the fourth component S – A is, at it should be, equal and contrary to the velocity of A relatively to S, that is S – A eastwards, because the contrary is westwards; both of them are realities. The bundle of velocities which radiate from C cannot be combined, because they belong to one point C.
The calculation of the velocity VCA is easy to perform by projection of the polygon of velocities on two axes OX and OY, fig. 22, of which the first has the same direction as vCB and the second one the same as vBR. The equations of projection will the be :
Vx
= vCB
cos 0+vBR
cos 90 + vRS
cos
135 +vSA
cos 90 =
and
Vy =vCB cos 90+ vBR cos 0 +vRS cos 45 + vSA cos 0 = 15 + 5√2/2 + 10 ≈ m/s
Further VCA = √V²x + Vy² ≈ 497,3 m/s
Vx and Vy can evidently be considered as velocities, but not of the point C. They would be components of VCA if A and B were at rest relatively to S and if vCB = Vx and vBR = Vy..
c. Shooting of three spread objects in one shot
As known, shoot—screens and chronographs re used for the determination of the velocities of projectiles in shooting. The holes in the screens determine the direction of the velocities, and with knowledge of the measured times it is easy to calculate how great the velocities are. Doubt of the reality of the motions of the projectiles regarded thereby cannot arise. With only one projectile in such a case, as shown in fig. 23, three different objects can be hit, which at the moment of firing occupy spread positions.
Upon the ground is placed a gun G, and before it three shot—screens 1, 2, 3. In a boat B, stand three other screens 4, 5, 6, and in the boat C further three 7, 8, 9. Every one of these groups is equipped with instruments necessary for time—determinations.
The primary velocities relatively to the earth may be : vPE, vBE and vCE of the projectile P, the boat B, and the boat C respectively.
The resultant velocities of the projectile relatively to B and C are the VPB and VPC respectively – see the figure !
Anybody will here find that the projectile has three mutual real motions and that there are three dangerous lines to be hit upon, namely I, on the ground or on an object at rest there, II and B, and III on C.
These three motions are all relative and therefore realities. To say that they are not realities, because they are relativities, would be an absurdity and a mechanic which contends that would not be rational; it would not satisfy the pretensions of reality. If however anybody thinks that the motion of P on the lines II and III only are apparent and that there is no danger in taking a place on II in B or on III in C, and he is foolhardy enough to do so, he will then in the most sensible manner be convinced of these realities.
f. Straight—lined and curvilinear motion of light ; great velocities. The paths of the light from a luminous body is straight—lined relatively to the same , if the surroundings consist of either concentric layers of homogenous substance or no substance at all, that is, a space filled with only geometrical points. They are thereby straight—lined because there is no cause for deviation neither in one direction nor in another. Should the light on the other hand have to pass a medium of variable density and in layers that are not concentric to the light--source, the it will be deflected.
Another cause of curvilinear motion of the light—rays is, that the light—source has a velocity of variable direction relatively to the latter illuminated body. The motion of the light relatively to the latter is thereby combined and curvilinear.
Light—sources near the earth have in general such inconsiderable velocities in comparison to that of the light there v = 300 000 km/s, that the velocities of the former can be neglected. But as for the light from the stars the matter is different. The stars are passing round the earth in 24 hours at a distance of several light—years. As a light—year is a distance of 365x24x60x60xv km the stars have relatively to the earth exceedingly great velocities – and furthermore in curved paths.
Since the motion of the light is composed of the motion of the light relatively to the star is composed of the motion of the light relatively to the star and the motion of the star relatively ton the earth, the motion of the star—light relatively to the earth is curved, and the curves evidently are, as regards the stars at constant distances from the polar axis and which are traversing in planes perpendicular to its axis and lying between the poles, Archimedean spirals.
Now, certainly, it can be asked : are these motions of the stars realities or are they only apparent? – Some authors are of the latter opinion and have tied to prove the truth of it. Their reasons, however, really are of the kind that they deserve a moment of attention.
The German authors Littrow & Weis, for instance, say in their works Wunder des Himmels (Berlin 1886) among other things :
”Und wie viele Tausende von ebenso großen und wohl noch größeren Körpern finden wir am Himmel, von welchen allen dieselbe höchst unwahrscheinliche Voraussetzung gelten müsste; dass das Grosse um das viel Kleinere, dass der ganze Himmel mit allen seinen zahllosen Gestirnen, deren Größe ebenso unbegreiflich ist wie ihre Entfernung, sich um diese kleine Kugel, um einen Punkt bewegen soll, der gegen das übrige Welt als ein wahres Nichts gelten muss, und dies alles bloß darum, weil diesen Punkt unser Wohnort ist, dessen Existenz die Bewohner jener unzähligen und von ihm so weit entfernten Himmelskörper nur in den seltensten Fällen kennen.“ Och vidare ; „…Alle ohne Ausnahme werde von dieser Kraft mit einer Regelmäßigkeit dem Weltraume fortgeführt , dass auch nicht einer derselben eine Sekunde zu früh oder zu spät kommt, als ob ihre ganze Bestimmung nur die wäre, unsere Uhren zu regulieren und uns das alle Nächte wiederkommende Schauspiel ihrer einförmigen Prozession aufzuführen.“
These authors evidently think that by such argumentations, and similar, they have giving binding proofs of their opinion that the motion of the stars relatively to the earth only are apparent, but as a mater of fact they have proved nothing of the kind. It is true that the velocities in question are so great that they cannot be clearly comprehended by any human being, but that is no reason for denying their reality. If nothing else were necessary but to deny the reality of a phenomenon, in order to prove that it is not real, then many, and many such would have to be declared apparent, though their reality is unquestionable.
Here a little example will be given, which in all its simplicity can in a certain degree serve to illustrate the matter. The distance between the two Swedish towns Ystad and Haparanda is about 1 500 km. To cover this twice this distance in only one hundredth of a second a velocity of 300 000 km/s would be necessary. Certainly nobody can comprehend so great a velocity, but that is no reason for denying that the light has such a velocity relatively to the luminous body from which it is emanating.
It is further true, that a multitude of bodies in the heavens are much greater than our earth, but that is no reason for denying the reality of their motions relatively to the same. This body is great enough to bear a weightless coordinate—system with axes of unlimited length, representing a space of infinite extension in every direction, a geometrical system of points, that is, a system in which the slightest resistance cannot exist, no matter how great the bodies passing through it may be and whichever velocities they may have – see 4 b!
It must in this special case be remembered, that the question is not, whether the one of the bodies regarded is greater or smaller than the other, or to what purpose the universe has been created, whether the purpose of this creation only be to regulate our watches, or to show the multitude of brilliant lights in heaven. No, the special question is not at all like that, it is only the following : do not stars change their positions relatively to the earth ? That is in all its simplicity the question. Interpretations such as those of the authors cited do not contribute to solve this problem.
With proofs such as L.&W. have given the conclusions certainly can be that because he distances of the stars are ‘unbegreiflich’ they are not realities, and because the greatness of the stars is ‘unbegreiflich’ their motions relatively to the earth cannot be realities, and so on. Well certainly such conclusions can be made, especially in ‘popular’ publications, but they are not tenable. It is however, obvious that everybody has a right to raise objections to what he finds unclear, but is then not sufficient to say that he does not understand the matter, he must give reasons founded on facts for his opinion, if it s going to be taken for good.
Concerning the velocities of the stars relatively to the earth it must be remembered, that an object whichsoever is either at real rest or in real motion relatively to every other object at every moment of time; a third possibility does not exist. And further, that the velocity of the light from a star relatively to the earth is before the light reaches the earth another one than the the velocity of the same light relatively to the star, because the latter is only a component of the former.
The centre P of a star passes at an arbitrary moment of time the position p upon a line fixed at the earth, but at the next moment P passes another similar point. P then is not at rest relatively to the earth; it thus has a real motion relatively to the earth. The real motions always occur in couples, and so it is in this case, as in all others: P and the star are in real motion relatively to the geometrical point p, which has no extension in any direction, and relatively tp the earth connected with p, and p and the earth have a real motion relatively to the star. The star is moving in the earth—space, the earth in space belonging to the star. Upon a structure erected on earth and reaching to the star, if such an erection were possible, the star would write its way relatively to the earth in a writing of fire.
Thus the matter stands!
The two light—paths from at star S, which we have to regard, the one relatively to S and the other relatively to the earth E can be graphically shown.
The former is a straight line connecting S an E, the latter consists of spiral—windings round E, one for every period of 24 hours because S makes one revolution round E during this time, thus for every year 365 such spiral-parts one after another.
The velocity VLE of the light L relatively to E is at every moment of time composed of two components, namely, the velocity vLX = 300 000 km/s of the light passing the point X of the connecting line SE mentioned, and the velocity vXE of X relatively to the earth.
Such compositions are, however, in effect nothing new. About two hundred years ago Bradley has shown that the motion of the star—light in a telescope was composed in the same manner (aberration of the light). He certainly did not formulate his opinion in the same words as we have used here, but in the essential he mentioned the fact.
The guide for the compositions is :
L – E L – X X – E
The parallelogram fig. 24 b shows the connection of these velocities. The two velocities of the light have at the star the initial values : the first component vLS = 300 000 km/s and the resultant VLE = √v²LS + v²SE Fig 24 b
the second component has at the same place the value
vSE = 2πr = 2π n 365x24x60x60 x vLS ≈ 2 300 n vLS km/s t 24x60x60
if the distance r between E and S is n light—years and t the time for a revolution of the star around the earth. This value is thus several thousand times greater than the velocity of the light relatively to the light--source.
The second component in fig, 24 is, however, at the same time as the distance X diminished to zero at E. Therefore t is to be noted that no matter how many years the star—light requires to arrive at the earth and no matter how many spiral--windings it has to pass in the earth—space, its velocity at the earth will always be vLE = 300 000 km/s.
As the astronomers have discovered stars at distances from the earth of several hundred light--years, these stars have in the earth—space velocities which must be regarded as unlimited. Therefore, it is not possible to determine in figures an uppermost limit of the velocities in the universe.
The opinion of H.A.Lorentz and other authors, that the velocity of the light relatively to the light—source, 300 000 km/s is the greatest velocity in the universe, is thus contrary to the reality. And the theory of relativity by A. Einstein, which is founded hereupon and for the rest upon a number of absurdities, is erroneous.
The motion of a star relatively to the earth is not to be confounded with its motion in the hypothetic medium ‘ether’. The velocity of a star in this medium may perhaps be very small if any at all, though it has an enormous velocity in the completely weightless an resistance--less earth-space, not to be confounded with the earth—atmosphere.
The velocity of the stars relatively to the earth would have to be expressed in very great numbers if the units of length and time were kilometres and seconds respectively, but these units are altogether to small for this case. More suitable units are light—years for the length and day (à 24 hours) for the time. Then these velocities would be expressed by the formula :
vSE = 2 π r
however the result remaining the same as before.
The reality of this velocity depends on the greatness of the star—distances, r, from us ; the velocity is an immediate and necessary consequence of that distance. For human beings it is quite impossible to comprehend either the one or the other of them.
In this connection it may bee of some interest to recollect, that the angular velocity of the stars relatively to the earth is very small, only one twenty--fourth of that of the minute—hand of an ordinary watch. If, therefore, this hand were thought extended to a length of a star—distance, the terminal point of the same would have a velocity relatively to the watch twenty-four times as great as the enormous velocity of the star relative to the earth.
As to the path of the star—light relatively to the earth it may suffice to draft only an extremely little part of thereof.
[Referring to the authors Polhem-publication ;”Realitäten, Abstraktionen, Fingierungen und Fiktionen in der theoretischen Mechanik, Stockholm 1911, Otto Figur also has treated the problem of the deflection of the light in his little publication “Erdrotation und Lichtfortpflantzung”, Berlin 1913]
Fig. 24 shows a little more than one winding of the same. E represents the earth, XE part of the connecting—line between S and E, Y the position of X relatively to E when the light—element in question arrives at E. The terminal velocity of the star—light relatively to E is then = vLE and its direction YE tangential to the spiral at E.
If an observer intercepts the light at E, he will in his conception locate the star upon the tangent to the spiral at E, that is in the prolongation of the line EY, and the star also really will lie there, if the star—distance is an exactly even number of light—days, but in other eventualities the star will lie upon a straight line through E which deviates from 0 to 360 from the tangent mentioned. The light which the observer sees has been emitted from the star several years before, and the light that issues from it at the moment of observation arrives at E several years after the moment in which it left the light—source.
What has been said above is independent of the hypothesis of the transmission of light, as to whether it be an emanation or an undulation, because in none of the said notions the components and the resultant depend thereof ; the motions are all founded on facts.
It is evident that a disturbance will be caused if the starlight has to in an oblique direction through a layer of a medium of variable density, for instance when it passes through the atmosphere of the sun; the two paths, both that relatively to the earth and relatively to the star, will then acquire other directions, and thus the light also relatively to the star will be deflected.
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