UC-NRLF LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class Ar A FEW CHAPTERS IN ASTRONOMY. A FEW CHAPTERS IN ASTRONOMY BY CLAUDIUS KENNEDY, MA. OF THE UNIVERSITY OF LONDON : TAYLOE AND FEANCIS, EED LION COURT, FLEET STEEET. 1894. FLAMMAM. FEINTED BY TAYLOR AND FRANCIS, BED LION COURT, FLEET STREET. PEEFACE. THE mathematical discussions in this little book are quite elementary, and geometrical in character, except that, in three instances, two differentiations and an integration of the most rudimentary kind have been used. In a very few cases, the results of analysis have been simply accepted; and even of these, few as they are, some are given only to verify conclusions already arrived at independently. November 15, 1894. 192462 CONTENTS. Page PREFACE .... v CHAPTER I. ON A VISUAL ILLUSION AFFECTING CERTAIN ASTRONOMICAL PHENOMENA. Difference between sphere of vision and plane of vision Middle of moon's illuminated limb apparently pointing above sun Deceptive appearance of curvature in meteor paths Danger of making the radiant of a very sparse meteor system higher than the reality Danger, when using alignments only, of making the position of a very faint object lower than the reality Possible modification, by this illusion, of the apparent curvature of the very long tail of a comet CHAPTER II. THE EFFECT OF THE EARTH'S ROTATION ON CERTAIN MOVING BODIES. Some brief historical notes Resolution of earth's rotation into V, or that about the vertical line, as axis, and M, or that about the horizontal meridional line Effect of Von horizontally moving bodies. Some instances, including logarithmic spiral described by homing pigeon over the sea. Foucault's Pendulum postponed to Chapter IV Effect of M on vertically moving bodies. Body of high density dropped from a height ; resistance Vlii CONTENTS. Page of air being taken as unimportant. Experiments of Guglielmini, Benzenberg, and Reich. Body of very low density falling in air Effect of M on the rate of a perfectly free pendulum Effect of Fand M together. Some instances. Their effect on knife-edge pendulum. Their effect on projectiles postponed to next Chapter NOTES 11 OHAPTEE m. DEVIATION OF PBOJECTILES FROM THE ROTATION OF THE EARTH. This effect is relatively very small Whole shift of point of fall of projectile from rotation of earth is compounded of three shifts, viz. (a) the (purely) longitudinal shift, (b) the (purely) transverse shift, and (c) the westward shift Whole alteration of range is (a) combined with longitudinal component of (c) ; and whole deflection is (6) combined with transverse component of ( c ) Formulae for the various shifts in terms of r, the range, h, the height of trajectory, t, the time of flight, and 8, the angle of descent Formulae thus expressed almost quite as applicable to ballistic as to parabolic trajectories; while those for parabolic trajectories in terms of initial velocity, elevation of discharge, and ff, are altogether inapplicable to ballistic trajectories Tables of deviations NOTES 34 CHAPTER IV. FOUCAULT'S PENDULUM. Discussed separately, though belonging to Chapter II Its beha- viour a dynamical problem, and by no means a mere kinematical one Interferences with its desired performance, from mode of suspension, from its own inherent nature, and from resistance of air Pre-eminently important to keep its amplitude of oscillation, both angular and linear, as small as practicable Insufficiently careful experiments with it quite worthless, or rather delusive. Mr. Bunt's later experiments specially successful NOTES 60 CONTENTS. IX CHAPTER V. ON THE POSITION OF THE DYNAMICAL HIGH TIDE RELATIVELY TO THE CELESTIAL TIDE -PRODUCING BODY. Page Magnitude of lunar tidal forcesThe tide a wave ; though a forced one Motion of water in a wave ; especially in a tidal wave Tangential, or horizontal, tidal forces greatly more important than radial, or vertical, ones. The latter conspire with the former, with a very slight exception producing un- important modification. The general result is almost as if the forces were wholly tangential ; they shall be taken so The equation, v^ldg^ taken as granted If undisturbed water be " shallow," that is of less than the lunar critical depth, 12-76 miles, the free tidal wave could not keep up with the moon. If it be " deep," that is of greater than the critical depth, the free tidal wave would go faster than the moon Position, relatively to moon, of lunar dynamical high tide ? Four answers to this, according as water is " shallow," or " deep," and without, or with, friction. Case of water of critical depth discussed further on- The two answers for frictionless water can be given by means of a \ery simple consideration, viz., for " shallow " water the tide must be in such a position that the tidal forces shall be working with gravity, so as to accelerate the oscillation of the water, which means that low water is under the moon ; and vice versa, for " deep " water Friction with " shallow " water shifts high tide forwards, and with " deep " water, backwards Discussion of case in which the water is of critical depth Shift, whether forwards or backwards, of high tide by friction is greater as the coefficient of friction is greater ; but no finite magnitude of friction could make shift as much as 45 For this and another independent reason the crest of the dynamical high tide cannot be, under any circumstances, 45 behind the moon Solar tides NOTES .... (59 CHAPTER VI. THE "HORIZONTAL" PENDULUM. This a convenient name ; though the Pendulum's rod need not, and its plane of oscillation must not, be horizontal. Different b X CONTENTS. Page modes of suspension described, with some reference to their com- parative advantages Mode of obtaining the sensibility of the instrument without having to depend on the accuracy of working of the setting screw NOTES 93 CHAPTER VII. THE MOON'S VARIATION. Magnitude of solar disturbing forces producing the Variation Diagram of the Var. The Var. in elongation and in radius- vector The pure Var. orbit, relatively to earth and line joining earth and sun, is a compound epicyclic curve, with deferent and first and second epicycles Its radius of curvature at syzygies and at quadratures It is very slightly flatter, at syzygies and at quadratures, than an ellipse with the same principal axes Apparently new geometrical proof that the tangential disturbing forces, by themselves, would produce an oval Var. orbit with its least axis in syzygies, and that the radial forces, by themselves, would do the same The solar tangential forces produce, by their direct immediate action, only 4/llths of the Var. in elongation ; while the tangential component of the earth's attraction on the moon produces the remaining much greater part Some mistakes easily made concerning the Var. NOTES 104 CHAPTER VIII. THE MOON'S PAEALLACTIC INEQUALITY. Magnitude of the solar disturbing forces producing this in- equality Diagram of the P.I. P.I. in elongation and in radius- vector The P. I. orbit, relatively to earth and line joining earth and sun, is a peculiar epicyclic curve Its radius of curvature at conjunction and at opposition The existence of this inequality pointed out, and its magnitude estimated, by Newton ; though it had not yet been detected in his time by observation Some apparent paradoxes connected with the P. I. ; one being that the acceleration and retardation of the moon's motion are always CONTENTS. xi Page contrary to what the solar tangential forces are endeavouring to produce While the system of disturbing forces causing the Var. has two axes of symmetry, one in syzygies and the other in quad- ratures, that causing the P. I. has only one such axis, in syzygies ; a most important dynamical difference In the P.I. orbit the tangential component of the earth's attraction on the moon is never less than 3 '31 times as great as the opposing solar tan- gential force at the same point, and generally much more than this Thus the immediate cause of the P.I. in elongation is the earth's own attraction NOTES . . 132 A FEW CHAPTERS ASTRONOMY. ERRATA. Page 25, Hue 6 from bottom, for FDH read *HD. ' line 3, to HF read DP. 30, line 3, for X read C', tw.ee. 32 line 7 from bottom, for H read r. " 19 ' f r Cltivetvoiution of the sun round the from in beyond a certain limit. _ .. ^^oi_, vein i e is at our point of view. But ordinary observers, and even astronomers themselves, are not in the habit of referring terrestrial objects around them to the sphere of vision. Such objects are referred to what writers on perspective call the plane of vision at right angles to the line of sight, which the eye, as it were, always carries about with it. There are different reasons for this. The idea of the plane of vision is, in some respects, simpler than that of the sphere of vision, and presents itself more immediately to the observer ; and this the more readily as it is always of limited angular extent. Besides this, all ordinary drawings and pic- tures are made on plane surfaces, for different obvious reasons. A FEW CHAPTERS ASTRONOMY. CHAPTER I. ON A VISUAL ILLUSION" AFFECTING CERTAIN ASTRONOMICAL PHENOMENA. IN considering the visual positional relations of the heavenly bodies to each other and to various given lines and planes, it is of course necessary to regard them as projected on an imaginary spherical surface whose centre is at our point of view. But ordinary observers, and even astronomers themselves, are not in the habit of referring terrestrial objects around them to the sphere of vision. Such objects are referred to what writers on perspective call the plane of vision at right angles to the line of sight, which the eye, as it were, always carries about with it. There are different reasons for this. The idea of the plane of vision is, in some respects, simpler than that of the sphere of vision, and presents itself more immediately to the observer ; and this the more readily as it is always of limited angular extent. Besides this, all ordinary drawings and pic- tures are made on plane surfaces, for different obvious reasons. B 2 ON A VISUAL ILLUSION AFFECTING When the eye is stationary the angular extent of distinct vision is quite small. Even if the eye be allowed to move, while the head still remains stationary, the angular range of vision, or the extent of the field of view which can be attained without too much disturbing effort, though much greater than before, is still small ; and therefore in such cases the difference between the plane of vision and the sphere of vision may be practically of very little importance. But it would be otherwise if the plane of distinct vision could be made larger, for then its own perspective would sensibly affect the question. We need not, however, go into this ; for if we would compare two objects whose horizontal angular distance is too great for them to appear in the same limited field of view, we turn the head, or perhaps the whole body, round a vertical axis from one to the other ; and we see each by turns in its own separate limited plane of vision, and usually with a very indistinct idea of the geometrical relations between those different planes. This is the main cause of the illusion now in question. We may just mention here the most striking instance of such illusion, returning to it further on, for explanation. It has the advantage of being easily observed every month. If the crescent moon be not less than two or three days old, the sun being near setting, the middle of her illuminated limb, which, of course, is turned directly towards the sun, will seem to point decidedly above the sun. As the angular distance of the moon from the sun increases, the apparent discrepancy becomes more marked, and, as the present writer happens to know, competent astronomers and mathematicians, who of course are perfectly aware of the real conditions of the case, will acknowledge that they cannot divest themselves of the feeling that they actually see the middle of the moon's illuminated limb to be pointing several degrees above the sun. The unequal raising of the sun and the moon by refraction has no share in producing the illusion ; for its effect is to raise CERTAIN ASTRONOMICAL PHENOMENA. 3 the sun, which is at the horizon, more than it does the moon, which is higher in the sky, and so to diminish the illusion to a slight extent. Every great circle of the sphere of vision is, to the eye viewing it simply and unconnectedly, a right line, because the eye is in the plane of it. The horizon is such a great circle, and it is to the eye a right line ; a straight edge held in the hand can be applied to it and will fit at any place. Not only this, but the horizon presents itself to the mind as everywhere a right line ; for a reason which we shall mention presently. But if we could draw any other great circle on the sphere of vision, or on the vault of heaven, though it, also, would be to the eye viewing it unconnectedly perfectly straight in every part, it would intersect the horizon in two opposite points, while having some elevation above the horizon at its middle part; therefore to the mind of the beholder, who is so habituated to dealing with lines &c. as drawn on plane surfaces, and knows that two right lines, as existing on such surfaces, cannot meet in two points, that line would present itself as an arch. Its inclination to the horizon would be continually varying, as ifc was followed by the eye from end to end. A person standing opposite the middle of a very long, straight, horizontal, archi- tectural feature, or other such line, of sufficient height, can only with difficulty divest himself of the idea that as he turns his head from side to side, he sees that line as a curve with its concavity downwards ; since near him it presents itself as horizontal ; but on each side as sloping down to the horizon. If we suppose a straight line to be traced on the sky at a considerable altitude, we should not refer its direction, at any point, immediately to the horizon, perhaps quite out of sight. But we should do what would be equivalent : that is, refer its direction, at any point, to the vertical at that point ; and we always have a pretty accurate consciousness of the vertical from the sense that we have of the direction of gravitation. As we B2 4 ON A VISUAL ILLUSION AFFECTING consider, in succession, at not too great an altitude, the imaginary vertical great circles which we trace for ourselves in succession while turning the body round, we take them as parallel to each other, because they have been similarly related to the successive outlooks. But they are actually converging ; the consequence is that a straight line or great circle traced across them would cut them at very perceptibly different angles ; even within a comparatively short distance. It cuts at different angles lines which in one sense have the same direction, or are parallel after their own fashion, being all at right angles to the horizon, and therefore it seems to be curved. But why does the horizontal great circle look straight, whilst another inclined to it looks curved ; both being straight ? The reason is that, in looking round, the observer turns his head, or his whole body, on a vertical axis, so that every point of the horizon has the same directional relation to his outlook, as he faces it standing erect. The horizon, therefore, naturally becomes the most general line, or plane, to which the position of an object is referred. But this is not the case with the other great circle. As the beholder turns round on his vertical axis, which is inclined to the plane of that great circle, every point of that circle has a different relation of direction to his outlook, as he stands erect ; consequently when the eye is made to run round it in the ordinary way it seems curved. But if, while holding himself up straight, he were bound to a post which then was inclined until it w'as at right angles to the plane of the great circle now in question, and if the post were then rotated on its axis, the familiar horizon being hid from view, that great circle, as he was turned round, would appear to him to be straight, for the same reason that the horizon ordinarily does so ; and if he fixed his attention very strongly on that line, then the horizon, if uncovered, would seem to him to be curved as he was being turned round. That this would be so can be proved by an easy experiment. Stand near the corner of a long enough room, or lobby, or passage, and view the opposite cornice, CERTAIN ASTRONOMICAL PHENOMENA. 5 or the juncture of wall and ceiling, running along the length of the room &c. The cornice will be everywhere straight to the eye ; yet its visual inclination to the horizon increases con- tinually as the eye follows it from the nearer to the further end. Hold a straight rod vertically, with both hands, so as to visually cross the cornice ; now while the arms remain rigid and unmoved, in order that the rod shall be always vertical, turn the whole body from side to side briskly enough, keeping the eye and attention strongly fixed on the visual intersection of rod and cornice ; and the cornice will appear to be curved with the concavity downwards, on account of the continual change of its visual angle of intersection with the vertical rod. A few trials may be necessary in order to catch the effect, as it is considerably diminished by the knowledge that the cornice is straight, and by the observer's involuntarily comparing consecutive portions of it with each other ; which latter cannot be done in the case of an imaginary straight line on the sky. To return to the case of the crescent moon referred to already. Suppose that our latitude is nearly that of London, say 51J, and that the young moon is 45 from the sun, which is setting ; and, to obtain a mean condition, let the time be an equinox and the moon on the celestial equator ; the altitude of the moon will be 26. If the straight line along which the middle of her illuminated limb points towards the sun could be traced on the sky. its inclination to the horizon would be, of course, at the sun 38| (the'complement of the latitude) ; but at the moon it would be about 29|. That straight line therefore is steeper at the sun than at the moon by 9. If persons who had not considered the matter, and even some others, when off their guard, tried to trace that line by the eye, they would start from the moon at a a downward slope of 294, and preserve that slope as well as they could, until reaching the horizon ; just as they would do if dealing with a straight line on a plane surface directly facing them. This of course will carry them many degrees above the 6 ON A VISUAL ILLUSION AFFECTING sun. But if the observer were in some unaccustomed attitude, say half reclining and looking obliquely over his shoulder, so as to obscure his sense of the vertical or horizontal direction, and 'if all known horizontal and vertical lines were properly con- cealed from view, and if he had a good eye for straightness and symmetry, he would doubtless be able, having started in the proper direction from the moon, to continue his trackless course until hitting off the sun. Perhaps the simplest, and for some persons the most striking, exhibition of this deception would be when the moon is in the first quarter, or " half moon," and the sun is setting. Suppose the altitude of the moon to be, at the time, m degrees. The terminator, or boundary of light and shade on the moon, is straight and vertical, and the middle of the illuminated limb is pointing horizontally, and yet directly at the sun which is setting m degrees lower down. If we try to follow by the eye the direction in which it points, we shall be tempted to trace for ourselves an imaginary line on the sky everywhere horizontal and having always the same distance from the horizon, as we should do in a diagram on a plane surface ; and the result will be that our production of a line, which really points directly at fche sun, will pass m degrees above the sun. (Such a line, if traced on the sky, would be a small circle of the celestial sphere, and, paradoxical as it -sounds, everywhere convex towards the straight horizon.) In this case the illusion is obvious, and felt at once to be something that requires explanation ; besides which it is not calculated to lead to any ulterior mistake. But there is another exhibition of this illusion which is not of so innocent a character ; it does not manifestly betray itself as an illusion, and it has given rise to misconception. It is a seeming phenomenon which by ordinary persons is not con- sidered to require explanation, because it appears at first sight to depend so evidently on another principle. Even those who CEKTAIN ASTRONOMICAL PHENOMENA. / must be aware of the actual circumstances in this case, have not, so far as wo know, given any warning on the subject, at least in print. Every one must have noticed what seems like the well-marked curvature of the path of an ordinary meteor or shooting-star, whether a sporadic one, such as may be seen on every clear night, or one belonging to a shower, provided its apparent path be not too near the vertical. This apparent phenomenon was, as the present writer can testify, very strikingly displayed (if this be not a bull) by the shower of Andromedids *, or Bielids, on November 27, 1872. Certain others also remarked the same, as anyone must have done. Any pictures (not diagrams on a star map) that we see of meteoric showers invariably give a decided curvature, concave downwards, to the luminous tracks. To most persons there is no difficulty about this ; but quite the contrary. One of the observers just referred to, speaking of that shower of Andromedids a couple of days after its occurrence, remarked how interesting it was to see the curvature of the trajectories of the celestial projectiles due to gravitation. But a moment's consideration will show that this is quite a mistake. The nearest point of any of these visible tracks was probably not less than forty-five miles distant, the track itself being many miles in length. Now the very longest period of visibility that we can allow to any of those meteors is two seconds, in which time one of those bodies would fall, considering the resistance of the air, less than 64 feet. But a linear deflec- tion of 64 feet would be quite insensible to the eye in such luminous tracks as we have mentioned ; even supposing it to be at right angles to the apparent track, which will but seldom happen. The case, of course, is quite different of a large meteorite which is seen by an observer to fall to the ground, not far off, after having been visible for a longer time. The illusion now in question is clearly due to the constant change of the * These are sometimes cabled " Anclromedes," as though the name of the constellation were Androme. 8 ON A VISUAL ILLUSION AFFECTING inclination to the horizon of the sensibly straight luminous tracks of the meteors. It is true that, unless the direction of the motion of a meteor is parallel to that of the earth, when the meteor enters the earth's atmosphere the resistance of the air will not only pro- duce a violent retardation of its velocity, but will cause a deflection and curvature in the path of the meteor relatively to fixed space. But this curvature will not be visible to the observer. This is easily seen thus: Suppose the meteor to be visible, even before entering the atmosphere, the observer would see only its motion relative to the earth, the air, and himself, all regarded as stationary ; when the meteor, with this apparent motion, enters the apparently stationary atmosphere there is nothing to cause any perceptible change in the direction of its motion ; no curva- ture visible to the beholder will be produced by the resistance of the air in the path of the meteor. (Nor will there be any change in the position of the apparent radiant produced by said resistance. We mention this because the contrary has been directly contended for.) The reason why the seeming curvature in a meteor's track is not greater than it is seems to be this, that the eye is not only comparing the track with the vertical, or the horizontal, at every point, but it is also to some extent comparing contiguous lengths of the track with each other ; and this tends to correct and diminish the illusion. For this reason the more rapid the flight of the meteor, the less will be the appearance of curvature in its path, for in such cases the visible path approaches more nearly to the condition of a luminous line seen at once from end to end, the parts of which can be more readily compared with each other. This was well illustrated by many of the quick-moving Perseids of August 10, 1883 *. * There is a detail of this illusion which is worth mentioning. It appeared to be very noticeable with a large proportion of the meteors of Nov. 27, 1872. Near the end of visibility, the apparent downward curva- CERTAIN ASTRONOMICAL PHENOMENA. The illusion of which we now speak may easily lead some persons into error when endeavouring to fix upon the radiant point of a very sparse meteoric system. If the insufficiently-experienced observer has not been for- tunate enough to catch with his eye any of the few meteors pretty near to the radiant point, he will, in producing the visible parts of the meteor-tracks backwards, almost certainly pass above the radiant, and so fix its position higher than it ought to be. Or, if on the look-out for meteors belonging to a certain known radiant, he might easily refer thereto some sporadic meteors really coming from a different origin at a lower altitude, when perhaps it might be important to know that, in fact, none belonging to the radiant were to be seen on that night. Conversely, when endeavouring to fix the position of a visible but very faint object, say a new comet, by using alignments with known stars at considerable angular distances from the object, he may easily do the opposite ; that is. assign to it a position lower than the true one. From his alignments the very faint object might be found again on the following night by himself, though perhaps not by another, whose skill in allowing for the illusion now in question might be either greater or less than his. This illusion might, with some persons, slightly affect the apparent curvature of a comet's tail, if very long. Some years ture of the meteor's path seems to increase somewhat rapidly, as in the ballistic trajectory of a projectile, caused by the resistance of the air. This also is represented in some pictures of meteoric showers. But though gravitation, with the resistance of the air, would really produce such an effect, it is utterly impossible, for the reason given above, that it could have been perceptible to the eye. The deception may be due, in some way, to the fact that the eye is following the apparently curved path of a luminous point whose velocity is being, for two distinct reasons, ever more and more rapidly retarded. This seeming phenomenon gives rise to another mis- apprehension. It makes the meLeors look, at the end of their luminous tracks, as though they were no farther off than the falling stars of a rocket. 10 VISUAL ILLUSION AFFECTING ASTRONOMICAL PHENOMENA. ago there was a difference of opinion between two correspondents in a popular scientific periodical respecting the curvature of the (long) tail of the great comet of 1882. This was, in all proba- bility, produced by the cause above mentioned. This might be not unimportant, in view of the conclusions as to the composi- tion of comets' tails drawn by Bredichin and others from their curvature in connection with the known motions of the comets. But a comet's tail, being a visible and permanent object during the observation, so that different parts of it can be compared directly with each other, is much less liable to be affected by the illusion now in question. CHAPTER II. THE EFFECT OF THE EARTH'S ROTATION ON CERTAIN MOVING BODIES. IT was believed by Aristotle and by Ptolemy that the earth's rotation, if it existed, should affect the motion of certain freely moving bodies. Galileo also perceived that this must be so, while rejecting the particular effects contemplated by them, at least by the latter. (See Chapt. III., Note A.) Newton was the first to point out that freely falling bodies must deviate to the east of the vertical, on account of the rotation of the earth ; and he suggested that experiments should be made with these in order to obtain direct proof of that rotation. Such experiments were tried by Hooke, in 1680, but with an insufficient height of fall. In 1836 Edward Sang, C.E., of Edinburgh, showed that the earth's rotation could be demonstrated by means of what is now called the gyrostat ; but he did not carry out any experiments therewith. In 1837 the subject was discussed, in connection with the flight of projectiles, by Poisson. It came much more prominently before the general public when Poucault exhibited his famous Pendulum to the French Academy in Eeb. 1851. Shortly afterwards he devised, for himself, and actually per- formed, the experiment with the gyrostat which had been proposed fifteen years before by Sang. A common and popular explanation of the deflection of projectiles, currents of air, &c., from the rotation of the earth, is that if, in our JS". latitudes, a body be moving southwards it is all the while passing over ground which has a greater velocity eastwards, from the rotation of the earth, than the 12 THE EFFECT OF THE EARTH'S ROTATION ground which it has started from, or has lately crossed, and that therefore it is left behind a little towards the west, or the right hand, by the surface of the ground beneath it ; and that, for corresponding contrary reasons, when moving northwards it will gain on' that surface towards the east, or still to the right hand. This is, of course, perfectly true ; but the particulari- zation of the meridional direction is often intended to imply, what is sometimes directly declared, viz., that the above state- ments are not applicable to bodies moving in the east and west direction. It is strangely forgotten that if a point on the solid ground south or north of an observer is moving towards his left, when he faces it, relatively to him as centre, with a certain angular velocity, a point on the ground east or west of him must be doing the very same ; and that therefore a sufficiently free Jiorizontally-moving body must be left behind, to the right in N., and to the left in S., latitudes, in whatever azimuth direction it may be going; and that, other things being equal, ita apparent deflection must be the same for all azimuths of motion. The period of the earth's rotation is, of course, a sidereal (not a solar) day ; this contains 86,164 seconds of mean solar time. The angle described in one second of solar time is, then, 360/86,164, or 15-04 seconds of arc, which in circular measure is 27r/86,164, or 1/13,713*; this then represents the earth's angular velocity of rotation, which we shall denote by u). The resolution and composition of rotations is among the first elements of rigid dynamics. The two components of the earth's rotation with which we are now concerned are F, or that about the vertical line at the locality in question as axis, whose angular rate is w sin X, X being the latitude of the place, and M\ or that about the horizontal meridional line at the locality as axis, whose angular rate is w cos X. (See NOTE A.) * It is interesting to note that 13,713 is, itself, the mantissa of its own logarithm to five decimal places. But we need not attach anj mystical significance to this coincidence. ON CERTAIN MOVING BODIES. ] 3 We may give here a practical illustration of the existence of these two components of the earth's rotation. If in N. latitudes a star close to the horizon be observed with a telescope whose eye-piece is furnished with a micrometer scale, the star will be found to have a motion in the horizontal direction towards the right (whatever vertical motion it may have compounded there- with) ; and this horizontal motion will be found to be the same for all stars close to the horizon in whatever azimuth direction they may be : and the angular rate of the horizontal motion will prove to be that of the earth's rotation multiplied by the sine of the latitude ; this being due to the earth's component rotation V. Similarly, if one observes any stars close to the prime vertical, or the great circle passing through the zenith and the E. and W. points on the horizon, he will find that they all have the same rate of motion along that great circle (whatever other motion they may have compounded therewith) ; and this angular rate of motion along the prime vertical will prove to be that of the earth's rotation multiplied by the cosine of the latitude ; this being due to the earth's component rotation M. Of sufficiently free bodies, those which are moving horizontally are affected by the component F", by which the surface of the ground at the place of observation rotates in its own (instan- taneous) plane. Those which are moving vertically, whether upwards or downwards, are affected by the component rotation Mi by which the surface of the ground is always being tilted over eastwards. We shall first consider those which are influenced by the component V. It may be best to begin with an imaginary case, for illustration. Suppose a body started to slide on a perfectly frictionless, even, horizontal surface, or floor, in a vacuum. If the floor were stationary the body would, of course, describe, from inertia, a right line thereon with a uniform velocity v. But as the floor is always rotating in its own (instantaneous) plane 14 THE EFFECT OF THE EARTH ? S ROTATION with the angular Telocity o> sin \, and as there is no connection between the floor and the body which would make the latter partake of the rotation, it will not do so; but it and its radius- vector will be left behind, and that line, if visible, would appear to rotate about the point of starting, watch-wise in IN", latitudes, and counter- watch-wise in S. latitudes, with the uniform angular velocity w sin X, while being itself described by the body with the uniform linear velocity v. Consequently the body would describe about the point of starting, as pole, a spiral of Archimedes, whose equation would be r = : 0. If the body were started (i) sin A. from the middle of the floor, with so small a velocity that it would not reach the edge of the floor for a few days, it would present the curious phenomenon of revolving (with an ever widening orbit) round the point of starting for no apparent reason. We must, however, content ourselves with the consideration of masses moving horizontally under more ordinary conditions. The winds afford a familiar instance. The explanation of the direction of the trade-winds and cyclones is now pretty generally known, and needs only to be mentioned. The greater heating of the air by the sun in the neighbourhood of the thermal equa- tor causes that air to ascend, which occasions an indraught of the lower air both from the !N". and from the S. For a non-rotating earth, the general direction of these would be meridional ; but the rotation of the earth causes an apparent turning to the right on the north side, and to the left on the south side, of the equator ; thus producing the N.E. trades on the north side, and the S.E. trades on the south side, of that line. A sufficient local extra heating of the air in N. latitudes causes, in a similar way, an indraught of the lower air from all sides; the component rotation V causes the converging masses of air to pass in N. latitudes to the right of the centre of the super- heated area, which produces a vortex turning in the opposite direction to that of the hands of a watch lying face upwards on the table ; the corresponding result in south latitudes being a vortex ON CEETAIN MOVING BODIES. 15 turning the other way, or with the hands of the watch. Such vortices being called cyclones. Ocean currents must be very considerably affected by compo- nent rotation F; but these are subject to a variety of other important influences of which we shall mention only prevailing winds, land barriers, and mutual interference. It would be generally impossible to distinguish the effect now in question from others, and useless to speculate thereupon ; except perhaps in one apparently simple instance, with which, as it happens, we are practically concerned. It can hardly be doubted that it is largely in consequence of component rotation V that the warm Gulf Stream bears so strongly on the coast of north-western Europe. It may be worth while to advert to the following : There are five great ocean vortices. The two in N. latitudes, viz., that in the N". Pacific and that in the N. Atlantic, both turn watch-wise. The three in S. latitudes, viz., that in the S.E. Pacific, that in the S. Atlantic, and that between S. Africa and Australia, all turn counter -watch -wise. It seems highly probable that all this is, at least, promoted by the earth's rotation. If a great ocean vortex were due to an extensive current movement of the water, produced somehow under the condition of the earth's rotation, the direction of its turning would be such as we have just mentioned, and opposite to that of a cyclone in the same latitudes produced as above described. The course of the flight of migrating birds is probably some- times affected by component rotation F". But as the consideration of their case would involve some speculation, let us propose to ourselves another one which might actually occur. The keeper of a light-house several miles out from the coast has some homing pigeons, bred by himself, which are well acquainted with the district. One is let go from a point on the coast ; it starts at once to return directly to the light-house; the bird is guided solely by his sight of the light-house, and the water being per- fectly smooth, he is without means of knowing that he is always edging sideways to the right of the instantaneous straight line 16 THE EFFECT OF THE EARTH ? S ROTATION from himself to the light-house : he will keep his head always pointed directly towards the light-house, and to do this he must be continually turning very slowly towards the left, doubtless without perceiving that he is doing so. The forward velocity of his flight is uniform, and his involuntary sideward motion to the right will go on increasing, until the resistance of the per- fectly still air to that motion becomes great enough to prevent any further increase therein ; the sideward velocity then has reached its final magnitude, and becomes constant, like the forward velocity. The bird's visible course, or that relative to the surface of the earth, will then become, quam prox., a loga- rithmic spiral described backwards towards its pole, which is at the light-house. (See NOTE B.) Let A be the point at which this has taken place. If now the latitude be 51 30', that of London, and the distance from A to the light-house be 10 miles, and the bird's velocity of flight be at the mean for such cases, say 800 yards per minute, or 40 feet per second, and his weight 14 oz., and the coefficient of sideward shifting 9-8, which we have good reason to believe is pretty nearly correct ; then his greatest departure to the right of his intended straight line of flight from A to his home will be just about 70 yards, at the distance of 3-68 miles from the light-house. If this departure seems somewhat small, let us remember that it has taken place in spite of the bird's constant (unconscious) endeavour to avoid it, and in spite of the lateral resistance of the air. Probably there is always a sensible deviation of this kind when a bird is travelling to a sufficiently distant intended goal. His flight, however, being generally over the land, the sight of the more prominent objects in view would make him more or less aware of his sideward shifting, and thus suggest to him to make some allowance for it by directing his head to the proper side of the goal, or the left in jSf. latitudes ; but the amount of angular allowance necessary would depend on the velocity of flight, and on the latitude, and also on the bird's own weight and his coefficient of sideward shifting ; and it seems very unlikely that OX CERTAIN MOVING BODIES. 17 instinct, much as it can do, would enable him to make due allowance on account of these ; though it would doubtless enable him to provide against a cross wind. We see, then, that the familiar expression, " As straight as the crow flies," should not be lightly used, without distinctly postulating the condition that the bird is making due correction for the rotation of the earth. In the case of a railway engine and train running along a perfectly straight reach of the line, the rails being perfectly level with each other, the sideward shift is prevented by the resistance of the right-hand rail in 1ST. latitudes, and of the left-hand rail in S. latitudes ; and it is said that the right-hand rail and the flanges of the right-hand wheels get more wear in N. latitudes, on this account, than the others. This is undoubtedly so. We know already (see NOTE B) that the expression for the pressure P against the right-hand rail is W; ...... (1) in which v is the velocity in feet per second and W the weight of the moving body (see NOTE B). For an engine going at 30 miles an hour, or 44 feet per second, in the latitude of London 51 30', this would be l/6410th part of its weight, and if this weight were 30 tons the whole pressure would be 10'48 Ib. ; and this would have to be distributed among all the right-hand wheels. The effect, then, is so small that it must be undistin- guishable ; as it would be altogether overborne and masked by a gentle "cross wind, or by a difference of level between the rails, say 4*71 feet apart, of only 1/lOOth inch ; or by a gentle curve in the line of 60 miles radius ; not to mention other causes of unequal wear on the two rail?. It is said also that rivers in N. latitudes must, for the same reason, wear away their right-hand banks slightly more than their left ones. This is undoubtedly true ; but the effect is utterly imperceptible ; not only because the greatest velocity of a river flow is relatively so small, but also on account of the c 18 THE EFFECT OF THE EAETH*S KOTATIOBT incalculably greater effect of various other causes of inequality in the erosion of the banks. Among moving bodies influenced by V must be mentioned the famous Foucault's Pendulum; but this is deserving of a chapter to itself, which we shall give it. We now come to moving bodies which are affected only by the earth's other component rotation M, that is to say those moving vertically, whether upwards or downwards. Just as in N. latitudes, a sufficiently free body, projected or moving away horizontally from before a spectator standing vertically, will deviate towards his right, so if gravitation could be neglected, to a spectator lying horizontally in the meridian in jST. latitudes, with his feet to the south, a body projected away from before him in the plane of the prime vertical will deviate to his right, owing to the rotation of that plane in itself with the angular velocity w cos \. Gravitation alters the case, except for a ver- tical discharge or movement. If the observer lie on his back and discharge the projectile vertically upwards its deflection towards his right is one to the west. If he lie face downwards, say at the edge of a mural cliff, and discharge, or simply drop, the body downwards, its deflection towards his right is one to the east. We shall here discuss the latter, viz., a body dropped from a height. That such must deviate to the east of the plumb-line is easily seen otherwise thus. A point on the surface of the ground is moving eastwards, from the rotation of the earth, with the linear velocity Etu cos X, R being the earth's radius ; but a point directly over it, at the height A, is moving eastwards with the velocity (R-t-A)w cos \. The latter is therefore moving east- wards faster than the former with the additional velocity Tim cos X ; consequently a body simply dropped from the upper point will, at its fall, have left the lower point behind it towards the west. This deviation, 3, of the body towards the east, if the resistance UNIVERSITY OF ON CEKTAIN MOVING BODIES. 19 of the air to the falling body be neglected, is given by the equation 3=ffowcosX, ....... (2) which is, for same h, one fourth of the Westward Shift in the Chapter on the Deviation of Projectiles (since, for same A, this t is half the other). (For proof see NOTE C.) It being in a vacuum 7i=|<7tf 2 ; and the above expression for this deviation can be written |A / w cos X, as it usually is. A body dropped from a height must have also, as is evident, and as was pointed out by Hooke, a very small deviation towards the south ; this is not produced by Jlf, but by the horizontal (southward) component of the centrifugal force of the earth's rotation being greater at the top of tjie height of fall than at the ground (except at the equator, where both are zero). Its magnitude, which is easily obtained geometrically, is only JAZ'V sin 2\, or | w 2 sin 2/\ ; neglecting the resistance of the J air. The presence of w 2 in it shows, at a glance, that it must be excessively small for all practicable experiments. In that of Guglielmiui, mentioned below, it would be less than 1 /30,000th of an inch. (The formulas now given agree with that of Prof. Bartholomew Price obtained analytically.) If, in the analytical discussion of the deviation of a falling body from the vertical, quantities of higher (i. e. smaller) orders of magnitude than the first are neglected, this component of it does not emerge into view. A body projected vertically upwards is not affected in this manner, either in its ascent or descent. Experiments have been carried out by various persons to detect the deviation of falling bodies from the vertical owing to the rotation of the earth. For instance by Guglielmini, in 1792, in a tower at Bologna (lat. 44 30'), with a height of fall of 241 feet; by Benzenberg, in 1803, in a tower at Hamburg (lat. 53 33'), with a fall of 234 feet, and, in 1804, in a coal- mine at Schlebusch, Westphalia (lat. 51 25'), with a fall of c2 20 THE EFFECT OF THE EARTH'S ROTATION 262 feet ; and by Reich, in 1 832, in a mine near Freiberg, Saxony (lat. 50 53'), with a fall of 488 feet. These experi- ments, especially those of Reich, were, as far as regards the eastward deviation, satisfactory, considering the delicacy of their nature and the great difficulty of avoiding various causes of in- accuracy, some of which could produce disturbances often very much greater than the deviation to be determined. (See NOTE D.) It is self-evident that, the height of fall being given, the a stward deviation will be greater if the time of falling can be made so in a proper manner. Therefore, for given h, the east- ward deviation is greater in resisting air than in a vacuum. ( But we shall find in Chapt. III. that this last is not the case with the westward deviation of the point of fall of a body discharged vertically.) This suggests a more convenient method of carrying out such experiments as the above. By making the falling body descend slowly enough we can obtain an eastward deviation, <), large enough to be satisfactorily determined, \vith very much smaller heights of fall than those mentioned above. The falling body might be a sort of parachute, very easily designed and con- structed, which, like a shuttle-cock, would be kept rotating about its vertical axis by the resistance of the air. If this were such that it would descend with a uniform velocity, v, of three feet per second, it would have, with a fall of only 80 feet in the latitude of London, a deviation, , to the east of just over one inch, allowing a little for the lateral resistance of the air. This deviation is 17 times as much as if the fall had been in a vacuum, and probably 14 times as much as that of a bullet let fall in air. In this case the equation is 2=^oi cos X; (3) the 2/3 of equation. (2) being Inow unity (see NOTE E). Since t is here 7i/v, this value of o is to cos X. Therefore, for the same parachute, the deviation varies as 7i 2 . Observe the last paragraph of NOTE E. ON CERTAIN MOVING BODIES. 21 The parachute might be an inverted cone, about three inches in diameter, composed, say, of tracing-paper, and furnished with two very small wings opposite each other and set obliquely so as to cause rotation. If the vertical angle of such cone be 90, or a little less, it will descend steadily with the velocity mentioned. It should, of course, be made to descend within a chimney-like box, to protect it from movements of the air ; and this should be in a suitable place inside a building ; so that there might be no convection currents within the box, caused by inequalities of temperature on different sides of it. It would probably be impossible, except by accident, to make the parachute so sym- metrical about its axis that it would not be slightly deflected from its proper line of fall by the resistance of the air. But because of its rotation it would descend in a cylindrical helix of very small diameter, the axis of which would be the mean line of descent and the actual line in a vacuum. If a large enough number of experiments were instituted, in which the parachute was made to start with the same side in different azimuths, the small errors arising from the semidiameter of the helix would be self-compensating. The very small lateral resistance of the air would, of course, slightly diminish the lateral deviation from the rotation of the earth. A free pendulum (that is one free to swing in any direction, like Foucault's Pendulum, and unlike a knife-edge pendulum, or that of a clock) is affected, as to its rate of oscillation, by its sharing in component rotation M. It is, whether it be hanging at rest, or oscillating, rotating about the meridional horizontal line through its point of support, with the angular velocity o) cos A. There is, therefore, a downward centrifugal force, as we shall express it, acting on the pendulum at its centre of mass, which, taking the mass as unity, is Vw 2 cos 2 /\ ; r being its mass- radius, or distance of the centre of mass from the axis of rota- tion. It is evident that if the plane of vibration be E. and W., this centrifugal force, though apparently conspiring with g, will not increase the rate of vibration, because it is always directed 22 THE EFFECT OF THE EAETH's ROTATION along the pendulum rod ; it is not parallel with the direction of g, except incidentally, at the instant when the pendulum is at the lowest point. Consequently the period of a free pendulum swinging E. and W. is not affected by its rotation with M But if the plane of swing be in the meridian, the centrifugal force due to the rotation of that plane about the horizontal N. and S. line through the point of suspension is always parallel to the direction of g, and not in the line of the pendulum rod ; except at the instant when the pendulum is at the lowest point. It is always proportional to the distance of the centre of mass from the said axis of rotation ; but if the amplitude of swing of the pendulum be very small, as it ought always to be in the scientific use of the pendulum, this never differs sensibly from r. The pendulum, therefore, is oscillating, not simply under g acting at the centre of mass, but also under the parallel, con- spiring, and sensibly constant centrifugal force rw 2 cos 2 X, acting at the same centre (the mass is still taken as unity). It is very easy to see that if the plane of vibration be not in the meridian, but inclined thereto at the azimuth angle 2, we shall have for the time t of the vibration of the free pendulum, not t = TTA /- but (see NOTE F) V g , \ 1 IT- cos2 X cos 2; r , quam procc. . (4) 9 I % J If then the free pendulum's radius of oscillation I be that of a seconds pendulum, it will gain, in consequence of its own rota- 2 tion with M, 5 cos 2 X cos z sec. in every swing. It being a J Foucault's Pendulum, its plane of vibration will rotate relatively to the material surface of the ground once in 24 sidereal hours / sin X. Therefore its rate of gaining is constantly varying from zero to its maximum, and back again, with a period of 12 sidereal hours / sin X. If the pendulum is oscillating meridionally at the equator ON CERTAIN MOVING BODIES. 23 (where it will retain its azimuth of oscillation), so that the gain shall be greatest, and if r be 37 inches, which is perhaps a fair . mean value of it, the gain will be at the rate of one second in 125 years. Of course the practical unimportance of this does not detract from its dynamical interest. At the poles of the earth, where cos 2 \ vanishes, the vibration period of the free pendulum is unaffected by the rotation of the earth. We now come to moving bodies which are affected by both components, V and M, of the earth's rotation. Some of the movements of the atmosphere and of the ocean, must be modified by V and by M at once ; each making its own special contribution to the whole effect. There is a phenomenon which must be largely due to both components of the earth's rotation acting together as auxiliaries. There would appear to be, in equatorial regions, a continuous current from E. to W. in the upper parts of the atmosphere, at the height of 20 miles or so. The peculiar sunsets which began with the great eruption of Krakatoa, in 1883, passed thence successively westward round the equator. It was evident that their cause was, at first, of limited extent, and that it was travelling in the direction mentioned. Before it became too diffused and widely spread, several passages of it round the equator could be distinguished, showing that it completed the circuit of the equator in about 13 days. It seems impossible to account for this but by the great cloud of fine dust from that unusually violent eruption ; such dust being known to be capable of producing such effects. That dust must have been carried by a continuous current in the upper air over the equator from E. to W., at the rate of 76 miles per hour. It is obvious, from what we have seen respecting the trade-winds, that Fand M would both conspire to produce this current, helped, no doubt, by the daily revolution round the earth of the sun's heating effect on the atmosphere. We now turn to the pendulum swinging on knife-edges. This 24 THE EFFECT OF THE E\KTH's ROTATION is affected by M, as to its rate of oscillation, precisely in the same manner as the free pendulum, considered above, which has for the instant the same azimuth of oscillation ; but, unlike the latter, its rate is affected by V also. The plane of its oscillation rotates about the vertical line through its position of rest with the angular velocity w sin X. This produces, in this pendulum, a centrifugal force directed away from the pendulum's position of rest, and opposing gravity. Then for very small amplitudes of oscillation, we have for the time t of the knife-edge pendulum, as affected by both components, or the whole, of the rotation of the earth (see NOTE G) (5) (J If always made to swing in the meridian, it will gain at the equator at the same rate as a free pendulum so swinging which has the same I and r ; and it will lose at the poles at that same rate (though of course the free pendulum will not do so) ; and at latitude 45 its rate will be unaffected by its rotation with the earth. In general, in order that the knife-edge pendulum should be unaffected by its rotation with the earth, its plane of vibration should have such an azimuth z that cosz = tan 2 X. This relation is, of course, impossible in latitudes higher than 45 ; therefore in such latitudes the knife-edge pendulum must always swing, because of its rotation with the earth, more slowly than is due to the length of its radius of oscillation. We see that two pendulums with the same calculated Z, or radius of oscillation, at the same locality, and with parallel planes of oscillation, will not go together with perfect accuracy, on the rotating earth, unless they have also the same r, or mass-radius. If the pendulum be a straight uniform rod, it will have the same I, or calculated radius of oscillation, viz., two thirds of its whole length, whether it be swinging about one end, or about a point of trisection ; but its r will be three times as great in the former case as in the latter ; and the rate of gaining will also be greater in the same proportion. ON CEKTAIN MOVING BODIES. 25 We see also that, in consequence of its rotation with the earth, the point of suspension and the actual centre of oscillation of a pendulum are not interchangeable ; except under the condition that the centre of mass is halfway between those two points, which, of course, is a quite possible condition. The present matter would be of no practical importance in the ascertainment of the value of g by pendulum experiments. Still it should not be passed over altogether without notice ; it ought to be at least mentioned, if only for the purpose of pointedly excluding it from consideration. A difference of one hundredth of an inch in the height of the barometer would be taken account of in obtaining the value of g by the pendulum ; and it is by no means self-evident beforehand that the rotation of the instrument with the earth has less effect on its rate of vibration than that apparently quite insignificant item of consideration. The apparent course of a projectile is affected by both com- ponent rotations, V and M. But it will be better to consider this separately in the next chapter. NOTE A, from p. 12. In Pig. 1 let the circle be the outline of the earth, P its north pole, and C its centre. Let D be the situation of the place of observation at a certain instant, and PDA the meridian line of said place, DEG being its parallel of latitude. Suppose that the rotation of the earth about its axis PC, in the direction indicated by the arrow, would carry the place of observation from D to E in one second of time. Draw tangents at D and E to the surface of the earth in the meridian planes of those points, meeting the production of the earth's axis in H, and complete the diagram. The angle FDH is evi- dently the latitude of D, or X. In one second the earth has turned through the angle ACB, or DFE, or w. But the hori- zontal N. and S. line through the place of observation, now at E, has turned only through the angle DHE. Now the angles DHE and DFE, being both exceedingly small and with the same sub- 2b THE EFFECT OF THE EARTH S ROTATION tense, as we may call it, they are inversely proportional to their radii HD and jiF, or directly as sin \ to 1. Therefore in one second the face of the ground at the place of observation has turned in its own plane through o> sin A. Again, CD and CE produced are vertical lines at D and E. Therefore, in the same time, the vertical line at the place of observation has turned eastwards about the horizontal N". and S. line, as axis, through DOE. Now DCE and ACB, being both exceedingly small and with equal radii, they are to each other directly as their subtenses, or as FD to CA, that is as cos X to 1. Therefore in one second the vertical line at the place of obser- vation has turned eastwards about the N. and S. horizontal line at that place as axis, through w cos X. NOTE B, from p. 16. First let us prove the following, to be used again in p. 17. A perfectly free body is moving horizon- ON CERTAIN MOVING BODIES. 27 tally, always directly away from its starting-point. Its radius- vector, or the line from that point to itself, will have, in N. latft., a uniform angular velocity of deflection to the right, relatively to the ground beneath, the magnitude of which is w sin X. Now let the body have the uniform velocity v along its radius-vector r, so that r = ^, t being the length of the time of the movement. The velocity of the linear sideward shifting of the body is rw sin X, or vtw sin X ; it therefore increases uniformly with the ime, that is with a constant acceleration, which we shall call a. The linear space described in the first second of time under this constant acceleration is v u) sin X. Therefore a = 2twsinX, per sec., per sec. Multiplying the right side of this equation by m, the mass of the body, and the left side by the equivalent W/# (W being the weight of the body and g gravity), we have for ma, or the apparent sideward pull on the body, or E, 9 The rightward sidling of the body, relatively to the ground beneath it, is as though it were produced by a constant force or pull P, of the magnitude now given. And if that sideward shifting be stopped by some impediment (such as the right-hand rail in the case of a railway train in N. lats.), the forward v remaining the same, the body will continue to press against the impediment with that force F. Now if the impediment be that of the resistance of the air, the body's, in this case the pigeon's, sideward motion will at first increase, until the consequently increasing resistance of the air to that motion becomes =F. The sideward velocity then becomes uniform, like the bird's forward velocity along the radius-vector. Let s be the sideward shift in one second when this has taken, place. Then s/v is the tangent of the angle between the tangent to the curve and the radius-vector, at any point of the curve. That angle is then constant ; and this is a distinguishing property of the logarithmic spiral. 28 THE EFFECT OF THE EARTH'S KOTATION" n s rdQ s dr Ur thus -= , as is evident, or dd= : whence o 6= ~ log r+ C ; C being a constant which we do not now want to determine. Thus when the sideward velocity becomes uniform, but not until then, the curve settles into a logarithmic spiral whose pole is at the starting-point. Now suppose the bird to do the opposite, viz., to fly towards a given point with the velocity v, always turning so as to keep his head directly towards the point, notwithstanding his con- tinual shifting rightwards from the rotation of the earth. It is easily seen that he will describe a similar spiral backwards; the polo being at the goal-point. In Fig. 2, Aa and be are intended Fig. 2. to represent v, ab and cd to represent s. As in p. 16, A is not the pigeon's starting-point on his homeward flight ; but the point at which his sideward shifting has become constant. For clearness this figure and the next have been drawn altogether out of scale. L is the light-house. o The equation 6= logr+(?, though perfectly accurate if the problem, as stated, be regarded as one of abstract kinematics, will, for certain obvious dynamical reasons, not be realizable in the concrete case of the pigeon for distances too near the pole. If the logarithmic spiral A bd were produced backwards towards the pole L, it would make an infinite number of turns round the pole before reaching it ; which, in accordance with the statement of the problem, would have to be described by the bird in a very short time. Near the pole the bird could not, and would not if he could, do as we have proposed for him. But for the distances therefrom with which we are now concerned we cannot doubt ON CEKTAIN MOVING BODIES. 29 that he would do so ; and his departure from the logarithmic spiral due to his inertia (for there would be such) would be quite insensible. Let us assume that the resistance of the air to the transverse velocity is proportional to the square of that velocity, and there- fore to s 2 . The resistance due to the final transverse velocity being, as we have called it, F, s=ZC/y/F ; in which K is a con- stant. "We know the value of F from the above ; that of K can be ascertained only by experiment*. It would appear that it is just about 9-8, if the weight of the bird be expressed in ounces. The approximate correctness of this has received a certain satis- factory confirmation. We have then g _o. A'" ^n XW P , nr Q .a /* X 40 x sin 51 30' JI4 V g V 13713x32-2 which is 0-4366 ft. ; and s/v, the tangent of the tangential angle, is 0-0109, or 1/92, very nearly. We neglect the quite unimportant effect of the difference of latitude between A and the light-house. The greatest departure of the pigeon to the right of AL is easily obtained very approximately in this case. Since s/v is so very small, it differs very slightly indeed from the circular measure of the angle 38', of which it is the tangent, and also from the sine of that angle. If DB, Fig. 3, be the greatest distance of the curve from LA, the tangent at D is parallel to LA., and DLA is equal to what we have called the tangential angle of the curve. DB, which we wish to ascertain, is LB- or (as DLA is so very small) LD - , quam prox. Now v ; * * Experiments were made with a falling inverted cone of light paper esti- mated as presenting to the air through which it moved a horizontal areal section equivalent (not equal) to that of the side aspect of a flying homer. The coefficient K ^s the number of feet fallen through by the cone in one second, after attaining its final velocity, divided by the square root of the number of ounces in its weight. 30 THE EFFECT OF THE EAKTH's ROTATION let be the angle between LA and the selected axis or prime vector, wherever that may be, and & the angle between LD and the same ; then we have 61= - log LA + K, and 0' = - log LD + K. Therefore 00', or angle DLA, or q. pr., - thus - =- log! ; whence log =1, which is the logarithm v v L.JD LD of the base of the system of logarithms, viz. : the Naperian. Thus LA/LD=that base ; and LD is 10 miles /2-7182*, or 3-68 miles, and DB is this X - (i. e. by -%) which is 70 yards, very nearly. Fig. 3. NOTE C, from p. 19. Though the following geometrical proof of this, by R. A. Proctor, is on the same lines as that given in Chapt. III., NOTE C, for another deviation, we may consider it here on account of the use to be made of it in the next NOTE to this. In Pig. 4, let bed be the surface of the earth and C its centre, and let ab be the height of the fall. The body, ready to drop from a, has been describing the continuation of ea beyond a, with a uniform areal velocity about C. When let go it describes the (absolute) curve ad under the force of gravitation directed to (7, and therefore with the same areal velocity about C as it had before. The curve ad, though really an ellipse with the centre of the earth in one focus, is sensibly a parabola. Suppose that when the body has reached d, the top of the height of fall has ON CEETAIN MOVING BODIES. 31 reached e; draw eO. We can see quite easily, a priori, that cd and cf are so exceedingly small, relatively to ab and be, that the proportional difference between ec and ef may be neglected with- Fig. 4. out sensible inaccuracy. Now the areas a Ce and aCd are equal, as describable in the same time ; and therefore taking away the part common to both, aef is equal to fCd. Then, since abce is sensibly a rectangle, and, as we have said, ec may be taken for ef without appreciable error, we have, from a well- known property of the parabola, ^ab x bc= |Tt x cd ; or ^ARw cos \t =|Ro ; E being the earth's radius. That is to say, 3 = f faw cos \. Q.E.D. NOTE D, from p. 20. This Guglielmini must not be confounded with the distinguished physicist, with the same surname, also of Bologna, who died in the year 1710. He described his above experiments in a work De motu terrce diumo, Bologna, 1792, quoted by Delambre in Astron. Theor. et Prat. torn. ii. p. 192. Benzenberg described his experiments in a book Versuclie uber das Gesetz des Falles, Dortmund, 1804, and in Versuclie uber die Umdrehung der Erde neu berechnet, Diisseldorff, 1845. For an account of Reich's Fallversuche uber die Umdrehung der Erde, see Poggendorff's Annalen, vol. xxix. 1833, p. 494, as also Houel's De deviatione meridionali corjporum libere cadentium, 32 THE EFFECT OF THE EARTH'S ROTATION Utrecht, 1839. This experiment has been tried also at Verviers in Belgium, and doubtless elsewhere. E, from page 20. This can be readily seen thus. In Fig. 4 the curve ad is sensibly a parabola ; but now as the velocity of descent is uniform, ad is sensibly a right line (but of course much longer than before for the same ab). The area aCd is still equal to aCe ; because the resistance of the air on which it depends is sensibly (though not accurately) a central force, though directed from C ; and aef, which we have agreed to take as aec, is now one half of ab x be, instead of one third of it ; consequently the equation (2) becomes 3 = ^wcos X. Q.E.D. Of course the parachute, after being let go, will not attain its final and constant velocity until it has fallen a short distance ; in the present case about one foot. This will make the resulting deviation less than what is given in formula (3), just demon- strated ; but, for a fall of 80 feet, or more, the difference is so small, proportionally, as to be quite unimportant. NOTE F, from p. 22. The absolute centrifugal force being, as we have said, rw 2 cos 2 X, if the plane of vibration be inclined to that of the meridian at the azimuth angle z, the effective part of the c.f . will be ro> 2 cos 2 X cos z, and the pendulum will oscillate under g + rw 2 cos 2 X cos z, acting at the centre of mass and parallel to g. Therefore the time of vibration of the free pendulum is not HA / -, but v; which can be written, quam prox., as equation (4) in text. NOTE G, from p. 24. It is easy to see that for very small amplitudes of oscillation the tangential component of the c. f., now in question, acting on the centre of mass away from the position of rest of the pendulum, is no 2 sin 2 X sin 0. This then acts at the same point, and according to the same law of distance ON CERTAIN MOVING BODIES. 33 from the point of rest, as the tangential component of gravity 9 or cj sin d. Therefore, while in p. 22, and in NOTE F, no 2 cos 2 \ cos z had to be added to g, now rw 2 sin 2 A must be subtracted from their sum, making g + r w 2 (cos 2 X cos z sin 2 A). Therefore the time of vibration of the knife-edge pendulum, as affected by its rotation with both V and M, is rw a (cos a A cos z sin 2 A)' which can be written, quam prox., as equation (5) in text. CHAPTER III. DEVIATION OF PKOJECTILES FKOM THE ROTATION OF THE EAETH. THIS interesting subject, though coming under the heading of the last chapter, seems worthy of having a chapter to itself. It is treated imperfectly in elementary books, &c., which cannot afford to give it the amount of space that could be desired. A sometimes important factor of the question, viz., the westward shift ,of the point of fall of the projectile from the earth's rotation, is usually overlooked ; and this sometimes gives occasion to certain incorrect statements (see footnote, p. 42) : besides which, in the works just referred to the alteration of the range of the projectile's flight by the rotation of the earth is neglected altogether. (See NOTE A.) The present subject, though a very interesting one in itself, is of but little practical importance. The effects with which we are now concerned are so overborne and masked by other disturb- ances of accuracy in the intended flight of projectiles, that they may be not even mentioned in a modern text-book of gunnery. They are, however, recognized by the Boyal Artillery Institution. It is hardly necessary to observe that the deviation now in question is, unlike the others, only apparent, and relative to us ; like the rising and setting of the sun. It is not the projectile which departs from its course in a certain direction, but the earth which turns beneath it in the opposite direction. The principle concerned in the deviation of projectiles from DEVIATION OF PROJECTILES. 35 the rotation of the earth depends on the existence of the two components of the earth's angular movement of rotation, which we have considered in Chapter II. The component of the earth's rotation which has the vertical line at the place of dis- charge as its axis we have called component rotation V, its angular velocity being o o O O ^l O5 d d ss -a CO O Th O i> 6 o o J2 rH To co rl K & i G\ QO 05 t^ Tfl CO O 05 II II 00 O C5 ?? a o o o ^ O5 O2 OQ o o o o CM CM I 3 8 1^ <^) 00 O^ S o T f cb rH rH cb rH N rH ^H iH rH rH rH + + + 1 1 1 H 11 II II II II II II II o O -J ^ Q o ^O >O (^> iO (^^ $ be O 05 CO rH 00 rH CM 1^ ^ ^>^ >^ ^.^ t^ ^^ ^^ cr 1 2,GO 00 oo 00 GO GO op 00 00 W CO cb cb cb cb cb cb cb cb 1 * -ti ^f> r^ , ^ rCj i I o g *G o 1 i 4 co o s CO o 10 CO *-- g f CO o o IQ ^ cb r-l I o CO cb o CO CO cb I M "4* 1 1 1 _l T L- 4^ ^O ~ 1 ' s PH 1 " H II II II II II H 11 ft i o o CN o fe r i o ,--xO ^ 05 rH 11 CM (M CO CO CO .^ QQ O!} 02 ' QQ CO "sT 8 8 8 8 o o 8 if *p 8 S s s 10 o 10 ^ cb cb cb ^ cb cb o ^ s p4 w p4 ^ ^ ^ ^ vz *N ^ OQ 5 fa ^*" GQ 4^ 48 DEVIATION OF PROJECTILES FKOM We may now give a diagrammatic illustration, Figs. 5, 6, 7, 8, for diverse azimuths of discharge, but with the same trajectory in all four cases, viz., that selected for the above tables. But lat. 15 N". is now selected in order that the three shifts may, for convenience, not differ too much in magnitude. We could, of course, take the shifts of the point of fall of the projectile in any order we please : but it will be convenient to begin, as above, with (a), the purely longitudinal shift. The principle of con- struction is the same, and the lettering correspondent in all four figs. The thick line Im is the latter part of the range for a non- rotating earth ; m being the end thereof. The dotted line mn is the purely longitudinal shift, whether an increase or a decrease of range. The dotted line no is the purely transverse shift, to the right, the lat. being N. The dotted line op is the westward shift. And the double line mp is the whole shift compounded of the others. The letters Z, m, ra, o, p, taken in alphabetical order, enable "the reader to compare these four diagrams at a glance. From m draw me due east, whether the lat. be N. or S., its length representing the value given by formula (1), which is, in this case, 58'3 yards (this me is the purely longitudinal shift for E. and for W. firing) ; from e draw en at right angles to above range ; mn is the (purely) longitudinal shift, its value being 58-3 yds. X sin z, formula (2). From n draw no in the line of m, that is at right angles to the range, and towards the right hand in N. lats., and towards the left in S. lats., its length repre- senting the value given by formula (3) ; this is the (purely) transverse shift, the magnitude of which is, in this case, 25*7 yds. From o draw op due west, whether the latitude be N". or S., its length representing the value given by formula (4) ; this is the westward shift, the magnitude of which is, in this case, 42-0 yds. Then the double line mp represents, in magnitude and direction, the whole shift of the point of fall of the projectile compounded of the three shifts just mentioned. The whole, or net, longi- tudinal" shift is sensibly the orthographic, .projection of mp on mn ; and the whole or net transverse shift is the distance of p THE ROTATION OF THE EARTH. 49 cn 50 DEVIATION OF PROJECTILES FROM from mn, since both of these shifts are so very small relatively to the range. On the scale of these Figs, the thick line repre- senting the undisturbed range ending at m should be 41 -5 feet long. For a given trajectory, as we see, the lines no, op, and the line of construction me are constant for all azimuths of pro- jection ; but not so any other lines. In Figs. 5 and 6 the actual net transverse shift is to the left, though the latitude is north. It would, of course, be greater if the direction of discharge were due N. For azimuth of discharge 142 10', or 37 50' E. of N., formula (8) becomes zero, and there is no lateral deflection. The proportionally very small effect of the resistance of the air on our formula is still neglected. The following distances and times of flight with the Martini- Henry Eifle and Bullet, fired so as to have the range of 1000 yds., are taken from Mackinlay's Text-book of Gunnery, 1887, p. 159 ; the weight of the bullet being !! oz. and its diameter 0-45 inch (the angle of elevation about 2 31', muzzle velocity 1353 ft. per sec. ; these two items, however, do not now concern us). The deflections here given are the (purely) transverse ones, formula (3). They are calculated for lat. 51 31', N. The deflection, formula (6), involved in the westward shift, has been disregarded ; as it is relatively very small in such flat trajectories. Even at the distance of the full 1000 yds., with height of tra- jectory 45-5 ft., it would not amount, even at its maximum for N. and for S. firing, to th inch. But, for this same trajectory, the increase of range for E. and the decrease for'W., firing would be as much as about 2*3 yds. Distance, yards. 200 400 600 800 1000 Time of flight, seconds. Deflections, to right, inches. 0-501 0-21 1-104 0-91 1-787 2-20 2-548 4-20 3-395 6-96 THE EOTATION OF THE EAETH. O~L The diminution of these deflections from the lateral resistance of the air is evidently exceedingly small. It would appear that in the last deflection, where it is greatest, it would only be able to diminish by 1 the digit in the second place of decimals. A, from p. 34. Aristotle contemplated a connection between the earth's rotation, if it existed, and the movement of certain projectiles. He argued (De Coelo^ II, 14, 6) that since, as he believed, a heavy body projected vertically upwards falls back on the point of discharge, the earth must be without rotation. He gives no hint of what the effect of the earth's rotation would be in this case, if it existed. But Ptolemy con- tended {Almagest , I, 7) that if the earth rotated with the enormous eastward linear velocity of its surface involved (except, of course, in very high latitudes) in a globe of its si/e turning completely round in one day, flying birds and projectiles could never get eastward of their point of departure, but would be left a long way behind to the westward of that point. It was reserved for Galileo to give the now so obvious refutation of this objection of Ptolemy's, which he does in his Systema Cosmicum. Galileo, however, seems to have considered the connection between the motion of projectiles and the rotation of the earth, not for its own sake, but merely with the object of removing what was regarded by many as a most serious difficulty in the way of the system of Copernicus. His mind was fixed so strongly on this important object that he did not care to go, as fully as he might and could have done, into the question with which we are now concerned. In page 225 of the London edition, 1663, when disproving the supposed effect, according to Ptolemy's ideas, of the earth's rotation, if it existed, on the motion of a body dropped from a height, he ignores altogether the real deviation from the vertical that must be produced in the fall of such a body by that rotation ; and in page 239 he categorically and distinctly declares that a cannon ball discharged vertically would fall back on the E2 52 DEVIATION OF PEOJECTILES FROM mouth of the cannon, notwithstanding the rotation of the earth. Now, as we have said, Aristotle's words, taken as they stand, mean only that the earth's rotation would prevent a body dis- charged vertically from falling back on the point of discharge. Thus, then, if we judge them simply by what they say, Aristotle was right and Galileo wrong on this point ! But it is greatly to be feared that if we could cross-examine Aristotle and get him to be more explicit, he might commit himself undesirably ; and on the other hand, it would not be fair to take Galileo at his word on this point ; because we have reason for knowing, from the very work now referred to, that he was better on the present question than he here represents himself to be. His attention was so wholly engrossed with proving that the eastward trans- lation of the surface of the earth with everything on it has no effect, relatively to us, on the motion of projectiles &c., that he here disregards the angular tilting of that surface towards the east ; although he does not do this elsewhere. NOTE B, from p. 37. The proof of this is quite simple. Let us first suppose that we are at the equator, and that the discharge is due E. Let , Fig. 9, be the point of discharge ; ag the Fig. 9. Wn ^^ ^^ h position of the surface of the ground (whose curvature may be now neglected) at the instant of discharge ; afg the trajectory, which would intersect the surface at g, if the earth did not rotate; all the position of the surface of the ground at the instant of fall of the projectile at Ti. We are now concerned solely with the rotation of the surface of the ground about a horizontal N. and S. axis at a, perpendicular to the plane of the paper. The angle gah is wt, and very small, even for the longest THE ROTATION OF THE EAIITH. 53 attainable time of flight of a projectile. Draw gi perpendicular to ah ; ih is the increase of range now under consideration. Now ih is so very small relatively to ag and ah that these two lines may be taken, without sensible error, as having the pro- portion of equality. Let the angle of descent ghi be . Then hi, the increase of range now in question, is gi cot ; but gi (as the angle gai is so very small) is rut. Therefore (for firing due E. at the equator) hi = rut cot 8. For azimuth z, we must take, as is evident, r sin z, instead of r ; and for latitude X, we must take, as we know, o> cos X, instead of w. Hence for any latitude and azimuth, this alteration of range, hi, = rwt sin z cos X cot c ; in which r may be regarded without any sensible proportional error as being ah, the actual range. As with this demonstration, so with the diagram Fig. 9, it is only very approximately correct. The trajectory afh would not rigorously coincide with the supposed one afg, as far as it goes ; the former would not be a simple prolongation of the latter, though exceedingly near thereto. NOTE C, from p. 40. The following proof of this (for a vacuum) by Mr. R. A. Proctor, appeared some years ago in the London English Mechanic. First take the case of a projectile discharged vertically at the equator. Let aped, Fig. 10, be the surface of the earth, whose w curvature and eastward translation must now be recognized. The lines drawn perpendicular thereto at a, e, and d meet at the 54 DEVIATION OF PROJECTILES FROM centre, (7, of the earth. Let a be the position of the point of discharge at the instant of discharge ; cibe the orbit described by the projectile about the centre, (7, of the earth ; bp its greatest height above the surface of the ground ; the orbit is an ellipse differing insensibly from a parabola. Let e be the position of the point of fall at the instant of fall ; d the position of the point of discharge at the same instant, which will be, as we know, ahead of e. The projectile, having been moving in the backward prolongation of the line ae with a uniform velocity, describing equal areas in equal times about C, has received, at or, an impulse along the radius-vector Ca. If it were quite free it would move uniformly in its new direction of motion, still describing areas about (7, per unit of time, equal to the former. But it is acted on by the force of gravity directed to C ; this, however, leaves it still describing, about that point, areas the same as before. Therefore the area a Ceb area aCd, and area abe = area eCd. That is, from a property of the parabola, %ae x bp * = ^Rxed; R being earth's radius. But though the difference, ed, between ae and ad cannot be ignored, it being the very subject of investigation, yet as it is relatively so exceedingly small, ae and ad have very nearly the proportion of equality ; so that we can, with very small error, write ad for ae, in our last equation. Hence, very approximately, f ad x bp= \ R x ed. But as we are at the equator, ad=ll sin X, the plane of oscillation is left behind and will seem to the observer, who is unconscious of his own motion along with the earth, to have a rotation, with that rate, in the opposite direction, or that of the motion of a watch lying face upwards on the table. We may here note that a reader must be sometimes puzzled by a statement which is often inconsiderately made without any qualification, though nothing wrong be really intended by it. He will find it stated that the Pendulum oscillates always 11 in the same plane" (italics not ours), and that the plane of oscillation " remains always parallel to itself," and that it "always retains its own direction," and that it " is fixed," and that it " has fixity of position," &c. This is so only in the respect just FOUCATTLT'S PENDULUM. 61 mentioned, viz. that it does not partake of the earth's component rotation V, nor turn at all about the vertical line as axis. But the plane of oscillation participates, after its own fashion, in the earth's component rotation M about the horizontal meridional line at the place of observation. "When that plane is in the meridian, or N. and S., it turns about said line, as axis, with the angular velocity w cos X; when it is at right angles to the meridian, or E. and W., it does not turn about that line at all ; at that time it really does, though for a very short period, " retain its own direc- tion." In general, if z be its azimuth or inclination to the plane of the meridian, its rate of turning about the horizontal N. and S. line is w cos X cos z ; the angle z always varying and increasing with the time. It is then inconvenient and, for learners, mis- leading to speak without reservation of the plane of oscillation as " remaining always parallel to itself," when it has, in reality, the peculiar varying angular movement just described. However, we are free, now, to disregard this movement, as it does not sensibly affect the present question. Foucault communicated an account of his Pendulum to the French Academy on February 3, 1851, which appears in the Comptes Rendus for that date. A description of it taken from his own paper will be found also in Phil. Mag. 1851, first half, p. 575, and in Edinb. New Phil. Journ. vol. li. 1851, p. 101. Though the main principle of this Pendulum, as propounded by Foucault and stated above, is simple enough and to be called a kinematical one, the complete theory of it, even for a vacuum, presents an exceedingly difficult dynamical problem, one indeed apparently incapable of complete solution. This problem has been investigated by many able mathematicians, from 1851 downwards ; perhaps the latest paper on the subject is that by M. De Sparre, " Sur le Pendule de Foucault" presented to the French Academy and reported on in the Comptes Rendus, April 13, 1891. The causes of disturbance in the desired performance of this 62 FOUCAULT'S PENDULUM. Pendulum are of several quite different kinds, which, however, cannot be kept altogether separate, on account of their interaction. The first kind is connected with the setting-up of the instru- ment. It is obvious that there should be the greatest practicable equality of freedom in all directions at the point of suspension, whether the Pendulum be supported by a cord, or wire, yielding by its flexibility or its elasticity ; or whether it be by a fine point, say of steel, resting on a very hard smooth surface, say of agate. Deficiency of accuracy in this respect will be of less importance, the greater the length of the Pendulum. There should be of course very great steadiness and rigidity in the supporting structure ; unless this have perfectly equal elasticity in all horizontal directions, a condition not to be easily attained. If the Pendulum be heavy, which for certain reasons it ought to be, and if it be suspended from a beam there will be some small elastic yielding in the transverse, with almost none in the longitudinal direction of the beam. In order to obtain great length in the Pendulum, which is desirable for certain reasons, it has been hung in church-towers, sometimes surmounted by spires. But the elastic swaying of such structures at a con- siderable height from the ground under the varying pressure of a moderate wind is very appreciable, and in some cases might quite annul the advantage derivable from the great length of the Pendulum. That the instrument should be safe from the direct interference of the movements of the air, it should, as a general rule, be confined in a draught-proof case. The dis- turbances referred to, so far, may be almost quite avoided by the exercise of very great care and accuracy. The second kind of disturbance is inherent in the very nature of the Pendulum itself. Suppose it to be set swinging on a non-rotating earth ; if the oscillations were exactly in a plane they would, of course, remain so, and the plane would remain stationary. But if they were not in a plane, the bob would describe, in a vacuum, what may be called an ellipse, whose axis- major would continually rotate in the same direction as that in FOUCAULT'S PENDULUM. 63 which the bob was describiog the curve. If I be the length of the pendulum and a and b the semi-axis major and minor of the ellipse, both relatively very small, then on a non-rotating earth and in a vacuum, a would accomplish a complete rotation in the time of a whole vibration, or two complete swings of the pendulum (that is ~l 8 Z 2 - sees.) multiplied by n -^ very nearly. That is to say, the angular movement of the axis-major in one second would be, in circular measure, ^ *1 g j^~ very nearly. See articles in Phil. O I 12 Mag. 1851, second half, and Williamson and Tarleton's Dynamics, p. 214 (see also NOTE A). This result is only approximate, though very closely so, for great enough Z, or small enough ab. It would obtain also on the rotating earth, though of course in combination with the effects of the rotation. In order to keep this disturbance as small as may be, Z should be as great and the product ab as small as possible without practical disadvantage. If it were practicable to keep b at zero, that would, of course, be sufficient to keep the above expression for this angular movement so, likewise ; but we shall find that this is not practicable, though it can be approached to pretty nearly. There is another unavoidable source of interference with the desired performance of this Pendulum ; which is that, as we have seen, it is affected, though very slightly, by the earth's component rotation M about the horizontal meridional line at the place of observation, and that, therefore, its behaviour is not altogether independent of the azimuth of its mean plane of oscillation. It may be that certain variations in the rate of rotation of that plane, as described by some experimenters, have been, to some extent, due to this circumstance. Let us note the following for the sake of illustration ; though it is sensibly quite unimportant. The rate of that angular movement (in a vacuum) of the line of apses mentioned above is, as we have seen, proportional to \/<7 cceteris jparibus ; y being the whole downward acceleration, in- 64 POFCAULT'S PENDULUM. eluding that of the centrifugal force from the rotation of the instrument connected with M. But we have seen that when the Pendulum is swinging N. and S. the downward centrifugal force is a maximum, and when the Pendulum is swinging E. and W. that force is zero. Therefore, if this effect could exist by itself, the line of apses would move very slightly faster when near N. and S. than when near E. and "W. As another illustra- tion, we may observe that the behaviour of this Pendulum is not entirely independent of the azimuth of oscillation with which it is started. We shall meet with still another illustration further on. The gyrostat, when used to prove the rotation of the earth, is quite free from such complications as those now referred to. The third kind of interference with the desired performance of this Pendulum is that arising from the resistance of the air. For very small velocities, this resistance would be directly pro- portional to the velocity, very nearly ; if there were not anything to prevent this. But there is something to prevent this ; for as the amplitude of swing must be kept small and the axis-minor of the ellipse exceedingly small, the Pendulum is always moving in air which has been already disturbed by itself. If it were moving in a wide enough ellipse to avoid this, the resistance of the air, if acting by itself, would cause a retrograde movement of the apses of the ellipse ; but in the case of a quite small axis- minor this would be lessened by the movement of the air following in the wake of the bob. There is then reason for believing that, in this case, this effect of the resisting air is unimportant. See NOTE B. But there is another which, though it is indirect, is of much more consequence. While the axis-minor is small, but appreciable, the stream of air following in the wake of the bob in one swing will not act centrically and directly against the bob in its return ; but it is evidently always tending to turn it away from the axis-major ; this is strongest while the bob is descending towards the axis-minor, and the effect is to increase the axis- minor. This tendency must grow with the growth of its own 65 result, until the ellipse becomes wide enough for the cause to cease. This is, no doubt, one reason why the axis-minor (unless it be exceedingly small) grows larger, at first absolutely, and then relatively, during the continuance of an experiment with this Pendulum. It would therefore be impossible to calculate the effect of the resistance of the air on the behaviour of the instrument, as the precise conditions of it are unknown and altering continuously with the lapse of time. To diminish as much as possible the relative importance of the air, the bob must be, of course, as large as convenient and of high density. It should also be very homogeneous and carefully turned in a lathe and suspended accurately in its axis of figure. We have seen that, besides the precautions necessary in the making and mounting of the Pendulum, there is the very impor- tant one of starting it properly, so as to have as small an axis- minor of its path as possible. For this purpose the plan has been generally followed of starting the Pendulum by drawing it to one side by a thread attached to a stationary object, and when the Pendulum has come to rest of severing the thread by burning it. But, on account of the rotation of the earth, the centre of the bob will in this case pass to the right of the point of rest in northern latitudes. The plan has therefore been adopted of projecting it from the point of rest with the view of making it swing to and fro through that point. But supposing that it did this at first, it would describe, relatively to the table beneath ifc and to the accompanying air, a series of loops all described in the same direction (and therefore not " figures-of-8," as sometimes called), and the tangential resistance of the air near the outer ends of the loops, although excessively small, would, by con- tinued action in the same direction and by accumulation of effects, cause the axis of the bob to pass to the right of the central point of rest. If the linear amplitude of oscillation were too large, this might well have a quite sensible effect. It is therefore all important, in experiments with this instru- 66 FOUCAULT'S PENDULUM. ment, to use as small an amplitude of oscillation as practicable ; in order to diminish, as much as possible, three quite different causes of disturbance noted above. This was not sufficiently attended to at first. It should be remembered that any roughish experiments with Foucault's Pendulum are necessarily quite delusive. In conse- quence of insufficient guarding against the causes of disturbance, it has happened, even with some experiments considered worthy of being described in a scientific journal, that the line of apses has actually gone the wrong way ! This has, not unnaturally, given occasion to certain persons, including the famous "Parallax" to ridicule the principle of this Pendulum altogether. The later experiments of Mr. Thomas Gr. Bunt, of Bristol, described by himself in different papers in the Phil. Mag. for 1851 and 1852, were carried out with unusual care to minimise the causes of disturbance, and they were, for that reason, specially successful. He started with a linear amplitude of swing of only one inch on each side of the point of rest. He mentions that (the axis-minor of the ellipse described being always kept very small) all his Pendulums had two nodal lines nearly at right angles to each other, at which the direction of revolution of the bob in the ellipse changed to the opposite. This affords another illustration of the fact that this Pendulum is not altogether indifferent to the azimuth of its mean plane of oscillation. An interesting table of results obtained by various experi- menters with Foucault's Pendulum will be found in pp. 44, 45 of Kev. Dr. Haughton's Manual of Astronomy. NOTE A, from p. 63. That the axis-major of the ellipse must rotate (in a vacuum) in the direction in which the Pendulum describes the curve can be seen quite easily without analysis. The force directed to the point of rest, under which the Pendulum is oscillating, is accurately g sin ; d being the angular distance from the point of rest. Therefore when is very small, the FOUCAULT'S PENDULUM. 67 Pendulum is moving under a central force which is very nearly indeed directly proportional to the linear distance ; it therefore describes very nearly a fixed " central ellipse." But, from the exigencies of the experiment, 6 cannot be allowed to be exceed- ingly small ; and therefore the force, which is proportional to sin 0, varies, as is evident, more slowly than the distance, whether linear or angular, from the point of rest; and the deficiency in the central force, owing to this, which is at first excessively small, increases with the distance from the point of rest and at a much higher ratio. This causes a progressive motion of each end of the axis-major ; because in the neighbourhood of the apse, where the deficiency is greatest, the central force takes longer to stop the rising of the bob from the centre of force and to pull it round the apse than it would do if it were accurately proportional to the distance ; the bob will not attain its apse, and begin to turn back again, until it has passed the position of the last preceding corresponding apse. For a corresponding contrary reason, the said deficiency in the central force, as occurring near the ends of the axis-minor, would tend to produce a retrograde motion of each of those points. But the former tendency is greater than the latter ; since the said deficiency is greater at the ends of the axis-major than at those of the axis-minor, in a much higher proportion than the distances from the centre of the ellipse. The importance of this consideration is enormously enhanced by the fact that the axis-minor must be always kept very small. The whole result is consequently a progressive rotation of the ellipse. NOTE B, from p. 64. That the resistance of the still air, if it could act separately, would cause an angular movement of the axis-major in the direction contrary to that in which the bob describes the ellipse, can be seen in a similar manner. See Fig. 12, in which the axis-minor is, for clearness, made greatly too large in proportion. Whilst the bob is going from D to A, the resistance of the air, which is tangential to the curve, tends F2 68 FOUCAULT'S PENDULUM. to make A regress ; because it causes the bob of the pendulum to cease rising from E, and begin to turn downwards, sooner than it would do without that resistance ; that is before it has reached the last preceding position of A. But whilst the bob is going from A to B, the tangential resistance tends, in a corresponding manner, to make B progress. The former effect, however, exceeds the latter ; because whilst the bob is rising from D to A its velocity and the consequent resistance of the air are at their maximum at first ; but whilst the bob is going from A to B the velocity and the resistance only reach their maximum at last. The whole result will be that the " ellipse " would rotate retro- gressively if the resistance of the still air were the only disturber of the elliptic motion. This is corroborated by the experiments of Mr. Alexander Gerard. However, if the ellipse be narrow enough, the last-mentioned effect will evidently be increased by the resistance of the wake -stream ; so that the whole effect may be quite small. CHAPTER V. ON THE POSITION OF THE DYNAMICAL HIGH TIDE RELATIVELY TO THE CELESTIAL TIDE-PRODUCING BODY. As is often done for simplicity, we shall consider only the tides that would be produced in a canal of uniform depth and of uni- form width running right round the earth's equator and returning into itself ; and we shall suppose the tide-producing heavenly body to be always in the plane of the equator. We shall, moreover, confine our attention, at first, to the tides caused by the moon. We need not do more than remind the reader that the lunar tidal forces are directed as the outer broken-line arrows in Figs. 15 and 16, the moon being away to the right, and that they consist only of the differential attraction of the moon on the water of the ocean, or the difference, both as to magnitude and direction, between her attraction at the centre of the earth and at the various parts of the superficial ocean. The tangential tidal force at a point on the earth's surface having the angular distance from the moon is f ^ sin 20y ; and the radial force at that point is f "03 (cos 20 -|- 3)7 ; r being the earth's radius, M the moon's mass, R the moon's distance from the earth, and y the unit of gravitation. These forces are, then, inversely pro- portional to R 3 . The differential tidal force is at its maximum directly under the moon, where it is all radial, and where it is only about l/29th of the moon's whole attraction at the distance 70 ON THE POSITION OP THE DYNAMICAL HIGH TIDE of the earth, or about l/8,400,000th of g, or the earth's attrac- tion at its surface. If the earth always kept the same side turned towards the moon, the lunar tidal forces would, of course, produce one tidal protuberance in the water on the side of the earth next the moon, and another on the opposite side. The protuberances would be stationary on the earth, and the discussion of their magnitude &c. would be one of hydrostatics only ; they are therefore called statical tides, or equilibrium tides. But as the earth rotates under the moon, the actual case in our equatorial canal would be very different. The two tidal protuberances and intervening depressions, in order to keep up with the moon, would have to sweep right round the canal in the mean period of 24 hours 50-5 minutes, at the 1 rate of 1003-5 miles per hour. This they would do, not after the manner of a tremendous torrent moving bodily along with that enormous velocity, but in the style of a smooth ground-swell in the sea, whose gentle wave-forms may be travelling onwards with a con- siderable speed, although the individual particles of the water are only moving backwards and forwards, for short distances, with quite small velocities. This is the manner in which the actual tides in our oceans really do travel. We are therefore concerned with a dynamical question, and have to do, not with <; statical," but with " dynamical," tides. The present subject is one on which it is very easy to go wrong ; it contains several instances of what any person insufficiently acquainted with it would naturally regard, at first sight, as apparent paradox. Let us begin by noting briefly the way in which the water moves in a travelling wave, or water-undulation. Anyone can observe this for himself when watching sufficient wind-waves on the sea ; although such surface undulations differ importantly in certain respects from tidal ones, whose disturbances extend to the bottom of the ocean. See Fig. 13, which represents two waves moving towards the right. The upper dotted arrows RELATIVELY TO THE CELESTIAL TIDE-rRODUCiyG BODY. 71 show the directions of the movement of the various parts of the water. The lower arrows the directions of the gravitation forces due to the disturbance of level. On the crest of the wave the water is moving horizontally forwards with the greatest velocity; at the bottom of the trough the water is moving horizontally backwards with the greatest velocity. At the points of mean Fig. 13. level, halfway up the slopes of the wave-ridge, the water is moving neither forwards nor backwards, but on the front slope, vertically upwards ; while proceeding to form the upper part of the ridge by addition in front ; and on the hinder slope, verti- cally downwards ; while withdrawing from the hinder part of the wave-ridge. In a wind-wave each particle of water moves in a fore-and-aft vertical circle ; in a tide-wave in a very elon- gated ellipse with minor axis vertical ; this axis diminishing as we descend, until it vanishes at the bottom. The progress of the wave form is produced by continual addition of water in front, and subtraction of water behind. It is very easily seen that the velocity of the wave-form, though so entirely different from that of the particles of the water, will be proportional, cceteris paribus, to the latter ; and also that for a given velocity of the wave-form, its magnitude will increase or diminish in the same proportion as the velocity of the particles of water. Such a wave, having been started by some cause, would, on the cessation of that cause, continue to move onwards of itself, at its own proper rate, in consequence of the forces occasioned by the disturbance of level. There would be the unbalanced weight of the part of the wave projecting above mean level, and the unbalanced deficiency of weight in the part below mean level resulting in an upward pressure in that part, Fig. 13. Ifc is evident that the said pressure and deficiency of pressure is 72 ON THE POSITION OP THE DYNAMICAL HIGH TIDE proportional to the volume of water above, and deficiency thereof below mean level ; that is to say (the oscillations being rela- tively small), proportional to the greatest heights and depressions of the .water. The forces are then always proportional to the distance from the position of rest ; as in the case of a common pendulum oscillating with a relatively small linear amplitude ; and the oscillations are therefore isochronous, or performed in equal times, whatever be their magnitude; if this be always relatively small. Of course the forces will be, cceteris paribus, proportional to f , a (cos2e + g), and to that for the tangential force, 126 THE MOON'S VARIATION. OQT>3 viz., 3 sin 2e ; the earth's attraction on the moon being unity. Let us substitute in these for e, which will involve an exceedingly small inaccuracy. Then, to use Fig. 31 for a different purpose, if we take the radius of the circle therein to represent the common coefficient in these expressions, and if am be one third of said radius, fm will represent, on the same scale, the radial disturbing force for assumed 0, and ef the tangential disturbing force, and em will represent, quam prox., both in magnitude, direction, and sense (but of course not in position), the whole disturbing force acting on M. Its magnitude is said radius of the circle X^ V 10 + 6 cos 20; and its inclination to EC is , sin 20 tan" 1 - cos D, from p. 111. This can be seen as follows from equation (5). E and being as in text, let p be the radius of curvature at the points in question, and e an indefinitely small elongation of the moon, for which equations (5) and (3) coincide. Now /o=arc 2 /2 (fall from tangent). But, for syzygies, arc 2 =R 2 (l-(7) 2 sin 2 e, and 2 fall =2 { R(1 ~ a) -K(l - C cos 2e) } \ cos e } \ By the addition and subtraction of Ccose within the large parentheses this becomes ((l-<7)(l-cos0)-2 Rcos cose f _ -)(l-cos ' P -2 \ (l-C'Xl-costfj 26\1 cos'^ Dividing above and below by 1 cos e, and then making e=0, we obtain the result in text. THE MOON'S VARIATION. 127 In an ellipse whose semi-axes major and minor are a and >, respectively, the radius of curvature at the apses is , and that a* a at the ends of the axis-minor -7- Therefore if the Yar. orbifc o ' were an ellipse with the same principal axes, p would be at syzygies R ^-, and at quadratures RX- L. Taking R as 1 6 1 -j- C unity, and C as 00738, we find the following values for p : - At syz. At quadr. In Var. orbit 1-0230, 0-9787. In ellipse 1-0224, 0-9781. which verifies the anticipation in text that the Yar. orbit is very slightly flatter than an ellipse, with the same principal axes, both at syzygies and at quadratures. NOTE E, from p. 115. This will be sufficiently seen from the following. It is a well-known principle (confining our attention now to the ellipse) that if a body be projected from a given point in presence of a given centre of attractive force having the law of gravitation, with a given velocity not too great for the description of an ellipse about that centre, the ellipse described by the body will have the same axis-major, whatever be the direction of discharge. Now let the body be at first describing a circular orbit aid, Fig. 32, with radius r about the centre of force c, in the direction of the arrow. At the point a the direction of the body's motion is changed outwards, say by the angle ; and it proceeds to describe an ellipse gae, of which a focus is at c, and whose semi-axis-major is equal to r, the radius of the circle. Since ca, drawn from the focus, is equal to the semi-axis-major of the ellipse, a is at the end of the axis-minor thereof. Conse- quently a line drawn through c parallel to the new direction of motion at a gives the direction of the axis-major containing the apogee of which we are in quest. The geocentric angular distance of the apogee from a is 90 0; and /being the centre 128 THE MOON S VARIATION. of the ellipse the height of the apogee above the circle is equal to c/, or r sin ft. The same Pig. can be used (by supposing the body to be describing the circle in the opposite direction) to show that if the change of the direction of the body's motion at a had been Fig. 32. downwards, and of magnitude 0, the resulting perigee at g would be 90 + from a, and fall below the circle at that point, r sin 6. Therefore, whether the deflection at a be upwards or down- wards, the new orbit will be an ellipse whose axis-major is 2r, and axis- minor 2r cos ; and if be very small, the apses are distant from a by 90 very nearly. We need not pursue this any further, because the actual conditions are slightly different from what we have just con- sidered. The moon is not simply deflected outwards without change of velocity by the radial forces near syzygies ; though the condition nearly approaches this, as the radial disturbing forces are so very small. But the difference of conditions is evidently in favour of an apogee both higher and nearer to quadrature than what we have been contemplating. The deflection does THE MOON S VAEIATION. 129 not indeed occur at a single point ; but it may be regarded as the result of a large number of exceedingly small, outwardly directed, radial impulses, whose magnitudes are very nearly equal at equal distances on each side of syzygy. The whole effect is, therefore, different from that of a single impulse at syzygy equal to the sum of the others; but, as regards our present purpose, the difference is quite unimportant. NOTE F, from p. 116. We have seen, p. 106, that the radial disturbing force vanishes at the four points of the moon's orbit distant 54 44' from syzygies, marked 0000 in Pig. 33. The arc BO is slightly less than two thirds of AO ; and, the changes of the moon's velocity being small, her times of describing OB Fig. 33. O and AO are still more nearly in the same ratio. Again, we have seen above that the inward radial force at B is half the outward radial force at A ; and therefore, as it is easy to see, the average inward force on each side of quadrature is somewhat less than half that on each side of syzygy. In consequence of the outwardly exceeding so much the inwardly directed radial impulses, the earthward pull on the moon is, on the whole, diminished ; and therefore the mean distance of the moon from the earth is by them increased. 130 , from p. 118. It will be found from simple considera- tions similar to the above, mutatis mutandis, that if the gravita- tion attraction varied, not according to its actual law, but according to that other law of force so frequent in nature, viz., directly as the distance, the solar disturbing forces, if turned on to act on an originally circular lunar orbit, would make that orbit into what might be called an " ellipse," with the earth at its centre, whose axis-major, in the line of the second and fourth octants, would be continually increasing, whilst its axis-minor would be continually decreasing (more rapidly), until the moon came into collision with the earth. The instantaneous ellipse would be always writhing ; especially towards the conclusion of its history. In this case there would be no baffling action between the solar and the terrestrial forces. The solar tangential forces would have the same positions, relatively to the sun, as they have now ; though, of course, their directions would be reversed. The earth's attraction, owing to its now supposed law, would make the lunar orbit, when disturbed, a central ellipse, if free to do so ; and, as is evident, the solar tangential disturbing forces would fall in with this and go on increasing the ellij_,ticity. The radial disturbing forces, always directed inwards, would be proportional to the moon's radius-vector, like the earth's attraction, and would therefore conspire therewith. There would be, moreover, this seemingly curious result, that, supposing the sun's distance to be always very large in com- parison with that of the moon, the Var. forces would not sensibly alter with the sun's distance ; instead of being, as they actually are, inversely proportional to the cube of that distance. NOTE H, from p. 119. This will be seen thus. In Pig. 34, E is the earth's place, r the moon's radius-vector for the point a, in question, in the Yar. orbit, of which the curve ac represents a portion, and de an indefinitely small alteration of e, the moon's elongation. Let be the angle between the radius-vector and the curve at a, or the tangent thereto. Then, R, the moon's THE MOON'S VAKIATIOtf. 131 mean radius-vector, being taken as unity, we have r=l-C r cos2e (p. 110) and dr=+2Csin2 de. The earth's attraction at the point in question is ; the mean attraction being unity. But - = 1 + 2 C cos 2e, quam procc. This multiplied by cos is the terrestrial tangential force. Fig. 34. to S Now cos = cot 0, q. pr. ; as 6 differs so very slightly from a right angle. Draw cb perpendicular to Ea. Then cot 0= ^-=1^1 2C r sin2ec? Therefore the terrestrial tangential force is (1 + 2(7 cos 2e)2(7sin 2e(l + C cos 2e) = -gV sin 2e(l +^ cos 2e). Q. E. D. This is never less than -^ sin 2e ; while the solar tangential force is y^sin2e (p. 105). Therefore the proportion of the terrestrial to the solar tangential force, at any point in the lunar Variation orbit, is always at least as high as 1 20 to 68, or as 7 to 4, very nearly. [ 132 ] CHAPTER VIII. THE MOON'S PAEALLACTIC INEQUALITY. WE now turn to the moon's Parallactic Inequality, whose scheme of solar disturbing forces and of changes of velocity, &c., are indicated in Fig 35. In the Variation scheme the disturbing forces, both tangential and radial, on the sunward side of the moon's orbit and those on the opposite side are regarded as equal, which, however, they evidently are not ; the former being slightly greater, and the latter slightly less, than the mean. To remedy this we must now add to those on the sunward side the necessary differential forces having the same direction ; and we must subtract from the Yar. forces, both radial and tangential, on the off side of the orbit, the same differential forces ; or, in other words, join with them the said differential forces having the contrary direction. These constitute the P.I. forces, with which we now have to do. The inwardly directed radial disturbing forces at B and D, in the Variation orbit, Fig. 26, are not affected by the difference between the sunward and the other side of the Var. orbit, and we have put no arrows at those places in Fig. 35. The sun being over A, the disturbing forces, with which we now have to do, are represented by the arrows drawn with broken lines. The terrestrial tangential forces, to be mentioned presently, are omitted to avoid confusion. They are directed oppositely to the solar ones ; they are, however, at their maximum at both quad- ratures, while the solar ones vanish at those points. All vanish at syzygies. THE MOON'S PARALLACTIC INEQUALITY. 133 The P.I. forces, being second differences, or differences between what were themselves only differential forces, are exceedingly small. It is easy to find that, taking the earth's mean attraction OT>4 on the moon as unity, the solar P.I. tangential force =6 - 4 CT>4 J^-U x (sin c sin 3 e), and that the radial force = 6^rra(cos 3 e | cos e) ; all the letters here having the same meaning as they have in the expressions for the Variation forces in p. 105. (See NOTE A.) The Fig. 35. greatest P.I. forces are the radial ones at syzygies ; and these are only l/23,300th of the earth's mean attraction on the moon. They are about l/259th of the Var. radial forces at the same points ; these being also at their maximum at those points. The P.I. perturbations, like the Var. ones, can be calculated and considered by themselves. Since these two sets of disturb-' ances are both very small, they can be combined like two sets 134 THE MOON'S PARALLACTIC INEQUALITY. of " small oscillations " by simple superposition. "We shall, therefore, as we did with the Yar., neglect now all other deformations of the moon's orbit and inequalities in her motion, and suppose the moon's undisturbed orbit and the sun's relative annual orbit round the earth to be both circular, and the angular velocities therein uniform, and the P.I. forces to be the only disturbing forces acting on the moon. The lunar orbit round the earth resulting from this we shall call the P.I. orbit. Let e be the moon's actual elongation, as above, in the pure P.I. orbit, and her mean elongation, or that in the undisturbed circular orbit, both reckoned eastwards up to 360 ; the sun's angular motion of apparent revolution round the earth being sup- posed, as in Chapt. VII., constant, for simplicity. Then we have e=0 2' 5" sin ....... (1) The moon's Parallactic Inequality in elongation is, then, 2' 5" sin 0. We have adopted the coefficient 2' 5" from the latest investigations of the American astronomers ; Hansen gives a smaller value for it, viz., 2' 1". Observations might be made on this equation (1) corresponding to those in NOTE A of the preceding chapter on the Variation ; but they are probably unnecessary. Let r be the moon's actual, and H her mean, radius-vector, or distance from the earth. Then we have (2) T> The Moon's P.I. in radius-vector is, then, +3520 Prom these two equations, in combination, may be derived a simple geometrical construction for obtaining the moon's position in space, for any assumed 0, or mean elongation. First let us note that the coefficient 2' 5", in equation (1), is, in circular measure, j^. Whence the moon's linear departure, back- wards or forwards, from the line of her mean radius-vector is THE MOON'S PAHALLACTIC INEQUALITY. 135 T> sin0, q. pr. ; her departure from her mean distance T> from the earth being, as we have seen, + 3530 COS ^' In Fig. 36, E is the place of the earth, and ES the direction of the sun. The construction is as follows : Draw E to represent, in magnitude and position, the mean radius-vector, at some given time, whose length is 238,820 miles ; SEa is 0, and a is the moon's mean, position. Draw ab sunward, making the angle with the production of Ea (that is parallel to ES), and of magnitude to represent 2(1650+3520)^, or 106 miles; then draw 6M making the angle 2 with ab (and with ES), and of length to represent ^(1^3^)^ or 38 miles ; then M is the moon's true place for assumed 0. (See NOTE B.) Fig. 36. Thus we see that the moon's P.I. orbit, considered as described about E, and relatively to ES regarded as stationary, is a peculiar epicyclic curve ; Ea is the radius of the deferent circle turning progressively with its constant angular velocity. It carries, as in Fig. 36, the line ab, which remains parallel to ES and itself, which also carries at its end the radius 6M of the 136 THE MOON'S PAKALLACTIC INEQUALITY. epicycle, which radius rotates progressively with twice the angular velocity of E#. If we take the step EF, from E sunwards, equal to ctb, then the P.I. orhit will be, relatively to F, a simple epicyclic with the same deferent and epicycle having the same simple proportion of their angular velocities. The movement of the moon in her P.I. orbit can be represented in another manner, which, of course, is the same at bottom, but has its own interest. It follows quite easily from equations (1) and (2). (See NOTE C.) It is often said simply that the moon's P.I. in elongation from the sun is proportional to the sine of her elongation. The differ- ence involved between this and equation (1) is exceedingly small and, as regards our present purpose, insensible. Let us then take leave to write equation (1) thus e = 2' 5" sine (3) According to this, the moon is at her mean place in elongation at both syzygies, most behind it at first quadrature, and most before it at last quadrature. If in equation (2) we substitute cos e for cos 0, the inaccuracy is again insensible. When, then, we write the equation thus r=K(l+ 3520 cos e) = B(l + c cose), ... (4) it becomes a convenient polar equation of the P.I. orbit referred to the (annually rotating) line of conjunction, as prime vector. This equation is always, quam prox., correct, and at syzygies and at quadratures quite so. According to this, the moon is at her mean distance from the earth at both quadratures, at her greatest distance at conjunction, and at her least at opposition. These differ from the mean by only about 67*8 miles. The P.I. orbit, as given by equation (4), differs very little indeed from a circle which has been first shifted bodily sunwards -p by the distance QKOI or 67'8 miles, retaining quite unaltered THE MOON ? S PARALIACTIC INEQUALITY. 137 its syzygy diameter, and then drawn out at right angles to that diameter until it becomes wider by lie 2 , or only 102 feet. (See NOTE D.) The greatest width is very near, and on the sunward side of, the line of quadratures. In the drawing out, the circle becomes flattened a little at both syzygies ; but very slightly more at opposition than at conjunction. It is easy to obtain geometrically the radius of curvature at conjunction, viz., R ~TTT> an( ^ ^ na ^ a t opposition, viz., R^j c being the coefficient ^5. The former is less than the latter (though by only about 1-4 inch} ; but both exceed R, the mean radius-vector and radius of curvature. (See NOTE E.) As with the Var. diagram, so now, we can fill in all the writing in the P.I. diagram, Fig. 35, p. 133, when we know simply from equation (3) that the moon is most behind her mean place at first quadrature and most before it at last quad- rature ; she being of course at her mean place at both syzygies. Among these conclusions let us note particularly that the moon's velocity is least at conjunction, greatest at opposition, and at its mean at both quadratures. We must return to this hereafter. The reader will perceive better the differences between the P.I. and the Var. diagrams by comparing them for himself, than by reading our description of them. We may, however, draw his attention to the following point. If we start from C in both diagrams, we shall find that, as regards the writing only, the four reaches, or divisions (constituting one half) of the Var. orbit from C to A, correspond, respectively, to the four reaches (con- stituting the whole) of the P.I. orbit. An elegant explanation of the production of the Parallactic Inequality by the disturbing forces now in question will be found in Airy's Gravitation, p. 68, which we shall not reproduce here. (See NOTE P.) The existence of this lunar inequality was pointed out by Newton. It is very interesting to find that he had determined 138 THE MOON'S PARALLACTIC INEQUALITY. dynamically its amount with a wonderful closeness of approxi- mation ; though it had not been, in his time, detected by observation. The value that he gave to the coefficient was 2' 20" ; this is too large ; the reason being that he went on the supposi- tion that the sun's parallax was 10", which we now know to be greater than the true magnitude. This brings us to the connection between the sun's parallax and this lunar inequality, which was named by Newton from its dependence on the ratio between the sun's and the moon's parallax. Newton could only derive the magnitude of the P.I. longitude coefficient from the then supposed magnitude of the sun's parallax. But now that the said coefficiont is obtainable by observation, it can be used for solving the inverse problem, viz., obtaining the parallax of the sun. Different formulae have been given for the connection of the two quantities, which formulas are, of course, very approximately the same at bottom. They come to this, that under the actual conditions of magnitude of the quantities concerned the sun's parallax is almost exactly one fourteenth of the P.I. longitude coefficient. But besides this, the interest of this lunar inequality is greatly increased by its having several apparent paradoxes connected with it. This circumstance has not attracted the attention it deserves ; and the neglect of it has given rise to certain errone- ous statements. Of the seeming paradoxes we shall mention five, to be dealt with by eqns. (3) and (4) and NOTE E. 1. Since the Var. forces produce the inequalities indicated in Fig. 26, p. 105, the reader might naturally expect that the increase of those forces in the sunward half of the moon's orbit, by the addition of the similarly directed P.I. forces, should increase the inequalities in the moon's motion there ; and, cor- respondingly, that the diminution of those forces on the off side of the orbit, by applying to them the oppositely directed P.I. forces, should diminish the inequalities there. But these are both the reverse of the truth. THE MOON'S PAEALLACTIC INEQUALITY. 139 2. Since the Yar. orbit is compressed at A and C by the influ- ence of the Var. forces, the reader would naturally expect that the just mentioned increase of the forces on the sunward side of the orbit, by the addition of the P.I. forces, should increase the compression there ; and, correspondingly, that the diminution of the forces on the off side of the orbit, by the subtraction of the P.I. forces, should diminish the compression there. But these are both the reverse of the truth. 3. When the reader has teachably accepted, from equation (4), the position that the effect of the P.I. forces is to elongate the originally undisturbed orbit towards the sun and to compress it on the opposite side, he will loyally endeavour to carry out his newly acquired knowledge, and will conclude that the orbit is more flattened on the off side from the sun, and less flattened on the side next the sun, than elsewhere. But as respects the sunward side this is the reverse of the truth. (See again NOTE E.) 4. The reader will most naturally, and even commendably, think that the moon would be gaining, or losing, velocity, in the P.I. orbit, according as the P.I. tangential disturbing forces are directed with, or against, her motion, respectively ; and there- fore that her velocity is greatest at conjunction. He will be confirmed in this expectation by seeing that such happens to be the case in the Var. orbit ; see Fig. 26. He will think also that the moon's velocity must be least at opposition, since she has been opposed by the solar tangential force all the time of her passing from conjunction to opposition. But all this is the reverse of the truth. The moon always quickens or slackens her pace in apparent defiance of the solar P.I. tangential forces. (See NOTE G.) 5. It would be reasonable enough to expect that since the P.I. forces are proportional to the inverse fourth power of the sun's distance, the P.I. in longitude should also be proportional to the same, or at least pretty nearly so. Such, however, is by no means the case. The P.I. in longitude is inversely propor- 140 tional to a complicated function of the sun's distance ; which function is much nearer to the cube than to the fourth power thereof. So that if there were an alteration of the sun's mean distance, the P.I. and the Yar. in longitude would change at not very different rates. We shall mention further on what some might regard, at first sight, as another apparent paradox ; but it is only kinematical in character. The reader, if he accepts our statements, will probably begin in despair to imagine that the name which has been given to this scheme of lunar inequalities is a mistake for " Paradoxical Inequality." The general explanation of the above apparent paradoxes is two-fold. In the first place, as we have noted already, the solar disturbing forces, whether of the Var. or the P.I., produce their respective inequalities of the moon's motion in elongation in two quite different ways, viz., by their direct local influence on the moon's velocity in the various parts of her orbit, and also by what we may call their indirect general influence in deforming the orbit, and thus creating tangential components of the earth's attraction on the moon, which are actually greater than the solar tangential forces. In the second place, unlike the case of the Variation (see p. 118, above), the P.I. system of solar disturbing forces has but one axis of symmetry passing through the earth, that of the line of syzygies. This involves a most important difference as to the dynamics of these two schemes of lunar inequalities, as considered in NOTE F. It so happens, as we have seen, that, in the case of the Var. orbit, the created terrestrial tangential forces always act along with the solar tangential forces ; and thus, in the usual elemen- tary treatment of the Variation, they are not prominently noticed, or are even disregarded altogether ; although they are, even in that orbit, more important than are the solar ones themselves, as to their direct local action. THE MOOD'S PAKALLACTIC INEQUALITY. 141 But in the case of the P.I. orbit, the relative importance of the terrestrial tangential forces is much more striking, for two reasons. The small solar P.I. forces have, by accumulation of effects, deformed the orbit to such an extent (very small, however, absolutely) that the terrestrial tangential forces created thereby are much greater, proportionally, than the solar tan- gential forces. The former are always equal to the latter multiplied by 3'31 sec 2 e. They are, therefore, never less than 3*31 times as great as the latter ; and when the moon is not far from quadratures, very much more, proportionally, than this. (See NOTE H.) And as the greater terrestrial, act always against the smaller solar, ones, the singular result follows that the inequalities in the moon's velocity and in her elongation, now under consideration, are the opposite of what the solar tangential forces, with which we are now engaged, are endeavouring to effect by their direct local action. So that, paradoxical as it sounds, it is strictly true that the terrestrial tangential forces are the immediate cause of the moon's P.I. in elongation, and that this lunar perturbation would be greater, but for the hindrance of the direct local action of the solar tangential forces. Thus the moon's Parallactic Inequality presents a peculiarly interesting dynamical problem. NOTE A, from p. 133. The verification of these expressions for the disturbing forces, though of a simple character, is a little troublesome. If the reader should undertake it, let him beware not to stop at the first approximation, which would give the numerical factor 7 instead of 6 ; which latter is sensibly accurate. SR 4 The value of the coefficient 6^ is 0-0000858. These expressions show that the P.I. forces are, quam prose., inversely proportional to the fourth power of the sun's distance from the earth. It might seem just at first sight that they are also directly proportional to the fourth power of the moon's distance from the earth. But they are proportional only to the 142 THE MOON'S PAEALLACTIC INEQUALITY. second power thereof. The E 4 comes in on account of the earth's mean attraction on the moon being here taken as unity. The trigonometrical factor in the expression for the tangential force can be written | sin 2e cos e ; that for the radial force can be written J cos 2e cos e. This gives the interesting result that, at the elongation e, the tangential force divided by the radial force = tan 2e. NOTE B, from p. 135. In Fig. 37 the points marked a, 6, M, are the same as those similarly marked in Fig. 36. Ea being the moon's mean radius-vector, necessarily drawn vastly too Fig. 37. to 3. short relatively to the other lines, and a the moon's mean place, draw ac making the angle with the production of Ea (and parallel with ES) to represent ^- E ; then draw cd parallel to Ea, and sensibly pointing backwards to the earth ; draw ad at right angles to Ea and cd ; then ad is j^E sin $. Take e so that ae may represent g^E ; draw eM. parallel to ad ; then cZM is g^E cos 0, and M is the moon's true place. Now ec is ^(r50~35lo)' ^i gect ^ i n &> an d draw 6M. Then be and 6M are both iB/r^-sk). and 6 is ^i^-f ^(tSR-sns). < THE MOON'S PAEALLACTIC INEQUALITY. 143 JLj-fg-Lg). The angle Mfo==2Mc&, or 20; whence the statement in text follows. NOTE C, from p. 136. (See Fig. 38.) As before, E is the place of the earth, and ES the direction of the sun, and the points marked a, e, M are the same as those similarly marked in Fig. 37. Let Ea be the moon's mean (both as to position and magnitude) radius-vector ; so that a is the moon's mean position. With centre a and radius g^B, describe the circle Fig. 38. toS shown in the Fig. Draw the radius ae parallel to ES, making the angle hae equal to aES, or 0. Through e draw/M perpen- dicular to Ea ; then of is ~- Q ~R cos 0, or (since E/ is not sensibly different from Ee) the change in the length of the moon's radius- vector, for assumed ; and fe is ~R sin 0. Now if /M be to fe in the proportion of the two P.I. coefficients 2' 5", or, in. circular measure, ~, to ^- , which is 15 to 7, very nearly, then M is the position of the moon for her assumed mean elongation 0. 144 THE MOON'S PARALLACTIC INEQUALITY. The point M describes, round a as centre, an ellipse which is as though it were rigidly attached to Ea, and therefore rotates about its centre once a month progressively ; its semi-axes major and minor, ay and a/*, being jggyR and g^R, respectively ; and as ae rotates, relatively to ah, with the moon's mean angular velocity in elongation (not in longitude), the ellipse is described retrogressively in a synodical month. At the time of conjunction the moon is at the point h of the ellipse, and farthest from the earth ; at the time of opposition she is at i in the ellipse, and nearest to the earth ; and she describes the ellipse with a simple harmonic motion relatively to each of the principal axes of the (rotating) ellipse. We have been considering the matter from the standpoint of an observer on the earth. But it is only as seen from the earth that the moon makes a complete circuit round a. Since the above ellipse, which is described once in a synodical month retrogressively, rotates progressively once in the same time, or otherwise more simply, since ab is greater than 6M, the moon never makes any circuit round a relatively to fixed space, or as viewed by a spectator looking at right angles to the plane of her orbit. She is always more or less nearly on the sunward side of a. Here, then, is the seeming kinematical paradox, as some might regard it at first sight (only), to which we have already alluded ; viz., that in describing the P.I. orbit the moon is always nearly on the same side, speaking roughly, of her mean place ! The explanation of this is that a is the moon's mean place relatively only to the earth about which she is revolving. The P.I. has, moreover, its own seeming kinematical paradox precisely similar to that of the Yar. considered at p. 121. In this we have contemplated the P.I. as existing by itself; but if we consider it as superposed on the Yar., a must be regarded as the moon's position in the Yar. orbit. The inaccuracy involved in doing this is quite insensible. The P.I. disturbing forces are too complicated to be introduced with advantage into Fig. 38. THE MOON'S PARALLACTIC INEQUALITY. 145 NOTE D, from p. 137. This will appear thus: Adopting equation (4), we have r=R(l-f ccose) ; c being the coefficient gJjjQ. The ordinate y at any point of the lunar P. I. orbit is r sin e, or R (sin e + c cos e sin e), which is R (sin e + |c sin 2e). Therefore cZ?/ = R (cos e + c cos 2e)de. For y a maximum, cos e= ccos 2e= c(2 cos 2 e 1). This quadratic equation gives which is and this is c, quam prox. ; and sin e = N/l c 2 , which is 1- |c 2 , q. pr., on account of the exceeding smallness of c. Hence y (or r sin e), at its maximum, is E(l -f c 2 )(l ^c 2 ), and this is E(l -f |c 2 ), q. pr. Therefore, taking R as 238,820 miles, that maximum diameter is longer than the syzygy diameter, or 2R, by Re 2 , or 102 feet (Q.E.D.). Said maximum diameter passes very nearly indeed through the middle point of the syzygy diameter, and conse- quently between the centre of the earth and the sun, and thu does not coincide with the line of quadratures. NOTE E, from p. 137. The radius of curvature p at conjunction may be obtained thus : Let be an indefinitely small e or elongation ; then we have by equation (4), for the radius-vector at conjunction, R (1 + c); c being the coefficient ^ ; as in last NOTE. The radius of curvature p = arc 2 /2 (f all from tangent). But arc 2 = R 2 (l+c) 2 sin 2 *, and 2R (1+c cos e ccos 2 e). cose 146 Therefore E(l + c) 2 (1 + cos e) cos e When e vanishes this becomes E. ^ . Similarly, the radius JL -7" ^j(? Q _ c \2 of curvature at opposition p becomes E^ -- L 9 as in text. 1 ^(5 These evidently differ very slightly indeed from E, and from each other. Taking E as 238,820 miles, and c as 0-000284, we find p p = l*39 inch. NOTE P, from p. 137. If we gave the proof here we should have to do it ab initio, which would require much space. But we may make the following observations on the subject. The mode of production of the P.I. orbit is exceedingly different from that of the Yar. orbit. The scheme of Yar. forces is symmetrical relatively to the line of syzygies and to that of quadratures ; consequently they go through their period of change in half a synodical lunation. But the scheme of P.I. forces is symmetrical relatively to the line of syzygies only ; and consequently their period is a whole lunation. Now a focal ellipse and the scheme of changing of the gravitation forces therein are symmetrical about one axis only, viz. the apsidal, and the period of the changing forces is that of one revolution of the body about the centre of force. It is evident, therefore, that if there were nothing in the conditions of the case to prevent it, the P.I. forces, when turned on to act on an originally circular lunar orbit, would go on increasing indefinitely the deformation of the orbit, whatever the character thereof might be. A moment's consideration will show the nature of the deformation. Since tbe tangential forces are proportional to sin e sin 3 e, the magnitudes of those belong- ing to the lower two arrows in Fig. 39 vary symmetrically on each side of the quadrature D ; they are equal at equal distances THE MOON S PARALLACTIC INEQUALITY. 147 on both sides of that point. Therefore, since they are so ex- ceedingly small, their impulses are very nearly equivalent, for our present purpose, to a short sufficiently strong tangential impulse acting at D. Therefore, as they are acting with the moon's motion, they tend to produce an apogee at the opposite side of the orbit very near B. Similarly, as the forces belonging to the two upper arrows in the same Fig. are acting against the moon's motion, their impulses would produce a perigee very near D. The outwardly-directed radial forces on the sunward side of the orbit tend to produce an apogee near B, and the inwardly- Fig. 39. directed ones on the other side a perigee near D. The conditions of the focal elliptical orbit lend themselves compliantly to this ; and if the disturbing forces cease to act, the deformation of the orbit would continue (with a very slight alteration). The consequence is that if the P.I. forces could continue to act, without the sun's relative revolution round the earth, the eccentricity of the orbit would go on increasing to a result which could not easily be followed out ; but probably until the moon fell upon the earth ; the line of apses remaining in quadratures and fixed in space. But this latter is prevented by the sun's relative annual revolution round the earth, which would diminish the eccentricity, and thereby give to the line of apses a pro- 148 THE MOON'S PARALLACTIC INEQUALITY. gressive angular movement, relatively to space, which would, at first, be slower than the sun's ; so that the sun would be over- taking it. By the time that the sun had overtaken it, the angular movement of the axis, which had been increasing, though all the while less than that of the sun, would be brought up to equality with that of the sun ; and it would thenceforth continue pointing to the sun. Thus, owing to the baffling conditions of the sun's relative revolution round the earth, the result of the action of the disturbing forces is exceedingly different from what it- would otherwise be. Nevertheless these forces are always tending to produce their own proper effect, which is to make an apogee very near B, and a perigee near D. But the very small effect that they can produce in one lunation, when compounded with that at conjunction, is only sufficient to cause the latter to be always ahead of its position in the preceding lunation, and to keep it moving progressively with the sun. The P.I. orbit has been, for convenience, and indeed in accord- ance with precedent, roughly spoken of as an ellipse ; it being intended that the earth is at the focus, and that the apsidal diameter (in syzygies) is the axis-major, with the apogee in conjunction. The velocity of the moon in the P.I. orbit would accord very nearly indeed with this ; but the actual " ellipse " is one of a rather peculiar kind, in that its axis-major is slightly less than its width. t NOTE G, from p. 139. This follows from the scheme of P.I. forces in Fig. 35, p. 133, and from what is told us by equation (3), as mentioned in p. 137, taken in connection with the prin- ciples of the apparent kinematical paradox in Chapt. VII., p. 121. But as some persons may feel a difficulty in accepting this, it may be well to put the argument together here, though it be a little repetition. By equation (3), the moon is at her mean place at conjunction ; therefore she is there moving either with greatest or least velocity which ? At the preceding quadrature D she is, by said equation, most before her mean place, and therefore THE MOON'S PAEALLACTIC INEQCTALITT. 149 moving with mean velocity ; but, since she is back at her mean place at conjunction, she has been losing velocity in the quadrant preceding conjunction. Therefore she is going with least velocity at conjunction. Similarly, on comparing first quadrature with opposition, we find that the moon is going with greatest velocity at opposition. Thus her velocity is at the maximum at opposition, at the mean at last quadrature, and at the minimum at con- junction ; that is, she has been losing velocity all through that semi-orbit ; though the solar tangential force has been all the while acting in consequentia , or along with her motion. Similarly she gains velocity all through the other semi-orbit ; though the solar tangential force has been acting in antecedently or against her motion. We have, so far, been content with the general law of the change of the moon's angular velocity in the P.I. orbit ; but the exact law can be obtained by differentiating equation (3). The coefficient, in that equation, as expressed in circular measure, oeing ^, the actual vel.=the mean do. X (1 j^ cos e). This equation shows, at a glance, that the velocity is least at con- junction, greatest at opposition, and at its mean at quadratures ; the opposite of what the solar tangential forces would cause by their immediate local action. NOTE H, from p. 141. In Fig. 40, E is the earth's place, the curve ac a portion of the moon's P.I. orbit, and de an indefinitely small increase of , the moon's elongation. Let 6 be the angle between the moon's radius-vector and the curve at a, or the tangent thereto. Then E, the moon's mean radius-vector, being taken as unity we have, for the actual radius-vector at a, by equation (4), T 1 -h c cos e ; whence dr csin e de. The earth's mean attraction on the moon being, as well as R, taken for unity, the earth's attraction at a is -3, or 1 2c cos e, 150 THE MOON'S PAEALLACTIC INEQUALITY. quam prooc., or sensibly 1. This multiplied by cos 6 is th earth's tangential force at a. Now cos 0=cot0, q. pr., as 6 differs so very slightly from a db dr csinede nght angle. But cot 6 = 5=^' and this - (rj : eeo , t)