GC 30/ A z UC-NRLF IlllilP 11 1 $B E7D bD3 ELEMENTARY THEORY THE TIDES. T.K.ABBOTT. LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementarytheoryOOabborich ELEMENTARY THEORY OP THE TIDES: THE FUNDAMENTAL THEOREMS DEMONSTRA TED WITHOUT MATHEMATICS, AND THK INFLUENCE ON THE LENGTH OF THE DAY DISCUSSED. BY T. K. ABBOTT, B.D., D.Litt., It FELLOW OF TRINITY COLLKOE, DUBLIN. f I VN1VER81TY SfcoHO edition, revised. LONGMANS, GREEN, AND CO LONDON, NEW YORK, AND BOMBAY. 1901. S£N£fiAL Ft'rs^ Edition, 1888. Second Edition, igoi. In Italian. By Prof. Edoardo de Ferrari {-with additional elucidations by the author), 8° pp. 67, Pistoia, igoi. Printed at The Dublin University Press, by Ponsonby & Weldrick. PREFACE The substance of the following pages has already ap- peared, partly in the Philosophical Magazine, 1871, 1872, and the Quarterli/ Journal of Mathernatics, 1872, and partly in Hennathena, 1882. Hitherto correct statements about the Tides have been confined to treatises which employ the resources of the higher mathematics. Other works almost without exception* repeat such erroneous statements as that the place of high water without friction would be under the moon, and that high water is retarded by friction. No apology then is needed for the publication, in a more accessible form, of the present Essay, in which the fundamental theorems are deduced from elementary physical principles without the use of mathematics, except for quantitative calculations. The problem of the influence of the Tides on the length of the day is discussed in a similar method. * The only exception with which I am acquainted is Stubbs' edition of Brinkley's Astronomy, in which the reasoning of this Essay is adopted. 142654 IV PREFACE. For the benefit of readers who may wish to see the latter problem analytically treated, I have given in an Appendix the substance of Sir Q-eorge Airy's investi- gation. In this second edition, I have introduced some correc- tions and further elucidations, but have omitted the Appendix. The subject of the first part of this Essay has been treated with more detail by the late Dr. Samuel Haughton in a paper entitled " Mathematical Principles of Tidal Theory and Observation," published in Her- mathena, 1879, p. 563. The present Essay has been translated into Italian by Professor Edoardo de Ferrari, of Empoli. ELEMENTARY THEORY OF THE TIDES. The attractive force of the sun or moon on the solid mass of the earth is the same as if the latter were concentrated at the centre. But the attractive force, on movable par- ticles at the surface, is greater than this on the side nearest to the moon (or sun), and less on the opposite side. This excess and defect constitute the tide-producing force. Confining ourselves to the moon : — The direction of the tide-producing force is ahmys tangen- tialy and towards the line joining the centres of the Earth and moon. The following (Newton's) construction represents this force in direction and magnitude. Fig. I. I Let xl (fig. ^) be perpendicular to EM, and let hn = mn = El. Then if JUE represent the attractive force of the moon at the centre E^ wn will represent the whole B 2 THE TIDES. disturbing force in magnitude and direction. The proof is as follows : — Let it be borne in mind that ME is about sixty times the radius of the earth. Hence, if we consider Ml = Ma?, the error cannot exceed — = IjKqq^^ P^^^- ^^^ the error is greatest when x coincides with D (fig. 2). But in that case MD' = MJE' + r^ = {QOrY +r^ = 3601 r' .-. MB = ryMoi = r (eO + ^^ nearly. Therefore r ME i./i)-il/£=j2^=^7g^^^ nearly. Again, Mn, Mm, Ml, ME, being arithmetical proportionals with a difference varying from 0 to ^th, may be regarded as geometrical proportionals ; the greatest possible error being about the same as before. Hence, Mn : Ml or Mx : : MP or Mx^ : ME"^, i.e. as the moon's force at E : force at x ; therefore, if Mx represent the moon's force at x, Mn will represent the force at E in magnitude and direction, and the difference or disturbing force will be represented in magnitude and direction by xn. In order to have a fixed scale we must represent the force at the centre by ME. On this scale xn is in the nearer hemisphere too small, and in the more remote too large, in the proportion of 3 1 Mn to ME. This error is at most ^ths = nTrth. This will be considered by-and-by, but for the present it may be overlooked. The vertical component of xn is xh. The tangential TIDAL FORCE. ^ OF THt VNIVERSITY CALifOHHi> aciuaiiy exceeaea tne torce ot gravity. And, as will be seen by-and-by, it is too minute to produce any indirect effect. The tide-producing force then always acts towards JEM (in the direction of the arrows in fig. 2). From this we can deduce theorems relating to the place of high and low water, &c., without requiring to determine the magnitude of the force, which will be hereafter taken into account. At present we need only observe that it is very small compared with gravity. First, then, let us consider the case of water limited to an equatorial canal. The moon being supposed in the equator, we shall establish the following theorems : — I. If there were no friction it would be low water under the moon, and high water in quadratures. II. Friction accelerates the times of high and low water. III. In addition to the oscillatory motion of the water there is a constant current produced by the action of the moon. IV. The effect of friction on this is to increase the length of the day. H 2 THE TIDES. ERRATA Page 3, line 4 should read — Ik = El sin MEx = Ex cos MEx sin MEx = Jr sin 2 MEx, MD-ME = ME 120 2 X 60 X 60 nearly. Again, Muy Mm, Ml, ME, being arithmetical proportionals witli a difference varying from 0 to F=th, may be regarded as geometrical proportionals ; the greatest possible error being about the same as before. Hence, Mn : Ml or Mx : : MP or Mx^ : ME^, i.e. as the moon's force at E : force at x ; therefore, if Mx represent the moon's force at x, Mn will represent the force at E in magnitude and direction, and the difference or disturbing force will be represented in magnitude and direction by xn. In order to have a fixed scale we must represent the force at the centre by ME. On this scale xn is in the nearer hemisphere too small, and in the more remote too large, in the proportion of 3 1 Mn to ME. This error is at most TrTrths = rrTrth. This will be considered by-and-by, but for the present it may be overlooked. The vertical component of xn is xh. The tangential TIDAL FORCE. 3 component of xn is equal and parallel to nhy the perpendicular on the radius, and is, therefore, pro- portional to Ik, which is one- third of nh. Note that Ik = El sin MEi*^ Ex cos Jf^sin ME^= Jr gin 2 J/"^ that is to say, the tangential force is proportional to sin 2 (angle from moon). The same reasoning applies to the dotted letters in the further hemisphere. The vertical component being in the same line as gravity (either in the same or an opposite direction) cannot directly produce any motion. In fact, it could not do so unless it actually exceeded the force of gravity. And, as will be seen by-and-by, it is too minute to produce any indirect effect. Tlie tide-producing force then always acts towards EM (in the direction of the arrows in fig. 2). From this we can deduce theorems relating to the place of high and low water, &c., without requiring to determine the magnitude of the force, which will be hereafter taken into account. At present we need only observe that it is very small compared with gravity. First, then, let us consider the case of water limited to an equatorial canal. The moon being supposed in the equator, we shall establish the following theorems : — I. If there were no friction it would be low water under the moon, and high water in quadratures. II. Friction accelerates the times of high and low water. III. In addition to the oscillatory motion of the water there is a constant current produced by the action of the moon. IV. The effect of friction on this is to increase the length of the day. H2 THE TIDES. I. — Without friction it tvould be low ivater under the moony and high icater in quadratures. I suppose the moon to be fixed, and the earth rotating in the direction ^-S CD, carrying the ocean with it. That the ocean is so carried is a fact of experience. Now, in the course of one lunar day every particle of the ocean is subjected to precisely the same forces, acting in the same order of succession and for the same periods, being accelerated for about one quarter of a day, viz., while passing from B io C '^ then retarded for a quarter, from C to />, and so on. The variation in the amount of the force does not concern us, being the same for every particle. This being so, it is obvious that those particles will be Fig. 2. moving faster which have been for a longer time acted on by an accelerating force, and the velocity will be a maximum when the accelerating force has acted during its full period, viz. through one quadrant. On the other hand, those particles will be moving slower which have been longer acted on by a retarding force, and the absolute velocity will be a minimum when the retarding force has acted during its full period, or through one quadrant. The maximum velocity is therefore at A and (7, the minimum at ^ and D. Secondly, it is clear that the tide will be rising where PLACE OF HIGH WATER IN EQUATORIAL CANAL, 5 ^SLch. portion of water is moving faster than that just in -advance of it ; or, in other words, where water is flowing in faster than it flows out. Where this process has gone on for the maximum time the tide will be highest. On the other hand, the tide will be falling where the water is moving slower than that in advance of it — or, in other words, is flowing out faster than it flows in. Where this has continued for the maximum time the tide is lowest. Now consider any point s in the quadrant BC. The water now passing s has been subject to an accelerating force during the whole time since it passed 5, longer therefore than any particles behind it, as at r. It is there- fore moving faster ; and as the water in the space r s is thus flowing out faster than it flows in, the tide is falling. This is the case through the whole quadrant BC. At C the force changes and becomes a retarding force. The particle at ?/ has been subject to this retarding force longer than one behind it, as at j^*, and is therefore moving slower. Here, therefore, water is flowing in faster than it flows out, and the tide is rising ; and this holds through the quadrant CD. What is said of these quadrants holds also of tliose opposite to them ; tlie tide is falling all through DA and rising through AB. Hence it is highest at B and D, lowest at A and C. Where will the tide be falling fastest ? Clearly where the difference of velocity between r and s is greatest, i.e. where the amount of force to which the water at s has been subject since it passed r is greatest — in other words, where the force is at its maximum, viz. at / (fig. 3), 45° from C. Similarly it will be rising fastest at that point in the quadrant CD where the force is greatest, viz. at g, 45° from C. 6 THE TIDES. On the whole, then, the water in the supposed equatorial canal assumes the form of an ellipse ; and as it is the earth that is rotating, this ellipse does not change its absolute position except with the moon's motion ; only the water accompanying the rotating earth moves fastest at A and M B Fig. 3. C, and is there lowest; and slowest at B and D, and is there highest. Relatively to the earth it is moving westward from e to /and from g to h; eastward from /to ^ and from h to e. At Ay B, C, D, the particles are in their mean places ; at ^,/, g, h, they are farthest from their mean places, and change the direction of their relative motion. This is represented in fig. 8, where the inside ^^^^r^/ t i ^sr Z) AC Fig. 4. arrows show the direction of the earth's motion ; the out- side arrows that of the relative motion of the water. The path of any one particle may be represented by fig. 4, where the letter A indicates the position of the particle when its mean place is at A in fig. 3. FRICTION ACCELERATES HIGH WATER. 7 II. — Friction accelerates the times of high and low water. The theorem that the effect of friction is to accelerate the time of high and low water admits of an equally simple proof. As the water approaches (7, the tangential force diminishes gradually to zero at C. Therefore it must have been equal to the force of friction at some point n (fig. 1), after which friction prevails and the velocity diminishes. It is therefore low water at n. Approach- ing D, the ocean is moving slower than the earth ; there- fore here friction tends to accelerate it, while the retarding force is decreasing to zero. The two forces, then, must be equal at same point o, after which the velocity again increases. It is high water therefore at o. It sounds paradoxical to say that friction ** accelerates " high water. The paradox is only apparent. Friction checks the motion, so that the water stops rising or falling sooner than it otherwise would ; and thus we may speak of the phase of high or low water being accelerated. The preceding proof assumes that the ocean is carried round by the earth in its rotation. This amounts to supposing that it has not assumed a position of equili- * It is important to observe that we are not entitled to assume that when the tide is rising fastest the water is flowing in from both sides. This is by no means evident. The rate of rise depends on the difference in velocity between two successive parts of the ocean, and this may be greater when the two velocities have the same sign than when they have different signs. Taking into consideration the rotation of the earth, the assumption amounts to this — that the tide is rising fastest where the velocity of the ocean is just equal to that of the eai-th. This is certainly not evident : in fact it would not be true if the tangential force did not decrease at the same rate on both sides of each of the four maxima. It ought not, therefore, to be assumed, but deduced. 8 THE TIDES. It is a priori an admissible suggestion that the ocean is in a state of equilibrium under the moon's action, ^.e. that it is absolutely at rest (relatively to the moon), while the earth rotates. But this would imply an apparent move- ment of the whole body of water with a velocity equal and opposite to that of the earth's rotation, i.e. at the equator there would be an apparent current of about 1000 miles per hour. As this does not correspond to the fact, the hypothesis is practically inadmissible ; but when friction is considered it appears theoretically inadmissible also. For in this case friction would continually act in the same direction, and its effect would be to make the east- ward forces preponderate ; so that although the ocean should be supposed at rest at first, it would ultimately be dragged round by the earth. The actual form of the earth, moreover, in which the equatorial ocean is inter- rupted by continents, would render this equilibrium of the ocean impossible. III. — There is a constant current westward produced by the moon^s disturbing force. This occurs from two causes. First, the water in the supposed equatorial canal has now taken the form of an ellipse ; and, in consequence of friction, the places of greatest elevation are not at B and 2), but somewhere in the quadrants BA, CD. Now, the moon's tangential force, hn, is, cceteris paribus, proportional to Ex the distance of the particles attracted from the centre of the earth. It follows that it is greater in the quadrants BA, CD, than in the other two; but in the former the force is CONSTANT WESTWARD CdRRENT. 9 retarding ; in the latter it is accelerating ; therefore the retarding force exceeds the accelerating, and produces a permanent westward motion. Secondly, the water having reached its mean place at n^ and passed it with its greatest eastward velocity, it is, when it reaches C, eastward of its mean place, i.e. it is nearer to g. On the whole way before reaching g it is nearer to that point than if there were no friction ; but on passing g it begins to move westward ; but its eastward excursion having been shortened by friction, it begins this motion to the west of where it would otherwise be. At o it again arrives at its mean place, which, without friction, it would not reach until D. Thus, in the whole quadrant CD, the particles are nearer to g than if friction had not operated. But the tangential force is greater the nearer the particles are to g, being proportional to sin 2 (angle from moon) = cos 2 (angle from/ or g) ; hence the force in tlie quadrant CDy which is a retarding force, is increased. After passing its mean place at o, the water going westward is, on arriv- ing at D, west of its mean place ; and until it reaches h it continues to be west of the place which it would have occupied had friction not operated, i.e. friction withdraws it from h. At h its westward excursion is stopped, and it begins to return eastward. But now from h to A it is east- ward of the place due to it without friction. Thus through- out this quadrant the particles are brought farther from h by friction. But here the force is accelerating. Therefore the force in the accelerating quadrants is diminished, while that in the retarding quadrants is increased, and hence again a balance of retarding force, and therefore a current westward. Or thus : — Without friction, the quadrant 10 THE TIDES. fg, throughout which the water is moving faster than the earth, has its middle point at C ; and the following quadrant in which it is slower has its middle point at D. These quadrants are, therefore, equally divided between the accelerating and the retarding quadrants. With friction, the middle points being displaced to n and o respectively, the water is moving faster than the earth throughmore tlian half the quadrant BC, and slower through more than half the quadrant CD ; and similarly in the opposite quadrants. But BC, DE are the accelerating quadrants, and CD, AB the retarding quadrants. Therefore the water is exposed for a longer time to the retarding than to the accelerating force. We have here, therefore, a vera causa which may pos- sibly be effective in retarding the earth's rotation. An attempt will presently be made to estimate the maximum amount of this effect. On the Quantitative Valuation of the Tidal Disturbance, To determine more precisely the magnitude of the dis- turbing force : — The moon's attraction at x : force of gravity : : moon's mass earth's mass , 1 „- , ,, j^^^ • -a » or nearly = ^^ : 81 J ; there- fore the whole attraction of the moon (represented in fig. 1 by ME) = -^j^ — ^2- ^^^ 0^ ^^® same scale tlie greatest tide-producing force is represented by-r (the greatest value HEIGHT OF TIDE. 11 1 IfTP of Ik being - r), i.e. by -jj-. (This we shall call H. ) The- greatest tangential force then is 9 9 ^ 40x81^x60^ 11,736,000 365,000' nearly. The tangential force at any given point ia = - H^m 2w (w being the angle from the moon) . Neglect- ing the effect of pressure, the effect of the moon's action through one quadrant : the effect of this maximum con- tinued for the same period : : 1 : Jzr (this appears from the construction in fig. 5). The number of seconds in the mean lunar day being 89432, the velocity generated in one- fourth of this time is- 22358 1 „ , , 365000-x-T;; = 26 ^''^ '''^'^^' This is the difference between the greatest eastward and the greatest westward velocity ; therefore the greatest eastward velocity is — , and the greatest westward velocity is also 35- As the same amount of water passes through a given section in a given time, the increase in height : total depth of the sea : : relative westward velocity of the water : earth's velocity of rotation (relatively to the moon). The last is about 1470 feet per second. Hence , . ... .._ depth of sea 1 depth of sea the rise of the tide = ^^^^f^- x ^ = 76440-' For a sea of three miles in depth this would give for the 22 lunar tide 2*4 inches. The solar tide is about -=^ of this. OF THE YER8ITY or 12 THE TIDES. The following is a geoinetrical construction for the velocity and height at any place: — Round the radius OB describe a circle. Since the angle at BcO is right, Be is equal to the perpendicular from CE, i.e. to xl in fig. 1, and cp equal to Ik ; so that the tangential disturbing force at a is proportional to the perpendicular cp. -m: If aa^ be the space passed over in the rotation of the earth in one second, the force acting on the water may be supposed unchanged while it passes from aioa^ ', and its effect during that interval {i.e. in this quadrant, the retardation) will also be proportional to cp or its double cf and to the time: that is, to aa\ or the angle at 0, aOa\ Calling H the moon's greatest tide-producing force, r the earth's radius, and r the angular velocity = :, the retardation seconds in lunar day Now the angle at 89432' r r 0 = the angle at /, being in the same segment ; and this angle multiplied \ry cf= the small perpendicular cd, or pp\ CONSTRUCTION FOR VELOCITY AND HEIGHT. 13 which is parallel and equal to it. Therefore the whole retardation since leaving B is proportional to the sum of all the ahscissae pp' — that is, to Bp'. It is H ~ — - — . r Zir This represents the defect from the greatest eastward velo- city ; and after passing its mean value at the middle point 8 it represents a velocity which, relatively to the earth, is westerly. The velocity of the current relatively to the earth is represented hy ;;s. We shall now show that the height of the tide at ci above its lowest point is also proportional to Bp\ If at any point in the supposed canal a thin section be taken, the quantity of water entering this section in a given time is proportional to the product of the depth and the velocity. If the water flows in a little more rapidly than it flows out, it is clear that the increase in the quantity contained in the section, and therefore the increase in depth, will be proportional to the difference between these two velocities and to the whole depth ( -i-^ — -; — ^ ,. ). ^ \ length 01 section / This holds as long as the change is small compared with the whole depth. If this be supposed uniform throughout the canal, the increase in it (that is, in the height of the tide) at a' is therefore proportional to the retardation ; and since the tide began to rise at B, where the velocity began to diminish, it follows that B2)' is also proportional to the height of the tide at a above its lowest point. It is easy to deduce from this construction the cor- responding formulae. For, if OB = r, we have Bp=r[l-QO^^(jj), Butj!?s=-J^r-^j9=|r(2cos'w-l)=^rcos2w. And since sB is proportional to the mean height, the defect 14 THE TIDES. from this height is proportional to ps, and therefore to €0S 2w. The effect of pressure with such a tide will be extremely small. As it operates to send the water away from its position of greatest elevation, it will so far assist the moon's force without changing the place of high water. To estimate the effect of friction, the moon's force being = - jB" sin 2a>, the velocity v, undisturbed by friction = Fcos2a> (F being the greatest velocity). If the dis- placement caused by friction is very small, we may take this value of «? as a sufficient approximation. The friction, assumed proportional to velocity = Vf cos 2a>, and the velocity begins to be checked when Vf cos 2w = - jffsin 2w . ^ Vf 365000. __. . . or, tan 2(u = - -^ = p^r — /= - 7000 /nearly. When the displacement (which we shall call S) is sensible, we shall have as follows : since the velocity is periodic and is a maximum, positive or negative, when w = S ; and is = 0, when w - S = ± 45°, ± 1 35°, we may assume v=V cos 2 ( w - S) . Then friction = fV cos 2 (w - S) and the net amount of force = - iT sin 2a> -fV cos 2 (w - ^). Hence H cos 2to /Fsin 2 ( ^ COS 2X ^ and total rise or tall = -— ; — cos 2m. r ^r- This is = cos 2X x half the total rise or fall in the equa- torial canal with the moon in the equator. After passing 45° latitude, the decrease in the circles of latitude becomes important. If we assume our meridional canal to be of uniform width, then the canals will gradually overlap, the tide thus diminishing until at the pole, as is obvious, there will be no tide. MOON'S DECLINATION CONSIDERED. 21 Let us now consider the case of the moon having a declination, which for simplicity I shall suppose less than 22° 30'. This limitation will not affect our results. We shall, as before, take the two components separately. With respect, then, to the component which acts parallel to the equator. Near the equator itself the considera- tions previously applied still hold good. Next consider a place a, whose polar distance is less than the moon's declination, to which therefore the moon is circumpolar, and (with the assumed declination) alternately north and Fig. 7. south of the zenith. If ahccl be the circle of rotation of such a place, the distance of the moon from any point in this circle is less than that from the earth's centre. If> then, the direction of the rotation be ahcd it is obvious that the water will be accelerated through the whole semi- circle, ahc, and retarded through cda. The same reasoning as already employed will show that it will be low water at c and high water at a. Now take an intermediate place whose circle of rotation is Imno. Here the water is retarded and rising from I to in and from nio o\ and accelerated 22 THE TIDES. and falling from m to n and from o to I, and the interval olm is less than mno. Hence the tide is lowest at n and not so low at /, and it is high water at m and o. Hence we have a diurnal tide in addition to the semi-diurnal, this diurnal tide becoming of more importance as we recede from the equator until the co-latitude = moon's declination^ when the semi-diurnal tide disappears. At the equator the meridional component acts during half a rotation towards the equator, and during the other half from it, and in each case is an elevating force, which, as before, has its greatest effect 90° from the moon. At all places whose latitude is not greater than the moon's declina- tion there is a permanent accumulation. In the circle ahcd this component is directed towards the north at a and towards the south at c, the points of change being where the great circles from M touch abed. This gives rise to a north and south oscillation. The southerly force being the greater, there will be a residual depression of the water in this region. The depressing force, however, varies, being greatest at a and at c?,* while the elevating force is greatest where the great circle tangents from M meet the circle. Hence, by 4° and 5° the tide will be lowest at the latter points and high at the former, and there will be a diurnal tide, as in the former case. Com- bining this with the former result, the effect of both components together will be to give high water at a. * If the moon's declination were greater than 22° 30', c might be less than 45° from M, in which case the force there would be an elevating one. Again, at a place whose latitude was greater than 22° 30', and less than th& moon's declination, the moon's least nadir distance (•= IN) would be greater than 45°, and the force depressing. MOON'S DECLINATION CONSIDERED. 23 It is not necessary to enter into a detailed examination of the state of things at intermediate places. It is not difficult to see that, as long as the moon's declination is small, there will be an accumulation effected by the me- ridional component extending from the equator to about lat. 45°, and that, as the moon's declination increases, tlie accumulation becomes less at the equator and greater towards 45°, until the declination reaches 45°. With a declination greater than 45° there would be an accumu- lation at the poles ; and obviously, if the moon were at the pole, the ocean would take the form of a prolate spheroid. The place of high water at any latitude, as far as this is due to the meridional component, would be easily found ; but the proportionate effect of the meridional and equatorial components depends partly on the latitude and partly on the moon's declination ; and it does not come within the scope of the present essay to solve this problem. It is sufficient to observe that the importance of the meridional component increases with the declination as well as with the latitude. If the moon were at the pole this force would be alone ; and, whatever the declina- tion, it alone produces an effect at the pole. The same reasoning applies, mutatis mutandis, to the solar tide. It was remarked, on p. 10, that the disturbing force is slightly greater on the side nearer the moon than on the remoter side. The effect of this inequality is to produce a small diurnal tide. 24 THE TIDES. On the Effect of the Tides on the Length of THE Day. § 1. — Historical. In the year 1754 the Berlin Academy proposed, as the subject for a prize essay, the question, " Does any Cause exist tending to Eetard the Rotation of the Earth? '' What the result of the competition was I do not know ; but the question led to the publication by Kant of a short essay, in which he suggested that such a retarding cause existed in the tides. He worked out this suggestion in a rough way, there being, as he truly said, no ascertained data on which any trustworthy calculation could be built. Laplace examined the question from the historical side, with the help of the records of ancient eclipses, and came to the conclusion that the period of rotation had not altered. Eecently, in consequence of the improvement of the lunar tables, astronomers have seen reason to re-open the question. It has been inferred from the records of ancient eclipses that the day is lengthening at the rate of one second in two hundred thousand years. At first sight this may seem to be an amount too small to leave any trace in history. It must be remembered, however, that in calculating what part of the earth's surface came into the shadow of a given total eclipse, say 2500 years ago, we have to "unwind" 2500 times 365i(= 913,125) rotations, and a difference amounting to an eightieth of a second between the first and last of these would in the whole period have a very considerable effect.* M. Delaunay * About 100 minutes: see Ball, ** Elements of Astronomy," p. 377. I UNIVERSITY I EFFECT ON LENGTH OF DAY. 25 attributes the retardation to the moment of the moon's disturbing force on the tidal prominences. He started from the assumptions that without friction it would be high water under the moon and anti-moon, and that friction retards the time of high water. Both these assumptions were erroneous ; but they so far counteracted one another as to leave the place of high water in the same quadrants as the true theory, viz. in the quadrants east of the moon and anti-moon, in which the moon's force is retarding. Sir George Airy corrected these errors, and working out the equations, found two terms wliich indicate a constant current westward — one term (the smallest) depending on the vertical, and tlie other on the horizontal, displacement of the water.* In my own Essay on the Theory of the Tides {Quarterly Journal of Mathematics ^ 1872, imdi Philosophical Magazine), the effect of friction was indicated, but there was no at- tempt to estimate it quantitatively. I am not aware that any attempt has been made to solve this problem ;t and indeed it would be absurd to pretend to do so with any degree of accuracy. What I propose to do is to estimate the effect so far as to enable us to form a judgment as to the actual importance of the tides as a cause retarding the earth's rotation. It will be convenient first to prove the following propo- sition respecting the effect of obstacles : — * Monthly Notices of the Royal Astronomical Society, 1866, p. 221. t See, however, Lord Kelvin, Philosophical Journal, 1866, p. 533. He mentions also a Paper by Wm. Fernel, Astronomical Journal of Cambridge, U.S.A., December 8, 1853, which I have not seen. 26 THE TIDES. § 2. — Obstacles which check the motion of the xmter towards a certain point retard the time of high water, and increase the height. If the obstacle is a complete barrier, the tide will rise as long as the motion of the water is toxoards it, and will fall as long as the motion i^from it. Hence, at 45° east of quadratures it will be high water on the east of such an obstacle, and low water on the west of it. The influence of this on the time of high water at other places will extend as far as the pressure is felt. An obstacle not sufficient to stop motion altogether will produce a similar effect, but of course much smaller, in consequence of the continuity of the surface. If the obstacle be such as to destroy half the velocity of the water, then high water on its east side would be 30° after quadra- tures. In both cases the height would obviously be increased. It appears from this that the effect of such obstacles is in both respects the reverse of that of friction. § 3. — Effect of the moment of the moon^s attraction on the tidal prominences in an equatorial canal with the moon in the equator. This is the way in which the retardation was supposed by Delaunay to be produced, and Thomson and Taithave adopted the same view.* * The statement that the earth rotates in a "friction collar," "vrhich seems to put the matter in a nutshell, obviously assumes that the passage of the tidal wave is the passage of a mass of M^ater. But this is true only so far as there is a residual westward current, which is certainly not self- evident. ATTRACTION ON TIDAL PROMINENCES. 27 Now, in order to estimate the greatest effect possible, let us suppose that the greatest elevation is in the middle of the quadrant, i.e. 45° before quadratures ; and further that the elevation is not diminished by friction. Let £"= the moon's greatest liorizontal force. lo = angle from the moon. e = greatest elevation. Then the tangential force at any point = J^sin 2(o, and the elevation = e sin 2a>. For it is proportional to the velocity, which as we have seen (p. 14) = Fcos 2 (w - ^), and if S = 45° this = Fsin 2a>. Multiplying by the element of the equator, and the unit of width, we get the moment = jETe sin' 2(u x rdu x r. Now sin* 2w is always positive, varying from 0 to 1 and from 1 to 0. Substituting its equivalent ■2-(l - cos 4w) we divide the moment into a periodical part i Her^ cos 4(odw which varies from - iHer^dat to + -J- Her'^div, and therefore produces no permanent effect, and a constant part which is = ^ ffer'^dw. Summing this round the circle, and multiplying by the coefficient of friction, we have for the whole moment Heirr'^f. Taking the density of the earth as 5, the moment of inertia of the equatorial section of the earth is - Trr*. Dividing the former by the latter, we have for the angular (negative) acceleration -— r-- . TVT rr 1 -. depth of sea ^°"' ^=365000"°'^ " = --76440-- If we assume the depth of the sea to be 3 miles, the angu- lar (negative) acceleration becomes nearly = ^^ ., ..,;; . 93 billions X r 28 THE TIDES. Multiplying by the number of seconds in 100,000 years . 1 / (about 3 billions), we obtain ^- nearly. Now, the velocity of the earth's surface at the equator relatively to the moon is about 1470 ; the angular velocity therefore is r Tj- 1 earth's velocity ence, — /= -^^^ /. If the earth's velocity is diminished in this propor- tion, the length of the lunar day will be increased by 89432 ---^T^Tr / seconds = nearly 1*96 / seconds. To find the effect on the actual (solar) day we must take the angular velocity = , when the last fraction will become / or, approximately, 1.836 seconds. The solar tide need not here be taken into account since its effect on the pro- minences due to the moon is alternately positive and negative. If the displacement instead of being 45° is = § then the elevation will be (as a first approximation) e sin 2S and the retardation 1*7/ sin 2S seconds. Now, in the case supposed, /is exceedingly small, the friction being ultimately that of water on water. Hence we conclude that in an unobstructed equatorial canal the effect of the moon's attraction on the tidal prominences in retarding the rotation would be quite insignificant, even on the supposition above adopted, that the place of high water is 45° before quadratures. If this place were affected only by friction, the displacement could never reach 45**, for tan 28 = 7000 /and /must be less than unity. The EFFECT OF RESIDUAL CURRENT WESTWARD. 29 elevation also is diminished by friction in the proportion of cos 2B to 1. Remembering this, it is easy to see that the figure just given should be multiplied by ^ sin 48, so that with the greatest possible displacement, the result would be practically 0. In order that S should be = 22 J°, since tan 28 = 7000 /, / should be = (which is far beyond itsvalue). We should then have 1*83/Jsin 48 = nearly. The retardation in this case would therefore be less than one second in 700 million years. It will be seen here- after that /is millions of times less than this. There is another way of viewing the matter, which does not introduce /. The following consideration explains this : — § 4. — Of the effect of the residual current westward due to the change in the time of high icater. The constant force found above, = \IIe sin 28, produces an accumulating westward tendency in the water. This once impressed will continue to increase until the increase is checked by friction, that is, until friction becomes equal to this constant force. This occurs when ^He sin 28 x ^ x/= ^He sin 28, that is, when t = -. After this the velocity of this west- ward current is constant, the constant force being expended on counteracting friction. Therefore when we take a sufficiently long time we may assume that the total moment 30 THE TIDES. (of the water) is ultimately not affected by the coefficient /. In fact the retarding force fv is then = \He sin 28. This being premised, I shall now examine the question from the point of view suggested by Airy. § 5. — Effect of the changes in the disturUng force due to the displacement of the water. By substituting, in the expression for the disturbing force, the altered value of the ordinate of the water for the original value {x-v X, for cc). Airy finds that the expres- sion contains a constant term dependent on the distance of high water from quadrature. The source of this con- stant term may be understood from the following observa- tion : — The particles are in their mean place at the moment of high water and at that of low water ; at the former they are travelling W. with their greatest velocity; at the latter they are travelling E., also with their greatest velocity. Now, the place of high water being W. of quadrature, and the water moving W., it follows that when the water reaches quadrature, approaching the moon, it is behind, or west of the place which, without friction, it would have occupied. On the other hand, at syzygy it is in advance, or E. of its place. In both cases the disturb- ing force is diminished by this displacement, the force being greater the nearer the particles are to the middle point of the quadrant. In other words, H sin 2ai is diminished throughout, a> being increased when over 45°, and dimi- nished when less than 45°. In the following quadrant, i.e. after passing the moon, the opposite change takes EFFECT OF DISPLACEMENT OF WATER. 81 place, since the particles enter it E. of the place they would otherwise occupy, and leave it W. of their place. Now, the former quadrant is that in which the moon's force is accelerating, the latter that in which it is retarding. The same observation applies to the other two quadrants. Thus the accelarating and retarding forces are no longer in equipoise, the latter predominating. To calculate the effect :— The maximum excursion of the water without friction in the case of a canal three miles deep would be about 136 feet. For the greatest velocity = ^— (feet per second). Now if this continued for one- 22356 fourth of a day the space passed over would be — ^^r— = 430 nearly. With the varying velocity ^ cos 2oi the space traversed is less than this in the proportion of 1 to ^tt (as in the calculation of the velocity in p. 11). It is therefore = 272 feet nearly. This is the double excursion. There- fore the maximum excursion on either side is 136 feet. Assume that this is undiminished ; and assume, as before, that it is high water 45° "W. of quadratures. Then we may assume the displacement at each point to be 136 cos 2(o ; and the moon's force being ff sin 2(u, the change in the disturbing force due to this displacement 1 *^fi = 2H cos 2cD X cos 2(0 per second. r ^ 1 o^ The constant part of this = fl" — • r Putting for H its value ^ , and calculating the 32 THE TIDES. effect continued for one lunar day (89280 seconds), we have 11 136 100 TTT X > or 77- — ttt: , nearlv. 46 r ' 65 millions "^ This acts on the whole mass of the canal. Introducing the moments, as in p. 27, we have as the acceleration for one day 200 mass of canal ^ — . — ^ Qb millions mass of equatorial section of earth x r With the assumed depth of sea, the latter factor = ttttt^ • ^ ooOO X r Hence the daily angular acceleration = 1 1 "325000 "" 3300 X r' Multiplying by the lunar days in 100,000 years (about 33 millions) we have nearly as before ^y • This gives a retardation of about 1*83 seconds in 100,000 years. Adding the solar tide we have as the total 2-6 seconds. This is on the hypothesis that the elevation is not diminished. Introducing the necessary correction we have, as in p. 28, 2-6 i sin 4g. For the reason before stated, it is unnecessary to multiply this by the coefficient of friction. There is a third way of viewing this cause. Owing to the displacement of the place of high water, since that is the point where the water is moving fastest westward, the water is a longer time in the retarding quadrants than in the others, e.g. on the previous hypotheses it is 136 feet behind its place on entering the accelerating quadrant, ACTUAL STATE OF EARTH. 33 and 136 feet in advance on leaving it. It is, therefore, in 272 that quadrant about ^^ seconds = about '18" short of a quarter of a lunar day. This would give a result similar to that already found. The preceding calculations are obviously applicable to the case of a globe uniformly covered with water, since each section parallel to the equator would give the same results. The meridional wave would have no effect on the rotation. It is not worth while to extend our calculation to the case of the moon not being in the equator. The nett result would be to diminish the retardation. § 6. — Application to the actual state of the earth's surface. In attempting to apply the preceding results to the actual condition of things on the earth's surface, the fol- lowing points must be noted : — First. — On the earth as it actually is the effect of friction proper on the tides is trifling compared witli that of ob- stacles. Against these the tidal current impinges, and in addition the increased elevation gives the moon an in- creased pull, which, if acting towards the obstacle, exerts its full moment on the earth, but only for a fraction of a day. Secondly. — The existence of a retarding influence de- pends, as we have seen, on the place of high water being in what I have called the retarding quadrants, i.e. less than six hours in time later than the moon's meridian passage. If this condition is violated, the influence might D 34 THE TIDES. be accelerating. Suppose a continent whose coasts run N. and S. (as those of America may be roughly said to do) ; then if it is high water on the east coast less than six hours after the moon, the effect of the pull just mentioned is retarding ; if it is high water on the west coast more than six hours after the moon, the effect is to accelerate. In other cases the direct effect is nil. Now, owing to the great irregularity of distribution of land and water, theory will not help us in determining the times of high water ; but on consulting the Tables founded on observation we find, for example, the following re- sults : — In the open part of the Pacific Ocean high water is about 30° before quadratures ; farther from the equator at both sides it is at quadratures ; farther still it is 30° after quadratures. (Confining ourselves to the direct effect of the moon's action on the tidal prominences) The effect on — East coast of China, .... None. »> India, .... >» >> Australia, >> >> Africa, .... Retarding. >» S. America, n j> N. America, . IN'one. West coast of S. America (Peru), . Accelerating )) N. America (California, &c.), . >> >> Australia, . . . . >» (It should be observed, that for the present purpose we ought to take the time of high water where the depth of ACTUAL STATE OF EARTH. 35 the sea begins sensibly to diminish in approaching the coast, but this we are unable to do for want of data.) These instances are sufficient to show the difficulty (per- haps amounting to impossibility) of determining whether there is any preponderance of retardation at all. At all events, however, it is clear that the retardation, if any, must fall very far short of the maximum. It is to be remembered, further, that in the case con- sidered above of a globe uniformly covered with water, each section of the globe parallel to the equator has its own tidal current to encounter its own inertia, and hence the result in the case of the equatorial canal was applicable to the entire globe. But in the case of the earth it is not so, and this would still further diminish the retar- dation. On the whole, it would appear that no certain conclusion as to the amount of retardation of the earth's rotation by the tides can be drawn from theoretic considerations. I am not aware that any estimate of the value of the coefficient / has been published. The following has been kindly communicated to me by the late Professor Fitz Gerald :— Let ju = the coefficient of viscosity of water. Then x being the ordinate of the water, let (h being the depth in centimetres) = g =f^) 36 THE TIDES. With a depth of 3 miles, h = 483000 nearly .*. h^ = 233000 millions nearly. The value of /x at a temperature of 10° C. = 50°F. is = '013257, 15°C. = 59°F -011503, 20° C, = 68°F -010164. If we take the third of these values, we find 1 / nearly = Then sin 2S = tan 23 = 7000/ 3 "82 billions 1 546000 millions If we introduce this value into the formulse for the retardation given above, we arrive at the result that if the displacement is due to friction alone, the consequent retardation would be in fact nil. Supposing, liowever, that from whatever cause the displacement amounted to 15° (and we have seen on p. 34 that over a certain limited region it reaches 30°), the formula on p. 32 would give 1-15 seconds in 100,000 years. When we consider how far the actual state of things is from the hypothesis of a globe uniformly covered with water 3 miles deep, with the moon always in the equator, the fact that the retar- dation deduced from eclipses is about two-fifths of this is certainly curious. But we cannot attribute much im- portance to this coincidence. (With the value of / here given it would take about 12,000 years to produce the effect described in § 4 so as make this formula applicable) . THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. mr^ JUN 10 1943 16^e^!)'53J^^. ffB-t REC.CIR mz 3 '80 JUHIO'65^ REC D LD JUN? '65-8pl^ U|t_lii3l f:>r^^ . mu '67-2 f>- . •■»^.r» 9.1—1 nnm-7 'a a ys 09263