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CHAPTEP V EARTH TIDES W. D. LAMBERT U. S. Coast and Geodetic Survey The ticle-producin~ forces of the sure and moon are not confined to the surface of the earth but act throughout its mass. Since in nature there is Do absolutely rigid substance, there must be some yielding to these forces. This yielding or deformation of the mass of the earth- considered separately from the water on its surface is called an earth tide. For brevity let us speak of the "solid" earth, meaning thereb:- all the rest of the earth except the atmosphere and the hydrosphere, and not thereby denying the possibility that the substance composing the central portions of the earth may in many respects act like a very dense liquid under high pressure.* In addition to the deformation due directly to the action of the tide-producing forces in the " solid '' earth, there are obviously deformations transmitted downward from the surface and caused by the varying, tidal load of the ocean water. These are known as secondary earth tides and contribute greatly to the complication of the subject. lTor the sake of simplicity, therefore, we shall consider at first only the primary earth tides; that is, the deformations of the " solid " earth as they would be if the hydrosphere were absent; the atmosphere may be disregarded in this connection. The oceanic tides of nature are not of the simple kind so often shown in diagrams illustrating the nature of tidal force; that is, there is no smooth tidal bulge with its maximum elevations at the two antipodal points where the tide-producing body is ire the zenith and the nadir. This fact is expressed by saying that the oceanic tides are not of the equilibrium type.! One reason for this is that the oceans are too shallow; their shallowness makes their natural periods of oscillation rather ion, in con~parison with the- diurnal and semi-diurnal tidal forces. However, the depth would not have to be increased many fold before it would be * See The constitution of the interior of the earth, a forthcoming bulletin of this series. ~ Note the distinction between tides of the equilibrium type, and the equilibrium tide in the strict sense. (See p. 22.) The height h of the latter is given by h= V/g, where ~ is the potential of the tidal forces in question and g the ac- celeration of gravity. The equilibrium height of tide in this strict sense has previ- ously been used for comparison purposes (see p. 26) and will hereafter be so used in this chapter. 68
EAR TH TIDES 69 sufficient to give ocean tides of the equilibrium type. Now the earth tides have a certain analogy to tides in a very deep ocean, an ocean extending clear to the center of the earth. We may therefore expect the primary earth tides to be of' the equilibrium type, and in fact they are.* Some idea of the short period required for the earth as a whole to adjust it,sel:f to equilibrium with deformations of the tidal type may be obtained from the following: An impulse, such as an earthquake distur- bance, takes a little over 20 minutes to travel along, a diameter, or say 41 minutes for the round trip. A liquid sphere of the size and mean density of' the earth would have a natural period of 94 minutes for deformations of the tidal type. An incompressible elastic sphere of the size and mean density of' the earth and the rigidity of steel would have a natural period of 35 minutes. If we disea.rd the simplifying as- sumption of incompressibility the period is increased by six minutes only for an assumed rigidity equal to that of steel and by a little less for a Breather rigidity. Thus from any point of' view it is clear that the earth has time to adjust itself to the instantaneous field of force of the semi- diurnal and diurnal tides and a forttori' to the force-fields of the long- period tides. The primary earth tides are therefore of the equilibrium type. Friction might conceivably alter the phase, so that the maximum height of the tide would occur a little alter the moon's meridian pa.ssa¢,e,! but it will be seen in a later chapter that the lag in phase due to friction must be very small. The amount of' tidal rise and fall in the " solid " earth is a question of' great geophysical interest, for it is closely connected with the properties of the substance of' which the earth is composed. Of course we cannot stand outside the earth and measure its expansion and contraction with ~ giant pair of calipers, but we are obliged to infer the rise and fall indi- reGtly :t by a process presently to be described; for the moment let us think of the tidal rise and fall in the solid § earth as if it. could be measured directly. NVe can easily compute what the tidal rise and fall would be with refer- e:~:e to the center if the earth were entirely destitute of rigidity, that is, * The methods by which earth tides are observed will be treated latter. ~ This means that in space the tidal bulge is in advance of the moon, counting in the direction of the earth's rotation and of the moon's orbital motion. In time there is a lag in phase. ~ The rise and fall of the ocean tides is observable only relative to the adjacent land; that is, both the water surface and the land surface change their distances from the center of the earth and the observed change in the height of the water is the (a.l,,ebraic) difference of the changes in their distances from. the center of the earth. § From here on the quotation marks enclosing the word solid used in this con- nect.ion will be omitted, but may be supplied mentally by the reader.
JO FI GURE OF THE EAR TH if the only resistance due to change of form were that due to gravitation. This statement rests on two premises: 1) that the ellipticity of the earth due to rotation represents ~ deformation of' tidal type; 2) that any elastic stresses due to the rotation that may originally have been present have so decayed, owing to the lapse of time and to the fact that they often exceed the resisting, power of the earths substance, that the resulting, ellipticity due to rotation will be the same as if the earth's substance had no rigidity whatever; that is, as if it were liquid. The first premise may be proved mathematically; the second premise seems highly plausible from what we know of materials under stress and at such high t,empera- tures as we suppose to prevail within the earth, but it, cannot be asserted as an absolute, demonstrable truth. The mere rotation of the earth would impart an ellipticity equal to 1/~80 to the originally spherical equipo- tential surfaces surrounding a perfectly rigid sphere of mass equal to that of the earth and having, for the radius of the outer equipotential surface the mean radius of the earth. This corresponds to the equilibrium tide in the strict sense of the word. (See footnote, ante.) The actual ellipt,icit,v is 1/297. This increase is due to the self-attraction of the equatorial bulge produced by the rotation. The increase would be even larger if the earth were homogeneous, for the ellipticity would then be 1/232. The ratio of the a.ctualyieldin~, (tide) to the equilibrium tide in the strict sense is a pure number which we shall denote by h. Here for the fluid or plastic yielding of' a body like the earth 7~= 1l/397=~.9~. If, on the other hand, the earth opposes to the deformation elastic resis- tance as well as gravitational resistance, then the value of h will be much smaller. If' we treat the earth as of uniform density and incompressible and ascribe to it the modulus of rigidity A, the value of h will be 5 1 + ~ Papa (1) where a is the radius of tile earth, ,~ gravity at its surface and p its mean density.: For ,u = 8.0 x 1011 c.~.s. units, which represents about the rigidity of steel and corresponds to an estimate formerly much in Vogue, h = 0.783, for ,u = 20 x 10~t c.g.s. units 7~_ 0.380, which corresponds roughly to * Note that if ,u = 0, which represents the case of fluid yielding' h= 5/2, which is the value for a homogeneous earth, i. e., `~ 1/232 1/580
EAR TH TIDES ?'1 what is actual!: observed. The amplitude of the equilibrium lunar semi- diurnal tide, 11~`, at the equator is about 24 cm. (10 inches); the actual amount by which the surface of the earth rises above its mean position or falls below it is then 0.3x0 inches=a~ inches for ]~2- Since M2 is the; largest of the partial tides. (see p. 42) and since it attains its maximum range at the equator, it is evident that the actual rise and fall of the earth's surface due to the primary earth tide is generally a matter of a few inches. It is not intended to imply that it is beyond our mathematical power to dispense with the simplifying assumptions of incompressibility and of uniform density and elasticity; the assumptions underlying the calcu- lation can be brought into closer conformity with the presumed actual conditions prevailing within the earth but only at the expense of an enormous increase ire the mathematical difficulties of the problem, an increase attended by a smaller change in the values of h, and of similar related quantities, than might at first be expected. Herglotz s made computations that allowed for the decrease in density from center to surface; the assumptions of uniform modulus of rigidity and of incompressibility were retained. Schweyclar ''3~'4~'5 made a number of computations on the subject on various hypotheses of variable density, and rigidity; he also included the effect of oceanic tides in an ocean assumed to cover the entire earth, but retained throughout the assumption of incompressibility. This assumption greatly simplifies the calculation for a gravitating body as large as the earth, for with so large a com- pressible body the methods of the ordinary mathematical theory of elasticity are inadequate. Love i~ showed how to allow for the compressi- bility of a gravitating, sphere, but did not introduce the complications of variable density and rigidity. Hosl~ins 9 included compressibility and variable density and rigidity in his calculations. The numerical calcula- tions made in most of these papers except Schweydar's-were made under the influence of the idea that the earth is about as rigid as steel and the effects of higher rigidities were not as a rule considered. None of the papers previously cited develops the comparatively modern idea, based chiefly Ol:1 seismological evidence, of a liquid core with a radius somewhat greater than half the radius of the earth. This has been done by Je~reys,~° ii who has, however, made use of various simplifying, assump- tions. Thus there has been so far no complete mathematical solution that takes full account of modern ideas regarding the constitution of the earth. Historically speaking, the first determination of the earth titles. came by way of the long-period ocean tides. A tide in the water can be observed only with reference to the land. This observed quantity may be thought
~2 FIGURE OF THE EAR TH ., of as the difference between the rise and fall of the water with respect to the center of the earth and the rise and fall of the land with respect to the same point. Tides of long, period (a fortnight or a month) were used because it was assumed that they could be treated theoretically, whereas the diurnal or semi-diurnal tides were well known to be beyond any known mathematical treatment at all adequate to the purpose in hand. The theoretical treatment actually used for the lon~,-period tides was fairly simple. If TY represents the potential of the lon<,-period tidal forces, the height of the "equilibrium tide" (referred to the center) in a non-attra.ctin~ ocean covering an unyielding globe is simply W/q, ~ v ~ v ~ ~ v , v where g Is the acceleration of gravity. But the earth does yield and its deformation gives rise to a ~,ra.vitational potential of the same type as W; this additional potential may therefore be written 1cW where k is ~ pure number like 1-',; its value, like the value of h, depends on the distri- bution of density within the earth and for the case of homogeneity we have k=3/a, Tether the yielding be elastic or plastic in character. For the actual earth we seem to have more nearly 7~=:h.* The " equilibrium height" of the oceanic tide (referred to the center) instead of bein, 1V/g is (~+k) 1~/g. The tide in the earth (still referred to the center) is, as we halve seen, h.W/g, so that the observed differential tide of the sea referred to the land is (1+7c-h) W/g. We know W/g from theory, we determine the whole expression (~+k-h) W/g from observation and thence we deduce 1+k-h. A determination from the long-period tides was published in Thomson and Tait's Natural Philosophy and in revised form in Sir George Darwin's Collected Scientific Papers.i The observa- tions of lon~-period tides at all available ports gave for 1+7~-h a value a little greater than two-thirds. The Indian ports, where long series of tidal observations were available but which afforded only a small range of latitude gave a value nearer to unity.! The value 1 + 1~-7~ = 2/3 cor- responds under the simpli.lyin~, assumptions on which formula ( 1 ), page 70, was derived to a modulus of rigidity equal to 7.3 x 10~i, or a trifle less than that of steel. There are various corrections that might be made or objections that might be offered to this method. The sel:~-attraction of the water is not provided for, neither is the effect of the continents on the amplitude and phase of the tide. Either correction is a simple matter by itself. The * In this case there is no theoretical reason why the ratio of k to h should be absolutely identical for elastic and for plastic yielding, but apparently the ratio is not very different in the two cases. ~ A value of 1 + k-It equal to unity corresponds to an absolutely unyielding earth, a value equal to zero to an absolutely plastic earth. The nearer the value is to unity the greater the modulus of rigidity.
EAR TH TIDES ~3 self-attraction of the tidal water assumed to cover the entire globe may be allowed for by means of a simple theorem of the theory of spherical harmonics; a further mathematical advance has been made by Street '0 in allowing for the pressure of the varying load of tidal water on the surface of the globe.* If we omit the self-attraction of the water, the presence of the conti- nents may be allowed for by the "corrected equilibrium theory" of Darwin and Turner. Neither of the corrections just mentioned is large, but unfortunately their joint effect cannot be found by applying the principle of the superposition of small corrections. The fundamental difficulty, however, lies in the question of whether the oceanic tides are really of the equilibrium type. It has been letdown since the time of Laplace that tides of quite another type (see p. 26) are possible Ott a rotating globe covered with a frictionless liquid. It had usually been assumed without much examination that in the case of nature friction and the presence of the continents would prevent the existence of such a type of tide and would make the actual tide approximate rather closely to the equilibrium type, the friction being sufficient to prevent existence of the non-equilibrium tide just mentioned, without being enough to introduce any marked lag in the phase. These assumptions, however, are no longer so generally accented as they were. and so the method based - - 1 ~ , on them tor ctetermlulng the earth tides is no longer considered a satisfactory. It should be said, however, that the results obtained are ire general agreement, in spite of these theoretical difficulties, with those obtained by the method next to be described. This method consists in determining the earth tides by the changes they produce in the direction of the vertical. These changes may be re- ferred to the direction of the earth's axis or to the ground old which the instrument for detecting them rests. The latter method is more common. If the changes were larger, the: might be measured by the deflections of an ordinary plumb line or pendu- lum, but since they are of the order of only 0'.'0l, and since an ordinary plumb line or pendulum is disturbed by earth tremors and by the various thermal effects of the sun's heat, special precautions must be taken. The so-called " horizontal pendulum " is more sensitive than the orcli- nary penclulum. A familiar analogue to the horizontal pendulum is an ordinary gate hinged to an unsteady post. Any slight wobble in the gate post causes the gate to swing through a large arc. The Darwin brothers, * Street's published numerical calculations are made with so low a modulus of rigidity as to have little application to the case of nature. Moreover, his defini- tion of equilibrium tide is not the one here adopted, which appears to be the pre- srailing one. This difference in definition is likely to cause confusion.
~4 FI GURE OF THE EAR TH George and I-Iorace, conducted a long series of experiments with one form of the horizontal pendulum. Their memoirs 3' 4 may be consulted for references to earlier experiments. Di:lliculties with the constructio and installation of the apparatus prevented them from obtaining any very satisfactory results. The first results that could fairly be called satisfactory were obtained by von Rebeur-Paschwitz.~4 He was followed by other experimenters, Heeker,` Orloff',~9 and Schweydar,~ who int.ro- cluced various improvements. In general, the rigidity of the earth inferred from their results was about that of steel. One peculiarity that showed to a. ~rea.t.er or less degree in all their experiments was. that the actual ellipse cleseribed by the plumb line was not similar to the theo- retical ellipse for the plumb line on a rigid earth, as would naturally lie expected; the actual ellipse was found to be not only smaller than the theoretical ellipse, as was to be expected, but flatter also. This was interpreted as meaning, that the earth was stiffer in one cl.irectio~n than in another, and various explanations were attempted, all of them more ingenious than satisfying,; Loved showed that the supposed causes of this peculiarity were quantitatively insufficient. The most reasonable explanation is that it is the result of' the secondary effect of the oceanic tides; these latter of course are not in phase with the earth tides nor do they vary with latitude and longitude according, to any simple law. NVhat is known of the tides in the open ocean makes the explanation seem qualitatively reasonable and the observed effect seems of the right order of magnitude but no precise quantitative check has been made. An entirely different kind of apparatus for measuring, the same quantity was devised by Michelson and Gale. Any small body of water like a Lyle tends to set itself perpendicular to the plumb line at its middle point; as the plumb line swings to one side or the other the water at one side of the Lyle rises; and at the opposite side falls. Tidal oscillations of this sort have actually been found in Lal~e Superior. To the eye they are masked by chan<,es of level resulting -from inflow of water, winds and variations in level due to non-uniform barometric pressure over the lolls. Harmonic analysis of observations extending over several years was used to reveal these tidal oscillations. Michelson and Gale, is got rid of' the various disturbing factors by using horizontal pipes buried in the ,rouncl. Two pipes, each five hundred -feet long,, one in the meridian ancl one in the prime vertical, were installed in the grounds o:l' the Yerkes Observatory at NVilliams Bay, Wisconsin. The rise ancl fall of the water at one end of the pipe due to a change in the direction of the plumb line is evidently equal to the horizontal motion of a plumb line with a length of 2~0 feet, hall' that of the pipe; if we combine the observations at the two ends we have a fluctuation equal to the movement of a plumb
~ ~s bn~ h~e hundreJ feet lo~g. E~cn vitb this len~T1 tbe line~r displ~ce- ments ~re of conFsc ver~ minute; heing oT tbe order of ~ ver~ Te~ ten- thous~ndths of ~n incb. Tbe rise ~] f~M of tbe v~ier in tbe cnds of the pipes ~s re~] ~t f~st ~ith ~ microsco~c ~nd l~tcr h~ me~ns of ~ speci~ll~ devised inte~ferometer ~pp~r~tus. Ibe cur~es derive] from thc otscr~c] re~dings ~rc surprisi~gly smootb ~] corrcspon] closel~ ~tcr ~plyi~g suit~tle ~edurtion I~ctor to thc tbeorctic~1 cur~cs Ior ~ rigid c~tb. In ~ l~ter form of tbe ~r~tus inst~lled ~t F~s~icn~; C~liforni~: it h~s been IoUnd possitl~ by incre~sing thc scnsitivencss of the intederomcter ~rtus to sborten tbo pTpes use] ~n] st]1 ret~in tbe ~ccur~cy of tbe . . . . OIlglO~I 8.ppR.T~lUS. In ~e e~TEcr rcports on thc work it v~s st~te] tb~t tbe ~eJUction f~ctor Ior thc e~st-vcst iTspl~cc~cnt di~erc] Irom tb~t Io~ tbe nQTth- soUtb displ~ccmcnt; in tbis respcct thc rcsUlts ~ppc~rc] to rcsemBlc tbosc ott~inc] in ~UTOpC vith ~e boTizoGt~1 pCD0UlU~. ~Ut it RppC~TC] l~tcr tb~t ~n cr~or t~] tccn m~ic i~ thc cslcUl~tion oI thc thcorctic~1 dcHection ~n] tb~t tbe reJUction I~cto1 v~s pr~ctic~ll~ ~c s~e IOT totb directioDs. Ibc tiJe w~s ~n~l~sc] into its 1Un~r sc~i-dTUr~l; sol~r scmi-diUrn~l, n~r diUrn~1 ~] 1Un~r Ior~i~Uy bic Ior totb dircctions; thc l~gs i~ pb~sc ~crc only ~ Ic~ Jegrees ~n] thc v~rio~s rciUction I~ctors vere ^# e-. Ibe mo~ulUs oI rigidity Jeduce] Irom the preLmin~ry results v~s 8.0 x lOli c. g. s. units; ~ little gre~ter th~n tb~t oI steel ~d tbe coe~rient of viscosit~ TO.9 x 101° c. g. s. u~its; ~lso ~bo~t th~t of steel. Tbis m~Les the time of rel~x~tion (see Cb~pter VT) utout ~ dsy ~nd ~ b~lF ~o corresponding resUlts ~ere given ~hen the deOnitive reJUctions vere publisbeJ; ~n] tbere ~re re~so~s. Ior helieving (see Ch~pte~ YI on Ii]~1 Friction) tb~t tbe time oI rel~x~tion is mucb too sm~lt tb~t i~ tb~t tbe soli] p~rt oT tbe e~rtb ss ~ ~bole possesses murb more ne~ll~ perIect el~stici~ Ior Iorces oI tbe or~in~ry tid~1 periods tb~n tbe coc~cient of viscosity ~tove given woUl] i~]ic~te. Ibe l~gs in pb~se ~re prot~hl~ m~inly Jue to tbe seconJ~ ehects oI tbe oce~nic tiJes, ~bicb ~re Ielt to iist~nces Irom tbe co~st tb~t ~t hrst si~t might seem sUrprising. Ibe IortnigbJ~ tide, ~bicb; bovever; v~s too sm~ll to Be vcry ~ccur~tely determi~eJ; b~] ~ neg~tive l~g; interprete] solely ~s tbe resUlt of Iriction tbis ~oUl] be impo~sible. Ibe trUe expl~n~tion is no donh-~p~rt Iro~ errors of otserv~tion-tbe secon]~ry e~ects of tbe oce~nic tides. Tbe gre~test presc~t nee] in e~rth-tiJe Jetermin~Iions is not ~ more ~ccUr~te metbod of ~e~sUrin~ th~m th~n is obt~ine] ~itb ~icbelson ~n] O~le~ ~ter-l~veI ~n) interIerometer Jpp~l~tUS; tUt ~ hetier kno~leige of the oce~n tiJea ~n] ~ more ~dequ~te mutbPm~tic~1 tbeoly for tbe eEmin~tion of tbeir second~> e~ect~ Only ~ben ~e b~ve tbese, sb~G ve Be ~hle to Jete~mine ~ccU~tely tbe el~stic ~ield1ng of tbe soli] p~lt of tbe os~tb.
76 FI GURE OF THE EAR TH Of course on the coast, and especially on coasts where oceanic tides are large over extensive areas, these secondary effects are large; they snag considerably exceed the primary effects of the earth tides. At Pasa- clena in one component the observed earth-tide is two and one-]:~al£ times the theoretical value; at a point in Korea about four times. The opinion has sometimes been expressed that local geological con- cl.itions may have considerable effect on earth tides; among such conditions are supposed to be the presence of extensive beds of rock near the surface having a modulus of rigidity markedly different from the average and the nearness of faults or volcano vents. Observations of earth tides have not been sufficient to test this idea in practice. It seems, however, to the writer that it would be difficult to separate these effects front the secor~clary effects of the oceanic tides and that their magnitude is probably exag- gerated. Earth tides are made from the center of the earth outward. It is generally supposed that to within perhaps 100 kilometers of the surface the physical properties of the earth-substance are the same at any given level, whether the point in question be under an ocean or under a continent. Now each shell of the earth beginning with a small central core may be considered as receiving from below a certain amount of swelling or sinking due to the tides below it and as making its own contribution to the swelling, or sinking and passing on the net result to the shell next above it.* The earth tide increases from the center outward,! and when a level of 100 kilometers below the surface is reached the earth tide has attained nearly the same value as at the surface. The crust above the 100-kilometer level does little more than ride upon the earth below it. For these reasons it seems to the writer that conditions at tl~.e surface can have little effect on earth tides.+ From certain points of view the best determination of 7c and Or is that of Schweydar.''4 This is not because the horizontal-pendulum apparatus used by Schweydar gives better results than the apparatus of Michelson and Gale but because by a combination of ingenuity and favorable cir- cumsta.nces the secondary effects of the ocean tides appear to have been snore nearly eliminated. Schwevclar obtains .. .. 1+7v-16=0.841 § ~ It is equally true of course that the shells above affect those below as well as those above. ~ The law of increase depends, on the constitution of the earth, both as to density and to elastic constants. :~: Perhaps discontinuitics in structure, such as those due to a, fault or ~ volcano merit may introduce peculiarities into the observed earth tides. The writer has seen no definite mechanism suggested for an effect of this sort. § Schweydar reverses the significance of these symbols. In this article the original notation of Love and Larmor is adhered to.
EAR TH TIDES The value of 1 + 7~-7~ obtained by Michelson and Gale is 0.690, which corresponds to a more yielding, earth than Schwe,vdar's Azure. The writer has made some rough preliminary calculations that tend to show that an allowance for the secondary effects of the oceanic tides would tend to raise Michelson and Gale's value to a figure approximating, Schweydar's. On account of mathematical difficulties and the uncertainty of the data., namely, the ra.~¢,es and lunitida.l intervals in midocean, the calculations were not pushed to a definite numerical conclusion. The value of 1+7e-h is all that observations of earth tides alone will give. To obtain 7~ and h separate!: or to draw inferences about the rigidity of the earth, observations of some other nature.must be used or hypotheses must be made. The value of 7e may be deduced from tile length of the Eulerian period in the variation of latitude. But even Title the values of 7~ and A, hypotheses must be made in order to deduce the elastic constants of the earth.* The modulus of rigidity of S.6xI0'i deduced from Michelson ancl Gale's work is based on the hypothesis of an earth composed of incompressible material and uniform as to density and elasticity. Schweydar~s values are based on the following, laws for the variation of density and rigidity: Density Here P=P0~l-3(t) pO = density: at center of eat th = l O.1, ,8 = 0.764, a pure number, r = distance from center of earth a=radius of earth. The modulus of rigidity ,~ is given by / r a= 30.8 x 10~ <1 - 0.90 .> c.~,.s. units (2) (3) 30.S x 103i is obviously the central clensity. The substance is assumed to lie incompressible, an assumption that greatly simplifies calculations con- cerning ~ large gravitating, mass lilts the earth. These assumptions give 7~ = 0.270 7~=0.429 (zl-) * The hypothesis. should reproduce the assumed values of k and h, but rather dissimilar hypotheses will give the same values of k and h, especially if a margin of observational error be allowed.
78 FI CURE OF THE EAR TH The values of 7~ give an Eulerian period of the latitude variation nearly agreeing, with the observed one. If uniform rigidity is assumed throughout, Schweydar finds a rigidity equal to 17.6 x 10~i c.¢,.s. units, more than twice that of steel. Schwey- dar's assumptions do not correspond with recent ideas about the internal constitution of the earth. According to these the nucleus with a radius equal to more than half the radius of the earth is fluid, or nearly so. Outside the nucleus lies a shell the rigidity of which increases with the depth below the outer surface. The transition in physical properties between the shell and the nucleus is rather abrupt. Schweyclar's formula makes the rigidity at the center a maximum instead of nearly zero, as it would be according to the hypothesis of a fluic1 nucleus. Moreover, accord- ing to seismological evidence the rigidity in the outer shell is rather higher than Schweyclar's formula makes it. The quantities k and 1~ are merely measures in some sort of the average rigidity. As has been said, no computation on earth tides has (so far as the writer knows) been published that takes adequate account of recent ideas regarding, the constitution of the earth. The nearest thin<, to such a computation is the work of Jeffreys is already referred to. This tends to show that there is no serious conflict between these modern ideas and the value of 7~ and h determined from earth tides and the variation of latitude. Both the horizontal pendulum and the ~ichelson-Gale apparatus measure the deflection of the vertical with reference to the crust of the earth. The change in the direction of the vertical might be conceivabl\ referred to the axis of the earth, and this is, in fact, exactly what is clone when observations for latitude or longitude are made. The effect of any change in the direction of the instrument due to a tide in the crust of the earth is removed when the spirit levels of the transit instrument or zenith telescope are read and the proper correction appliecl. However, the smallest angular change that can be detected by an observation for latitude and longitude is much larger than the corresponding quantity in an observation with the horizontal pendulum or the Michelson-Gale apparatus, so that trustworthy conclusions can be derived only from long series of observations. The observations of the International Latitude Service have been analyzed by Pr%byllok. ~ The yielclin~ of the earth comes out about the same as from observations especially made to cietect earth tides. The quantity determined, however, is not 1+7~-A, as for observa- tions with the horizontal pendulum or the )Tichelson-Gale apparatus, but i+7~-1.
EARTH TIDES ~9 Here k is the same quantity as before, but I is a quantity dependent on the linear elastic displacement along the meridian of the place of observa- tion. Przbyllok apparently overlooks the fact that different quantities are obtained from the ordinary earth-tide observations and from latitude observations. ~ + k-h is normally less than unity, whereas ~ + k-I is greater than unity at least in the case of a homogeneous earth. Ilere again the secondary effects of the disturbing tides come in and there is the same difficulty in making the proper correction. The photographic observations for the variation of latitude made at Gaithersburg, Maryland, simultaneously with the regular observations of the International Latitude Service have been discussed by Stetson.~5 He finds a lunar effect, that is, an apparent earth tide, much larger than that riven by the visual observations at the same place in the hands of ~ it, ~ ~ ~ 1 ~ · ~ ~ ~ 1 ~1 _ 1~ 1 L:~ Lid ~1 l~rzbyllok. The earth tide iounct by Stetson Is several times that pre- clicted by theory. No adequate explanation has so far been given. REFERENCES 1. Darwin, G. H. Attempted evaluation of the rigidity of the Earth based on tides of long period. Thomson and Tait's Natural Philosophy, 2nd ea., sect. 848, or Scientific papers, 1: 340. Cambridge, 1907. 2. , and Turner, H. H. On the correction to the equilibrium theory of tides for the continents. Proc. Roy. Soc., 40:303 (1886); or Scientific papers, 1: 328. Cambridge, 1907. 3. , Horace, and others (report of a committee). On an instrument for detecting and measuring small changes in the force of gravity. Brit. Assoc. Rept. for 1881, p. 93; or Scientific papers, 1: 389. Cambridge, 1907. , and others (Second Committee Report). The lunar disturbance of gravity; variation in the vertical due to elasticity of the earth's surface. Brit. Assoc. Rept. for 1882, p. 95; or Scientific papers, 1: 430. Cambridge, 1907. 5. Ehlert, R. Die Horizontalpendelbeobachtungen im Meridian zu Strassburg i. e., Gerlands Beitr. Geophysik., III:(1898), p. 131. 6. Haid, M. Gezeiten und Starrheitskoeffizient n der festen Erde. Verhandl. der 17ten Allgememeinen Konferenz der Internationalen Erdmessung. I Tell. Hamburg, 1912. 7. Decker, O., and Meissner, O. Beobachtungen an Horizontalpendeln uber die Deformation des Erdl~orpers. II Heft. Veroffentl. d. Konigl. Preuszischcn Geodatischen Inst. not. 49. Potsdam, 1911. 8. Herglotz, G. i:;ber die Elastizitat der Erde bei Berucksichtigung ihrer variablen Dichte. Z. Math. Physik., 52: 275 (1905). 9. Hoskins, L. M. The strain of gravitating sphere of variable density and elas- ticity. Trans. Am. Math. Soc., 21: 1 (1920). 10. Jeffreys, Harold. The rigidity of the Earth's central core. Monthly Notices Roy. Astron. Soc. Geophysical Supplement, 1: 371 (1926). 11. . The Earth, its origin, history and physical constitution. 2nd ed. Cambridge (Eng.), 1929. ~.
80 FIGURE OF THE EARTI-I 12. Lambert, W. D. Les marees de l'ecorce terrestre et leur relations avec let autres branches de la geophysique. Bull. geodesique? No. 5: 45 (1925). 13. . Earth tides. Travaux de la Section de geodesic. Rapports generaux etablis a ['occasion de la 3'l'e assembler generate a Prague. Paris, 1929. 14. Larmor, J. The relation of the Earth's free precessional notation to its resis- tance against tidal deformation. Proc. Roy. Soc., London' 82 A: 89 (1909). 15. Love, A. E. H. The yielding of the Earth to disturbing forces. Proc. Roy. Soc. London, 82 A: 73 (1909) . 16. . Some problems of geodynamics. (Adams Prize Essay), Cambridge, 1911. 17. Michelson, A. A. Preliminary results of measurements of the rigidity of the Earth. Astrophys. J., 39: 105 (1914). 18. , and Gale, H. G. The rigidity of the Earth. Astrophys. J., 50: 330 (1919) . 19. Orloff, A. Beobachtungen uber die Deformation des Erdkorpers. Astro- nomische Nachrichten, 186: 80 (1910). 20. Przbyllok, E. Uber die M~-Tide der Lotbewegung. Astronomische Nachrich- ten, 218: 85 (1923). 21. von Rebeur-Paschwitz, E. Horizontalpendel-Beobachtungen auf der haiser- lichen Universitats-Sternwarte zu Strassburg, 1892-1894. Gerlands Beitr. Geophysik, 2: 2;11 (1895) . 22. Schweydar, Mr. Ein Beitrag Or Bestimmung des Starrheitsl;oeffizienten der Erde. Gerlands Beitr. Geophysil;? 9: 41 (1908). 23. . Untersuchungen uber die Gezeiten der festen Erde. Veroffentlichung des Koniglich. Pireuszischen Geodatischen Institutes, No. 54, Potsdam, 1912. 24. . Lotschwankung und Deformation der Erde durch Flutlfrafte. Zen tralbureau der International Erdmessung. No. 38. Berlin, 1921. 25. Stetson, H. T. The variation of latitude and the moon's position. Nature, 123: 127 (1929); or Science, 19: 17 (1929). 26. Street, R. O. Oceanic tides as modified by a yielding Earth. Monthly Notices, Roy. Astron. Soc, 1: 292 (1925) .
