mi ftsjjijji GIFT Or Astronomical Society of the Pacific A* ARY ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC SPHEROIDS, AND O.V THE OCEAN TIDES UPON A YIELDING NUCLEUS. BY G. H. DARWIN, M.A., FELLOW OF TRINITY COLLEGE, CAMBRIDGE. From the PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY. PART I. 1870. LONDON : HAIMtlnON AM> SONS, PRINTERS IN ORDINARY 10 HEIi MAJKSTY, ST. MARTIN'S LANE. ASTRONOMY tcv * AsUon. Oopt. PHILOSOPHICAL TRANSACTIONS. I. On the Bodily Tides of Viscous and Semi-elastic Spheroids, and on the Ocean Tides upon a Yielding Nucleus. By G. H. DARWIN, M.A., Fellow of Trinity College, Cambridge. Communicated by J. W. L. GLAISHER, M.A., F.R.S. Received May 14, Read May 23, 1878. IN a well-known investigation Sir WILLIAM THOMSON has discussed the problem of the bodily tides of a homogeneous elastic sphere, and has drawn therefrom very important conclusions as to the great rigidity of the earth. * Now it appears improbable that the earth should be perfectly elastic ; for the con- tortions of geological strata show that the matter constituting the earth is somewhat plastic, at least near the surface. We know also that even the most refractory metals can be made to flow under the action of sufficiently great forces. Although Sir W. THOMSON'S investigation has gone far to overthrow the old idea of a semi-fluid interior to the earth, yet geologists are so strongly impressed by the fact that enormous masses of rock are being, and have been, poured out of volcanic vents in the earth's surface, that the belief is not yet extinct that we live on a thin shell over a sea of molten lava. Under these circumstances it appears to be of interest to inves- tigate the consequences which would arise from the supposition that the matter constituting the earth is of a viscous or imperfectly elastic nature ; for if the interior is constituted in this way, then the solid crust, unless very thick, cannot possess rigidity enough to repress the tidal surgings, and these hypotheses must give results fairly conformable to the reality. The hypothesis of imperfect elasticity will be prin- * Sir WILLIAM states th&C M. LAME had treated the subject at an earlier date, but in an entirely different manner. I am not aware, however, that M. LAME had fully discussed the subject in its physical aspect. MDCCCLXX1X. B 06318 MR. DARWIN ON THE BODILY TIDES OP VISCOUS AND SEMI-ELASTIC cipally interesting as showing how far Sir W. THOMSON'S results are modified by the supposition that the elasticity breaks down under continued stress. In tlii.s paper, then, I follow out these hypotheses, and it will be seen that the results are fully as hostile to the idea of any great mobility of the interior of the 'sts is that of Sir W. THOMSON. The ftnly. terrestrial evidence of the existence of a bodily tide in the earth would be "tlia't'the 'ocean tides would be less in height than is indicated by theory. The subject of this paper is therefore intimately connected with the theory of the ocean tides. In the first part the equilibrium tide-theory is applied to estimate the reduction and alteration of phase of ocean tides as due to bodily tides, but that theory is acknowledged on all hands to be quite fallacious in its explanation of tides of short period. In the* second part of this paper, therefore, I have considered the dynamical theory of tides in an equatorial canal running round a tidally-distorted nucleus, and the results are almost the same as those given by the equilibrium theory. The first two sections of the paper are occupied with the adaptation of Sir W. THOMSON'S work* to the present hypotheses ; as, of course, it was impossible to repro- duce the whole of his argument, I fear that the investigation will only be intelligible to those who are either already acquainted with that work, or who are willing to accept my quotations therefrom as established. As some readers may like to know the results of this inquiry without going into the mathematics by which they are established, I have given in Part III. a summary of the whole, and have as far as possible relegated to that part of the paper the comments and conclusions to be drawn. I have tried, however, to give so much explanation in the body of the paper as will make it clear whither the argument is tending. The case of pure viscosity is considered first, because the analysis is somewhat simpler, and because the results will afterwards admit of an easy extension to the case of elastico-viscosity. I. THE BODILY TIDES OF VISCOUS AND ELASTICO-VISCOUS SPHEROIDS. 1. Analogy between the flow of a viscous body and the strain of an elastic one. The general equations of flow of a* > viscous fluid, when the Affects of inertia are neylccted, are * His paper will be found in Phil. Trans., 1863, p. 573, and 733-737 and 834r-846 of THOMSON ami TAIT'H 'Natural Philosophy,' edit, of 1867. SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 3 (1) where x, y, z are the rectangular coordinates of a point of the fluid ; a, ft, y are the component velocities parallel to the axes ; p is the mean of the three pressures across planes perpendicular to the three axes respectively ; X, Y, Z are the component forces acting on the fluid, estimated per unit volume ; v is the coefficient of viscosity ; and d 2 d~ 'd 3 V 2 is the Laplacian operation TI Besides these we have the equation of continuity ~~r+~^ I 7^ Also if P, Q, R, S, T, U are the normal and tangential stresses estimated in the usual way across three planes perpendicular of the axes (2) } efa/J Now in an elastic solid, if a, /3, j be the displacements, mfyi be the coefficient of dilatation, and n that of rigidity, and if 8= +-{- ; the equations of equilibrium are -- ax f dy (3)* Also ?-'<*) dz and S, T, U have the same forms as in (2), with n written instead of v. 1 n Therefore if we put p=-(P+Q+R), we have p= (TO j)8, so that (8) may be written * THOMSON and TAIT'S ' Nat, Phil.,' 698, eq. (7) and (8). 2 B MR. DAKWIX ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC n '"-3 Also i , Q= &c., 11= are auxiliary functions defined by . . . . (6) In the case considered by Sir W. THOMSON of an elastic sphere deformed by bodily stress and subject to no surface action, we have to substitute in (5) and (6) only those surface actions which are equal and opposite to the surface forces corresponding to the first part of the solution ;* but in the case which we now wish to consider, we must add to these latter the components of the normal traction gw^. <&; corresponding to A;_ I} B,-_i, C;_! and Vj, *; + ., corresponding to A i+1 , B,- +1 , C,- +1 . Then by the formulae (6) 2t + l Thus Then by (5) we form a corresponding to A;^, 'Bi^, C,-_ 1; and also to A; +1 , B,- +1 , C,- +1 , and add them together. The final result is that a normal traction S,- gives, ] J (7) / ~j _j and symmetrical expressions for ft' and y '. a, ft', y are here wiitten for a, ft, y to show that this is only a partial solution, and v is written for n to show that it corresponds to the viscous problem. If we now put S,= gwcri, we get the state of flow of the fluid due to the transmitted pressure of the deficiencies and excesses of matter below and above the true spherical surface. This constitutes the solution as far as it depends on (iii). There remain the parts dependent on (i) and (ii), which may for the present be classified together ; arid for this part Sir W. THOMSON'S solution is directly applicable. The state of internal strain of an elastic sphere, subject to no surface action, but under the influence of a bodily force of which the potential is W,-, may be at once adapted to give the state of flow of a viscous sphere under like conditions. The solution is MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC (i- _ _ ~ ,\ the longitude. Let p. &, v be the velocities radially, and along and perpendicular to the meridian respectively. Then the expressions for p, rs, v will be precisely the same as those for a, /3, y in (8), save that for we must put - ; Cl'X (JLT e d d . d d for , . ^ ; and for , ill] rsin vd

and there is no further state of flow, for the fluid has assumed the form which it would have done if it had not been viscous. This result is of course in accordance with the equilibrium theory of tides. If S, be zero, the equation shows how the inequalities on the surface of a viscous globe would gradually subside under the influence of simple gravity. We see how much more slowly the change takes place if i be large ; that is to say, inequalities of small extent die out much more slowly than wide-spread inequalities. Is it not possible that this solution may throw some light on the laws of geological subsidence and upheaval ? 4. Digression on the adjustments of the earth to a form of equilibrium. In a former paper I had occasion to refer to some points touching the precession of a viscous spheroid, and to consider its rate of adjustment to a new form of equilibrium, * I write " exp." for " e to the power of." SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 11 when its axis of rotation had come to depart from its axis of symmetry.* I propose then to discuss the subject shortly, and to establish the law which was there assumed. Suppose that the earth is rotating with an angular velocity w about the axis of z, but that at the instant at which we commence our consideration the axis of symmetry is inclined to the axis of z at an angle a in the plane of xij. and that at that instant the equation to the free surface is r=a< l+~~r(q [cos a cos 0-\- sin a sin cos <]-) > where in is the ratio of centrifugal force at the equator to pure gravity, and therefore tara equal to . 9 Then putting i=2 in (12), and dropping the suffixes of S, s, cr, s= ~r~(^ [ J). 4: \0 ] We may conceive the earth to be at rest, if we apply a potential i A o n "- o/i 5=-] 2 )exp(Kt) > * L\ 3 / V / J Now a;p( K<)] cos 2 ^+exp( Kt)(cos a cos = cos 2 [1 sin 2 a exp( /<)]+ sin 2 a sin 2 cos 2 < exp(-Kt) sin a cos a sin 6 cos cos < exp(-Kt). Therefore the Cartesian equation to the spheroid at the time t is, __ a 3_5mr z g/ 1 _ gin2 a /^^^a gm 2 a K B p (_^)+2 az sin a cos aexp(- K ()} or * " On the Influence of Geological Changes on the Earth's Axis of Rotation," Phil. Trans., Vol. 167, Part I., sec. 5. C 2 12 MB. DARWIN OX THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC sm 3 a p(-*)}+y*+* l -[l+Y(l -sin 2 a e;cp(-irf))} + 5m sin a cos a xz e.rp( K() =a 2 Let a' be the inclination of the principal axis at this time to the axis of z, then , sin '2a .'/if *<) tan 2a = r- T-; 7 - 7, 12 sin- a <:<])( Kt) If a be small, as it was in the case I considered in my former paper, then a.'=aexp( xt) and -= KO.'. Therefore the velocity of approach of the principal axis to the axis of rotation varies as the angle between them, which is the law assumed. Also *=~ , so that K (the v of my former paper) varies inversely as the coefficient 1 U of viscosity, as was also assumed. 5. Bodily tides in a viscous earth* The only case of interest in which S, of equation (11) is a function of the time, is where it is a surface harmonic of the second order, and is periodic in time ; for this will give the solution of the tidal problem. Since, moreover, we are only interested in the case where the motion has attained a permanently periodic character, the exponential terms in the solution of (11) may be set aside, Let S 2 =S cos (vt+ij), and in accordance with THOMSON'S notation.t let j* =g, and =l ; and therefore - =-. . 19u r Then putting i=2 in (11), and omitting the suffix of cr for brevity, we have (14) It is evident that a must be of the form A cos (vt+'B), and therefore A{ vt sin (v+B) + g cos (vt+~B) } =aS cos (vt+y) 1 In certain cases the forces do not form a rigorously equilibrating system, but there is a very small couple tending to turn the earth. The effects of this unbalanced couple, which varies as the square of g (J> wil1 b considered in a succeeding paper on tho " Precession of a Viscous Spheroid." (Read before tho Royal Society, December 19th, 1878.) t 'Nat. Phil.', 840, eq. (27). SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 13 CT or if we put tan e = , ; sec e cos (y-|-B+e)=:aS cos (vt-\-ij), Hence A=-S cos e, and 6=17 e. Therefore the solution of (14) is, cr=-S cose cos (vt-\-r) e) ........ (15) n Where tan e= = 2gaw But if the globe were a perfect fluid, and if the equilibrium theory of tides were true, we should have by (13), relatively to the earth, and let the moon's terrestrial longitude, measured from the plane of xz, at the time t be (at. Then at the time t, the gravitation potential of the tide generating force, estimated per unit volume of the earth's mass is 3m which is equal to V^n Q cos 2 0)+T ^tcr'fsm 2 6 cos 2< cos 2a>t+ sin 2 6 sin 2 sin 4 v \O / 4 Cr The first term of this expression is independent of the time, and therefore produces an effect on the viscous earth, which will have died out when the motion has become steady ; its only effect is slightly to increase the ellipticity of the earth's surface. SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 15 The two latter terms give rise to two tides, in one of which (according to previous notation) S cos(^+7?)=T ~ t sin 2 6 cos 2< cos 2a>t, ~t G and in the second of which 3 in I TT diurnal tide is - , where tan = = . Also the height of the tide is less than the S cos 4 if \ z Now e, which depends on the frequency of the tide generating potential, will clearly be the same for both these tides ; and therefore they will each be equal to the corres- ponding tides of a fluid spheroid, reduced by the same amount and subject to the same retardation. They may therefore be recompounded into a single tide ; and since v will here be equal to 2, it follows that the retardation of the bodily semi- 2o>r 19u< = -- , 2Y\ By the same reasoning as in 1, we may put, S=y JL^ anc ^ ^ e ec L ua ti ns become dt t m Then supposing the substance to be incompressible, so that m is infinitely large com- pared to n, and therefore m-i-m \n is unity, the equations become and two similar equations. * Proc. Lond. Math. Soc., Dec. 14, 1876, p. 107-9. It seems to me that the hypothesis ought to repre- sent the elastico-viscosity of ice very closely. MDCCCLXXIX. D 18 MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC Now these equations have exactly the same form as those for the motion of a viscous fluid, save that the coefficient of viscosity v is replaced by n (7+1;) We may there- fore at once pass to the differential equation (11) which gives the form of the surface of the spheroid at any time. Substituting, therefore, in (11) for -, - (7+7;), we get V 71 ^* (tt I lffj i gwa ' t " 2 n ++ 7it-++ n This equation admits of solution just in the same way that equation (11) was solved ; but I shall confine myself to the case of the tidal problem, where i='2 and S 2 =S cos (vt-}-TJ). In this special case the equation becomes dcr . = cos i .p. 19w , 1 2 And it we put - M =-, tan \if=vt, and tt= " 2gwa k 5 This may be written da- , k vak -, , + -0-= : -b COS (Vi at t g sin i/r In the solution appropriate to the tidal problem, we may omit the exponential term, vt and assume cr=A cos (vt+B). Then if we put tan^= K d ^, and a , sin v a cos y ^- _ _ "?=^ ^ S sin ->|r 3 cos -v/r' so that S cos y Hence the bodily tide of the elastico-viscous spheroid is equal to the equilibrium tide COS *V of a fluid spheroid multiplied by - * and high tide is retarded by x$-i-v. The formula for tan x may be expressed in a somewhat more convenient form ; we 1 Qnvt have tan ib=v', and therefore tan v=tan ifr-f-r 2gwa SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 19 But nt is the coefficient of viscosity, and in treating the tides of the purely viscous 19y spheroid we put tan e=^ -- X coefficient of viscosity; therefore adopting the same notation here, we have tan x = tan t/'+tan e. If the modulus of relaxation t be zero, whilst the coefficient of rigidity n becomes infinite, but nt finite, the substance is purely viscous, and we have \p=0 and x =e > so that the solution reduces to the case already considered. If t be infinite, the sub- T , TT TT , cos y , sin y , stance is purely elastic, and we have W=T, x = ~ and since "==# , therefore 2 A 2 cos ^ sin TJT A,' . . a-= b cos (vl+ri). m But according to THOMSON'S notation* =-, so that ar= S cos (vt-\--n). which is 2gwa Z the solution of THOMSON'S problem, of the purely elastic spheroid. The present solution embraces, therefore, both the case considered by him, and that of the viscous spheroid. 9. Ocean tides on an elastico-viscous nucleus. If r = a+u be the equation to the ocean spheroid, we have, as in sec. (6), that the height of tide relatively to the nucleus is given by 2 9 u cr= ff and substituting the present value of 5S COS l|r If the nucleus had been rigid the rise and fall would have been given by H cos (vt +17), where H=- ff S ; therefore on the yielding nucleus it is given by 5 a TT sin (vi u- 3 we must consider the magnitude of the fraction 4 X -; . If w were the earth's true angular velocity, instead of its angular velocity relatively to the moon, then - - would W be the elliptic! ty of its surface if it were homogeneous. This ellipticity is, as is well known, -^. Hence the fraction, which is the criterion of the negligeability of inertia, is about -g. If, then, it be considered that this way of looking at the subject certainly exag- gerates the influence of inertia, it is clear that the neglect of inertia is not such as to materially vitiate the results giveii above. II. A TIDAL YIELDING OF THE EARTH'S MASS, AND THE CANAL-THEORY OF TIDES. In the first part of this paper the equilibrium theory has been used for the determi- nation of the reduction of the height of tide, and the alteration of phase, due to bodily tides in the earth. Sir W. THOMSON remarks, with reference to a supposed elastic yielding of the earth's body : " Imperfect as the comparisons between theory and observation as to the actual height of the tides has been hitherto, it is scarcely possible to believe that the height is in reality only two-fifths of what it would be if, as has been universally assumed in tidal theories, the earth were perfectly rigid. It seems, therefore, nearly certain, with no other evidence than is afforded by the tides, that the tidal effective rigidity of the earth must be greater than that of glass."* The equilibrium theory is quite fallacious in its explanation of the semidiurnal tide, but Sir W. THOMSON is of opinion that it must give approximately correct results for tides of considerable period. It is therefore on the observed amount of the fortnightly tide that he places reliance in drawing the above conclusion. Under these ' Nat. Phil.', 843. SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 23 circumstances, a dynamical investigation of the effects of a tidal yielding of the earth on a tide of short period, according to the canal theory, is likely to be interesting. The following investigation will be applicable either to the case of the earth's mass yielding through elasticity, plasticity, or viscosity ; it thus embraces Sir W. THOMSON'S hypothesis of elasticity, as well as mine of viscosity and elastico-viscosity. 11. Semidiurnal tide in an equatorial canal on a yielding nucleus. I shall only consider the simple case of the moon moving uniformly in the equator, and raising tide waves in a narrow shallow equatorial canal of depth A. The potential of the tide-generating force, as far as concerns the present inquiry, is, / \ 2 O2 with the old notation, ( - J- sin 2 cos 2($ wf) + e]+^( -I sin 2 cos 2(0-f e~lH -Ysiu 2 where F= -a=v ; also the wave length is half the circum- 2 ference of the earth's equator, or ira ; and let m=~. Then we have the following ct> transformations : - 24 MB. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC nix Also in the small terms we may put r=a. Thus the potential becomes V= const. +r/y+F cos[w(a; r<)-fe]+G sin[m(z vt)+e]. Again, to find the equation to the bottom of the canal, we have to transform the equation sin 2 0cos [2($ If -i/ be the ordinate of the bottom of the canal, corresponding to the abscissa x, this equation becomes after development y'=h E cos [ni(x ttf)-f e]. We now have to find the forced waves in a horizontal shallow canal, under the action of a potential V, whilst the bottom executes a simple harmonic motion. As the canal is shallow, the motion may be treated in the same way as Professor STOKES has treated the long waves in a shallow canal, of which the bottom is stationary. In this method it appears that the particles of water, which are at any time in a vertical column, remain so throughout the whole motion. Suppose, then, that x+g=x is the abscissa of a vertical line of particles PQ, which, when undisturbed, had an abscissa x. Let 77 be the ordinate .of the surface corresponding to the abscissa x'. Let pq be a neighbouring line of particles, which when undisturbed were distant from PQ a small length k. Conceive a slice of water cut off by planes through PQ, pq perpendicular to the length of the canal, of which the breadth is 6. Then the volume of this slice Now PQ=/i E cos \_m(x vt)+e] Y), and N= Hence treating E and rj as small compared with h, the volume of the slice is SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 25 But this same slice, in its undisturbed condition, had a volume bhk. Therefore the equation of continuity is r) = h f ~ E cos [m(x'vt) + e]. Now the hydrodynamical equation of motion is approximately dp _dV _d* dx' dx' dt* The difference of the pressures on the two sides of the slice PQyp at any depth is Nn X j-, ', and this only depends on the difference of the depressions of the wave- UrOij surface below the axis of x on the two sides of the slice, viz. at P and p. Thus dp _ df) dx'~ ~ g fa' Substituting then for 77 from the equation of continuity, and observing that - ~ is ' 7- fc very nearly the same as -^, we have as the equation of wave motion, x sm But , = m F sin [m (x vt) + e] + m G cos [m (x vt) + e]. dx So that In obtaining the integral of this equation, we may omit the terms which are independent of G, F, E, because they only indicate free waves, which may be supposed not to exist. The approximation will also be sufficiently close, if x be written for x' on the right hand side. Assume, then, that f= A cos [rn(x y) + e]+B sin [rn(x vt) + e], By substitution in the equation of motion we find -m 2 (y 2 (/A){Acos+Bsin}=w{Gco8(F Er/).siu}. And as this must hold for all times and places, MDCCCLXXIX. 26 MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC . G -Jar sine ~~ m (v- -gh) = 2(a-a> 2 - and if r=a-ftt be the equation to the tidal spheroid, we have, as in Part I., u= S - -4 * ) cos'2(ait)-{-gEcos[2( J \- and the equation to the relative tidal spheroid is r=a-\-u cr i j^cos2(< tat) Now in either the case of the dynamical theory or of the equilibrium theory, if E be put equal to' zero, we get the equations to the tidal spheroid on a rigid nucleus. A comparison, then, of the above equations shows at once that both the reduction of tide and the acceleration of phase are the same in one theory as in the other. But where the SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 27 one gives high water, the other gives low water. The result is applicable to any kind of supposed yielding of the earth's mass ; and in the special case of viscosity, the table of results for the fortnightly tide at the end of Part I. is applicable. III. SUMMARY AND CONCLUSIONS. In 1 an analogy is shown between problems about the state of strain of in- compressible elastic solids, and the flow of incompressible viscous fluids, when inertia is neglected ; so that the solutions of the one class of problems may be made applicable to the other. Sir W. THOMSON'S problem of the bodily tides of an elastic sphere is then adapted so as to give the bodily tides of a viscous spheroid. The adaptation is ren- dered somewhat complex by the necessity of introducing the effects of the mutual gravitation of the parts of the spheroid. The solution is only applicable where the disturbing potential is capable of expansion as a series of solid harmonics, and it appears that each harmonic term in the potential then acts as though all the others did not exist ; in consequence of this it is only necessary to consider a typical term in the potential. In 3 an equation is found which gives the form of the free surface of the spheroid at any time, under the action of any disturbing potential, which satisfies the condition of expansibility. By putting the disturbing potential equal to zero, the law is found which governs the subsidence of inequalities on the surface of the spheroid, under the influence of mutual gravitation alone. If the form of the surface be expressed as a series of surface harmonics, it appears that any harmonic diminishes in geometrical progression as the time increases in arithmetical progression, and harmonics of higher orders subside much more slowly than those of lower orders. Common sense, indeed, would tell us that wide-spread inequalities must subside much more quickly than wrinkles, but only analysis could give the law connecting the rapidity of the sub- sidence with the magnitude of the inequality.* * On this Lord RATLEIOH remarks, that if we consider the problem in two dimensions, and imagine a number of parallel ridges, the distance between which is X, then inertia being neglected, the elements on which the time of subsidence depends are gw (force per unit mass due to weight), v the coefficient of viscosity, and X. Thus the time T must have the form The dimensions of yw, v, \ are respectively ML" 2 !"', ML-'T" 1 , L ; hence V And x= 1, v = l, 2= 1, so that T varies as -- pwX If we take the case on the sphere, then when i, the order of harmonics, is great, X compares with T ; VI so that T varies as owa E 2 28 MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC I hope at some future time to try whether it will not be possible to throw some light on the formation of parallel mountain chains and the direction of faults, by means of this equation. Probably the best way of doing this will be to transform the surface harmonics, which occur here, into BESSEL'S functions. lu 4 the rate is considered at which a spheroid would adjust itself to a new form of equilibrium, when its axis of rotation had separated from that of figure ; and the law is established which was assumed in a previous paper.* In 5 I pass to the case where the disturbing potential is a solid harmonic of the second degree, multiplied by a simple time harmonic. This is the case to be considered for the problem of a tidally distorted spheroid. A remarkably simple law is found connecting the viscosity, the height of tide, and the amount of lagging of tide ; it is shown that if v be the speed of the tide, and if tan e varies jointly as the coeffi- cient of viscosity and v, then the height of bodily tide is equal to that of the equi- librium tide of a perfectly fluid spheroid multiplied by cos e, and the tide lags by a time equal to -. v It is then shown ( 6) that in the equilibrium theory the ocean tides on the yielding nucleus will be equal in height to the ocean tides on a rigid nucleus multiplied by sin e. and that there will be an acceleration of the time of high water equal to . 2v v The tables in 7 give the results of the application of the preceding theories to the lunar semidiurnal and fortnightly tides for various degrees of viscosity. A comparison of the numbers in the first columns with the viscosity of pitch at near the freezing temperature (viz., about 1'SxlO 8 , as found by me), when it is hard, apparently solid and brittle, shows how enormously stiff the earth must be to resist the tidally deform- ing influence of the moon. For unless the viscosity were very much larger than that of pitch, the viscous sphere would comport itself sensibly like a perfect fluid, and the ocean tides would be quite insignificant. It follows, therefore, that no very consider- able portion of the interior of the earth can even distantly approach the fluid state. This does not, however, seem to be conclusive against the existence of bodily tides in the earth of the kind here considered; for although (as remarked by Sir W. THOMSON) a very great hydrostatic pressure probably has a tendency to impart rigidity to a substance, yet the very high temperature which must exist in the earth at a small depth would tend to induce a sort of viscosity at least if we judge by the behaviour of materials at the earth's surface. In 8 the theory of the tides of an imperfectly elastic spheroid is developed. The kind of imperfection of elasticity considered is where the forces requisite to maintain the body in any strained configuration diminish in geometrical progression as the time increases in arithmetical progression. There can be no doubt that all bodies do possess an imperfection in their elasticity of this general nature, but the exact law Phil. Trans., Vol. 167, Part I., sec. 5 of my paper. * SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 20 here assumed has not, as far as I am aware, any experimental justification ; its adoption was rather due to mathematical necessities than to any other reason. It would, of course, have been much more interesting if it had been possible to represent more exactly the mechanical properties of solid matter. One of the most important of these is that form of resistance to relative displacement, to which the term " plasticity " has been specially appropriated. This form of resistance is such that there is a change in the law of resistance to the relative motion of the parts, when the forces tending to cause flow have reached a certain definite intensity. This idea was founded, I believe, by MM. TRESCA and ST. VENANT on a long course of experiments on the punching and squeezing of metals ;* and they speak of a solid being reduced to the state of fluidity by stresses of a given magnitude. This theory introduces a discontinuity, since it has to be determined what parts of the body are reduced to the state of fluidity and what are not. But apart from this difficulty, there is another one which is almost insuperable, in the fact that the differential equations of flow are n on -linear. The hope of introducing this form of resistance must be abandoned, and the investi- gation must be confined to the inclusion of those two other continuous laws of resistance to relative displacement elasticity and viscosity. As above stated, the law of elastico- viscosity assumed in this paper has not got an experimental foundation. Indeed, KOHLRAUSCH'S experiments on glasst show that the elasticity degrades rapidly at first, and that it tends to attain a final condition, from which it does not seem to vary for an almost indefinite time. But glass is one of the most perfectly elastic substances known, and, by the light of TRESCA'S experi- ments, it seems probable that experiments with lead would have brought out very different results. It seems, moreover, hardly reasonable to suppose that the materials of the earth possess much mechanical similarity with glass. Notwithstanding all these objections, I think, for my part, that the results of this investigation of the tides of an ideal elastico-viscous sphere are worthy of attention. There are two constants which determine the nature of this ideal solid : first, the coefficient of rigidity, at the instant immediately after the body has been placed in its strained configuration ; and secondly, " the modulus of the time of relaxation ol rigidity," which is the time in which the force requisite to retain the body in its strained configuration has fallen away to '368 of its initial value. In this section it is shown that the equations of flow of this incompressible elastico- viscous body have the same mathematical form as those for a purely viscous body ; so that the solutions already attained are easily adapted to the new hypothesis. The only case where the problem is completely worked out, is when the disturbing * " Sur 1'ecoulement des Corps Solidos," Mem. des Savants Etrangers, torn, xviii. and torn, xx., p. 75 and p. 137. See also ' Comptes Eendus,' torn. 66, 68, and Liouville's Journ., 2 me serie, xiii., p. 379, and xvi., p. 308, for papers on this subject. f POGGENDORFF Ann., vol. 119, p. 337. * 30 MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC potential has the form appropriate to the tidal problem. The laws of reduction of bodily tide, of its lagging, of the reduction of ocean tide, and of its acceleration, are somewhat more complex than in the case of pure viscosity ; and the reader is referred to 8 for the statement of those laws. It is also shown that by appropriate choice of the values of the two constants, the solutions may be either made to give the results of the problem for a purely viscous sphere, or for a purely elastic one. The tables give the results, for the semidiurnal and fortnightly tides, of this theory for spheroids which have the rigidity of glass or of iron the two cases considered by Sir W. THOMSON. As it is only possible to judge of the amount of bodily tide by the reduction of the ocean tide, I have not given the heights and retardations of the bodily tide. It appears that if the time of relaxation of rigidity is about one quarter of the tidal period, then the reduction of ocean tide does not differ much from what it would be if the spheroid were perfectly elastic. The amount of tidal acceleration still, however, remains considerable. A like observation may be made with respect to the accelera- tion of tide in the case of pure viscosity approaching rigidity : and this leads me to think that one of the most promising ways of detecting such tides in the earth would be by the determination of the periods of maximum and minimum in a tide of long period, such as the fortnightly in a high latitude. In 10 it is shown that the effects of inertia, which had been neglected in finding the laws of the tidal movements, cannot be such as to materially affect the accuracy of the results. [* The hypothesis of a viscous or imperfectly elastic nature for the matter of the earth would be rendered extremely improbable, if the ellipticity of an equatorial section of the earth were not very small. An ellipsoidal figure with three unequal axes, even if theoretically one of equilibrium, could not continue to subsist very long, because it is a form of greater potential energy than the oblate spheroidal form, which is also a figure of equilibrium. Now, according to the results of geodesy, which until very recently have been generally accepted as the most accurate namely, those of Colonel A. II. CLARKE t there is a difference of 6,378 feet between the major and minor equatorial radii, and the meridian of the major axis is 15 34' E. of Greenwich. The heterogeneity of the earth would have to be very great to permit so large a deviation from the oblate spheroidal shape to be either pennanent, or to subside with extreme slowness. But since this paper was read, Colonel CLARKE has published a revision of his results, founded on new data;| and he now finds the difference between the equatorial radii to be only 1,524 feet, whilst the meridian of the greatest axis is 8 15' west. This exhibits a change of meridian of 24, and a reduction of equatorial * The part within brackets [ ] was added in November, 1878, in consequence of a conversation with Sir W. THOMSON. f Quoted in THOMSON and TAIT, Nat. Phil., sec. 797. | Phil. Mag., August, 1878. SPHEROIDS, AND OX THE OCEAN TIDES UPON A YIELDING NUCLEUS. 31 ellipticity to about one quarter of the formerly-received value. Moreover, the new value of the polar axis is about 1,000 feet larger than the old one. Colonel CLARKE himself obviously regards the ellipsoidal form of the equator as doubtful. Thus there is at all events no proved result of geodesy opposed to the present hypothesis concerning the constitution of the earth. Sir W. THOMSON remarks in a letter to me that ""we may look to further geodetic observations and revisals of such calculations as those of Colonel CLARKE for verification or disproof of your viscous theory."] In the first part of the paper the equilibrium theory is used in discussing the question of ocean tides ; in the second part I consider what would be the tides in a shallow equatorial canal running round the equator, if the nucleus yielded tidally at the same time. The reasons for undertaking this investigation are given at the beginning of that part. In 11 it is shown that the height of tide relatively to the nucleus bears the same proportion to the height of tide on a rigid nucleus as in the equilibrium theory, and the alteration of phase is also the same ; but where the one theory gives high water the other gives low water. The chief practical result of this paper may be summed up by saying that it is strongly confirmatory of the view that the earth has a very great effective rigidity. But its chief value is that it forms a necessary first chapter to the investigation of the precession of imperfectly elastic spheroids, which will be considered in a future paper. * I shall there, as I believe, be able to show, by an entirely different argument, that the bodily tides in the earth are probably exceedingly small at the present time. APPENDIX. November 7, On the observed height and phade of the fortnightly oceanic tide. In the following note I attempt to carry out thejj^gestion concerning the fort- nightly tide made in the preceding/paper. The reports of the Tidal Courmiittee of thxBprfish Association for 1872 and 1876 contain the reductions of th^ tidal observations at a number of stations, into a series of harmonic tides, corresponding to tire theoretical harmonic constituents of the tide- generating forces of the moon jjjsfl sun. /The tide with which we are here concerned is the fortnightly decKnatioj The heights of the tkjesat various James are all expressed in the form E, cos (nt e), where E, is half the range of the tide in Englisja4et^n the " speed " of the tide, and e the retardation of phase, so that e-rt*-~is < ne " lag" of the tide. * Read before the Royal Society on December 19th, 1878. 32 MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC With the notation of the present paper n = 2l for the fortnightly tide, and fit is the " mean moon's " longitude from her node. The following are t/ie results, giving the place of observation, its N. latitude, and the years of observations. With respect to Brest and Toulon, E, is reduced to feet from centimetres,/s6 as to be made comparable with the other results : Ramggale, about 51" 21'. , Liverpool, 53" 40'. Hartlepool, 54 41'. 1857-58. 1858-59. 1859-60. 1866-67. 1858 59. 1859-60. 1860-61. 093 170 c -7 037 148 -8 -024 7-2 :> 036 340 "6 062 :M i ':',( 053 222-34 073 158 -62 Brct, 48 23'. Toulon, 43 7'. / Kurrnchee, 24 53'. Cat Is'and, Gulf of M< xico, 30 23'. 1875. 1853. 1868-69. 1869-70. 1870 71. ; ~ R -099 6 80-65 051 139-50 038 335-40 064 333-91 035 283'22 / / -043 136'69 Tn their 1 present form the observations do not appear do present any semblance of law, but when they are rearranged we shall be able to form some idea as to whether they are really quite valueless or not for the point under/consideration. The theoretical expression for the fortnightly tide ox an ocean covering the whole earth, according to the equilibrium theory, is - a sin- i(\ cos- ff) cos : Where T=~, 3=^, a= earth's radius, i the average obliquity/of the earth's axis t<> --C Oil the normal to the plane of the lunar orbit during the forbffight in question, 6 the colatitude of the place of observation. If we take i=23 28' the obliquity of the ecliptic, million feet, we find -, r asin 2 i=-207 foot 10 a So that the fortnightly tide should be expressible 207 (i sin 2 (lat.)) In THOMSON'S corrected equilibrium theory the second factor sVmld be /, , x - -- sin- (lat.) O SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. where (5 is a certain definite integral, depending on the distribution of land and water, but which has not yet been evaluated. The latitude of evanescent fortnightly tide is 36 15' if ( is zero ; and if we bear in mind that 1 , to 2 , o) s be the component angular velocities of the earth about the instan- taneous positions of ABC. Then we have, as usual, the geometrical equations, di - = i sm X r rft dv , , each multiplied by a simple time harmonic, which will involve n and fl. For brevity of notation nt, fit are written simply n, fl, but wherever these symbols occur in the argument of a trigonometrical term they must be understood to be multi- plied by t the time. We have cos PM= sin 9 cos MR+ cos 9 sin Mil sin MRQ and cos MR= cos MN cos NR+ sin MN sin NR cosi = cos fi sin (n) cos /2 + sin 9 cos (n) sin fl cos i-\- cos 9 sin fl sin i ^ sin 0{sin[0 (u /2)]+ sin[< (n-|-/2)]} +-^ sin 0cosi{sin[0 (, /2)] sin[< (H+^)]|+ cos sin /2 sin i Let p= cos^, g= wn- Then cos PM=j9 2 sin ^ sin [(j}(nfl)^-\-2pq cos sin fl-\-q* sin sin [< (-+/2)] . (2) Therefore cos 2 PM=ip 4 sin 2 ^{ 1 cos [20 2(n /2)]} +2^Y cos 2 ^(1 cos 2/2) sin 2 6>{ 1 cos [20 2(n+/2)]} + 2^> 3 g sin cos <9{cos (n) cos [0 (i 2/2)] } -\-2ptfsm Ocos 9 (cos[0 (n+2/2)] cos (<$>n)}+p~q~ sm- ^{cos 2/2 cos (22n)} Then collecting terms, and noticing that sin- ^+2 cos 2 0=i+l-6ji- cos 2 we have I'-TI" (3) Now if all the cosines involving be expanded, it is clear 'hat we have V consisting 452 MR. G. H. DARWJN ON THE PRECESSION OF A VISCOUS SPHEROID, of thirteen terms which, have the desired form, and a fourteenth which is independent of the time. It will now be convenient to introduce some auxiliary functions, which may be defined thus, <&(2n)=p* cos 2(n fl)+p~q z cos 2n+^q l cos 2(n+/2) ~j V(n)=2p s qcos (n 2/2) ^pq(p' (2n ITT), W(n \ir), X(2Q |TT) are functions of the same form with sines replacing cosines. When the arguments of the functions are simply 2n, n, 2/2 respectively, they will be omitted and the functions written simply <, ^, X; and when the arguments are simply 2n ^n, n^ir, 2/2 \TT, they will be omitted and the functions written 4>', ', X'. These functions may of course be expanded like sines and cosines, e.g., ( a)=V cos a-f ' sin a and V(n )=' cos a sin a. If now these functions are introduced into the expression for V, and if we replace the direction cosines sin 6 cos <, sin 6 sin (f>, cos 6 of the point P by 77, , we have (l-G^/)] - (5) f 2 7j 2 , 2r), , ^^, ^f'-f 1 ?" 2 2 ) are surface harmonics of the second order, and the auxiliary functions involve only simple harmonic functions of the time. Hence we have obtained V in the desired form. We shall require later certain functions of the direction cosines of the moon referred to A B C expressed in terms of the auxiliary functions. The formation of these functions may be most conveniently done before proceeding further. Let x, y, z be these direction cosines, then cos ~PM.=x+yr)+z whence cos 2 But from (5) we have on rearranging the terms, cos 2 PM-r= (5') AND ON THE REMOTE HISTORY OF THE EARTH. 453 Then equating coefficients in these two expressions (5') and (G) Whence z 2 ar= <& X -i(l also (7) xy= <& These six equations (7) are the desired functions of x, y, z in terms of the auxiliary functions. 2. The form of the spheroid as tidally distorted. The tide-generating potential has thirteen terms, each consisting of a solid harmonic of the second degree multiplied by a simple harmonic function of the time, viz. : three in <&, three in <&', three in % three in ^', and one in X. The fourteenth term of V can raise no proper tide, because it is independent of the time, but it produces a permanent increment to the ellipticity of the mean spheroid. Hence according to our hypothesis, explained in the introductory remarks, there will be thirteen distinct simple tides ; the three tides corresponding to <&' may however be compounded with the three in <&, and similarly the ' tides with the ^ tides. Hence there are seven tides with speeds""" [2u 2/2, 2;i, 2-j-2/2], [H 2/2, r>, ?i + 2/2], [2/2], and each of these will be retarded by its own special amount. The cos 2(n /2). The equilibrium value of the corresponding tide is found by putting - equal to this expression divided by -wr-%. Then if we suppose that there is a frictional resistance to the tidal motion, the tide will lag and be reduced in height, and according to the preceding definitions the corresponding tide of our spheroid is expressed by cr a - sin 2 6 cos 2< cos [2(n /2) 2eJ All the other tides may be treated in the same way, by introducing the proper E's and e's. Thus if we write (2n 2/2 2ej)+J5:_pY cos (2n 2e) +E. 2 \q* cos (2n+2/2 2e 2 ) ^,=E\2p s qcos(n 2/2 c',) E'2pq(p q*)cos(n e') # 2 2pf/co8 (n+ 2/2 e',,) \- (8) X, = E"3p*q* cos (2/2 - 2e") and if in the same symbols accented sines replace cosines, then, by comparison with (5), we see that AND ON THE EEMOTE HISTORY OF THE EARTH. 455 ~-WX. (0) This is merely a symbolical way of writing down that every term in the tide- generating potential raises a lagging tide of its own type, but that tides of different speeds have different heights and lags. This same expression may also be written .-2&V.-Zto#. . . . (9') Then if we put c b a c b a c= d= e= f= = * e X e (10) It is clear that Whence T (11) 1 . . . (12) Of which expressions use will be made shortly. 3. T/ie couples about the axes A, B, C caused by the moon's attraction. The earth is supposed to be a homogeneous spheroid of mean radius a, and mass w per unit volume, so that its mass M=^irwa 3 . When undisturbed by tidal distortion it is a spheroid of revolution about the axis C, and its greatest and least principal moments of inertia are C, A. Upon this mean spheroid of revolution is .superposed the tide-wave cr. The attraction of the moon on the mean spheroid produces the ordinary precessional couples 2r(C A.)yz, 2r(C A)zx, about the axes A, B, C respectively; besides MDCCCLXXIX. 3 N 456 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, these there are three couples, U, ,/tt, & suppose, caused by the attraction on the wave surface cr. As it is only desired to determine the corrections to the ordinary theory of preces- sion, the former may be omitted from consideration, and the attention confined to the determination of ft, ffll, jt. The moon will be treated as an attractive particle of mass m. Now cr as denned by (9) is a surface harmonic of the second order; hence by the ordinary formula in the theory of the potential, the gravitation potential of the tide- AA 3 wave at a point whose coordinates referred to A, B, C are r rrj, rt, is fmcaf-j cr or T\f(i f-jtr. Hence the moments about the axes A, B, C of the forces which act on a particle of mass m, situated at that point, are f ,- \n .., ], &c., &c. Then if this particle has the mass of the moon; if r be put equal to c, the moon's distance; and if 17, be replaced in cr by x, y, z (the moon's direction cosines) in the previous expres- sions, it is clear that \Mar(y- 277)1 & c - & c -> are the couples on the earth caused by the moon's attraction. These reactive couples are the required 3L, ffil, $.. Hence referring back to (12) and remarking that fJfa*=C, the earth's moment of inertia, we see at once that 1L 2r 3 - = [(c b)?/? d(y 3 z 2 ) ezy+fbf] (13) Where the quantities on the right-hand side are denned by the thirteen equations (7) and (10). I shall confine my attention to determining the alteration in the uniform precession, the change in the obliquity of the ecliptic, and the tidal friction; because the nutations produced by the tidal motion will be so small as to possess no interest. In developing HL and ffil I shall only take into consideration the terms with argu- ment n, and in ^t only constant terms ; for it will be seen, when we come to the equations of motion, that these are the only terms which can lead to the desired end. 4. Development of the couples H and Jtt. Now substitute from (7) and (10) in the first of (13), and we have 0_2 AND ON THE REMOTE HISTORY OF THE EARTH. 457 A number of multiplications have now to be performed, and only those terms which contain the argument n to be retained. The particular argument n can only arise in six ways, viz. : from products of terms with arguments 2(n /2), w 2/2; 2n, n; 2(n+/2), n+2/2; n 2/2, 2/2; n+2/2, 2/2 and from terms of argument n multiplied by constant terms. If < and 'SP, and & and ^' be written underneath one another in the various com- binations in which they occur in the above expression, it will be obvious that the desired argument can only arise from terms which stand one vertically over the other; this renders the multiplication easier. The ^, X products are comparatively easy. Then we have (a) -^^' = -^[-E l (y) <^V&' =same as (/3) (8) +|*V^ =same as (a) ( e ) -X.' = -i\_E"6p*q s sin (n 2e") -E"6p s q 5 sin (+2e")] (0 +&'* =+i['i6py sin (n-O-tf'^Y sin (n- e ' 2 )J (1?) Now put =Fsin n+Gcos. Then if the expressions (a), (/3) . . . () be added 9 T 2 up when n=-, and the sum multiplied by , we shall get F ; and if we perform the " 8 same addition and multiplication when n=0, we shall get G. In performing the first addition the terms () (8) do not combine with any other, but the terms (/3), (y), (), (r\) combine. Now -f Hence F : T = 8 cos 2 i pqpq cos 2e pj cos 2e 2 S^coBe'! ^^(^-^(pHg*-^^^ z -W+ 3 2' : ) sine'i+l^Or-r/) 3 sin sin 2e" .................. (16) And ......... (17) TT To find M it is only necessary to substitute n - for n, and we have ~= Fcosn+Gsinn (18) C Now there is a certain approximation which gives very nearly correct results and which simplifies these expressions very much. It has already been remarked that the three 3>-tides have periods of nearly a half-day and the three ^-tides of nearly a day, and this will continue to be true so long as fl is small compared with n ; hence it may be assumed with but slight error that the semi-diurnal tides are all retarded by the same amount and that their heights are proportional to the corresponding terms in the tide-generating potential. That is, we may put c 1 =e 2 =e and E^E^E. The similar argument with respect to the diurnal tides permits us to put e'j=e' 2 =e' and Then introducing the quantities P=pq 2 = cost, Q=2pq= sin i and observing that ? -9W+py+^)-W(f*-^)]=W(i-*e ! ) ^^ we have, AND ON THE REMOTE HISTORY OF THE EARTH. 459 -%Q-) cos 2t-E'PQ(l-IQ~) cos e'-%E"PQ* cos 2e -Q* sin 2t-E'PQ? sin e'+^'Q 3 sin 2e" 5. Development of the couple $. In the couple 3. about the axis of rotation of the earth we only wish to retain non- periodic terms, and these can only arise from the products of terms with the same argument. By substitution from (7) and (10) in the last of (13) ...... (20) Then as far as we are now interested, sn e x ' 'p 2 2 -< sn e.. ( sn e 3 Hence 4--ssJ^ sin 2e,+^4V sin 2e+ J B,o 8 sin 2e C 9 " 2/V/ sin \+E'2p-q-(p--q-)- sin e'+^ 3 2_pV sin e' 8 . . . (21) If as in the last section we group the semi-diurnal and diurnal terms together and put E l =^E i =^E, &c., and observe that then sin 2e+W(l-i-) sine' .... (22) 6. The equations of motion of the earth abouts its centre of inertia. In forming the equations of motion we are met by a difficulty, because the axes A, B, C are neither principal axes, nor can they rigorously be said to be fixed in the earth. But M. LIOUVILLE has given the equations of motion of a body which is changing its shape, using any set of rectangular axes which move in any way with reference to the body, except that the origin always remains at the centre of inertia. 460 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, If A, B, C, D, E, F be the moments and products of inertia of the body about these axes of reference at any time ; H 1( H 2 , H 3 the moments of momentum of the motion of all the parts of the body relative to the axes ; w 1; &>.,, a> 3 the component angular velocities of the axes about their instantaneous positions, the equations may be written (23) and two other equations found from this by cyclical changes of letters and suffixes.* Now in the case to be considered here the axes A, B, C always occupy the average position of the same line of particles, and they move with very nearly an ordinary uniform precessional motion. Also the moments and products of inertia may be written A-f a', B+b', C+c', d', e', f, where a', b', c', d', e', f are small periodic functions of the time and a'+b'-f-c'=0, and where A, B, C are the principal moments of inertia of the undisturbed earth, so that B is equal to A. Now the quantities a', b', &c., have in effect been already determined, as may be shown as follows : By the ordinary formulat the force function of the moon's action on the earth is - \-T(- -- 1), where I is the moment of inertia of the earth about c V 3 / the line joining its centre to the moon, and is therefore But the first three terms of I only give rise to the ordinary precessional couples, and a comparison of the last six with (11) and (13) shows that a'_V_c;_d'_e;_f _T _, a b c d e f g' Also in the small terms we may ascribe to tit lt &> 3 , o> 3 their uniform precessional values, viz. : ta 1 = II cos??, &>.>:= II sin n, o> 3 = n. When these values are substituted in (23), we get some small terms of the form a'n 2 sin n, and others of the form a'llw sin n ; both these are very small compared to the terms in U, and ffil the fractions which express their relative magnitude being U~ , Hn - and - . T T There is also a term nH 3 sin n, which I conceive may also be safely neglected, as also the similar terms in the second and third equations. It is easy, moreover, to show that according to the theories of the tidal motion of a homogeneous viscous spheroid given in the previous paper, and according to * ROUTE'S ' Rigid Dynamics ' (first edition only), p. 150, or my paper in the Phil. Trans. 1877, Vol. 167, p. 272. The original is in LIOUVILLE'S Journal, 2nd series, vol. iii., 1858, p. 1. t ROUTE'S 'Rigid Dynamics,' 1877, p. 495. AND ON THE REMOTE HISTORY OF THE EARTH. 461 Sir WILLIAM THOMSON'S theory of elastic tides, H 1( PL, H 3 are all zero. Those theories both neglect inertia but the actuality is not likely to differ materially therefrom. Thus every term where ., occur may be omitted and the equations reduced to dt dt . . . . (24) As before with the couples, so here, we are only interested in terms with the argument n in the small terms on the left-hand side of the first two of equations (24), and in non-periodic tenns in the last of them. Now for each term in the moon's potential, as developed in Section 1, there is (by hypothesis) a corresponding co -periodic flux and reflux throughout the earth's mass, and therefore the H 1; H 3 , H 3 must each have periodic terms corresponding to each term in the moon's potential. Hence the only term in the moon's potential to be con- sidered is that with argument n, with respect to H x and H 3 in the first two equations ; and H 3 may be omitted from the third as being periodic. Suppose then that Hj was equal to h cos n-\-h' sin n, then precisely as we found from H by writing n - for n we have H.j=A sin n h' cos n. Thus '-j-JiH. ** (IV wH^O, and the H's disappear from the first two equations. Next retaining only terms in argument n in d' and e', we have from (10) 2 - 1 =-[ F cos n-fG sin n], &>.,=-[ Fsin ;< Goes H] .... (28) But the geometrical equations (1) give di = cuj sin n+(a. 2 cos n (l-Jr . -:-- sm i= to, cos n to., sin n Therefore, as far as concerns non-periodic terms, di G d-Jr . . F -T t -- , ^sm? = - ........ (29) dt n at n If we wish to keep all the seven tides distinct (as will have to be done later), we may write down the result for and -^- from (15) and (16). But it is of more immediate interest to consider the case where the semi-diurnal tides are grouped together, as also the diurnal ones. In this case we have by (19) . . . (30) and since sin {= Q e"} . (31) ,\XD ON THE REMOTE HISTORY OF THE EARTH. In these equations P and Q stand for the cosine and sine of the obliquity of the ecliptic. Several conclusions may be drawn from this result. If e, e', c" are zero the obliquity remains constant. Now if the spheroid be perfectly elastic, the tides do not lag, and therefore the obliquity remains unchanged ; it would also be easy to find the correction to the precession to be applied in the case of elasticity. It is possible that the investigation is not, strictly speaking, applicable to the case of a perfect fluid; I shall, however, show to what results it leads if we make the appli- cation to that case. Sir WILLIAM THOMSON has shown that the period of free vibration of a fluid sphere of the density of the earth would be about 1 hour 34 minutes. * And as this free period is pretty small compared to the forced period of the tidal oscillation, it follows that E, E' , E", will not differ much from unity. Then putting them equal to unity, and putting e, e', e" zero, since the tides do not lag, we find that the obliquity remains constant, and This equation gives the correction to be applied to the precession as derived from the assumption that the rotating spheroid of fluid is rigid. This result is equally true if all the seven tides are kept distinct. Now if the spheroid were rigid its precession would be cos i, where e is the ellipticity of the spheroid. n-a The ellipticity of a fluid spheroid rotating with an angular velocity n is \ - - or \ ; J " but besides this, there is ellipticity due to the non-periodic part of the tide-generating potential. By (3) 1 the non-periodic part of V is \wrr-(\ cos 2 8)(\ p-(f); such a disturb- ing potential will clearly produce an ellipticity ^-(1 6p z q 2 ). If therefore we put e = , and remember that 6p*q 2 =% sin 2 i, we have, s Hence if the spheroid were rigid, and had its actual ellipticity, we should have - . .,..,.. /O0'\ l-ism-i) ....... (32) * Phil. Trans., 1863, p. 608. MDCCCLXXIX. 3 O 464 MR G. H. DABWDf OX THE PRECESSION OF A VISCOUS SPHEROID, Adding (32'; to (32), the whole precession is We thus see that the effect of the non-periodic part of the tide-generating potei. which may be conveniently called a permanent tide, is just such as to neutralise the effects of the tidal action. The result (32") may be expressed as follows : The precession of a fluid spheroid is the fame as that of a rigid one ichich has an ellipticity equal to that due to the rotation of the spheroid. From this it follows that the precession of a fluid spheroid will differ by little from that of a rigid one of the same elliptieity, if the additional ellipticity due to the non- periodic part of the tide-generating influence is small compared with the whole ellipticity. Sir WILLIAM THOMSON has already expressed himself to somewhat the s^me effect in an address to the British Association at Glasgow.* Since e = , the criterion is the smaUness of . It may be expressed in a different form; for -; is small when -f- n is small compared with e, and -n is the reciprocal of the precessional period expressed in days. Hence the criterion may be stated thus : The precession of a fluid spheroid differs by little from that of a rigid one of the same ellipticity, when the precessional period of the spheroid expressed in terms of its rotation is large compared with the reciprocal of its ellipticity. In his address, Sir WILLIAM THOMSON did not give a criterion for the case of a fluid spheroid without any confining shell, but for the case of a thin rigid spheroidal shell enclosing fluid he gave a statement which involves the above criterion, save that the ellipticity referred to is that of the shell itself; for he says, "The amount of this difference (in precession and nutation) bears the same proportion to the actual precession or nutation as the fraction measuring the periodic speed of the disturbance (in terms of the period of rotation as unity) bears to the fraction measuring the interior elliptic-it v of the shell" This is, in fact, almost the same result as mine. This subject is again referred to in Part III. of the succeeding paper. See ' Nature,' September 14, 1876, p. 429. The above statement of results, and the comparison with Sir WILLIAM THOMSON'S criterion was added to the paper on September 17, 1879. 7> OX THE REMOTE HISTORY OP THE EARTH. 465 8. The disturbing adion of the sun. suppose that there is a second disturbing body, which may be conveniently called the son.* * It is not at first sight obvious how it is physically possible that the son should exercise an influence on the moon-tide, and the moon on tike son-tide, so as to produce a secular change in the obliquity of the ecliptic and to cause tidal friction, for the periods of the son and moon about the earth are different. It MMM, therefore, interesting to give a physical mfaning to the expansion of the tide-generating potential; it will then be seen that the interaction with which we are here dealing most occur. The expansion of the pttfr-ntial given in Section 1 is equivalent to the following statement: The tide-generating potential of a moon of mass m, moting in a circular orbit of obliquity i at a ;fo* 1t ~~. <, is equal to the tide-generating potential of ten satrfKtm at the same distance, whose orbits, munscja, and angular velocities are as follows : 1. A satellite of mass cos* ^. moving in the equator in the same direction and with the same angular Telocity as the moon, and coincident with it at ike nodes. This gives the slow semi-diurnal tide of speed 2( Q). 2. A satellite of mass M sin 4 -, moving in the equator in the opposite direction from that of the moon, bet with the same angular Telocity, and niiMiAiit, with it at the nodes. This gives the fast semi-diurnal tide of speed 2(+Q). 3. A satellite of in inn 2sin J -cos 3 -. fixed at the moon's node. This gives the sidereal semi-diurnal z - tide of speed 2*. 4. A repulsive satellite of mass m.2 sin g cos 3 '-, moving in X. declination 45' with twice the moon's L _ angular Telocity, in the same direction as the moon, and on the cohire 90~ in advance of the moon, when she v in her node. A satellite of mass man i cos* ^ moving in the equator with twice the moon's angular velocity, and in tlie same direction, and always on the same meridian as the fourth satellite. (4) and (5) give the slow diurnal tide of speed 2Q. 6. A satellite of mass m 801*5 COS 4> movmg in X. decimation 45 with twice the moon's angular velocity, z _ but in the opposite direction, and on the colnre 90 in advance of the moon when she is in her node. A repulsive satellite of mass m. - sin* ^ cos-? , moving in the equator with twice the moon's angular L 1. - velocity, but' in the opposite direction, and always on the same meridian as the sixth satellite. (6) and (7) give the &st semi-diurnal tide of n+2Q. 8. A satellite of mass m sin icos t fixed in X. declination -to" on the colnre. 9. A repulsive satellite of mass m, 3 sin t cos i, fixed in the equator on the same meridian as the eighth satellite. (8) and (9) give the sidereal diurnal tide of speed n. 10. A ring of matter of mass M, always passing through the moon and always parallel to the equator. This ring, of course, executes a simple harmonic motion in declination, and its mean position is the equator. This gives the fortnightly tide of speed 2Q. Xow if we form the potentials of each of these satellites, and omit those parts which, being indepen- dent of the time, are incapable of raising tides, and add them altogether, we shall obtain the expansion for the moon's tide-generating potential used above ; hence this system of satellites is mechanically 3 O 2 466 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, II cosec i must henceforth be taken as the full precession of the earth, and the time may be conveniently measured from an eclipse of the sun or moon. Let m f , c / be the sun's mass and distance ; fl i the earth's angular velocity in a circular orbit ; and let It would be rigorously necessary to introduce a new set of quantities to give the heights and lagging of the seven solar tides : but of the three solar semi-diurnal tides, one has rigorously the same period as one of the three lunar semi-diurnal tides (viz. : the sidereal semi-diurnal with a speed 2w), and the others have nearly the same period ; a similar remark applies to the solar diurnal tides. Hence we may, without much error, treat JE, e, E ', e as the same both for lunar and solar tides ; but E'" ', e'" must replace JS", c", because the semi-annual replaces the fortnightly tide. Then if new auxiliary functions <& ,, X, be introduced, the whole tide-generating potential V per unit volume of the earth at the point ? rrj, ? is given by If then, as in (10), we put c-b=^+X,, &c., c -b,=d> /( +X, f , &c., the equation to the tidally-distorted earth is r=a+cr+cr / , where equivalent to the action of the moon alone. TJje satellites 1, 2, 3, in fact, give the semi-diurnal or * terms ; satellites 4, 5, 6, 7, 8, 9 give the diurnal or * terms ; and satellite 10 gives the fortnightly or X term. This is analogous to " GAUSS'S way of stating the circumstances on which 'secular' variations in the elements of the solar system depend ;" and the analysis was suggested to me by a passage in THOMSON and TAIT'S 'Nat. Phil.,' 809, who there refer to the annular satellite 10. It will appear in Section 22 that the 3rd, 8th, and 9th satellites, which are fixed in the heavens and which give the sidereal tides, are equivalent to a distribution of the moon's mass in the form of a uniform circular ring coincident with her orbit. And perhaps some other simpler plan might be given which would replace the other repulsive satellites. These tides, here called "sidereal," are known, in the reports of the British Association on tides for 1872 and 1876, as the K tides. In a precisely similar way, it is clear that the sun's influence may be analysed into the influence of nine other satellites and one ring, or else to seven satellites and two rings. Then, with regard to the interaction of sun and moon, it is clear that those satellites of each system which are fixed in each system (viz. : 3, 8, and 9), or their equivalent rings, will not only exercise an influence on the tides raised by themselves, but each will necessarily exercise an influence on the tides raised by the other, so as to produce tidal friction. All the other satellites will, of course, attract or repel the tides of all the other satellites of the other systems ; but this interaction will necessarily be periodic, and will not cause any interaction in the way of tidal friction or change of obliquity, and as such periodic interaction is of no interest in the present investigation it may be omitted from consideration. In the analysis of the present section, this omission of all but the fixed satellites appears in the form of the omission of all terms involving the moon's or sun's angular velocity round the earth. AND ON THE REMOTE HISTORY OP THE EARTH. 467 8 a j-=> o G f, --=-a? &c., - =-a^ 3 , &c. Also if x, y, z and x t , y,, z, be the moon's and sun's direction cosines, we have as in (7), 6p), &c, y?-z,*= t Then using the same arguments as in Section 3, the couples about the three axes in the earth may be found, and we have where in the first term x, y, z are written for 7), in cr-fcr,, and in the second term x,, y t , 2 / are similarly written for 77, . Now let fU, IL M/ =, &,,, indicate the parts of the couple IL which depend on the moon's action on the lunar tides, the sun's action on the solar tides, and the moon's and sun's action on the solar and lunar tides respectively, then C" C "*" C~" Then obviously f &c. C g As before, we only want terms with argument n in H MW/ , ffll mm/ , and non-periodic terms in $.,,,. The quantities a, b, &c., x, y, z with suffixes differ from those without in having fl / in place of fl, and it is clear that no combination of terms which involve l t and fl can give the desired terms in the couples. Hence, as far as &,,,, ffil,,,,,,,, 4HLn, are concerned, the auxiliary functions may be abridged by the omission of all terms involving fl or fl,. Therefore, from (4), we now simply have <5>= / =:p 2 2' 2 cos In, ^^^=1 %pq(p~ =82, and the unit of time the tropical year 365'242 m. s. days. Then we have in radians per m. s. day r=f X sT of 4ir 2 -7- (month) - T,=f of 47r-=- (sidereal year)-. Then it will be found that r* = -6598 degrees per million tropical years (39) These three quantities will henceforth be written u~, uj~, int t . For the purpose of analysing the physical meaning of the differential equations for - - -- - - and - ( ), no distinction will be made between -- and - , &c., for it is here only dt dt\nj' n QM O ' sought to discover the rates of changes. But when we come to integrate and find the total changes in a given time, regard will have to be paid to the fact that both T and n are variables. For the immediate purpose of this section the numerical values of 2 , u~, uu / given in (39), will be used. I will now apply the foregoing results to the particular case where the earth is a viscous spheroid. Let p=Hr , where v is the coefficient of viscosity. l\)v AND ON THE REMOTE HISTORY OF THE EARTH. Then by the theory of bodily tides as developed in my last paper E= cos 2e, E'= cos e', E"= cos 2e", E'"= cos 2e"' 471 P P P i Rigorously, we should add to these E } =. cos 26], E. z = cos 2e 2 , E\= cos e\, E'= cos e' 2 2(n-/2) . 2( + /2) . , -2/2 . ji + 2/2 ^ (40') tan 2ej=- -, tan 2%= , tan e x = , tan e t =s But for the present we classify the three semi-diurnal tides together, as also the three diurnal ones. Then we have di 7=[i sin i cos {(1 f sin 2 i) sin 4e+f sin 3 i cos i sin 2e'](i6 2 +u / 2 ) - s sin 3 { sin 4e"w c (it -j^- sin 3 i sin 4e'"t/ / 2 (^ sin 3 i cos i sin 4e+^ sin i cos 3 i sin 2e')ww / . Now i sin i cos i(l f sin 2 i)=^j sin 2i(5-J-3 cos 2t)=^ l j(5 sin 2i+f sin 4i) f- sin 3 i cos i = -$5 sin 2i(l cos 2t)=^f(2 sin 2i sin 4t) a . o ^-^ eir ^ TS sin 3 1=^(3 sin *' sin 3?), \ sin 3 1 cos =^(2 sin 2z sin 4i) sin i cos 3 i=-g- sin 2i(l+ cos 2i)=^-(2 sin 2i+ sin 4i). di If these transformations be introduced, the equation for -- may be written / eft : 9 (it 2 sin 4e"+M / 2 sin 4e'") sin i+3(tr sin 4e"+M / 2 sin 4e"') sin 3 + [(5 sin 4e+6 sin 2e')(^ 2 +w / 2 ) (4 sin 4e+8 sin 2e>,] sin 2i +[( sin 4e 3 sin 2e')(w 2 +w / 2 ) + (2 sin 4e 4 sin 2e')wjsill 4i (41) Then substituting for u and u t their numerical values (39), and omitting the term depending on the semi-annual tide as unimportant, I find MDCCCLXXIX. 64r = - 5-9378 sin 4e" sin t+ 1 '9793 sin 4e" sin 3i' at + {27846 sin 4e+2'3611 sin 2e'} sin 2i + {1-8159 sin 4e 3'6317 sin 2e'} sin 4i 3 P . . (42) -!<- MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, The numbers are such that is expressed in degrees per million years. The various values which is capable of assuming as the viscosity and obliquity vary is best shown graphically. In Plate 36, figs. 2 and 3, each curve corresponds to a given degree of viscosity, that is to say to a given value of e, and the ordinates di give the values of as the obliquity increases from to 90. The scale at the side (.tt of each figure is a scale of degrees per hundred million years e.g., if we had e=30 and i about 57, the obliquity would be increasing at the rate of about 3 45' per hundred million years. The behaviour of this family of curves is so very peculiar for high degrees of viscosity, that I have given a special figure (viz. : Plate 36, fig. 3) for the viscosities for which =40, 41, 42, 43, 44. The peculiarly rapid variation of the forms of the curves for these values of e is due to the rising of the fortnightly tide into prominence for high degrees of viscosity. The matter of the spheroid is in fact so stiff that there is not time in 12 hours or a day to raise more than a very small tide, whilst in a fortnight a considerable lagginw tide is raised. For e=44 the fortnightly tide has risen to give its maximum effect (i.e., sin 4e"= 1), whilst the effects of the other tides only remain evident in the hump in the middle of the curve. Between e=44 and 45 the.ordinates of the curve diminish rapidly and the hump is smoothed down, so that when e=45 the curve is reduced to the horizontal axis. By the theory of the preceding paper,* the values of c when divided by 15 give the corresponding retardation of the bodily semi-diurnal tide e.g., when e=30 the tide is two hours late. Also the height of the tide is cos 2c of the height of the equilibrium tide of a perfectly fluid spheroid e.g., when e= 30 the height of tide is reduced by one-half. In the tables given in Part I., Section 7, of the preceding paper, will be found approximate values of the viscosity corresponding to each value of e. The numerical work necessary to draw these figures was done by means of CRELLE'S multiplication table, and as to fig. 2 in duplicate mechanically with a sector ; the ordi- nates were thus only determined with sufficient accuracy to draw a fairly good figure. For the two figures I found 108 values of each of the seven terms of ~ (nine values at ^ of i and twelve of e), and from the seven tables thus formed, the values corresponding to each ordinate of each member of the family were selected and added together. From this figure several remarkable propositions may be deduced. When the ordinates are positive, it shows that the obliquity tends to increase, and when negative to diminish. Whenever, then, any curve cuts the horizontal axis there is a position of dynamical equilibrium ; but when the curve passes from above to below, it * "On the Bodily Tides of Viscous and Semi-elastic Spheroids," &c., Phil. Trans., 1879, Part I. AND ON THE REMOTE HISTORY OP THE EARTH. 473 is one of stability, and when from below to above, of instability. It follows from this that the positions of stability and instability must occur alternately. When =0 or 45 (fluidity or rigidity) the curve reduces to the horizontal axis, and every position of the earth's axis is one of neutral equilibrium. But in every other case the position of 90 of obliquity is not a position of equi- librium, but the obliquity tends to diminish. On the other hand, from e=0 to about 30 (infinitely small viscosity to tide retardation of two hours), the position of zero obliquity is one of dynamical instability, whilst from then onwards to rigidity it becomes a position of stability. For viscosities ranging from e=0 to about 42^ there is a position of stability which lies between about 50 to 87 of obliquity ; and the obliquity of dynamical stability diminishes as the viscosity increases. For viscosities ranging from e=30 nearly to about 42^, there is a second position of dynamical equilibrium, at an obliquity which increases from to about 50, as the viscosity increases from its lower to its higher value. But this position is one of instability. From = about 42 there is only one position of equilibrium, and that stable, viz. : when the obliquity is zero. If the obliquity be supposed to increase past 90, it is equivalent to supposing the earth's diurnal rotation reversed, whilst the orbital motion of the earth and moon remains the same as before ; but it did not seem worth while to prolong the figure, as it would have no applicability to the planets of the solar system. And, indeed, the figure for all the larger obliquities would hardly be applicable, .because any planet whose obliquity increased very much, must gradually make the plane of the orbit of its satellite become inclined to that of its own orbit, and thus the hypothesis that the satellite's orbit remains coincident with the ecliptic would be very inexact. It follows from an inspection of the figure that for all obliquities there are two degrees of viscosity, one of which will make the rate of change of obliquity a maximum and the other minimum. A graphical construction showed that for obliquities of about 5 to 20, the degree of viscosity for a maximum corresponds to about e=17^" : ', whilst that for a minimum to about e=40. In order, however, to check this con- clusion,. I determined the values of e analytically when i=l5, and when the fortnightly tide (which has very little effect for small obliquities) is neglected. I find that the values are given by the roots of the equation a?+10a?+ 13-660o: 20-412=0, where x=3 cos 4e. This equation has three real roots, of which one gives a hyperbolic cosine, and the * I may here mention that I found when e=17^, that it would take about a thousand million years for the obliquity to increase from 5 to 23^, if regard was only paid to this equation of change of obliquity. The equations of tidal friction and tidal reaction will, however, entirely modify the aspects of the case. 3 P 2 474 MR. G. H. DARWIN ON THE PRECESSION OP A VISCOUS SPHEROID, other two give 6=18 15' ande=41 37'. This result therefore confirms the geometrical construction fairly well. It is proper to mention that the expressions of dynamical stability and instability are only used in a modified sense, for it will be seen when the effects of tidal friction come to be included, that these positions are continually shifting, so that they may be rather described as positions of instantaneous stability and instability. : I will now illustrate the case where there is only one satellite to the planet, and in order to change the point of view, I will suppose that the periodic time of the satellite is so short that we cannot classify the semi-diurnal and diurnal terms together, but must keep them all separate. Suppose that =5/2; then the speeds of the seven tides are proportional to the following numbers, 8, 10, 12 (semi-diurnal); 3, 5, 7 (diurnal) ; 2 (fortnightly). These are all the. data which are necessary to draw a family of curves similar to those in Plate 36, figs. 2 and 3, because the scale, to which the figure is drawn, is determined by the mass of the satellite, the mass and density of the planet, and the actual velocity of rotation of the planet. Then by (16) and (29) we have di T 2 = [4p 7 is the ratio of the two momenta, and /u, is the ratio of the two momenta at the fixed moment of time, which is the epoch. In the similar equation expressive of the rate of change in the earth's orbital motion round the sun, it is obvious that the orbital moment of momentum is so very large compared with the earth's moment of momentum of rotation, that ^ is very large and the earth's mean distance from the sun remains sensibly constant (see Section 19). Then by (16) and (29), remembering that i C,,,'J -,-,- /I p= cos-, (/= sin-, -77= , and N=, 2t L dt n, t^Q we have q sin ^iEZphf(p~(f} sin 2e E. 2 p Hence (1650 + 248+^32 of 7'04)e=8 whence That is to say, the semi-diurnal tide only lags by the small angle 14'. But this is not the solution which is interesting in the case of the earth, for we know that the earth does not behave approximately as a fluid body. In the other solution, 2e and e' approach 90, so that p is small ; hence 4p p , , 2/ip 2p , . . /, 4/2p sin 4e= . - = -, sin 2e = -: a = -- very nearly, and sin 4e = * + 4i 2 n' a + n- n p- + 4/2~ n Hence we have Put --=x. so that x= cot 2e" : then substituting for its value - , we have 2/2 n Z7'32 whence -1655 = This equation has two imaginary roots, and one real one, viz. : '12858. Hence the desired solution is given by cot 2e"= '12858 ; and 2e"=^7r 7 20', and the corres- ponding values of 2e and c' are 2e=|7r 16', and e'=^7r 32'. If these values for e, e, e" be used in the original equation (63), they will be found to satisfy it very closely ; and it appears that there is a true retardation of the moon of 3"'l in a century, whilst the lengthening of the day would make an apparent acceleration of 7"'l, the difference of the two being the observed 4". With these values the semi-diurnal and diurnal ocean-tides are, according to the equi- librium theory of ocean-tides, sensibly the same as those on a rigid nucleus, whilst the fortnightly tide is reduced to sin 2e" or '992 of its theoretical amount; and the time of high tide is accelerated by ;L 77, or 6^ hours in advance of its theoretical time.* I -' - a L * In the abstract of this paper (Proc. Roy. Soc., No. 191, 1878) the height and lag of the bodily ti * < " / (Usin4e+Vsin'2e) 13 Rsm4e The equation (43) of friction becomes //'/'\ (66) (67) And by (.38), Section 14, the equation of reaction becomes (08) * Phil. Trans., 1879, Part I. AND ON THE REMOTE HISTORY OF THE E 1RTH. 487 This is the third of the simultaneous differential equations which have to be treated. The four variables involved are i, N, , t, which give the obliquity, the earth's rotation, the square root of the moon's distance and the time. Besides where they are involved explicitly, they enter implicitly in Q, R, U, V, W, X, Z, sin 4e, sin 2e', sin 4e". Q, R, &c., are functions of the obliquity i only, but P is a constant. Also sln A- 4n "/^ T mflf ] P spvpra l ' ' attempts to solve these equations by retaining the time as independent variable, and substituting for and N" approximate values, but they were all unsatisfactory, because of the high powers of which occur, and no security could be felt that after a con- siderable time the solutions obtained did not differ a good deal from the true one. The results, however, were confirmatory of those given hereafter. The method finally adopted was to change the independent variable from t to . A new equation was thus formed between N and which involved the obliquity i only in a subordinate degree, and which admitted of approximate integration. This equation is in fact that of conservation of moment of momentum, modified by the effects of the solar tidal friction. Afterwards the time and the obliquity were found by the method of quadratures. As, however, it was not safe to push this solution beyond a certain point, it was carried as far as seemed safe, and then a new set of equations were formed, in which the final values of the variables, as found from the previous integration, were used as the initial values. A similar operation was carried out a third and fourth time. The operations were thus divided into a series of periods, which will be referred to as periods of integration. As the error in the final values in any one period is carried on to the next period, the error tends to accumulate ; on this account the integration in the first and second periods was carried out with greater accuracy than would in general be necessary for a speculative inquiry like the present one. The first step is to form the approximate equation of conservation of moment of momentum above referred to. Let A= W sin 4e+X sin 2e', B=Z sin 2e'. Then the equations of friction (67) and reaction (68) may be written, (69) (70) We now have to consider the proposed change of variable from t to The full expression for - contains a number of periodic terms ; -^f also contains dN terms which are co-periodic with those in -j-. Now the object which is here in view etc MDCCCLXXIX. 3 B 488 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, is to determine the increase in the average value of N per unit increase of the average value of The proposed new independent variable is therefore not f, but it is the average value of ; but as no occasion will arise for the use of f as involving periodic terms, I shall retain the same symbol. In order to justify the procedure to be adopted, it is necessary to show that, if f(t) be a function of t, then the rate of increase of its average value estimated over a period T, of which the beginning is variable, is equal to the average rate of its increase estimated over the same period. Now the average value of f(t) estimated over the 1 f' +T period T, beginning at the time t is - f(t)dt, and therefore the rate of the increase d i f <+T i f +T of the average value is j. f(t)dt, which is equal to - f(t)dt ; and this last {(t J. J f Ji j i expression is the average rate of increase of f(t) estimated over the same period. This therefore proves the proposition in question. dN Now suppose we have = M+ periodic terms, where M varies very slowly; then M is the average value of the rate of increase of N estimated over a period which is the least common multiple of the periods of the several periodic terms. Hence by the above proposition M is also the rate of increase of the average value of N estimated over the like period. Similarly if -p=X + periodic terms, X is the rate of increase of the average value CD of estimated over a period, which will be the same as in the former case. But the average value of A" is the proposed new dependent variable, and the average value of f the new independent variable. Hence, from the present point of view, = . This argument is, however, only strictly applicable, supposing there are not periodic terms in or ~ of incommensurable periods, and supposing the periodic terms dt (tt are rigorously circular functions, so that their amplitudes and frequencies are not func- tions of the time. It is obvious, however, that if the incommensurable terms do not represent long inequalities, and if M and X vary slowly, then the theorem remains very nearly true. With respect to the variability of amplitude and frequency, it is only necessary to pos- tulate that the so-called periodic terms are so nearly true circular functions that the integrals of them over any moderate multiple of their period is sensibly zero, to apply the argument. Suppose, for example, $(t) cos (<+x(*)) were one of the periodic terms, then we have only to suppose that /(<) and x(t) var y s slowly that they remain sensibly constant n during a period - or any moderately small multiple of it, in order to be safe in t ^ assuming t//(<) cos (rt+xC*))^ ^ sensibly zero. Now in all the inequalities in N and Jo AND ON THE REMOTE HISTORY OP THE EARTH. 489 it is a question of days or weeks, whilst in the variations of the amplitudes and frequencies of the inequalities it is a question of millions of years. Hence the above method is safely applicable here. It is worthy of remark that it has been nowhere assumed that the amplitudes of the periodic ioequalities are small compared with the non-periodic parts of the expression. A precisely similar argument wiU be applicable to every case where occasion will arise to change the independent variable. The change will accordingly be carried out without further comment, it being always understood that both dependent and inde- pendent variable are the average values of the quantities for which their symbols would in general stand. * Then dividing (69) by (70) we have dN Now =- =sin 2 i- approximately. This approximation will be suffi- A w sin 4e Sln 4 e sin 2e' ^ ciently accurate, because the last term is small and is diminishing. For the same reason, only a small error will be incurred by treating it as constant, provided the integration be not carried over too large a field a condition satisfied by the proposed "periods of integration." Attribute then to i, e, e' average values, and put \ 2 lT , . 3 .sin2e' J y=Vo sm l ^Te (72) and integrate. Then we have This is the approximate form of the equation of conservation of moment of momentum, and it is- very nearly accurate, provided does not vary too widely. By putting /3=0, y=0, we see that the equation is independent of the obliquity, if there be only two bodies, the earth and moon, provided we neglect the fourth power of the sine of the obliquity. The equation of reaction (68) may be written (74) * In order to feel complete confidence in my view, I placed the question before Mr. E. .T. ROUTH, and with great kindness he sent me some remarks on the subject, in which he confirmed the correctness of my procedure, although he arrived at the conclusion from rather a different point of view. - 3 R 2 490 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, Also, multiplying the equation of obliquity (66) by , we have lo I O I tt 1 1 I to I J -'o and we get , tan?'(l -sin 2 i) 10 tan i (l J sin 2 i ) from which the value of i corresponding to ='88 may easily be found. It is here useless to calculate more than four values, because the function to be integrated does not vary rapidly. We have now got final values of i, N, t corresponding to = '88. Since the earth is supposed to be viscous throughout the changes, therefore its figure must always be one of equilibrium, and its ellipticity of figure e=N*e . Also since = ( -~ ) = A / -, where c is the moon's distance from the earth, therefore \UJ V C c 3 {c \ -=*[-), which gives the moon's distance in earth's mean radii. a \a/ The fifth and sixth column of Table IV. were calculated from these formulas. The seventh column of Table IV. shows the distribution of moment of momentum in the system; it gives p. the ratio of the moment of momentum of the moon's and earth's motion round their common centre of inertia to that of the earth's rotation round its axis, at the beginning of each period of integration. Table I. shows the values of e, e', e" the angles of lagging of the semi-diurnal, diurnal, and fortnightly tides at the beginning of each period. Tables II. and III. show the relative importance of the contributions of each term to the values of and Iog 10 tan i'(l -^sin 2 i) at the beginning of each period. CLL sT The several lines of the Tables II. and III. are not comparable with one another, because they are referred to different initial values of fl and n in each line. I will now give some details of the numerical results of each integration. The 492 MB. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, computation as originally carried out* was based on a method slightly different from that above explained, but I was able to adapt the old computation to the above method by the omission of certain terms and the application of certain correcting factors. For this reason the results in the first three tables are only given in round numbers. In the fourth table the length of day is given to the nearest five minutes, and the obliquity to the nearest five minutes of arc. The integration begins when the length of the sidereal day is 23 hrs. 56 min., the moon's sidereal period 27 '3217 m. s. days, the obliquity of the ecliptic 23 28', and the tune zero. First period. Integration from =1 to '88; seven equidistant values computed for finding the time, and four for the obliquity. For the obliquity the integration was not carried out exactly as above explained, in as far as that ,-log 10 tant was found instead of ~log 10 tan i(l ^sin 2 t), but the differ- ence in method is very unimportant. The result marked* in Table III. is Iog 10 tan L The estimated average value of i was 22 15'. The final result is #=1-550, t=20 42', 2 = 46,301,000 Second period. Integration from =1 to '76 ; seven values computed for the time, and four for the obliquity. The estimated average for i was 19. The final result N= 1-559,1=17 21', =10,275,000 Third period. Integration from f=l to 76 ; four values computed. The estimated average for i was 16 30'. The final result #=1-267, i=15 30', -^ = 326,000 Fourth period. Integration from =1 to 76 : four values computed. The estimated average for i was 1 5. The small terms in /8 and y were omitted in the equation of conservation of moment of momentum. All the solar and combined terms, except that in V in the equation of obliquity, were omitted. The final result #=V160, i=14 25', -=10,300 * I have to thank Mr. E. M. LANQLEV, of Trinity College, for carrying out the laborious computations. The work was checked throughout by myself. AND ON THE REMOTE HISTORY OF THE EARTH. 493 TABLE I. Showing the lagging of the several tides at the beginning of each period. Semi-diurnal Diurnal ,iir which will give the integral of the Jo a s! right-hand side from to 1 ; and this is equal to log tan 2 - log tan 2 |. The integration was divided into a number of periods, just as hi the solution of Section 15. The following were the results : AND ON THE REMOTE HISTORY OF THE EARTH. 503 First period. From =1 to '88; /u=4'0074; i=2Q 28'; 2V=1'5478. The term in -- in the expression for iV added '0012 to the value of N~. Second period. From =1 to 76; /*=2'2784; i=l7 4'; #== 1-5590. The term in added '0011 to the value of N. "o Third period. From f=l to 76; /i=l-1107; /=15 22'; #=1-2677. The term in - - added '0007 to the value of 2V. *g It may be observed that during the first period of integration diminishes, and iT/ reaches its minimum about the end of the period. During the rest of the integration it increases. If we neglect the solar action and the obliquity, it is easy to find the n n n i , . . . dN -SN minimum value of, ror = - and reaches its minimum when -rr= -- r- ; but n n H N d = -p. Therefore N= Now N= !+/*(]- ), and hence f = J . If n= 4, f ' O PJ ^^yf ='9375. This value of f is passed throvigh at near the end of the first period of integration. At this period there are 19 '2 mean solar hours in the day; 22^ mean solar days in the sidereal month ; and 2 87 rotations of the earth in the sidereal month. This result of 28 7 is, of course, only approximate, the true result being about 29.* The physical meaning of these results is given in a table below. At the end of the third period of integration the solar terms (those in ) have become small in all the equations, and as they are rapidly diminishing they may be safely neglected. To continue the integration from this point a slight, variation of method will be convenient. Our equations may now be written approximately J\T=1 +/*(!-) 2/2 1 1 -- sect In order to find how large a diminution of obliquity is possible if the integration be continued, we require to stop at the point where n cos ?'=2/2. Now the equation N=l+p(l g) may be written * The subject is referred to from a more general point of view in. a paper on the " Secular Effects of Tidal Friction," see ' Proc. Roy. Soc.,' No. 197^1879. MDCCCLXXIX. 3 T 504 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, If therefore we put x=/n, we must stop the integration at the point where ?i= 2a^ sec *', x being given by the equation 2.r* sec i_ n And if we assume i=14, x is given by i 1 1 J O/ -i I \ I-- -I i O r\ -^' because /*= l-^sn^nj. Now at the end of the third period of integration, which is the beginning of the new period, I found log =3-84753, log /x=9-82338 10, and log s=. r r3937S 10 The unit of time being the present tropical year. Hence the equation is aJ* 5690*4-19586=0 The required root is nearly ^75090, and a second approximation gives x=fl*= 16703 (16'51 would have been more accurate). But /2 *=8'61G. Hence we desire to stop the integration when 16703 Now /x='6659 ; hence when f='516, A T =1'322. In order to integrate the equation of obliquity by quadratures, I assume the four equidistant values, #=1-000, 1-107, T214, 1-321 And by means of the equation =1 f G ~ q =l (#-l)(l'502) the corresponding values of are found to be 1-000, -8393, -G786, '5179 Then by means of the formula = T^, the corresponding values of are found n n u ./Vf to be 0909, -1388, -2395, '4951 I assumed conjecturally four values of i lying between i =16 22' and i=14, which I knew would be very nearly the final value of i ; and then computed four equidistant d i values of Iog 10 tan -. . The values were 19381, '16230, '11882, '00684. The fact that the last value is negative shows that the integration is carried a little beyond the point when n cos i=2fl, but this is unimportant. AND OX THE KKMOTK HISTORY OF THE EARTH. 505 Combining these values by the rules of the calculus of finite differences, I find /=1359'. This final value of (viz.: '5179) makes the moon's sidereal period 12 hours, and the value of N (viz.: T321) makes the day 5 hours 55 minutes. These results complete the integration of the fifth period. The physical meaning of the results for all five periods is given in the following table : Sidereal day in m.s. Moon's sidereal period Obliquity of hours and minutes. in m.s. days. ecliptic. h. m. Initial 23 56 27-32 days 23 28' 15 28 18-62 20 28' 9 55 8-17 17 4' 7 49 3-59 15 22' * Final 5 55 12 hours 14 0' * It is worthy of notice that at the end of the first period there were 2 8 "9 days of that time in the then sidereal month ; whilst at the end of the second period there were only 197. It seems then that at the present time tidal friction has, in a sense, done more than half its work, and that the number of days in the month has passed its maximum on its way towards the state of things in which the day and month are of equal length as investigated in the following section. In the last column of the preceding table the last two results in the column giving the obliquity of the ecliptic (which are marked with asterisks) cannot safely be accepted, because, as I have reason to believe, the simultaneous changes of inclination of the lunar orbit will, after the end of the second period of integration, have begun to influence the results perceptibly. For this same reason the integration, which has been carried to the critical point where n fcos 2 = 2/2, and where changes sign, will not be pursued any further. Never- theless we shall be able to trace the moon's periodic time, and the length of day to their initial condition. It is obvious that as long as n is greater than /2, there will be tidal friction, and n will continue to approach /2, whilst both increase retrospectively in magnitude. I shall now refer to a critical phase in the relationship between n and fl, of a totally different character from the preceding one, and which must occur at a point a little more remote in time than that at which the above integration stops. This critical phase occurs when the free nutation of the oblate spheroid has a fre- quency equal to that of the forced fortnightly nutation. In the ordinary theory of the precession and nutation of a rigid oblate spheroid,, the fortnightly nutation arises out of terms in the couples acting about a pair of axes fixed in the equator, which have speeds n 2/2 and n+ 2/2. If and A be the greatest and least principal moments of inertia, then on integration these terms are 3 T 2 506 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, divided by - r< + 71^2/2 and give rise to terms in y and -~ sin i of speed 2/2. When " (\v ((( 2/2 is neglected compared with n, we obtain the formula, given in any work on physical astronomy, for the fortnightly nutation. C A Now it is obvious that if - n-j- = 2/2, the former of these two terms becomes XL infinite. Since in our case the spheroid in homogeneous r~ =e the ellipticity of the A. /i2 spheroid ; and since the spheroid is viscous e=^ . Therefore the critical relationship n 3 is \-n = '2n. When this condition is satisfied the ordinary solution is nugatoiy, and the true solution represents a nutation the amplitude of wliich increases with the time. 2/j The critical point where the above integration stops is given by - =cos i, and this critical point by =!-}-!- ; it follows therefore that - is little larger in the second case than in the first. Therefore this critical point has not been already reached where the integration stops, but will occur shortly afterwards. It is obvious that the amplitude of the nutation cannot increase for an indefinite time, because the critical relationship is only exactly satisfied for a single instant. In fact, the problem is one of far greater complexity than that of ordinary disturbed rotation. The system is disturbed periodically, but the periodic time of the disturb- ance slowly increases, passing through a phase of equality to the free periodic time ; the problem is to find the amplitude of the oscillations when they are at their maximum, and to find the mean configuration of the system some time before and some time after the maximum, when the oscillations are small. This problem does not seem to be soluble, unless we take into account the slow variation of the argument in the periodic disturbing term ; and when the argument varies, the disturbing term is not strictly a simple time harmonic. In the case of the viscous spheroid, the question would be further complicated by the fact that when the nutation becomes large, a new series of bodily tides is set up by the effects of inertia. I have been unable to make a satisfactory examination of this problem, but as far as I have gone it appeared to me probable that the mean obliquity of the axis of the spheroid would not be affected by the passage of the system through a phase of large nutation ; and although I cannot pretend to say how large the nutation might be, yet I consider it probable that the amplitude would not have time to increase to a very wide extent.""' * I believe that I shall be able to show in an investigation, as yet incomplete, that when this critical phase is reached, the plane of the lunar orbit is nearly coincident with the equator of the earth. As the amplitude of this nutation depends on the sine of the obliquity of the equator to the lunar orbit, it seems probable that the nutation would not become considerable. June 30, 1879, AND ON THE REMOTE HISTORY OF THE EARTH. 507 Throughout all the preceding investigations, the periodic inequalities have been neglected. Now a full development of the couples fL, Jffl, ^Ht, which are due to the tides, shows that there occur terms of speeds n 2/2, and n 4/2 in the first two, and of speeds 2/2 and 4/2 in the last. The terms in n 2/2 in H, and ffll will clearly give rise to an increasing nutation- at the critical point which we are considering, but they will be so very much smaller than those arising out of the attraction on the permanent equatorial protuberance that they may be neglected. The terms in n 4/2 are multiplied by very small quantities, and I think it may safely be assumed ' that the system wpuld pass through the critical phase where \ \-n=4fl with sufficient rapidity to prevent the nutation becoming large. If we were to go to higher orders of approximation in the disturbing forces, it is clear that we should meet with an infinite number of critical phases, but the coefficients representing the amplitudes of the resulting nutations would be multiplied by such small quantities that they may safely be neglected. 18. The initial condition of the earth and moon.* It is now supposed that, when the earth's rotation has been tracked back to where it is equal to twice the moon's orbital motion, the obliquity to the plane of the lunar orbit has become zero. Then it is clear that, as long as there is any relative motion of the earth and moon, the tidal friction and reaction must continue to exist, and n and fl must tend to an equality. The previous investigation shows also that for small viscosity, however nearly n approaches fl, the position of zero obliquity is dynamically stable. As n is approaching fl, the changes must have taken place more and more slowly in time. For if the earth was a cooling spheroid, it is unreasonable to suppose that the process of becoming less stiff in consistency (which has hitherto been supposed to be taking place, as we go backwards in time) could ever have been reversed ; and if it were not reversed, then the lunar tides must have lagged by less and less, as more and more time was given by the slow relative motion of the two bodies for the moon's attraction to have its full effect. Hence the effects of the sun's attraction must again become sensible, after passing through a phase of insensibility a phase perhaps short in time, but fertile in changes in the system. I shall not here make the attempt to trace the reappearance of these solar terms. It is, however, possible to make a rough investigation of what must have been the initial state from which the earth and moon started the course of development, which has been tracked back thus far. To do this, it is only necessary to consider the equa- tion of conservation of moment of momentum. * For further consideration of this subject, see a paper on the " Secular Effects of Tidal Friction," ' Proc. Roy. Soc.,' No. 197, 1879. The arithmetic of this section has been recomputed since the paper was presented. 508 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS S1MI KUOI 1>. When the obliquity is neglected, that equation may be written ' = 1 -f/x| 1 ( ^rM L "u I W / J and it is proposed to find what values of n would make n equal to fl. In the course of the above investigation four different starting points were taken, viz. : those at the beginning of each period of integration. There are objections to taking any one of these, to give the numerical values required for the solution of the above equation ; for, on the one hand, the errors of each period accumulate on the next, and therefore it is advantageous to take one of the early periods ; whilst, on the other hand, in the early periods the values of the quantities are affected by the sensi- bility of the solar terms, and by the obliquity of the ecliptic. The beginning of the fourth period was chosen, because by that time the solar terms had become insigni- ficant. At that epoch I found log n =3'84753, when the present tropical year is the unit of time, and /A='6659, ft, being the ratio of the orbital moment of momentum to the earth's moment of momentum; also log s=5'39378 10, s being a constant. Now put x s =n=n, and we have x*(l +fj.)n (r 'c+ 7=0 Then substituting the numerical values, .*' 11727.r+40385 = This equation has two real roots, one of which is nearly equal to j/ 1 11 727, and the other to 40385 -j- 11 727. By HOENEE'S method these roots are found to be 2T4320 and 3'4559 respectively. These are the two values of the cube root of the earth's rotation, for which the earth and moon move round as a rigid body. The first gives a day of 5 hours 36 minutes, and the second a day of about 55^ m. s. days. The latter is the state to which the earth and moon tend, under the influence of tidal friction (whether of oceanic or bodily tides) in the far distant future. For this case THOMSON and TAIT give a day of 48 of our present days;*' the discrepancy between my value and theirs is explicable by the fact that they are considering a heterogeneous earth, whilst I treat a homogeneous one. Since on the hypothesis of heterogeneity the earth's moment of inertia is about ^Mo-, whilst on that of homogeneity it is -f Mtr, and since the f which occurs in the quantity s enters by means of the expression for the earth's moment of inertia, it follows that in my solution p, has been taken too small in the proportion 5 : 6. Hence if we wish 'to consider the case of heterogeneity, we must solve the equation x* 12664x+48462 = 0. The two roots of this equation are such that they give as the corresponding lengths of the day, 5 hours 16 minutes and 40'4 days respectively. The remaining discrepancy (between 40 and 48) is doubtless due in part * 'Nat. Phil.,' 276. They say: " It is probable that the moon, in ancient times liquid or viscous in its outer layer or throughout, was thus brought to turn always the same face to the earth." In the new edition (1879) the ultimate effects of tidal friction are considered, AND OX THE REMOTE HISTORY OF THE EARTH. 501) to the crude method of amending the solution, but also to the fact that they partly in- clude the obliquity in one way, whilst I partly include it in another way, and I include a large part of the solar tidal friction whilst they neglect it. It is interesting to note that the larger root, which gives the shorter length of day, is but little affected by the consideration of the earth's heterogeneity. With respect to the second solution (5G days), it must be remarked that the sun's tidal friction will go on lengthening the day even beyond this point, but then the lunar tides will again come into existence, and the lunar tidal friction will tend in part to counteract the solar. The tidal reaction will also be reversed, so that the moon will again approach the earth. Thus the effect of the sun is to make this a state of dynamical instability. The first solution, where both the day and month ai'e 5 hours 36 minutes long, is the one which is of interest in the present inquiry, for this is the initial state towards which the integration has been running back. This state of things is one of dynamical instability, as may be shown as follows : First consider the case where the sun does not exist. Suppose the earth to be rotating in about 5^ hours, and the moon moving orbitally around it in a little less than that time. Then the motion of the moon relatively to the earth is consentaneous with the earth's rotation, and therefore the tidal friction, small though it be, tends to accelerate the earth's rotation ; the tidal reaction is such as to tend to retard the moon's linear velocity, and therefore increase her orbital angular velocity, and reduce her distance from the earth. The end will be that the moon falls into the earth. This subject is graphically illustrated in a paper on the " Secular Effects of Tidal Friction," read before the Royal Society on Jime 19, 1879. Secondly, take the case where the sun also exists, and suppose the system started in the same way as before Now the motion of the earth relatively to the sun is rapid, and such that the solar tidal friction retards the earth's rotation ; whilst the lunar tidal friction is, as before, such as to accelerate the rotation. Hence if the viscosity be very large the earth's rotation may be accelerated, but if it be not very lai'ge it will be retarded. The tidal reaction, which depends on the lunar tides alone, continues negative, and the moon approaches the earth as before. Thus after a short time the motion of the moon relatively to the earth is more rapid than in the previous case, whatever be the ratio between solar and lunar tidal friction. Hence in this case the moon will fall into the earth more rapidly than if the sun did not exist, and the dynamical instability is more marked. If, however, the day were shorter than the month, the moon must continually recede from the earth, until it reaches the outer limit of a day of 56 m. s. days. There is one circumstance which might perhaps decide that this should be the direction in which the equilibrium would break down ; for the earth was a cooling * From here to the end of the section a good many alterations have been made since the paper was presented. July 5, 1879. 510 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, body, and therefore probably a contracting one, and therefore its rotation would tend to increase. Of course this increase of rotation is parti)- counteracted by the solar tidal friction, but on the present theory, the mere existence of the moon seems to show that it was not more than counteracted, for if it had been so the moon must have been drawn into and confounded with the earth. This month of 5 hours 36 minutes corresponds to a lunar distance of 2'52 earth's mean radii, or about 10,000 miles; the month of 5 hours 16 minutes corresponds to 2'39 earth's mean radii ; so that in the case of the earth's homogeneity only 6,000 miles intervene between the moon's centre and the earth's surface, and even this distance would be reduced if we treated the earth as heterogeneous. This small distance seems to me to point to a break-up of the earth-moon mass into two bodies at a tune when they were rotating in about 5 hours ; for of course the precise figures given above cannot claim any great exactitude (see also Section 23). It is a material circumstance in the conditions of the breaking-tip of the earth into two bodies to consider what would have been the ellipticity of the earth's figure when rotating in 5^ hours. Now the reciprocal of the ellipticity of a homogeneous fluid or viscous spheroid varies as the square of the period of rotation of the spheroid. The reciprocal of the ellipticity for a rotation in 24 hours is 232, and therefore the reciprocal of the ellipticity for a rotation in 5^ hours is (|i)- of 232=-igf 4 -X 232= 12 -2. Hence the ellipticity of the earth when rotating in 5^ hours is i^th. The conditions of stability of a rotating mass of fluid are as yet unknown, but when we look at the planets Jupiter and Saturn, it is not easy to believe that an ellipticity of y^th is sufficiently great to cause the break-up of the spheroid. A homogeneous fluid spheroid of the same density as the earth has its greatest ellipticity compatible with equilibrium when rotating in 2 hours 24 minutes.* The maximum ellipticity of all fluid spheroids of the same density is the same, and their periods of rotation multiplied by the square root of their densities is a function of the ellipticity only. Hence a spheroid, which rotates in 4 hours 48 minutes, will be in limiting equilibrium if its density is (|) 3 or ^ of that of the earth. If this latter spheroid had the same mass as the earth, its radius would be ^/4 or 1'59 of that of the earth. If therefore the earth had a radius of 6,360 miles, and rotated in 4 hours 48 minutes, it would just have the maximum ellipticity compatible with equilibrium. It is, however, by no means certain that instability would not have set in long before this limiting ellipticity was reached. In Part III. I shall refer to another possible cause of instability, which may perhaps be the cause of the break-up of the earth into two bodies. It is easy to find the minimum time in which the system can have passed from this initial configuration, where the day and month are both 5^ hours, down to the present * PBATT'S ' Fig. of Earth,' 2nd edition., Arts. (38 and 70. AND ON THE REMOTE HISTORY OF THE EARTH. 511 condition. If we neglect the obliquity of the ecliptic, the equation (57) of tidal reaction, when adapted to the case of a viscous spheroid, becomes i * sin 4e, Now it is clear that the rate of tidal reaction can never be greater than when n 46! = 1, when the lunar semi-diurnal tide lags by 22 obtain the minimum time by integrating the equation sin 46! = 1, when the lunar semi-diurnal tide lags by 22-|. Then since T=-~, we shall Whence _ft _ , , ]3 , "13 r^ 1 " Now =(-^| , and we have found by the solution of the biquadratic that the initial con- \ft / dition is given by /2*=2r4320 ; also with the present value of the month /2 '=4'38, the present year being in both cases the unit of time. Hence it follows that f is very nearly '2, and f 13 may be neglected compared with unity. Thus 1=^ |. lo TO" ftyt Now p.= 4-007 and ^ is 86,844,000 years. T o Hence t= 53,540,000 years. Thus we see that tidal reaction is competent to reduce the system from the initial state to the present state in something over 54 million years. The rest of the paper is occupied with the consideration of a number of miscellaneous points, which it was not convenient to discuss earlier. 19. The change in the length of year. The effects of tidal reaction on the earth's orbit round the sun have been neglected ; I shall now justify that neglect, and show by how much the length of the year may have been altered. It is easy to show that the moment of momentum of the orbital motion of the moon Q and earth round their common centre of inertia is -^7, where C is the earth's moment of inertia, and s=| f 1 L\ J 1 The moment of momentum of the earth's rotation is obviously Cn. The normal to the lunar orbit is inclined to the earth's axis at an angle i. Hence the resultant moment of momentum of the moon and earth is MDCCCLXXIX. 3 U 512 ME. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, The change in this quantity from one epoch to another is the amount of moment of momentum of the moon-earth system which has been destroyed by solar tidal friction. This destroyed moment of momentum reappears in the form of moment of momentum of the moon and earth in their orbital motion round the sun. Now at the beginning of the integration of Section 17, that is to say at the present time, I find that when the present year is taken as the unit of time, the resultant moment of momentum of the moon and earth is 11369 C. At the end of the third period of integration (after which the solav terms were neglected), and when the obliquity has become 15 22', I find the same quantity to be 11625 C. Hence the loss of moment of momentum is 256 C., or 102 '4 Ma*. Now at the present time the moment of momentum of the moon and earth in their orbit is (M+m)flc-=Ma 2 . ( -} fl ; - is clearly the sun's parallax, and with the v \a/ c t present unit of time fl, is 2ir. Hence the loss of moment of momentum is equal to the present moment of momentum of orbital motion multiplied by - (sun's parallax) 2 . But the moment of momentum of the earth's and moon's orbital motion round the sun varies as fl~ l ; hence the loss of moment of momentum corresponding to a change ?(-\ of fl, to fl, +8/2, is the present moment of momentum multiplied byi^r, whence it is clear that . -=3 - --- X (suns parallax)-. fl t 27T 1 + V But the shortening of the year is -^ of a year ; taking therefore the sun's parallax JZ, as 8"'8, we find that at the end of the third period of integration the year was shorter than at present by 2 X 365-25X86,400 seconds, which will be found equal to 277 seconds. Thus the solar tidal reaction had only the effect of lengthening the year by 2| seconds, since the epoch specified as the end of the third period of integration. The whole change in the length of year since the initial condition to which we traced back the moon would probably be very small indeed, but it is. impossible to make this assertion positively, because, as observed above, the solar effects must have again become sensible, after passing through a period of insensibility. AND ON THE REMOTE HISTORY OF THE EARTH. 513 20. Terms of the second order in the tide-generating potential. The whole of the previous investigation has been conducted on the hypothesis that the tide-generating potential, estimated per unit volume of the earth's mass, is tw 2 (cos 2 PM ),* but in fact this expression is only the first term of an infinite series. I shall now show what kind of quantities have been neglected by this treat- ment. According to the ordinary theory, the next term of the tide-generating potential is ' cos 3 PM-f cos PM) c c Although for my own satisfaction I have completely developed the influence of this term in a similar way to that exhibited at the beginning of this paper, yet it does not seem worth while to give so long a piece of algebra ; and I shall here confine myself to the consideration of the terms which will arise in the tidal friction from this term in the potential, when the obliquity is neglected. A comparison of the result with the value of the tidal friction, as already obtained, will afford the requisite information as to what has been neglected. Now when the obliquity is put zero (see Plate 36, fig. 1), cos PM = sin 9 sin (< w) where a> is written for n fl for brevity. Then cos 3 PM=f sm 3 sin (0- w) -1 sin 3 sin 3(<- w) and cos 3 PM f cos PM=- 2 % sin 0(1 5 cos 3 0) sin (< &>) ^ sin 3 0sin 3(< w). Then since m/7'\ 3 5 r 3 5 -(-) -=WT- - c c 2 c 3 therefore Vg-s-w-r^ ; & sin 3 sin 3(< ) If sin 3( to) and sin (< w) be expanded, we have V 3 in the desired form, viz. : a series of solid harmonics of the third degree, each multiplied by a simple time har- monic. Now if 1^83 cos (vt-\-if)) be a tide-generating potential, estimated per unit volume of a homogeneous perfectly fluid spheroid of density w, S 3 being a surface har- monic of the third order, then the equilibrium tide due to this potential is given by cr= S 3 cos (vt-\-rj), or =^ S 3 cos (vt-}-r)}. Hence just as in Section 2, the tide- 4^/ a J- U CJ * See Section 1 . 3 u 2 514 MB. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, generating potential of the third order due to the moon will raise tides in the earth, when there is a frictional resistance to the internal motion, given by -= - - - . &Fsin 3 6 sin 3 (0- w+/) +$F' sin (1 - 5 cos 2 6) sin (^-cu a J-U QC I Now , is a\* 3 Ma* Hence the moment about the earth's axis of the forces which the attraction of the distorted spheroid exercises on a pai'ticle of mass m, situated at r, 6, , is z 7- TT. Now if this mass be equal to that of the moon, and r=c, then - =- -J/a 2 =z-C, 11* 1C 1C where, as before, C is the moment of inertia of the earth. Hence the couple ffi 2 , which the moon's attraction exercises on the earth, is given by Jt,= -(7, where after differentiation we put 0= - and <=-+w. 7 c a

+/) - \F sin ^(1 ^ 5 cos=0) cos (<^>- Hence In the case of viscosity F=cos3f, F'=cosf Therefore Now if the obliquity had been neglected, the tidal friction j^, due to the term of the first order in the tide-generating potential, would be given by - - 1 =-| sin 4ei. Hence jEt 2 , M 8 /5 sin 6/+ sin 2/\ jElj \c/ \ siu4e] / That is to say, this is the ratio of the terms neglected previously to those included. AND ON THE REMOTE HISTORY OF THE EARTH. 515 Now according to the theory of viscous tides,* / , - + a> 22/Q sv tan 3f= - u= f f (3) 3 i__ n jt Cdfi '~~dt~~2 g dt~ ~% c" -* sin ^ cos t at g Now numerical calculation shows that at present '= -r , and since ?- sin i cos i is 5 10' C?i yi /a;v of the same order of magnitude as > (on which the changes of obliquity have been shown to depend), it follows that this term is fairly negligeable compared with those already included in the equations. As far as it goes, however, this term tends in the direction of increasing the obliquity with the time.t * " On the Influence of Geological Changes," &c., Phil. Trans, Vol. 167, Part I., page 272, Section 8. The notation is changed, and the equation presented in a form suitable for the present purpose. f In a paper in the ' Phil. Mag.,' March, 1877, I suggested that the obliquity might possibly be due to the contraction of the terrestrial nebula in cooling ; I there neglected tidal friction and assumed the con- servation of moment of momentum to hold good for the earth by itself, so that the ellipticity was con- tinually increasing with the time. I did not at that time perceive that this increase of ellipticity was antagonistic to the effects of contraction. Though the work of that paper is correct, as I believe, yet the fundamental assumption is incorrect, and therefore the results are not worthy of attention. AXD ON THE REMOTE HISTORY OF THE EARTH. 517 [It will however appear, I believe, that this secular change of ellipticity of the earth's figure will exercise an important influence on the plane of the lunar orbit and thereby will affect the secular change in the obliquity of the ecliptic. The investiga- tion of this point is however as yet incomplete.]* The other small term which I shall consider arises out of the ordinary precession, together with the fact that the tide -generating force diminishes with the time on account of the tidal reaction on the moon. The differential equations which give the ordinary precession are in effect (compare equations (26)) da> l C-A . . . jf = T - sin i cos i sin n (It O dcO a C A . jf= T sm i cos i cos n at O and they give rise to no change of obliquity if r be constant, but when t is small. Also ~TT~ =6=-] =^ Hence as far as regards the change of obliquity the equations may be written 37= ^-^- ( 37 ) sin i cos i t sin n at 8 V** >7 f' f I c ) sin i cos i t cos w , , at dt Then if we regard all the quantities, except t, on the right-hand sides of these equations as constants and integrate, we have . . . (>! = I 37 1 sin i cos l\wt cos n sin n} 3rJdf\ (t). 2 = sm i cos -ilnt sin + cos } 8 \ ( " / And if these be substituted in the geometrical equations (l) we have di 3r ./df = sin i cos t -r 5 - dt g \dt * Added July 3, 1879. 518 MR. G. H. DARWIX ON THE PRECESSION OF A VISCOUS SPHEROID, Now by comparing this with the small term due to the secular change of figure of the earth, we see that it is fairly negligeable, being of the same order of magnitude as that term. As far as it goes, however, it tends to increase the obliquity of the ecliptic. 22. The change of obliquity and tidal friction due to an annular satellite. Conceive the ring to be rotating round the planet with an angular velocity fl, let its radius be c, and its mass per unit length of its arc- , so that its mass is m. Let cl be &7TC the length of the arc measured from some point fixed in the ring up to the element c8Z ; and let fit be the longitude of the fixed point in the ring at the time t. Let 8V m. be the tide-generating potential due to the element ^~8/. Then we have by (5) Where the suffixes to the functions indicate that /2-M is to be written for fl. Then integrating all round the ring from Z=0 to Z=2ir it is clear that y = sin 2 6 cos 2< n + 2" * sin 6 cos 6 cos < which is the tide-generating potential of the ring. Hence, as in Section 2, the form of the tidally-distorted spheroid is given by (9), save that E, E^ E\, E z , E" are all zero. Also, as in that section, the moments of the forces which the tidally-distorted spheroid exerts on the element of ring are -.AMa/ da- jla\ *T~~*~' ' ' w ^ ere *"' /r l r > * r are P u * e< l uai to * ne rectangular coordinates of the element of ring, whose annular coordinate is I. Now if x, y, z are the direction cosines of the element, equations (7) are simply modified by fl being written fl-\-l. Hence the couples due to one element of ring may be found just as the whole couples were found before, and the integrals of the elementary couples from =0 to 2ir are the desired couples due to the whole ring. Now a little consideration shows that the results of this integration may be written down at once by putting E lt E. 2 , E\, -E" 2 , E" zero in (15), (16), and (21). Thus in order to determine the change of obliquity and the tidal friction due to an annular satellite, we have simply the expressions (33) and (34), save that rr, must be replaced It thus appears that an annular satellite causes tidal friction in its planet, and that the obliquity of the planet's axis to the ring tends to diminish, but both these AND ON THE REMOTE HISTORY OF THE EARTH. 519 effects are evanescent with the obliquity. Since this ring only raises the tides which are called sidereal semi-diurnal and sidereal diurnal, and since we see by (57), Section 1 4, that tidal reaction is independent of those tides, it follows that there is no tangential force on the ring tending to accelerate its linear motion. If, however, the arc of the ring be not of uniform density, there is a slight tendency for the lighter parts to gain on the heavier, and the heavier parts become more remote from the planet than the lighter. 23. Double tidal reaction. Throughout the whole of this investigation the moon has been supposed to be merely an attractive particle, but there can be no doubt but that, if the earth was plastic, the moon was so also. To take a simple case, I shall now suppose that both the earth and moon are homogeneous viscous spheres revolving round their common centre of inertia, and that the moon is rotating on her own axis with an angular velocity w, and that their axes are parallel and perpendicular to the plane of their orbit. Then the whole of the argument with respect to the earth as disturbed by the moon, may be transferred to the case of the moon as disturbed by the earth. All symbols which apply to the moon will be distinguished from those which, apply to the earth by an accent. Then from (21) or (43) we have and the equation which gives the lunar tidal friction is =-^81114^ (89) Now .M WO* and B' = ^ = ^='-a " s u' 5a w /'" So that 8' V'''V 9 Also (90) ( ' and therefore . . ^~a rt C a g w' MDCCCLXXIX. 3 X , Sill 4C 1 520 MK. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, Now the force on the moon tangential to her orbit, results from a double tidal reaction. By the method employed in Section 14, the tangential force due to the earth's tides is T-jtSf v-' T = =- sin 4e, and similarly the tangential force due to the moon's tides is riv ffi V T 2 i^ . , 1 = == -. sin 4e , r 2r g u-'-a and the whole tangential force is (T+T'). Hence following the argument of that section, the equation of tidal reaction becomes 7 1- > r~ * ~i C/E T" . < r '" . , P-~T. 2 sm 4ei+ sin 4e , Then taking the moon's apparent radius as 16', and the ratio of the earth's mass to ft, 7/7 that of the moon as 82, we have -- = 3'567 and =1'806 (so that taking w as 5^, the CL "tv specific gravity of the moon is 3), and hence ,= 11'64. At first sight it would appear from this that the effect of the tides in the moon was nearly twelve times as important as the effect of those in the earth, as far as concerns the influence on the moon's orbit, and hence it would seem that a grave oversight has been made in treating the moon as a simple attractive particle ; a little consideration will show, however, that this is by no means the case. Suppose that v', v are the coefficients of viscosity of the moon and earth respec- tively ; then the only tides which exist in each body being those of which the speeds are 2( becomes very nearly equal to fl, sin 4e'] becomes very small. Hence the term in the equation of tidal reaction dependent on the moon's own tides must have become rapidly evanescent. Now while this shows that the main body of our investigation is unaffected by the lunar tide, there is one slight modification of them to which it leads. In Section 18 we traced back the moon to the initial condition, when her centre was 10,000 miles from the earth's centre. If lunar tidal friction had been included, this distance would have been increased ; for the coefficient of x in the biquadratic *?/? ft ItTf/^^ (viz. : 11,727) would have to be diminished by - (cu w n ). Now - - is very nearlv ?r wo? J ' J ToVoth, and the unit of time being the year, it follows that we should have to suppose an enormously rapid primitive rotation of the moon round her axis, to make any sensible difference in the configuration of the two bodies when her centre of inertia moved as though rigidly connected with the earth's surface. The supposition of two viscous globes moving orbitally round their common centre of inertia, and one having a congruent and the other an incongruent axial rotation, would lead to some very curious results. 24. Secular contraction of the earth.'" If the earth be contracting as it cools, it follows, from the principle of conservation of moment of momentum, that the angular velocity of rotation is being increased. Sir WILLIAM THOMSON has, however, shown that the contraction (which probably now only takes place in the superficial strata) cannot be sufficiently rapid to perceptibly counteract the influence of tidal friction at the present time. * Rewritten in July, 1879. 3x2 522 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID. The enormous height of the lunar mountains compared to those in the earth seems. however, to give some indications that a cooling celestial orb must contract by a, perceptible fraction of its radius after it has consolidated."' Perhaps some of the contraction might be due to chemical combinations in the interior, when the heat had departed, so that the contraction might be deep-seated as well as superficial. It will be well, therefore, to point out how this contraction will influence the initial condition to which we have traced back the earth and moon, when they were found rotating as parts of a rigid body in a little more than 5 hours. Let C, C be the moment of inertia of the earth at any time, and initially. Then the equation of conservation of moment of momentum becomes And the biquadratic of Section 18 which gives the initial configuration becomes The required root of this equation is very nearly equal to (1+ju,) '\ " . Now y?=fl ; hence fl is nearly eqvial to (l+p) 1 . But in Section 18, when C was equal to C , it was nearly equal to (l+/t)ft . Therefore on the present hypothesis, the value * Suppose a sphere of radius a to contract until its radius is a + ia, but that, its surface being incom- pressible, in doing so it throws up n conical mountains, the radius of whose bases is 6, and their height Ji, and let 6 be large compared with h. The surface of such a cone is irb v /h !i + b^=7r(b 2 + ^/t 2 ). Hence the excess of the surface of the cone above the area of the base is $irk~, and 4:7ra-=^4:Tr(a+ca)~ + ^mrli'-. a n Therefore -- =7^(- a Lo\a Then suppose we have a second sphere of primitive radius a', which contracts and throws up the same Ka' w/7A2 -. *' g a A'i\ a number of mountains ; then similarly = I and -- ; =1 ; I. Now let these two spheres be a 16\a / a' a \/ia / the earth and moon. The height of the highest lunar mountain is 23,000 feet (GRANT'S 'Physical Astron.,' p. 229), and the height of the highest terrestrial mountain is 29,000 feet; therefore we may take -=if. Also - = -2729 (HERSCHEL'S 'Astron.,' Section 404). Therefore ~=M of '2729=-344, and ha It a (r ) ='1183 or ( - ) =8'45. Hence -=- =8| ; whence it appears that, if both lunar and terrestrial h a/ \IM / a a mountains are due to the crumpling of the surfaces of those globes in contraction, the moon's radius has been diminished by about eight times as large a fraction as the earth's. This is, no doubt, a very crude way of looking at the subject, because it entirely omits volcanic action from consideration, but it seems to justify the assertion that the moon has contracted much more than the earth, since both bodies solidified. AND ON THE REMOTE HISTORY OF THE EARTH. 523 /i of n as given in that section must be multiplied by -~ ; and the periodic time must be n multiplied by . But in this initial state C is greater than C ; hence the periodic ^o time when the two bodies move round as a rigid body is longer, and the moon is more distant from the earth, if the earth has sensibly contracted since this initial configuration. If, then, the theory here developed of the history of the moon is the true one, as I believe it is, it follows that the earth cannot have contracted since this initial state by so much as to considerably diminish the effects of tidal friction, and it follows that Sir WILLIAM THOMSON'S result as to the present unimportance of the contraction must have always been true. If the moon once formed a part of the earth we should expect to trace the changes back until the two bodies were in actual contact. But it is obvious that the data at our disposal are not of sufficient accuracy, and the equations to be solved are so com- plicated, that it is not to be expected that we should find a closer accordance, than has been found, between the results of computation and the result to be expected, if the moon was really once a part of the earth. It appears to me, therefore, that the present considerations only negative the hypothesis of any large contraction of the earth since the moon has existed. PART III. Summary and discussion of results.* The general object of the earlier or preparatory part of the paper is sufficiently explained in the introductory remarks. The earth is treated as a homogeneous spheroid, and in what follows, except where otherwise expressly stated, the matter of which it is formed is supposed to be purely viscous. The word " earth " is thus an abbreviation of the expression " a homogeneous rotating viscous spheroid ; " also wherever numerical values are given they are taken from the radius, mean density, mass, &c., of the earth. The case is considered first of the action of one tide-raising body, namely, the moon. To simplify the problem the moon is supposed to move in a circular orbit in the ecliptict that plane being the average position of the lunar orbit with respect to the * This part Las been altered in accordance with, the several additions and alterations occurring above. The results of subsequent investigations have modified the interpretation to be put on several of the results here obtained. I have, moreover, had the advantage of discussing several points with Sir WILLIAM THOMSON. July 9, 1879. t The effect of neglecting the eccentricity of the moon's orbit is, that we underestimate the efficiency of the tidal effects. Those effects vary as the inverse sixth power of r the radius vector, and if T be the 524 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, earth's axis. The case becomes enormously more complex if we suppose the moon to move in an inclined eccentric orbit with revolving nodes. The consideration of the secular changes in the inclination of the lunar orbit and of the eccentricity will form the subject of another investigation. The expression for the moon's tide-generating potential is shown to consist of 13 simple tide-generating terms, and the physical meaning of this expansion is given in the note to Section 8. The physical causes represented by these 13 terms raise 13 simple tides in the earth, the heights and retardations of which depend on their speeds and on the coefficient of viscosity. The 1 3 simple tides may be more easily represented both physically and analytically as seven tides, of which three are approximately semi-diurnal, three approximately diurnal, and one has a period equal to a half of the sidereal month, and is therefore called the fortnightly tide. Then by an approximation which is sufficiently exact for a great part of the investi- gation, the semi-diurnal tides may be grouped together, and the diurnal ones also. Hence the earth may be regarded as distorted by two complex tides, namely, the semi- diurnal and diurnal, and one simple tide, namely, the fortnightly. The absolute heights and retardations of these three tides are expressed by six functions of their speeds and of the coefficient of viscosity (Sections 1 and 2). When the form of the distorted spheroid is thus given, the couples about three axes fixed in the earth due to the attraction of the moon on the tidal protuberances are found. It must here be remarked that this attraction must in reality cause a tan- gential stress between the tidal protuberances and the true surface of the mean oblate spheroid. This tangential stress must cause a certain very small tangential flow,* and hence must ensue a very small diminution of the couples. The diminution of couple is here neglected, and the tidal spheroid is regarded as being instantaneously rigidly connected with the rotating spheroid. The full expression for the couples on the earth are long and complex, but since the nutations to which they give rise are exceedingly minute, they may be much abridged by the omission of all terms except such as can give rise to secular changes in the precession, the obliquity of the ecliptic, and the diurnal rotation. The terms retained represent that there are three couples independent of the time, the first of which tends to make the earth rotate about an axis in the equator which is always 90 from the nodes of the moon's orbit : this couple affects the obliquity to the ecliptic ; second, there is a couple about an axis in 1 1 fd/ periodic time of the moon, the average value of -j is fp | -pg. If c be the mean distance and e the eccen- ee tricity of the orbit, this integral will be found equal to -g .... 3ij~- If the eccentricity be small the average value of -5 is -g (l + ~ ^ e * 8 20 * n * s * 8 ^3 ^ c" 6 ' '^ aere are obviously forces tending tq modify the eccentricity of the moon's orbit. * See Part I. of the next paper. AND ON THE REMOTE HISTORY OF THE EARTH. 525 the equator which is always coincident with the nodes : this affects the precession ; third, there is a couple about the earth's axis of rotation, and this affects the length of the day (Sections 3, 4, and 5). All these couples vary as the fourth power of the moon's orbital angular velocity, or as the inverse sixth power of her distance. These three couples give the alteration in the precession due to the tidal movement, the rate of increase of obliquity, and the rate at which the diurnal rotation is being diminished, or in other words the tidal friction. The change of obliquity is in reality due to tidal friction, but it is convenient to retain the term specially for the change of rotation alone. It appears that if the bodily tides do not lag, which would be the case if the earth were perfectly fluid or perfectly elastic, then there is no alteration in the obliquity, nor any tidal friction (Section 7). The alteration in the precession is a very small fraction of the precession due to the earth considered as a rigid oblate spheroid. I have some doubts as to whether this result is properly applicable to the case of a perfectly fluid spheroid. At any rate, Sir WILLIAM THOMSON has stated, in agreement with this result, that a perfectly fluid spheroid has a precession scarcely differing from that of a perfectly rigid one. Moreover, the criterion which he gives of the negligeability of the additional terms in the precession in a closely analogous problem appears to be almost identical with that found by me (Section 7). I am not aware that the investigation on which his statement is founded has ever been published. The alteration in the pre- cession being insignificant, no more reference will be made to it. This concludes the analytical investigation as far as concerns the effects on the disturbed spheroid, where there is only one disturbing body. The sun is now (Section 8) introduced as a second disturbing body. Its independent effect on the earth may be determined at once by analogy with the effect of the moon. But the sun attracts the tides raised by the moon, and vice versa. Now notwith- standing that the periods of the sun and moon about the earth have no common multiple, yet the interaction is such as to produce a secular alteration in the position of the earth's axis and in the angular velocity of its diurnal rotation. A physical explanation of this curious result is given in the note to Section 8. I have dis- tinguished this from the separate effect of each disturbing body, as a combined effect. The combined effects are represented by two terms in the tide-generating potential, one of which goes through its period in 12 sidereal hours, and the other in a sidereal day""" ; the latter being much more important than the former for moderate obliquities to the ecliptic. Both these terms vanish when the earth's axis is perpendicular to the plane of the orbit. As far as concerns the combined effects, the disturbing bodies may be conceived to be * These combined effects depend on the tides which are designated as Kj and Kj in the British Asso- ciation's Report on Tides for 1872 and 1876, and which I have called the sidereal semi-diurnal and diurnal tides. For a general explanation of this result see the abstract of this paper in the ' Proceedings of the Royal Society,' No. 191, 1878. 526 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID. replaced by two circular rings of matter coincident with their orbits and equal in mass to them respectively. The tidal friction due to these rings is insignificant compared with that arising separately from the sun and moon. But the diurnal combined effect has an important influence in affecting the rate of change of obliquity. The combined effects are such as to cause the obliquity of the ecliptic to diminish, whereas the separate effects on the whole make it increase at least in general (see Section 22). The relative importance of ah 1 the effects may be seen from an inspection of Table III., Section 15. Section 1 1 contains a graphical analysis of the physical meaning of the equations, giving the rate of change of obh'quity for various degrees of viscosity and obliquity. Plate 36, figures 2 and 3, refer to the case where the disturbed planet is the earth, and the disturbing bodies the sun and moon. This analysis gives some remarkable results as to the dynamical stability or instability of the system. It will be here sufficient to state that, for moderate degrees of viscosity, the position of zero obliquity is unstable, but that there is a position of stability at a high obliquity. For large viscosities the position of zero obliquity becomes stable, and (except for a very close approximation to rigidity) there is an unstable position at a larger obliquity, and again a stable one at a still larger one." :: " These positions of dynamical equilibrium do not rigorously deserve the name, since they are slowly shifting in consequence of the effects of tidal friction ; they are rather positions in which the rate of change of obliquity becomes of a higher order of small quantities. It appears that the degree of viscosity of the earth which at the present time would cause the obliquity of the ecliptic to increase most rapidly is such that the bodily semi- diurnal tide would be retarded by about 1 hour and 10 minutes; and the viscosity which would cause the obliquity to decrease most rapidly is such that the bodily semi- diurnal tide would be retarded by about 2f hours. The former of these two viscosities was the one which I chose for subsequent numerical application, and for the consideration of secular changes in the system. Plate 36, fig. 4 (Section 11), shows a similar analysis of the case where there is only one disturbing satellite, which moves orbitally with one-fifth of the velocity of rotation of the planet. This case differs from the preceding one in the fact that the position of zero obliquity is now unstable for all viscosities, and that there is always one other, and only one other position of equilibrium, and that is a stable one. This shows that the fact that the earth's obliquity would diminish for large viscosity is due to the attraction of the sun on the lunar tides, and of the moon on the solar tides. It is not shown by these figures, but it is the fact that if the motion of the satellite * For a general explanation of some part of these results, see the abstract of this pu]>er in the ' Proceedings of the Royal Society,' No. 191,1878. AND ON THE REMOTE HISTORY OF THE EARTH. 527 relatively to the planet be slow enough (viz. : the month less than twice the day), the obliquity will diminish. This result, taken in conjunction with results given later with regard to the evolu- tion of satellites, shows that the obliquity of a planet perturbed by a single satellite must rise from zero to a maximum and then decrease again to zero. If we regard the earth as a satellite of the moon, we see that this must have been the case with the moon. Plate 36, fig. 5 (Section 12), contains a similar graphical analysis of the various values which may be assumed by the tidal friction. As might be expected, the tidal friction always tends to stop the planet's rotation, unless indeed the satellite's period is less than the planet's day, when the friction is reversed. This completes the consideration of the effect on the earth, at any instant, of the attraction of the sun and moon on their tides; the next subject is to consider the reaction on the disturbing bodies. Since the moon is tending to retard the earth's diurnal rotation, it is obvious that the earth must exercise a force on the moon tending to accelerate her linear velocity. The effect of this force is to cause her to recede from the earth and to decrease her orbital angular velocity. Hence tidal reaction causes a secular retardation of the moon's mean motion. The tidal reaction on the sun is shown to have a comparatively small influence on the earth's orbit and is neglected (Sections 14 and 19). The influence of tidal reaction on the lunar orbit is determined by finding the dis- turbing force on the moon tangential to her orbit, in terms of the couples which have been already found as perturbing the earth's rotation ; and hence the tangential force is found in terms of the rate of tidal friction. and of the rate of change of obliquity. It appears that the non-periodic part of the force, on which the secular change in the moon's distance depends, involves the lunar tides alone. By the considei'ation of the effects of the perturbing force on the moon's motion, an equation is found which gives the rate of increase of the square root of the moon's distance, in terms of the heights and retardations of the several lunar tides (Section 14). Besides the interaction of the two bodies which affects the moon's mean motion, there is another part which affects the plane of the lunar orbit ; but this latter effect is less important than the former, and in the present paper is neglected, since the moon is throughout supposed to remain in the ecliptic. The investigation of the subject will however, lead to interesting results, since a complete solution of the problem of the obliquity of the ecliptic cannot be attained without a simultaneous tracing of the secular changes in the plane of the lunar orbit. It appears that the influence of the tides, here called slow semi-diurnal and slow diurnal, is to increase the moon's distance from the earth, whilst the influence of the fast semi-diurnal, fast diurnal, and fortnightly tide tends to diminish the moon's dis- tance ; also the sidereal semi-diurnal and diurnal tides exercise no effects in this MDCCOLXXIX. 3 Y 528 MR. G. H. DARWIX OX THE PRECESSION OF A VISCOUS SPHEROID, respect. The two tides which tend to increase the moon's distance are much larger than the others, so that the moon in general tends to recede from the earth. The increase of distance is, of course, accompanied by an increase of the moon's periodic time, and hence there is in general a true secular retardation of the moon's motion. But this change is accompanied by a retardation of the earth's diurnal rotation, and a terrestrial observer, taking the earth as his clock, would conceive that the angular velocity of an ideal moon, which was undisturbed by tidal reaction, was undergoing a secular acceleration. The apparent acceleration of the ideal undisturbed moon must considerably exceed the true retardation of the real disturbed moon, and the difference between these two will give an apparent acceleration. It is thus possible to give an equation connecting the apparent acceleration of the moon's motion and the heights and retardations of the several bodily tides in the earth. Now there is at the present time an unexplained secular acceleration of the moon of about 4" per century, and therefore if we attribute the whole of this to the action of the bodily tides in the earth, instead of to the action of ocean tides, as was done by ADAMS and DELAUNAY, we get a numerical relation which must govern the actual heights and retardations of the bodily tides in the earth at the present time. This equation involves the six constants expressive of the heights and retardations ot the three bodily tides, and which are determined by the physical constitution of the earth. No further advance can therefore be made without some theory of the earth's nature. Two theories are considered. First, that the earth is purely viscous. The result shows that the earth is either nearly fluid which we know it is not or exceedingly nearly rigid. The only traces which we should ever be likely to find of such a high degree of viscosity would be in the fortnightly ocean tide ; and even here the influence would be scarcely perceptible, for its height would be '992 of its theoretical amount according to the equilibrium theory, whilst the time of high water would be only accelerated by six hours and a half. It is interesting to note that the indications of a fortnightly ocean tide, as deduced from tidal observations, are exceedingly uncertain, as is shown in a preceding paper,* where I have made a comparison of the heights and phases of such small fortnightly tides as have hitherto been observed. And now (July, 1879) Sir WILLIAM THOMSON has informed me that he thinks it very possible that the effects of the earth's rotation may be such as to prevent our trusting to the equilibrium theory to give even approximately the height of the fortnightly tide. He has recently read a paper on this subject before the Royal Society of Edinburgh. With the degree of viscosity of the earth, which gives the observed amount of secular acceleration to the moon, it appears that the moon is subject to such a true secular retardation that at the end of a century she is 3"'l behind the place in her orbit which she would have occupied if it were not for the tidal reaction, whilst the earth, considei'ed as a clock, is losing 1 3 seconds in the same time. This rate of retardation of the earth * See the Appendix to my paper on the " Bodily Tides," &c., Phil. Trans., Part I., 1879. AXD ON THE REMOTE HISTORY OF THE EARTH. 529 is such that an observer taking the earth as his clock would conceive a moon, which was undisturbed by tidal reaction, to be 7"'l in advance of her place at the end of a century. But the actual moon is 3"'l behind her true place, and thus our observer would suppose the moon to be in advance 7'1 3'1 or 4" at the end of the century. Lastly, the obliquity of the ecliptic is diminishing at the rate of 1 in 500 million years. The other hypothesis considered is that the earth is very nearly perfectly elastic. In this case the semi-diurnal and diurnal tides do not lag perceptibly, and the whole of the reaction is thrown on to the fortnightly tide, and moreover there is no peiceptible tidal frictional couple about the earth's axis of rotation. From this follows the remarkable conclusion that the moon may be undergoing a true secular acceleration of motion of something less than 3"'5 per century, whilst the length of day may remain almost un- affected. Under these circumstances the obliquity of the ecliptic must be diminishing at the rate of 1 in something like 130 million years. This supposition leads to such curious results, that I investigated what state of things we should arrive at if we look back for a very long period, and I found that 700 million years ago the obliquity might have been 5 greater than at present, whilst the month would only be a little less than a day longer. The suppositions on which these results are based are such that they necessarily give results more striking than would be physically possible. The enormous lapse of time which has to be postulated renders it in the highest degree improbable that more than a very small change in this direction has been taking place, and moreover the action of the ocean tides has been entirely omitted from consideration. The results of these two hypotheses show what fundamentally different interpreta- tions may be put to the phenomenon of the secular acceleration of the moon. Sir WILLIAM THOMSON also has drawn attention to another disturbing cause in the fall of meteoric dust on to the earth. * Under these circumstances, I cannot think that any estimate having any pretension to accuracy can be made as to the present rate of tidal friction. Since the obliquity of the ecliptic, the diurnal rotation of the eai'th, and the moon's distance change, the whole system is in a state of flux ; and the next question to be considered is to determine the state of things which existed a very long time ago (Part II.). This involved the integration of three simultaneous differential equations; the mathematical difficulties were, however, so great, that it was found impracticable to obtain a general analytical solution. I therefore had to confine myself to a numerical solution adapted to the case of the earth, sun, and moon, for one particular degree of viscosity of the earth. The particular viscosity was such that, with the present values of the day and month, the time of the lunar semi-diurnal tide was retarded by 1 hour and 10 minutes ; the greatest possible lagging of this tide is * 'Glasgow Geological Society,' Vol. III. Address "On Geological Time." 3 Y 2 530 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID 3 hours, and therefore this must be regarded as a very moderate degree of visco- sity. It was chosen because initially it makes the rate of change of obliquity a maximum, and although it is not that degree of viscosity which will make all the changes proceed with the greatest possible rapidity, yet it is sufficiently near that value to enable us to estimate very well the smallest time which can possibly have elapsed in the history of the earth, if changes of the kind found really have taken place. This estimate of time is confirmed by a second method, which will be referred to later. The changes were tracked backwards in time from the present epoch, a,nd for con- venience of diction I shall also reverse the form of speech e.g., a true loss of energy as the time increases will be spoken of as a gain of energy as we look backwards. I shall not enter at all into the mathematical difficulties of the problem, but shall proceed at onue to comment on the series of tables at the end of Section 15, which give the results of the solution. The whole process, as traced backwards, exhibits a gain of kinetic energy to the system (of which more presently), accompanied by a transference of moment of momentum from that of orbital motion of the moon and earth to that of rotation of the earth. The last column but one of Table IV. exhibits the fall of the ratio of the two moments of momentum from 4'01 down to '44. The whole moment of momentum of the moon-earth system rises slightly, because of solar tidal friction. The change is investigated in Section 19. Looked at in detail, we see the day, month, and obliquity all diminishing, and the changes proceeding at a rapidly increasing rate, so that an amount of change which at the beginning required many millions of years, at the end only requires as many thou- sands. The reason of this is that the moon's distance diminishes with great rapidity ; and as the effects vary as the square of the tide-generating force, they vary as the inverse sixth power of the moon's distance, or, in physical language, the height of the tides increases with great rapidity, and so also does the moon's attraction. But there is a counteracting principle, which to some extent makes the changes proceed slower. It is obvious that a disturbing body will not have time to raise such high tides in a rapidly rotating spheroid as in one which rotates slowly. As the earth's rotation increases, the lagging of the tides increases. The first column of Table I. shows the angle by which the crest of the lunar semi-diurnal tide precedes the moon ; we see that the angle is almost doubled at the end of the series of changes, as traced back- wards. It is not quite so easy to give a physical meaning to the other columns, although it might be done. In fact, as the rotation increases, the effect of each tide rises to a maximum, and then dies away ; the tides of longer period reach their maxi- mum effect much more slowly than the ones of short period. At the point where I have found it convenient to stop the solution (see Table IV.), the semi-diurnal effect haw passed its maximum, the diurnal tide has just come to give its maximum effect, whilst the fortnightly tide has not nearly risen to that point. AXD OX THE REMOTE HISTORY OF THE EARTH. 531 As the lunar effects increase in importance (when we look backwards), the relative value of the solar effects decreases rapidly, because the solar tidal reaction leaves the . earth's orbit sensibly unaffected (see Section 19), and thus the solar effects remain nearly constant, whilst the lunar effects have largely increased. The relative value of the several tidal effects is exhibited in Tables II. and III. Table IV. exhibits the length of day decreasing to a little more than a quarter of its present value, whilst the obliquity diminishes through 9. But the length of the month is the element which changes to the most startling extent, for it actually falls to iVth of its primitive value. It is particularly important to notice that all the changes might have taken place in 57 million years; and this is far within the time which physicists admit that the earth and moon may have existed. It is easy to find a great many verce causce for changes in the planetary system ; but it is in general correspondingly hard to show that they are competent to produce any marked effects, without exorbitant demands on the efficiency of the causes arid on lapse of time. It is a question of great interest to geologists to determine whether any part of these changes could have taken place during geological history. It seems to me that this question must be decided by whether or not a globe, such as has been considered, could have afforded a solid surface for animal life, and whether it might present a superficial appearance such as we know it. These questions must, I think, be answered in the affirmative, for the following reasons. The coefficient of viscosity of the spheroid with which the previous solution deals is given by the formula --tan 35 (see Section 11, (40)), when gravitation units of force .I. i_" )!/ are used. This, when turned into numbers, shows that 2'055xl0 7 grams weight are required to impart unit shear to a cubic centimeter block of the substance in 24 hours, or 2,055 kilogs. per square centimeter acting tangentially on the upper face of a slab one centimeter thick for 24 hours, would displace the upper surface through a millimeter relatively to the lower, which is held fixed. In British units this becomes, 13^ tons to the square inch, acting for 24 hours on a slab an inch thick, displaces the upper surface relatively to the lower through one-tenth of an inch. It is obvious that such a substance as this would be called a solid in ordinary parlance, and in the tidal problem this must be regarded as a rather small viscosity. It seems to me, then, that we have only got to postulate that the upper and cool surface of the earth presents such a difference from the interior that it yields with extreme slowness, if at all, to the weight of continents and mountains, to admit the possibility that the globe on which we live may be like that here treated of. If, therefore, astronomical facts should confirm the argument that the world has really gone through changes of the kind here investigated, I can see no adequate reason for assuming that the whole process was pre-geological. Under these circumstances it must be admitted that the obliquity to the ecliptic is now probably slowly decreasing; 532 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, that a long time ago it was perhaps a degree greater than at present, and that it was then nearly stationary for another long time, and that in still earlier times it was considerably less.""' The violent changes which some geologists seem to require in geologically recent times would still, I think, not follow from the theory of the earth's viscosity. According to the present hypothesis (and for the moment looking forward in time), the moon-earth system is, from a dynamical point of view, continually losing energy from the internal tidal friction. One part of this energy turns into potential energy of the moon's position relatively to the earth, and the rest developes heat in the interior of the earth. Section 16 'contains the investigation of the amount which has turned to heat between any two epochs. The heat is estimated by the number of degrees Fahrenheit, which the lost energy would be sufficient to raise the temperature of the whole earth's mass, if it were all applied at once, and if the earth had the specific heat of iron. The last column of Table IV., Section 15, gives the numerical results, and it appears therefrom that, during the 57 million years embraced by the solution, the energy lost suffices to heat the whole earth's mass 1760 Fahr. It would appear at first sight that this large amount of heat, generated internally, must seriously interfere with the accuracy of Sir WILLIAM THOMSON'S investigation of the secular cooling of the earth ;t but a further consideration of the subject in the next paper will show that this cannot be the case. There are other consequences of interest to geologists which flow from the present hypothesis. As we look at the whole series of changes from the remote past, the ellipticity of figure of the earth must have been continually diminishing, and thus the polar regions must have been ever rising and the equatorial ones falling ; but, as the ocean always followed these changes, they might quite well have left no geological traces. The tides must have been very much more frequent and larger, and accordingly the rate of oceanic denudation much accelerated. The more rapid alternations of clay and night J would probably lead to more sudden and violent storms, and the increased rotation of the earth would augment the violence of the trade winds, which in their turn would affect oceanic currents. Thus there would result an acceleration of geological action. The problem, of which the solution has just been discussed, deals with a spheroid of * In my paper " On the Effects of Geological Changes on the Earth's Axis," Phil. Trans. 1877, p. 271, I arrived at the conclusion that the obliquity had been unchanged throughout geological history. That result was obtained on the hypothesis of the earth's rigidity, except as regards geological upheavals. The result at which I now arrive affords a warning that every conclusion must always be read along with the postulates on which it is based. f ' Nat. Phil.,' Appendix. J At the point where the solution stops there are just 1,300 of the sidereal days of that time in the year, instead of 366 as at present. AND ON THE REMOTE HISTORY Ol 1 THE EARTH. 533 constant viscosity ; but there is every reason to believe that the earth is a cooling body, and has stiffened as it cooled. We therefore have to deal with a spheroid whose viscosity diminishes as we look backwards. A second solution is accordingly given (Section 17) where the viscosity is variable; no definite law of diminution of viscosity is assumed, however, but it is merely supposed that the viscosity always remains small from a tidal point of view. This solution gives no indication of the time which may have elapsed, and differs chiefly from the preceding one in the fact that the change in the obliquity is rather greater for a given amount of change in the moon's distance. There is not much to say about it here, because the two solutions follow closely parallel lines as far as the place where the former one left off. The first solution was not carried further, because as the mouth approximates in length to the day, the three semi-diurnal tides cease to be of nearly equal frequencies, and so likewise do the three diurnal tides; hence the assumption on which the solution was founded, as to their approximately equal speeds, ceases to be sufficiently accurate. In this second solution all the seven tides are throughout distinguished from one another. At about the stage where the previous solution stops the solar terms have become relatively unimportant, and are dropped out. It appears that (stiE. looking backwards in time) the obliquity will only continue to diminish a little more beyond the point it had reached when the previous method had become inapplicable. For when the month has become equal to twice the day, there is no change of obliquity ; and for yet smaller values of the month the change is the other way. This shows that for small viscosity of the planet the position of zero obliquity is dynamically stable for values of the month which are less than twice the day, while for greater values it is unstable ; and the same appears to be true for very large vis- cosity of the planet (see the foot-note on p. 500). If the integration be carried back as far as the critical point of relationship between the day and month, it appears that the whole change of obliquity since the beginning is 9^. The interesting question then arises Does the hypothesis of the earth's viscosity afford a complete explanation of the obliquity of the ecliptic ? It does not seem at present possible to give any very conclusive answer to this question ; for the problem which has been solved differs in many respects from the true problem of the earth. The most important difference from the truth is in the neglect of the secular changes of the plane of the lunar orbit; and I now (September, 1879) see reason to believe that that neglect will make a material difference in the results given for the obliquity at the end of the third and fourth periods of integration in both solutions. It will not, therefore, be possible to discuss this point adequately at present ; but it will be well to refer to some other points in which our hypothesis must differ from reality. I do not see that the heterogeneity of density and viscosity would make any very material difference in the solution, because both the change of obliquity and the tidal 534 ME. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, friction would be affected 2-><-ii'i ^ussii, and therefore the change of obliquity for a given amount of change in the day would not be much altered. Although the effects of the contraction of the earth in cooling would be certainly such as to render the changes more rapid in time, yet as the tidal friction would be somewhat counteracted, the critical point where the month is equal to twice the day would be reached when the moon was further from the earth than in my problem. 1 think, however, that there is reason to believe that the whole amount of contraction of the earth, since the moon has existed, has not been large (Section 24). There is one thing which might exercise a considerable influence favourable to change of obliquity. We are in almost complete ignorance of the behaviour of semi-solids under very great pressures, such as must exist in the earth, and there is no reason to suppose that the amount of relative displacement is simply proportional to the stress and the time of its action. Suppose, then, that the displacement varied as some other function of the time, then clearly the relative importance of the several tides might be much altered. Now, the great obstacle to a large change of obliquity is the diurnal combined effect (see Table IV., Section 15); and so any change in the law of viscosity which, allowed a relatively greater influence to the semi-diurnal tides would cause a greater change of obliquity, and this without much affecting the tidal friction and reaction. Such a law seems quite within the bounds of possibility. The special hypothesis, however, of elastico-viscosity, used in the previous paper, makes the other way, and allows greater influence to the tides of long period than to those of short. This was exemplified where it was shown that the tidal reaction might depend principally on the fortnightly tide. The whole investigation is based on a theory of tides in which the effects of inertia are neglected. Now it will be shown in Part III. of the next paper that the effect of inertia will be to make the crest of the tidal spheroid lag more for a given height of tide than results from the theory founded on the neglect of inertia. An analysis of the effect produced on the present results, by the modification of the theory of tides introduced by inertia, is given in the next paper. On the whole, we can only say at present that it seems probable that a part of the obliquity of the ecliptic may be referred to the causes here considered; but a complete discussion of the subject must be deferred to a future occasion, when the secular changes in the plane of the lunar orbit will be treated. The question of the obliquity is now set on one side, and it is supposed that when the moon has reached the critical point (where the month is twice the day) the obliquity to the plane of the lunar orbit was zero. In the more remote past the obliquity had no tendency to alter, except under the influence of certain nutations, which are referred to at the end of Section 17. The manner in which the moon's periodic time approximates to the day is an inducement to speculate as to the limiting or initial condition from which the earth and moon started their course of development. AND ON THE REMOTE HISTORY OF THE EARTH. 535 So long as there is any relative motion of the two bodies there must be tidal friction, and therefore the moon's period must continue to approach the day. It would be a problem of extreme complication to track the changes in detail to their end, and fortunately it is not necessary to do so. The principle of conservation of moment of momentum, which has been used throughout in tracing the parallel changes in the moon and earth, affords the means of leaping at once to the conclusion (Section 18). The equation expressive of that principle involves the moon's orbital angular velocity and the earth's diurnal rotation as its two valuables ; and it is only necessary to equate one to the other to obtain an equation, which will give the desired information. As we are now supposed to be transported back to the initial state, I shall hence- forth speak of time in the ordinary way ; there is no longer any convenience in speaking of the past as the future, and vice versd. The equation above referred to has two solutions, one of which indicates that tidal friction has done its work, and the other that it is just about to begin. Of the first I shall here say no more, but refer the reader to Section 18. The second solution indicates that the moon (considered as an attractive particle) moves round the earth as though it were rigidly fixed thereto in 5 hours 36 minutes. This is a state of dynamical instability ; for if the month is a little shorter than the day, the moon will approach the earth, and ultimately fall into it ; but if the day is a little shorter than the month, the moon will continually recede from the earth, and pass through the series of changes which were traced backwards. Since the earth is a cooling and contracting body, it is likely that its rotation would increase, and therefore the dynamical equilibrium would be more likely to break down in the latter than the former way. The continuous solution of the problem is taken up at the point where the moon has receded from the earth so far that her period is twice that of the earth's rotation. I have calculated that the heat generated in the interior of the earth in the course of the lengthening of the day from 5 hours 36 minutes to 23 hours 56 minutes would be sufficient, if applied all at once, to heat the whole earth's mass about 3000 Fahr., supposing the earth to have the specific heat of iron (see Section 16). A rough calculation shows that the minimum time in which the moon can have passed from the state where it had a period of 5 hours 36 minutes to the present state, is 54 miUion years, and this confirms the previous estimates of time. This periodic time of the moon corresponds to an interval of only 6,000 miles between the earth's surface and the moon's centre. If the earth had been treated as heterogeneous, this distance, and with it the common periodic time both of moon and earth, would be still further diminished. These results point strongly to the conclusion that, if the moon and earth were ever molten viscous masses, then they once formed parts of a common mass. We are thus led at once to the inquiry as to how and why the planet broke up. MDCCCLXXIX. 3 z 536 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, The conditions of stability of rotating masses of fluid are unfortunately unknown, and it is therefore impossible to do more than speculate on the subject. The most obvious explanation is similar to that given in LAPLACE'S nebular hypo- thesis, namely, that the planet being partly or wholly fluid, contracted, and thus rotated faster and faster until the ellipticity became so great that the equilibrium was unstable, and then an equatorial ring separated itself, and the ring finally conglomerated into a satellite. This theory, however, presents au important difference from the nebular hypothesis, in as far as that the ring was not left behind 240,000 miles away from the earth, when the planet was a rare gas, but that it was shed only 4,000 or 5,000 miles from the present surface of the earth, when the planet was perhaps partly solid and partly fluid. This view is to some extent confirmed by the ring of Saturn, which would thus be a satelh'te in the course of formation. It appears to me, however, that there is a good deal of difficulty in the acceptance of this view, when it is considered along with the numerical results of the previous investigation. At the moment when the ring separated from the planet it must have had the same linear velocity as the surface of the planet ; and it appears from Section 22 that such a ring would not tend to expand from tidal reaction, unless its density varied in different parts. Thus we should hardly expect the distance from the earth of the chain of meteorites to have increased much, until it had agglomerated to a con- siderable extent. It follows, therefore, that we ought to be able to trace back the moon's path, until she was nearly in contact with the earth's surface, and was always opposite the same face of the earth. Now this is exactly what has been done in the previous investigation. But there is one more condition to be satisfied, namely, that the common speed of rotation of the two bodies should be so great that the equilibrium of the rotating spheroid should be unstable. Although we do not know what is the limiting angular velocity of a rotating spheroid consistent with stability, yet it seems improbable that a rotation in a little over 5 hours, with an ellipticity of one-twelfth would render the system unstable. Now notwithstanding that the data of the problem to be solved are to some extent uncertain, and notwithstanding the imperfection of the solution of the problem here given, yet it hardly seems likely that better data and a more perfect solution would largely affect the result, so as to make the common period of revolution of the two bodies in the initial configuration very much less than 5 hours/ 5 ' Moreover we obtain no help from the hypothesis that the earth has considerably contracted since the shed- ding of the satellite, but rather the reverse ; for it appears from Section 24 that if the earth has contracted, then the common period of revolution of the two bodies in the * This is illustrated by my paper on " The Secular Effects of Tidal Friction," ' Proc. Roy. Soc.,' No. 1!'7, 1879, where it appears that the " line of momentum " does not cut the " curve of rigidity " at a very small angle, so that a small error in the data would not make a very large one in the solution. AXD ON THE REMOTE HISTORY OP THE EARTH. 537 initial configuration must have been slower, and the moon more distant from the earth. This slower revolution would correspond with a smaller ellipticity, and thus the system would probably be less nearly unstable. The following appears to me at least a possible cause of instability of the spheroid when rotating in about 5 hours. Sir WILLIAM THOMSON has shown that a fluid spheroid of the same mean density as the earth would perform a complete gravitational oscillation in 1 hour 34 minutes. The speed of oscillation varies as the square root of the density, hence it follows that a less dense spheroid would oscillate more slowly, and therefore a spheroid of the same mean density as the earth, but consisting of a denser nucleus and a rarer surface, would probably oscillate in a longer time than 1 hour 34 minutes. It seems to be quite possible that two complete gravitational oscillations of the earth in its primitive state might occupy 4 or 5 hours. But if this were the case, then the solar semi-diurnal tide would have very nearly the same period as the free oscillation of the spheroid, and accordingly the solar tides would be of enormous height. Does it not then seem possible that, if the rotation were fast enough to bring the spheroid into anything near the unstable condition, then the large solar tides might rupture the body into two or more parts ? In this case one would conjecture that it would not be a ring which would detach itself. It seems highly probable that the moon once did rotate more rapidly round her own axis than in her orbit, and if she was formed out of the fusion together of a ring of meteorites, this rotation would necessarily result. In Section 23 it is shown that the tidal friction due to the earth's action on the moon must have been enormous, and it must necessarily have soon brought her to present the same face constantly to the earth. This explanation was, I believe, first given by HELMHOLTZ. ln the process, the inclination of her axis to the plane of her orbit must have rapidly increased, and then, as she rotated more and more slowly, must have slowly diminished again. Her present aspect is thus in strict accordance with the results of the purely theoretical investigation. It would perhaps be premature to undertake a complete review of the planetary system, so as to see how far the ideas here developed accord with it. Although many facts which could be adduced seem favourable to their acceptance, I will only refer- to two. The satellites of Mars appear to me a most remarkable confirmation of these views. Their extreme minuteness has prevented them from being subject to any per- ceptible tidal reaction, just as the minuteness of the earth compared with the sun has prevented the earth's orbit from being perceptibly influenced (see Section 19) ; they thus remain as a standing memorial of the primitive periodic time of Mars round his axis. Mars, on the other hand, has been subjected to solar tidal friction. This case, however, deserves to be submitted to numerical calculation. The other case is that of Uranus, and this appears to be somewhat unfavourable to the theory ; for on account of the supposed adverse revolution of the satellites, and of the high inclinations of their orbits, it is not easy to believe that they could have 538 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, ETC. arisen from a planet which ever rotated about an axis at all nearly perpendicular to the ecliptic. The system of planets revolving round the sun present so strong a resemblance to the systems of satellites revolving round the planets, that we are almost compelled to believe that their modes of development have been somewhat alike. But in applying the present theory to explain the orbits of the planets, we are met by the great difficulty that the tidal reaction due to solar tides in the planet is exceedingly slow in its influence ; and not much help is got by supposing the tides in the sun to react on the planet. Thus enormous periods of time would have to be postulated for the evolution. If, however, this theory should be found to explain the greater part of the configura- tions of the satellites round the planets, it would hardly be logical to refuse it some amount of applicability to the planets. We should then have to suppose that before the birth of the satellites the planets occupied very much larger volumes, and possessed much more moment of momentum than they do now. If they did so, we should not expect to trace back the positions of the axes of the planets to the state when they were perpendicular to the ecliptic, as ought to be the case if the action of the satellites, and of the sun after their birth, is alone concerned. Whatever may be thought of the theory of the viscosity of the earth, and of the large speculations to which it has given rise, the fact remains that nearly all the effects which have been attributed to the action of bodily tides would also follow, though probably at a somewhat less rapid rate, from the influence of oceanic tides on a rigid nucleus. The effect of oceanic tidal friction on the obliquity of the ecliptic has already been considered by Mr. STONE, in the only paper on the subject which I have yet seen.* His argument is based on what I conceive to be an incorrect assumption as to the nature of the tidal frictional couple, and he neglects tidal reaction ; he finds that the effects would be quite insignificant. This result would, I think, be modified by a more satisfactory assumption. Ast. Soc. Monthly Notices, March 8, 1867. Phil. Trans. 1879 Plate 36 Pig. 1. Di cup-can skowifig the- rate, of changt. of obliquity for various degrees of viscosity of the- planet, , where, there are. two bodies. Fin. 4. ' shovtmg tiu r^u of change, of for varu>u,s be the colatitude and longitude of a point in the layer. Then consider a prismatic element bounded by the two cones 0, 0-\-B0, and by the two planes , +S<. The radial faces of this prism are acted on by the pressures and tangential stresses communicated by the four contiguous prisms. But the tangential stresses on these faces only arise from the fact that contiguous prisms are solicited by slightly different forces, and therefore the action of the four prisms, surrounding the prism in question, must be principally pressure. I therefore propose to consider that the prism resists the tendency of the impressed forces- to move tangentially along the surface of the sphere, by means of hydrostatic pressures on its four radial faces, and by a tangential stress across its base. This approximation by which the whole of the tangential stress is thrown on to the base, is clearly such as slightly to accentuate, as it were, the distribution of the tangential stresses on the surface of the sphere, by which the equilibrium of the layer a- is maintained. For consider the following special case : Suppose a- to be a surface of revolution, and V to be such that only a single small circle of latitude is solicited by a tangential force everywhere perpendicular to the meridian. Then it is obvious that, strictly speaking, the elements lying a short way north and south of the small circle would tend to be carried with it, and the tangential stress on the sphere would be a maximum along the small circle, and would gradually die away to the north and south. In the approximate method, however, which it is proposed to use, such an application of external force would be deemed to cause no tangential stress to the surface of the sphere to the north and south of the small circle acted on. This special case is clearly a great exaggeration of what holds in our problem, because it postulates a finite difference of disturbing force between elements infinitely near to one another. We will first find what are the hydrostatic pressures transmitted by the four prisms contiguous to the one we are considering. . Let p be the hydrostatic pressure at the point r, 0, of the layer cr. Then if we neglect the variations of gravity due to the layer cr and to V, p is entirely due to the attraction of the mean sphere of radius a. The mean pressure on the radial faces at the point in question is \gwv ; where a- is negative the pressures are of course tractions. We will first resolve along the meridian. The excess of the pressure acting on the face 0-\-B0 over that on the face (whose area is era sin 08]S0, or ^gwaa 2 sin and it acts towards the pole. The resolved part of the pressures on the faces (+S< and (whose area is craS0) along the meridian is 5 42 MR. G H. DARWIN ON PROBLEMS CONNECTED (^gto, and it acts towards the equator. Hence the whole force due to pressure on the element resolved along the meridian towards the equator is ^gwa808( sin 6 (whose area is a-a80), measured in the direction of increasing, is = gwaa ( Hence the force due to pressure perpendicular to the meridian is -8m~ d We have now to consider the impressed forces on the element. Since a- is a surface harmonic of the second degree, the potential of the layer of (\ 1 -) . Therefore the forces along and perpen- dicular to the meridian on a particle of mass 8m, just outside the layer a- but infinitely near the prismatic element, are |-Sm^and f^Sm- -, and these are also the forces 8 a dd 5 a siu0d(j>' acting on the element 8m due to the attraction of the rest of the layer tr. Lastly, the forces due to the external potential V are clearly 8m-- and a do Then collecting results we get for the forces due both to pressure and attraction, along the meridian towards the equator dV~ d and perpendicular to the meridian, in the direction of < increasing, -."; 1 d ^ 2 . ~ m(V Henceforward ^- will be written 3, as in the previous papers. WITH THE TIDES OF A VISCOUS SPHEKOJU. 543 Now these are the forces on the element which must be balanced by the tangential stresses across the base of the prismatic element. It follows from the above formulas that the tangential stresses communicated by the layer cr to the surface of the sphere are those due to a potential V gcr acting on the layer cr. V If be the earth's angular velocity of rotation relatively to the moon. Let A be the major axis of the tidal ellipsoid. Let AX=- sin- 9 cos 2(< ait - e). The former of these terms is not a function of the time, and its effect is to cause a permanent small increase of ellipticity of figure of the earth, which may be neglected. We are thus left with ^rr 2 sin 2 cos 2(< cot e) as the true tide-generating potential. Now if tan 2e= . where u is the coefficient of viscosity of the spheroid, then by (jaw the theory of the paper on " Tides," such a potential will raise a tide expressed by a>*)* ....... (2) Then if we put S=lTsin c 0cos2(< cat e) ........ (3) a fc and S g-=iTsin2esin 2 0sin2(< tot) (4) (S (J-)=Tsin 2esin 0cos 0si Ctv \ ft r:(S dM=Tsm 2esin 6 cos 2(\ 9 aJ sin# Multiplying these by wa?-, we find from (1) the tangential stresses communicated (t by the layer a- to the sphere. * " Tides," Section 5. WITH THE TIDES OP A VISCOUS SPHEROID. 545 They are T 2 wa z \ sin 4e sin" 6 cos 6 sin 4(< wt) along the meridian, B and T 2 ira~g sin 4e sin 3 #(l + cos 4(< ait)) perpendicular to the meridian. These stresses of course vanish when- e is zero, that is to say when the spheroid is perfectly fluid. In as for as they involve (frcot these expressions are periodic, and the periodic parts must correspond with periodic inequalities in the state of flow of the interior of the earth. These small tides of the second order have no present interest and may be neglected. We are left, therefore, with a non-periodic tangential stress per unit area of the surface of the sphere perpendicular to the meridian from east to west equal to ^wa~- sin 4e sin 3 6. The sum of the moments of these stresses about the axis Z constitutes the tidal frictional couple jj^t, which retards the earth's rotation. Therefore = \iuaf- sin 4ejj sin 3 6. a sin d.ci? sin 8d8d$ integrated all over the surface of the sphere, and effecting the integration we have ,~ 4?r r r 2 . Ji=: wr.-sm 4e. lo g But if C be the earth's moment of inertia, C= Therefore This expression agrees with that found by a different method in the paper on " Precession.' '* jg We may now write the tangential stress on the surface of the sphere as itm 9 ^ sin 3 ; Vv and the components of this stress parallel to the axes X, Y, Z are -izm 2 ^sin 3 0sin<, +>a 2 ^ sin 3 cos <,0 ..... (6) We now have to consider those effects of inertia which equilibrate this system of surface forces. The couple $. retards the earth's rotation very nearly as though it were a rigid * "Precession," Section 5 (22), when 1=0. ' 4 A 2 546 MR. G. H. DARWIN ON PROBLEMS CONNECTED body. Hence the effective force due to inertia on a unit of volume of the interior of JQ the earth at a point r, 0, is wr sin 6 77, and it acts in a small circle of latitude from (j west to east. The sum of the moments of these forces about the axis of Z is of course equal to jl, and therefore this bodily force would equilibrate the surface forces found in (6), if the earth were rigid. The components of the bodily force parallel to the axes are in rectangular co- ordinates. 7/-7r> WT, .......... (~) The problem is therefore reduced to that of finding the state of flow in the interior of a viscous sphere, which is subject to a bodily force of which the components are (7) and to the surface stresses of which the components are (6). Let a, ft, y be the component velocities of flow at the point x, y, z, and v the coefficient of viscosity. Then neglecting inertia because the motion is very slow, the equations of motion are (8) We have to find a solution of these equations, subject to the condition above stated, as to surface stress. Let a', ft', y, p be functions which satisfy the equations (8) throughout the sphere. Then if we put a=a'+a / , ft=ft'-{-ft l , y=y'-\-y / , p=p'-\-p,, we see that to complete the solution we have to find ft,, y t , p t , as determined by the equations =,.,. ,. . . . (h/ il: dx ay dz which they are to satisfy throughout the sphere. They must also satisfy cerbiin equations to be found by subtracting from the given surface stresses (6), components of surface stress to be calculated from ', ft', y, p.* We have first to find a, ft', y , p. Conceive the symbols in equations (8) to be accented, and differentiate the fii-st * This statement of method is taken from THOMSON and.TAii's ' Nat. Phil.,' 733. WITH THE TIDES OP A VISCOUS SPHEROID. 547 three by x, y, z respectively and add them ; then bearing in mind the fourth equation, we have V^'^O, of which _p'=0 is a solution. Thus the equations to be satisfied become Solutions of these are obviously ' o > 1 - ~c r ~y> ^ =-ro- l o . a i i s /) = id ~ T? sin 6 sin

A 7/> have components AS= - a 2 (-f- - sin 2 6) sin 6 sin <, B 3 = - ~ a 2 (f- sin- 6) sin 8 cos , C 3 =0. These are surface harmonics of the third order as they stand. Now the solution of Sir W. THOMSON'S problem of the state of strain of an incom- pressible elastic sphere, subject only to surface stress, is applicable to an incompressible viscous sphere, mutatis mutandis. His solution" 1 ''' shows that a surface stress, of which the components are A,-, B,, C,- (surface harmonics of the i' h order), gives rise to a state of flow expressed by and symmetrical expressions for ft, y. Where W and 3> are auxiliary functions defined by . . . . (16) In our case i=3, and it is easily shown that the auxiliary functions are both zero, so that the required solution is a=^ f (f_ sin 2 ff^ sin 6 sin>, ft= -~ f (f - sin 2 6) sin 6 cos +, y, = 0. If we add to these the values of a', ft', y from (10), we have as the complete solution of the problem, * THOMSON and TAIT'S ' Nat. Phil.,' 737, WITH THE TIDES OP A VISCOUS SPHEROID. 549 a= -- r s sin 3 6 sin A, /8= -r 3 sin 3 d cos <, y=0 . ... (17) ou O of O These values show that the motion is simply cylindrical round the earth's axis, each point moving in a small circle of latitude from east to west with a linear velocity ~I A sin 3 6, or with an angular velocity about the axis equal to -~-r 2 sin 2 0.* In this statement a meridian at the pole is the curve of reference, but it is more intelligible to state that each particle moves from west to east with an angular velocity about the axis equal to | 7r(a 3 r" sin 3 0), with reference to a point on the surface at the equator. The easterly rate of change of the longitude L of any point on the surface in weft 4F3, colatitude is therefore - cos 3 6. ou O Then since -^-= sin 2e cos 2e, and tan 2e=f --, therefore ' " C (17') This equation gives the rate of change of longitude. The solution is not applicable to the case of perfect fluidity, because the terms introduced by inertia in the equations of motion have been neglected ; and if the viscosity be infinitely small, the inertia terms are no longer small compared with those introduced by viscosity. In order to find the total change of longitude in a given period, it will be more convenient to proceed from a different formula. Let n, n, be the earth's rotation, and the moon's orbital motion at any time ; and let the suffix to any symbol denote its initial value, also let (7r) V / Then it was shown in the paper on " Precession " that the equation of conservation of moment of momentum of the moon-earth system is (18) Where /u, is a certain constant, which in the case of the homogeneous earth with the present lengths of day and month, is almost exactly equal to 4. By differentiation of (18) dn d ' The problem might probably be solved more shortly without using the general solution, but the general solution -will be required in Part III. f "Precession," equation (73), when i=0 and T'=O. 550 MR. G. H. DARWIN ON PROBLEMS CONNECTED t JT\ But the equation of tidal friction is -=7:. Therefore (it O (U fj, (.'//,, Now 7T f IT^ OL tea- ja ., -=-- cos-0. rf< 8v C Therefore f?L w 3 (19') All the quantities on the right-hand side of this equation are constant, and there- fore by integration we have for the change of longitude =Lin- 1 cos 2 0. But since co = /2 , and tan 2e =-f - ,", therefore in degrees of arc, UH^tv 180 19 ttp-A) ,. x AL = - /^Oon ~^ - cot ^olf"" 1 ) cos ~#- In order to make the numerical results comparable with those in the paper on " Precession," I will apply this to the particular case which was the subject of the first method of integration of that paper.* It was there supposed that e = 17 30', and it was shown that looking back about 46 million years had fallen from unity to '88. Substituting for the various quantities their numerical values, I find that AL=0-31 cos 3 0=19' cos 2 0. Hence looking back 46 million years, we find the longitude of a point in latitude 30, further west by 4f than at present, and a point in latitude 60, further west by 14' both being referred to a point on the equator. Such a shift is obviously quite insignificant, but in order to see whether this screwing motion of the earth's mass could have had any influence on the crushing of the surface strata, it will be well to estimate the amount by which a cubic foot of the earth's mass at the surface would have been distorted. The motion being referred to the pole, it appears from (17) that a point distant p from the axis shifts through , p 3 Bt in the time 8t. There would be no shearing if ov O * " Precession," Section 15. WITH THE TIDES OF A VISCOUS SPHEROID. 551 a point distant p + &p shifted through ^ p~(p-}-8p)Bt ; but this second point does 8v G shift throuh Hence the amount of shear in unit time is i w M Therefore at the equator, at the surface where the shear is greatest, the shear per unit time is With the present values of T and \ * o we shall find that there are no non-periodic terms in the expression giving the tan- gential stress along the meridian ; and that the only non-periodic terms which exist in the expression giving the tangential stress perpendicular to the meridian are precisely those whose effects have been already considered as causing secular distortion, and which have their maximum effect when the obliquity is zero. Hence the whole result must be (1) A very minute change in the permanent or average figure of the globe ; (2) The secular distortion already investigated ; (3) Small tides of the second order. The one question which is of interest is, therefore Can these small tides be of any importance ? The sum of the moments of all the tangential stresses which result from the above expressions, about a pair of axes in the equator, one 90 removed from the moon's meridian and the other in the moon's meridian, together give rise to the precessional and nutatioiial couples. Hence it follows that part of he tangential stresses form a non-equibrating system of forces acting on the sphere's surface. In order to find the distorting effects on the globe, WITH THE TIDES OF A VISCOUS SPHEEOID. 553 we should, therefore, have to equibrate the system by bodily forces arising from the effects of the inertia due to the uniform precession and the fortnightly nutation just as was done above with the tidal friction. This would be an exceedingly laborious process ; and although it seems certain that the tides thus raised would be very small, yet we are fortunately able to satisfy ourselves of the fact more rigorously. Certain parts of the tangential stresses do form an equibrating system of forces, and these are precisely those parts of the stresses which are the most important, because they do not involve the sine of the obliquity. I shall therefore evaluate the tangential stresses when the obliquity is zero. The complete potential due both to- the moon and to the diurnal rotation is lr 2 (n 2 +r)(i- cos 2 0)+i|rV sin 3 6 cos 2(<-w<-e), and the complete expression for the surface of the spheroid is given by Hence ~ =i( 2 -H(i- cos 2 6)+T cos 2e sin 2 cos 2(<- S g-=i Then neglecting r 3 compared with rn 2 , and omitting the terms which were previously considered as giving rise to secular distortion, we find (T (LI O\ tli war- 33! S ft- ) = wa~T^ sin 2e sin cos 0(4 cos 2 6) sin '2( at), a. do\ '('/ 9 o- rf / cr\ n z wa z - ( S ft ) = ira 2 Ti sin 2e sin 0(i cos 2 0) cos 2(). Hence the intensity of this normal traction is estimated by the quantity WO?^T sin 2e, and this affords a standard of comparison with the quantity wa?re sin 2e, which was the estimate of the intensity of the secondary tides. The ratio of the two is 2 R> T, U have symmetrical forms. There- fore, substituting in the expression for the work (which will be called ), and V ' '/ remembering that (f' 1 ' (ill * '. , we have Now from (22) ^($+()+(^ iv L\oay \y / \" / J and from (23) (25) Adding (24) and (25) and rearranging the terms 1 ilE 6 cos 4<-<^ + 8a 2 -5r 22 fr 2 sin 2 ^32a 2 -r 2 2G+ sin The first of these terms is periodic, going through its cycle of changes in six lunar hours, and therefore the average rate of work, or the average rate of heat generation, is given by . . (26) It will now be well to show that this formula leads to the same results as those given in the paper on " Precession." In order to find the whole heat generated per unit time throughout the sphere, we must find the integral |~^ sm OdrddcUjt, from r=a to 0, 9=-ir to 0, <=27r to 0. * THOMSON and TAIT, 'Nat, Phil.,' 670. WITH THE TIDES OF A VISCOUS SPHEROID. 557 In a later investigation we shall require a transformation of the expression for , and as it will here facilitate the integration, it will be more convenient to effect the transformation now. If Q.J, Qj, be the zonal harmonics of the second and fourth order, cos' 0--=fQo+i, oofl**=&Qa-fQH-|.* Now (8a--5r-)--fr' sin 2 0[32a 3 -(26 + sin- 6)r 2 ] = (8a a 5J- 2 ) 2 - r-[48a- V^ 3 i(32a 3 -28r 2 ) cos 2 fr 3 cos 1 ff] Qi (27) The last transformation being found by substituting for cos 3 6 and cos 4 6 in terms of Q, and Q 4 , and rearranging the terms. The integrals of Q., and Q t vanish when taken all round the sphere, and 320a 1 -5GOa-r 2 + 25 ( Jr 1 )y- sin edrdOd^=' i ^{^-^+^}= x| X 19, where C is the earth's moment of inertia, and therefore equal to - Hence we have r- sin 8drd0M=*- sin 2e.f X 19=-(r sin 2e) 2 C. ' X ,, But tan 2e= -- =2. , so that - =- cot 2e. (jaw .^uja- oou g o And the whole work done on the sphere per unit time is \ - sin 4e.CV Now, as shown in the first part (equation 5), if jfl be the tidal Motional couple Ji ^ 3 . -=l-sm4 e . Therefore the work done on the sphere per unit time is $$.<. It is worth mentioning, in passing, that if the integral be taken from ^a to 0, we find that '32 of the whole heat is generated within the central eighth of the volume ; and by taking the integral from fa to a, we find that one-tenth of the whole heat is generated within 500 miles of the surface. It remains to show the identity of this remarkably simple result, for the whole work done on the sphere, with that used in the paper on " Precession." It was there shown * TODHUNTER'S 'Functions of LAPLACE,' &?., p. 13 ; or any other work on the subject. 558 MR. G. H. DARWIN ON PROBLEMS CONNECTED (Section 16) that if n be the earth's rotation, r the moon's distance at any time, v the ratio of the earth's mass to the moon's, then the whole energy both potential and kinetic of the moon-earth system is Now c being the moon's distance initially, since the lunar orbit is supposed to be circular, Also /J2\i ' 1 **w ~t- Therefore according to the notation of the paper on " Precession." In that paper I also put -=sn (y n l . Therefore .= And the whole energy of the system is gC( 2 * r e r Therefore the rate of loss of energy is C(n-r +-zj/2 ). \ "'' ^^ / But = 77, and as shown in the first part (19), p-n (l -=-, also \=fl. at O (It ' Therefore the rate of loss of energy is j%(ii /2) or jiw, which expression agrees with that obtained above, The two methods therefore lead to the same result. I will now return to the investigation in hand. The average throughout the earth of the rate of loss of energy is ^co-^-gTra 3 , which quantity will be called H. Then o 2 i 3 . sin 4e.o. 49ra** M * g g Now lvn . V . 2o> _ -r 2 . a , , T 3 . t H '19' sin 2e) a*='^ > cot2e. sin 2 2e.twr=- 1 - L 9 -3wa 3 . -. sin4e.a>= / "3 Hence (26) may be written dE dt ml i ~"\ ; f ai~ / "'" " i "- \-" i ""* ~i\ i i i WITH THE TIDES OF A VISCOUS SPHEROID. 553 This expression gives the rate of generation of heat at any point in terms of the average rate, and if we equate it to a constant we get the equation to the family of surfaces of equal heat-generation. We may observe that the heat generated at the centre is 3-^g times the average, at the pole of the average, and at the equator -^ of the average. ^w **i The accompanying figure exhibits the curves of equal heat-generation ; the dotted line shows that of ^ of the average, and the others those of -|, 1, 1^, 2, 2-^, and 3 times the average. It is thus obvious from inspection of the figure that by far the largest part of the heat is generated in the central regions. The next point to consider is the effect which the generation of heat will have on underground temperature, and how far it may modify the investigation of the secular cooling of the earth. It has already been shown * that the total amount of heat which might be generated is very large, and my impression was that it might, to a great extent, explain the increase of temperature underground, until a conversation with Sir W. THOMSON led me to undertake the following calculations : We will first calculate in what length of time the earth is losing by cooling an amount of energy equal to its present kinetic energy of rotation. The earth's conductivity may be taken as about '004 according to the results given in EVERETT'S illustrations of the centimeter-gram-second system of units, and the temperature gradient at the surface as 1 C. in 27-^ meters, which is the same as 1 Fahr. in 50 feet the rate used by Sir W. THOMSON in his paper on the cooling of the earth.t This temperature gradient is - - degrees C. per centimeter, and since there are 31,557,000 seconds in a year, therefore in centimeter-gram-second units, * "Precession," Section 15, Table IV., and Section 16. t THOMSON and TAIT'S ' Nat. Phil.,' Appendix D. MDCCCLXXIX. 4 C MR. G. H. DARWIN ON PROBLEMS CONNECT III) The heat lost byl 44 earth per annum J = earth>s suvface " square centimeters x ^ x n JJ 3 x 31557 x 10 7 = earth's surface x45'9 (centimeter-gram-second heat units). Now if J be JOULE'S equivalent Earth's kinetic ener"vi t .- i ( '"i.' M/oo/M,i 2 \ ot rotation in heatl.=$ = () 2 f-^ , where C = fM"- uuits J ^ J V 3 I n 2 = earth's surface x (f) 2 e , where e =- =-54^ 3J ij = earth's sm-face x (5'5)x (6-37)xlOx(4) > for a= 6'37 x 10* centimeters. 3 x 4-34 x 10 l x 232 J = 4'34 x lO'gramcenthu. and v:=d%. = earth's surface x 1'2 x 1U 10 nearly. Therefore at the present rate of loss the earth is losing energy by cooling equivalent l - 2 x 10 10 to its kinetic energy of rotation in ^- =262 million years. If we had taken the earth as heterogeneous and C=JMa 2 we should have found 218 'million years. We will next find how much energy is lost to the moon-earth system in the series of changes investigated in the paper on " Precession." In that paper (Section 16) it was shown that the whole energy of the system is / a 5r f n -g-Ma 2 ! n 2 -^ -I, where v is earth -=- moon, r moon's distance, n earth's diurnal rotation. \ ^*/ Hence the loss of energy =-^Ma 2 n 2 | ( ) 1 -^ 2 ( ) , while n passes from n L\ n o/ """o V 1 ' r o/ J to n , and r from r to r . Now -T^ (-%V=^lr=8-84a, taking ,= 82, and -^- =232. 2i/ - 8v \5/i 2 a/ 32 x 82 5ft -a n D If D be the length of the day, ~~ = T1 P ; an d if n be the mgon's distance in earth's radii, then loss of energy = (=*) 1 8'84( J X earth's present k.e. of rotation. LV-L*/ \" "o/j But in the paper on "Precession" we showed the system passing from a day of 5 hours 40 minutes,* and a lunar distance of 2'547 earth's radii, to a day of 24 hours, and a lunar distance of 60 '4 earth's radii. * A recalculation in the paper on "Precession" gave 5 hours 36 minutes, but I have not thought it worth while to alter this calculation. WITH THE TIDES OF A 7ISCOUS SPHEROID. 561 Now 24 -h 5;* = 4 -23, and (2-547)- 1 (60'4)- l =-37G. Therefore the loss of energy=[(4'23) 2 1 '376 X8'84] X earth's present k.e. := 13 '57 X earth's present k.e. of rotation. Hence the whole heat, generated in the earth from first to last, gives a supply of heat, at the present rate of loss, for 13 '6 x 262 million years, or 3,560 million years. This amount of heat is certainly prodigious, and I found it hard to believe that it should not largely affect the underground temperature. But Sir W. THOMSON pointed out to me that the distribution of its generation would probably be such as not materially to affect the temperature gradient at the earth's surface ; this remarkable prevision on his part has been confirmed by the results of the following problem, which I thought might be taken to roughly represent the state of the case. Conceive an infinite slab of rock of thickness 2a (or 8,000 miles) being part of an infinite mass of rock ; suppose that in a unit of volume, distant x from the medial plane, there is generated, per unit time, a quantity of heat equal to j)[320a 4 ' 560crar-f 259x 4 ] ; suppose that initially the slab and the whole mass of rock have a uniform temperature V; let the heat begin to be generated according to the above law, and suppose that the two faces of the slab are for ever maintained at the constant temperature V ; then it is required to find the distribution of temperature within the slab after any time. This problem roughly represents the true problem to be considered, because if we replace x by the radius vector r, we have the average distribution of internal heat- generation due to friction ; also the maintenance of the faces of the slab at a constant temperature represents the rapid cooling of the earth's surface, as explained by Sii W. THOMSON in his investigation. Let 9- be temperature, y thermal capacity, k conductivity ; then the equation of heat- flow is V + 259**]. Let32o|=2L, 560^=12M, 259y = 30N, and let the thermometric conductivity /. /i. /t/ :=-. Then 7 d f = K f J^+LaV-MaV+N^-El. at ear- Let the constant Rr=(L M+N)a 6 , and put V/=5+Lo.*ic 2 MaV'+Nz 8 R =S-La 4 (a 2 -z 2 ) +Mrt z (a*-a!*) -.N(a 8 a^). Then when x= a, t|<=$. 4 c 2 562 MR. G. H. DARWIN ON PROBLEMS CONNECTED Since L, M, N, R are constants as regards the time, i/=V 2Pe~">''cos qx is obviously a solution of this equation. Now we wish to make ^=V, when x=a, for all values of t; since ^=5 when x=a, this condition is clearly satisfied by making q=(2i+l)^-. aBt Hence the solution may be written, J . (29) ' ' and it satisfies all the conditions except that, initially, when t=0, the temperature everywhere should be V. This last condition is satisfied if cos for all values between a;=a. The expression on the right must therefore be expanded by FOURIER'S Theorem ; but we need only consider the range from x=a to 0, because the rest, from =0 to a, will follow of its own accord. Let x o~ ! l et w b e written for - ; let M'.= , N'=-r and R'=R^T. /ft 2i ra" ra a, Tiien and tfiis has to be equal to SP 2(+ iCos (2i+l)x from X = TT to 0. ^ Since pcos (2i+l)x cos (2;'4-l)x^X = unless ,;' > and Therefore Now WITH THE TIDES OF A VISCOUS SPHEROID. 563 cos cos(2i+l)x+&c. t &C. If therefore y(x) be a function of x involving only even powers of x> This theorem will make the calculation of the coefficients very easy, for we have at once Substituting for E,', L, M', N' their values in terms of - we find p . v -,.^ - 1988 6216 -t2l-J-l ~~~ /i- . -. \i -i i I /f\- . t\n o I / t -j I w*J' Then putting for CT its value, viz.: -| of 3'14159, and putting i successively equal to 0, 1, 2, it will be found that So that the FOURIER expansion is 120-907 cos ^+1-107 cos ^-'048 which will be found to differ by not so much as one per cent, from the function 564 MR. G. H. DARWTN ON PROBLEMS CONNECTED .3 2_0 2 to which it should be equal. Then by substitution in (29) we have as the complete solution of the problem satisfying all the conditions ?-VA !W / r? The only quantity, which it is of interest to determine, is the temperature gradient at the surface, which is equal to when a?= = j = a. Now when x=a, Then if t be not so large but that K( j t is a small fraction, we have approximately z~ dx . . and since -=- * 7 This formula will give the temperature gradient at the surface when a proper value is assigned to J, and if t be not taken too large. With respect to the value of t, Sir W. THOMSON took K=400 in British units, the year being the unit of time ; and a=21 X 10 6 feet. Hence =- nearlv > and K( } =77^; if therefore t be 10" years, this fraction is - 2 ^. Therefore the \a& I -Ll/ solution given above will hold provided the time t does not exceed 1,000 million years. We next have to consider what is the proper value to assign to !). By (27) and (28) it appears that fta* is j-^rs f tne average heat generated WITH THE TIDES OP A VISCOUS SPHEROID. 565 throughout the whole earth, which we called H. Suppose that p times the present kinetic energy of the earth's rotation is destroyed by friction in a time T, and suppose the generation of heat to be uniform in time, then the average heat generated through- out the whole earth per unit time is Therefore -|~ . yMaX, 2 -:- earth's volume. yj L _ p -- 5JT ' g 5 J' /it 2/j Where e is the ellipticity of figure of the homogeneous earth and is equal to f - , which I take as equal to ^-$. Hence )o JT and dS- 16 x 85 /7r\ 3 w pe t ~dx~~ 9500 \2/ 7 ~J~r But y=sw, where is specific heat. Therefore >e n 1 t _ ~dx~ 9500 T JT' The dimensions of J are those of work (in gravitation units) per mass and per scale of temperature, that is to say, length per scale of temperature ; p, e , and s have no dimensions, and therefore this expression is of proper dimensions. Now suppose the solution to run for the whole time embraced by the changes considered in "Precession," then =T, and as we have shown p-= 13 '57. Suppose the specific heat to be that of iron, viz.: ^. Then if we take J = 772, so that the result will be given in degrees Fahrenheit per foot, we have __ 13-57x9 dx~ 950 X 232x772 26o' That is to say, at the end of the changes the temperature gradient would be 1 Fahr. per 2,650 feet, provided the whole operation did not take more than 1,000 million years. It might, however, be thought that if the tidal friction were to operate veiy slowly, 5G6 MR. G. H. DAEWIN ON PROBLEMS CONNECTED so that the whole series of changes from the day of 5 hours 36 minutes to that of 24 hours occupied much more than 1,000 million years, then the large amount of heat which is generated deep down would have time to leak out, so that finally the temperature gradient would be steeper than that just found. But this is not the case. Consider only the first, and by far the most important, term of the expression for the temperature gradient. It has the form |) (1 e~ pT ), when t=T at the end of the 1 e -pT series of changes. Now |) varies as T" 1 , and has its maximum value unity when T=0. Hence, however slowly the tidal friction operates, the temperature gradient can never be greater than if the heat were all generated instantaneously ; but the tem- perature gradient at the end of the changes is not sensibly less than it would be if all the heat were generated instantaneously, provided the series of changes do not occupy more than 1,000 million years. III. The forced oscillations of viscous, fluid, and elastic spheroids. In investigating the tides of a viscous spheroid, the effects of inertia were neglected, and it was shown that the neglect could not have an important influence on the results.* I shall here obtain an approximate solution of the problem including the effects of inertia ; that solution will easily lead to a parallel one for the case of an elastic sphere, and a comparison with the forced oscillations of a fluid spheroid will prove instructive as to the nature of the approximation. If W be the potential of the impressed forces, estimated per unit volume of the viscous body, then (with the same notation as before) the equations of flow are (da. dy ^+^+^ dx^d^ dz (30) The terms w( -,.+ &c. ) are those due to inertia, which were neglected in the paper \at I on " Tides." It will be supposed that the tidal motion is steady, and that W consists of a series of solid harmonics each multiplied by a simple time harmonic, also that W includes not only the potential of the external tide-generating body, but also the effective potential due to gravitation, as explained in the first part of this paper. * " Tides," Section 10. WITH THE TIDES OP A VISCOUS SPHEROID. 567 The tidal disturbance is supposed to be sufficiently slow to enable us to obtain a first approximation by the neglect of the inertia terms. In proceeding to the second approximation, the inertia terms depending on the squares and products of the velocities, that is to say, w( a _+/8 +y~r), ma y be neglected com- da. pared with w. A typical case will be considered in which W=Ycos (vt-}-e), where (tt Y is a solid harmonic of the i"' degree, and the e will be omitted throughout the analysis for brevity. Then if we write I = 2(i+l) 2 +l, the first approximation, when the inertia terms are neglected, is ^_ i ^d 1 Iv[\_2(i 1) 2(2t+l) ]dx 2i+l dx^ '\ Hence for the second approximation we must put da. ivv f ] . w~ = { \ sin ttf. en Iu [ J And the equations to be solved are dp, ,., . rfY' , WP fft ., -j- + w V -a+ -r- cos 'y<+ S STT-rf a 8 - 0/0 . , , , r 2 dx dx Iv [[_2(t 1) 2(2t+l) ]dx -J-+ &c. =0, -^H- &c. =0 dy dz These equations are to be satisfied throughout a sphere subject to no surface stress. It will be observed that in the term due directly to the impressed forces, we write Y' instead of Y ; this is because the effective potential due to gravitation will be different in the second approximation from what it was in the first, on account of the different form which must now be attributed to the tidal protuberance. The problem is now reduced to one strictly analogous to that solved in the paper on dm ' Tides;" for we may suppose that the terms introduced by w &c., are components of bodily force acting on the viscous spheroid, and that inertia is neglected. The equations being linear, we consider the effects of the several terms separately, and indicate the partial values of , /3, y, p by suffixes and accents. first, then, we have rfY' - cos vt=0. &c., &c. ax * "Tides," Section 3, equation (8), or THOMSON and TAIT, ' Nat. Phil.,' 834 (8). MDCCCLXXIX. 4 D 568 MR. G. H. JIAKWIN ON PROBLEMS CONNECTED The solution of this has the same form as in the first approximation, viz. : equation (31), with written for a, and Y' for Y. We shall have occasion hereafter to use the velocity of flow resolved along the radius vector, which may be called p. Then Hence Then observing that Y'-i-r ; is independent of r, we have as the surface value _ i+1 i(2i + l)T * Secondly, Ctj)r\ t / . tt'C/l* t-l fr -T / t' J- . f i +vV-a +- -577- smitf = 0, &c., &c (35) die Iu 2(t 1) dx This, again, may clearly be solved in the same way, and we have , wva- i(i + 2) fR(i + 2) (i+l)(2i + S) fl "jdY i .,.., rf /A7 . _ o; _ 1 \ I / /.\ n s= < vd -*f* 'j-'^^ ( Yr -' A I gin i't (3b) Pu 3 2(i l)\\_2(i 1) 2(2t- J - 1 ^ - ; O,-_LI' \"/ and and its surface value is ,_ . +3 ^(i + 2)(2t+l) Y . ^ 0= ' [2Iu(i-l)P ^ S] Thirdly, let So that U is a solid harmonic of the i"' degree multiplied by a simple time harmonic. Then the rest of the terms to be satisfied are given in the following equations : t+ 3) -+ &c. =0, -+ Ac. =0 ' . . . (40) These equations have to be satisfied throughout a sphere subject to no surface stresses. The procedure will be exactly that explained in Part I., viz.: put a.=a'-\-a. r , WITH THE TIDES OF A VISCOUS SPHEROID. 569 /3=fi'-}-/3 / , y=y -}-y,, pp'-\-p fl an ^ find a/ > P', j , p any functions which satisfy the equations (40) throughout the sphere, Differentiate the three equations (40) by x, y, z respectively and add them together, and notice that =o, \_ux\ ax/ ay\ j uz\ /j \^ux\ UM / uy\ / IK\ /j and that fl>y * ft')/ f]& lltHj *^// ivtv then we have V 2 _p'=0, of which p'=Q is a solution. Now if V,, be a solid harmonic of degree n, V V"V i( =TO(2n+m+ l)r Hence 4(2^ + 3) rfa; r 2t+5 2(2i + 5) (41) Substituting from (41) in the equations of motion (40), and putting ^'=0, our equa- tions become (42) of which a solution is obviously v 4 dx2i + 5 ' I ...... (43) It may easily be shown that these values satisfy the equation of continuity, and thus together with jp'=0 they are the required values of a', /3', y', p, which satisfy the equations throughout the sphere. The next step is to find the surface stresses to which these values give rise. The formulas (13) of Part I. are applicable 4 D 2 570 MR. G. H. DARWIN ON PROBLEMS CONNECTED Then remembering that We have rfr_ ( d . d , d \ > v[ r- 1 a = v x +v +ZT a va \ dr / \ dx J dy dzj 4 _ _ (45) Then adding (44) and (45) together, we have for the component of stress parallel to the axis of x across any of the concentric spherical surfaces, =-p^+f-l'4 by (13), Part I. And at the surface of the sphere, where r=a, .... (46) 2 The quantities in square brackets are independent of r, and are surface harmonics of orders i 1 and i-\-l respectively. Let r = A;_j A/ +1 , Where A. __ ._L_ a ,- +2 ,, +2 / r - 3 /-i +1 ~ a T (47) Also let the other two components G and H of the surface stress due to ', ft', y , p' be given by G=-B,_ 1 -B I+1 , H=-C,_ 1 -C (+1 ...... (47) Then by symmetry it is clear that the B's and C's only differ from the A's in having y and z in place of x, WITH THE TIDES OF A VISCOUS SPHEROID. 571 We now have got in (43) values of a, /B', y, which satisfy the equations (40) throughout the sphere, together with the surface stresses in (47) to which they correspond. Thus (43) would be the solution of the problem, if the surface of the sphere were subject to the surface stresses (47). It only remains to find /; /3 /; y it to satisfy the equations -th- wvs <=' -| /+&c -=' -f + &c - =o < 48 > throughout the sphere, which is not under the influence of bodily force, but is subject to surface stresses of which A/.j+A^, B/.j+B/^.^ !,_! +C,- +1 are the components. The sum of the solution of these equations and of the solutions (43) will clearly be the complete solution ; for (43) satisfies the condition as to the bodily force in (40), and the two sets of surface actions will annul one another, leaving no surface action, For the required solutions of (48), Sir W. THOMSON'S solution given in (15) and (16) of Part I. is at once applicable. We have first to find the auxiliary functions SP,_. : , <&, corresponding to A;_ l5 B/_j, C,-_i, and ^V, ,- +2 corresponding to A ;+1 , B/ +1 , C,- +1 . It is easy to show that and . +B _ - 2 We have next to substitute these values of the auxiliary functions in THOMSON'S solution (15), Part I. It will be simpler to perform the substitutions piece-meal, and to indicate the various parts which go to make up the complete value of a. / by accents to that symbol. First. For the terms in a. i depending on A,-_ 1 , ^,_2, <&,-, we have '. __ LI __ L ' v i - 3 [2(i-2)(i-l)(2i- dx i- _ff*f i(i+l)* dV_(i + iy- rfUl ~wl4(i-l)(i-2) dx 2(i-2~) dx] - rfU . . v 4(i-l) Jx ' (Note that i 2 divides out, so that the solution is still applicable when 1 = 2). 572 MR. G. H. DARWIN ON PROBLEMS CONNECTED Second. In finding the terms dependent on A,- +1 , ,-, 3> i+ . : it will be better to subdivide the process further. (i) a "= - (a 2 -?- 2 ) _^z(H-10(2t + 3), 2 ,rfU , . ~v 2I(2t-L^ ^ '>*- ,..> ,/, i r i-t W a , =~i T 7?97 Then since therefore This completes the solution for 7 . Collecting results from (49), (50), and (51), we have / , //i /// ,=/+, + a , Then collecting results, the complete value of as the solution of the second approximation is a=a -j-a ' + a' + a. So that it is only necessary to collect the results of equations (31), (with Y' written for Y), (36), (43), and (52), and to substitute for U its value from (39) in order to obtain the solution required. The values of /3 and y may then at once be written down by symmetry. The expressions are naturally very long, and I shall not write them down in the general case. The radial velocity p is however an important expression, because it alone is" necessary to enable us to obtain the second approximation' to the form of the spheroid, and accordingly I will give it. It may be collected from (33), (37), and by forming p' and p t from (43) and (52). I find then after some rather tedious analysis, which I did in order to verify my solution, that as far as concerns the inertia terms alone WITH THE TIDES OF A VISCOUS SPHEROID. 573 p= W - sin vt { &r* #a~ ir r Where _ , ~2.4(2t + 5)I' 4(*-l)(2*+5)I >& 2.4(t-l) (2i+l)I - If J be reduced to the form of a single fraction, I think it probable that the numerator would be divisible by 2{+l, but I do not think that the quotient would divide into factors, and therefore I leave it as it stands. In the case where i=2 this formula becomes which agrees (as will appear presently) with the same result obtained in a different way. I shall now go on to the special case where i=2, which will be required in the tidal problem. From (39) we have From (36) , ww? 4 17 , 3.7 WY 2 .rf _,,] . a '= - 4a 8 ~r*)-r =r-j-(Yr ) sm u 3 19 3 [_\ 2.5 /(fo 5 (fo x From (43) WV From (52) 4.4 Adding these expressions together, and adding , we get m ^ ( 53 ) u and symmetrical expressions for /3 and y. In order to obtain the radial flow we multiply a by -^ ft by -, y by -, and add, and find . . . (54) the e which was omitted in the trigonometrical term being now replaced. 574 ME. G. H. DARWIN OX PROBLEMS CONNECTED The surface value of p when r=a is 79 Y . (55) where /3 is given by (34). If we write j^-ir for e we see that a term Y siu (vt e) in the effective disturbing potential will give us 79 Y ^-e) ........ (56) Now suppose wtf^S cos vt to be an external disturbing potential per unit volume of the earth, not including the effective potential due to gravitation, and let r=a-\-a- / be the first approximation to the form of the tidal spheroid. Then by the theory of tides as previously developed (see equation (15), Section 5, " Tides") o-, S 19t>y = - cos e cos (vt e), where tan e= a g 2ya<>- Then when the sphere is deemed free of gravitation the effective disturbing potential is wri S cos vt JJ ) ; this is equal to wr~ sin e S sin (vt e). Then in proceeding to a second approximation we must put in equation (56) Y= wr'sine S. Thus we get from (56), at the surface where r=a, 79 . , , To find p we must put r=a-\- an( ^ ^ s ^ n height ( 1 + i^V" cos- e J of the /C \ equilibrium tide of a perfectly fluid spheroid. By the method employed it is postulated that -&Q- is a small fraction, because the MDCCCLXXIX. 4 E 576 MR. G. H. DARWIN ON PROBLEMS CONNECTED effects of inertia are supposed to be small. Hence x must be a small angle, and there will not be much error in putting X ~i~5~ >sm cos e > an( i sec X = 1- " Then we have for the lag of the tide (e+i- 5 % sin ecos ej, and for its height s (l+J& |cos 2 e). Let if) be the lag, then ?= C +T% sin e cos e, whence i =1) i^ 5 T5 sin 77 cos 77 very nearly. Also / i? \ cos e= cos 77! 1+1-5%- sin 2 77 ], and u> y=r sin 6 sin (< eat) z=-r cos Now consider the case when the viscosity is infinitely small : here e is small, and 38i* sin 2e=tan 2e=r~ :. 5g- 2 Plence - sin 2e -. which is independent of the viscosity. 38i> 5g 2 By siibstituting this value in (81), we see that however small the viscosity, the nature of the motion, by which each particle assumes its successive positions, always preserves the same character ; and the motion always involves molecular rotation. But it has been already proved that, however slow the tidal motion of a fluid spheroid may be, yet the fluid motion is always irrotational. Hence in the two methods of attacking the same problem, different first approxi- mations have been used, whence follows the discrepancy of 79 instead of 75. The fact is that in using the equations of flow of a viscous fluid, and neglecting inertia to obtain a first approximation, we postulate that w, w , w , are less im- portant than v V 2 , v V 2 /3, v V 2 y ; and this is no longer the case if v be very small. It does not follow therefore that, in approaching the problem of fluidity from the side of viscosity, we must necessarily obtain even an approximate result. But the comparison which has just been made, shows that as regards the form of the tidal spheroid the two methods lead to closely similar results. It follows therefore that, in questions regarding merely the form of the spheroid, and not the mode of internal motion, we only incur a very small error by using the limiting case when v=Q to give the solution for pure fluidity. In the paper on " Precession " (Section 7), some doubt was expressed as to the applicability of the analysis, which gave the effects of tides on the precession of a rotating spheroid, to the limiting case of fluidity ; but the present results seem to justify the conclusions there drawn. The next point to be considered is the effects of inertia in The forced oscillations of an elastic sphere. Sir WILLIAM THOMSON has found the form into which a homogeneous elastic sphere becomes distorted under the influence of a potential expressible as a solid harmonic of the points within the sphere. He afterwards supposed the sphere to possess the power of gravitation, and considered the effects by a synthetical method. The result is the equilibrium theory of the tides of an elastic sphere. When, however, the disturbing potential is periodic in time this theory is no longer accurate. 4 F 2 586 MR. G. H. DARWIN ON PROBLEMS CONNECTED It has already been remarked that the approximate solution of the problem of determining the state of internal flow of a viscous spheroid when inertia is neglected, is identical in form with that which gives the state of internal strain of an elastic sphere ; the velocities a, ft, y have merely to be read as displacements, and the coefficient of viscosity v as that of rigidity. The effects of mutual gravitation may also be introduced in both problems by the same artifice ; for in both cases we may take, instead of the external disturbing potential wr~Scosvt. an effective potential wr^S cos vt JJ-j, and then deem the sphere free of gravitational power. Now Sir WILLIAM THOMSON'S solution shows that the surface radial displacement (which is of course equal to cr) is equal to 5wa 3 /~ er\ -IS cost**-] (82) 19u \ s af ' If therefore we put (with Sir WILLIAM THOMSON) t= , we have -'= - cos vt. 2 ft This expression gives the equilibrium elastic tide, the suffix being added to the cr to indicate that it is only a first approximation. Before going further we may remark that Scosttf tt ff -= Scosv ..... (83) r+g When we wish to proceed to a second approximation, including the effects of inertia, it must be noticed that the equations of motion in the two problems only differ in the fact that in that relating to viscosity the terms introduced by inertia are dx d/3 dy . .. . , f . (Pa. d~$ d-y w- t w, ~77, whilst in the case of elasticity they are iv-^, w-^j, w . Hence a very slight alteration will make the whole of the above investigation applicable to the case of elasticity ; we have, in fact, merely to differentiate the approximate values for a, ft, y twice with regard to the time instead of once. Then just as before, we find the surface radial displacement, as far as it is due to inertia, to be (compare (55)) Tgpggs 79 Y i/ 2.3.19* r C< Vt> y and cos vt must be put equal to (the first approximation) Scosvt JJ -'. Hence by w 3 c s ft 5 79 t (57) and (83) the surface radial displacement due to inertia is 3 - S cos vt. v ^i.o.iJ" C H~ To this we must add the displacement due directly to the effective disturbing potential wr*(S cosvt jj-j, where a- is now the second approximation. This we know from (82) is equal to WITH THE TIDES OF A VISCOUS SPHEROID. 587 5 wa? cos vt % -]. Hence the total radial displacement is 5wa? / a- 5w 3 79 y 2 r wa? / a- 5w 3 79 y 2 r Q \ - (S cos vt tt\----.^ S cos vt ). 19v V 8 ' I9u lt>0 r + g / But the total radial displacement is itself equal to cr. Therefore and This is the second approximation to the form of the tidal spheroid, and from it we see that inertia has the effect of increasing the ellipticity of the spheroid in the proportion Analogy with (76) would lead one to believe that the period of the gravest vibration / 79 V of an elastic sphere is 2ir( ~ ; this result might be tested experimentally. \ lour / If (J be put equal to zero, the sphere is devoid of gravitation, and if t be put equal to zero the sphere becomes perfectly fluid ; but the solution is then open to objections similar to those considered, when viscosity graduates into fluidity. It is obvious that the whole of this present part might be easily adapted to that hypothesis of elastico- viscosity which was considered in the paper on " Tides," but it does not at present seem worth while to do so. By substituting these second approximations in the equations of motion again, we might proceed to a third approximation, and so on ; but the analytical labour of the process would become very great. IV. Discussion of the applicability of the results to the history of the earth. The first paper of this series was devoted to the consideration of inequalities of short period, in the state of flow of the interior, and in the form of surface, produced in a rotating viscous sphere by the attraction of an external disturbing body : this was the theory of tides. The investigation was admitted to be approximate from two causes (i) the neglect of the inertia of the relative motion of the parts of the spheroid ; (ii) the neglect of tangential action between the surface of the mean sphere and the tidal protuberances. 588 MR. G. H. DARWIN ON PROBLEMS CONNECTED In the second paper the inertia was still neglected, but the effects of these tan- gential actions were considered, in as far as they modified the rotation of the spheroid as a whole. In that paper the sphere was treated as though it were rigid, but had rigidly attached to its surface certain inequalities, which varied in distribution from instant to instant according to the tidal theory. In order to justify this assumption, it is now necessary to examine whether the tidal protuberances may be regarded as instantaneously and rigidly connected with the rotating sphere. If there is a secular distortion of the spheroid in excess of the regular tidal flux and reflux, the assumption is not rigorously exact ; but if the dis- tortion be very slow, the departure from exactness may be regarded as insensible. The first problem in the present paper is the investigation of the amount of secular distortion, and it is treated only in the simple case of a single disturbing body, or moon, moving in the equator of the tidally-distorted spheroid or earth. It is found, then, that the form of the lagging tide in the earth is not such that the pull, exercised by the moon on it, can retard the earth's rotation exactly as though the earth were a rigid body. In other words, there is an unequal distribution of the tidal frictional couple in various latitudes. We may see in a general way that the tidal protuberance is principally equatorial, and that accordingly the moon tends to retard the diurnal rotation of the equatorial portions of the sphere more rapidly than that of the polar regions. Hence the polar regions tend to outstrip the equator, and there is a slow motion from west to east relatively to the equator. When, however, we come to examine numerically the amount of this screwing motion of the earth's mass, it appears that the distortion is exceedingly slow, and accordingly the assumption of the instantaneous rigid connexion of the tidal protube- rance with the mean sphere is sufficiently accurate to allow all the results of the paper on " Precession " to hold good. In the special case, which was the subject of numerical solution in that paper, we were dealing with a viscous mass which in ordinary parlance would be called a solid, and it was maintained that the results might possibly be applicable to the earth within the limits of geological history. Now the present investigation shows that if we look back 45,000,000 years from the present state of things, we might find a point in lat. 30 further west with reference to a point on the equator, by 4f than at present, and a point in lat. 60 further west by 14^'. The amount of distortion of the surface strata is also shown to be exceedingly minute. From these results we may conclude that this cause has had little or nothing to do with the observed crumpling of strata, at least within recent geological times. If, however, the views maintained in the paper on " Precession " as to the remote history of the earth are correct, it would not follow, from what has been stated above, that this cause has never played an important part ; for the rate of the screwing of the WITH THE TIDES OF A VISCOUS SPHEROID. 589 earth's mass varies inversely as the sixth power of the moon's distance, multiplied by the angular velocity of the earth relatively to the moon. And according to that theory, in very early times the moon was very near the earth, whilst the relative angular velocity was comparatively great. Hence the screwing action may have been once sensible.""" . Now this sort of motion, acting on a mass which is not perfectly homogeneous, would raise wrinkles on the surface which would run in directions perpendicular to the axis of greatest pressure. In the case of the earth the wrinkles would run north and south at the equator, and would bear away to the eastward in northerly and southerly latitudes ; so that at the north pole the trend would be north-east, and at the south pole north-west. Also the intensity of the wrinkling force varies as the square of the cosine of the latitude, and is thus greatest at the equator, and zero at the poles. Any wrinkle when once formed would have a tendency to turn slightly, so as to become more nearly east and west, than it was when first made. The general configuration of the continents (the large wrinkles) on the earth's sur- face appears to me remarkable when viewed in connexion Avith these results. There can be little doubt that, on the whole, the highest mountains are equatorial, and that the general trend of the great continents is north and south in those regions. The theoretical directions of coast line are not so well marked in parts removed from the equator. * This result is not strictly applicable to the case of infinitely small viscosity, because it gives a finite though very small circulation, if the coefficient of viscosity be put equal to zero. By putting c=0 in (17'), Part I., we find a superior limit to the rate of distortion. With the present angular velocities of the earth and moon, -- must be less than 5 x 10~ 9 cos 2 6 in degrees per annum. etc It is easy to find when would be a maximum in the course of development considered in " Preces- ctt sion;" for, neglecting the solar effects, it will be greatest when i"(n Q) is greatest. .Now T 2 (?i Q) varies as [l+/i /*f -- ~'-f~ 3 ]IT li! > and this function is a maximum when n Taking ,1=4-0074, and '-^=27'32, we have r i -109'45f- 1 i3 o The solution of this is ='2218. With this solution will be found to be 5(3 million times as great as at present, being equal to 18' cos 3 per annum. With this value of f, the length of the day is 5 hours 50 minutes, and of the month 7 hours 10 minutes. This gives a superior limit to the greatest rate of distortion which can ever have occurred. By (19'), however, we see that the rate of distortion per unit increment of the moon's distance may be made as large as we please by taking the coefficient of viscosity small enough. These considerations seem to show that there is no reason why this screwing action of the earth should not once have had considerable effects. (Added October 15, 1879.) 590 MR. G. H. DARWIN ON PROBLEMS CONNECTED The great line of coast running from North Africa by Spain to Norway lias a decidedly north-easterly bearing, and the long Chinese coast exhibits a similar ten- dency. The same may be observed in the line from Greenland down to the Gulf of Mexico, but here we meet with a very unfavourable case in Panama, Mexico, and the long Californian coast line. From the paucity of land in the southern hemisphere the indications are not so good, nor are they veiy favourable to these views. The great line of elevation which runs from Borneo through Queensland to New Zealand might perhaps be taken as an example of north-westerly trend. The Cordilleras run very nearly north and south, but exhibit a clear north-westerly twist in Tierra del Fuego, and there is another slight bend of the same character in Bolivia. But if this cause was that which principally determined the direction of terrestrial inequalities, then the view must be held that the general position of the continents has always been somewhat as at present, and that, after the wrinkles were formed, the surface attained a considerable rigidity, so that the inequalities could not entirely subside during the continuous adjustment to the form of equilibrium of the earth, adapted at each period to the lengthening day. With respect to this point, it is worthy of remark that many geologists are of opinion that the great continents have always been more or less in their present positions. An inspection of Professor SCHIAPPARELLI'S map of Mars,""" I think, will prove that the north and south trend of continents is not something peculiar to the earth. In the equatorial regions we there observe a great many very large islands, separated by about twenty narrow channels running approximately north and south. The northern hemisphere is not given beyond lat. 40, but the coast lines of the southern hemisphere exhibit a strongly marked north-westerly tendency. It must be confessed, however, that the case of Mars is almost too favourable, because we have to suppose, according to the theory, that its distortion is due to the sun, from which the planet must always have been distant. The very short period of the inner satellite shows, however, that the Martian rotation must have been (according to the theory) largely retarded ; and where there has been retardation, there must have been internal distortion. The second problem which is considered in the first part of the present paper is concerned with certain secondary tides. My attention was called to these tides by some remarks of Dr. JULES GARRET,! who says : " Les actions perturbatrices du soleil et de la lune, qui produisent les mouvements coniques de la precession des e'quinoxes et de la nutation, n'agissent que sur cette portion de 1'ellipsoide terrestre qui excede la sphere tangente aux deux poles, c'est-a-dire, en admettant I'e'tat pateux de 1'inte'rieur, h peu pres uniquement sur ce * ' Appendice alle Memorie della Societa degli Spettroscopisti Italiani,' 1878, vol. vii., for a copy of which I have to thank M. SCHIAPPABELLI. t Societe Savoisienne d'Histoire et d'Archeologie, May 23, 1878. He is also author of a work, ' Lo Peplacement Polnire.' I think Dr. GARRET has misunderstood Mr. EVANS. WITH THE TIDES OP A VISCOUS SPHEROID. 591 que Ton est convenu d'appeler la croute terrestre, et presque sur toute la croute ter- restre. La croute glisse sur 1'interieur plastique. Elle parvient entrainer 1'inte'rieiir, car, sinon, 1'axe de la rotation du globe detneurerait parallele a lui-meme dans 1'espace, ou n'eprouverait que des variations iusignifiantes, et le phenomene de la precession des equinoxes n'existerait pas. Ainsi la croute et 1'interieur se meuvent de quantites in^gales, d'ou le deplacement geographique du pole sur la sphere. " Cette idee a ete emise, je crois, pour la premiere fois, par M. EVANS ; depuis par M. J. PfeocHE." Now with respect to this view, it appears to me to be sufficient to remark that, as the axes of the precessional and nutational couples are fixed relatively to the moon, whilst the earth rotates, therefore the tendency of any particular part of the crust to slide over the interior is reversed in direction every twelve lunar hours, and therefore the result is not a secular displacement of the crust, but a small tidal distortion. As, however, it was just possible that this general method of regarding the subject overlooked some residual tendency to secular distortion, I have given the subject a more careful consideration. From this it appears that there is no other tendency to distortion besides that arising out of tidal friction, which has just been discussed. It is also found that the secondary tides must be very small compared with the primary ones ; with the present angular velocity of diurnal rotation, probably not so much in height as one-hundredth of the primary lunar semi-diurnal bodily tide. It seems out of the question that any heterogeneity of viscosity could alter this result, and therefore it may, I think, be safely asserted that any sliding of the crust over the interior is impossible at least as arising from this set of causes. The second part of the paper is an investigation of the amount of work done in the interior of the viscous sphere by the bodily tidal distortion. According to the principles of energy, the work done on any element makes itself manifest in the form of heat. The whole work which is done on the system in a given time is equal to the whole energy lost to the system in the same time. From this consideration an estimate was given, in the paper on " Precession," of the whole amount of heat generated in the earth in a given time. In the present paper the case is taken of a moon moving round the earth in the plane of the equator, and the work done on each element of the interior is found. The work done on the whole earth is found by summing up the work on each element, and it appears that the work per unit time is eqxial to the tidal frictional couple multiplied by the relative angular velocity of the two bodies. This remarkably simple law results from a complex law of internal distribution of work, and its identity with the law found in " Precession," from simple considerations of energy, affords a valuable confirmation of the complete consistency of the theory of tides with itself. Fig. 2 gives a graphical illustration of the distribution in the interior of the work done, or of the heat generated, which amounts to the same thing. The reader is referred to Part II. for an explanation of the figure. Mere inspection of the figure MDCCCLXXIX. 4 G 502 MR. G. H. DARWIN ON PROBLEMS CONNECTED shows that by far the larger part of the heat is generated in the central parts, and calculation shows that about one-third of the whole heat is generated within the central one-eighth of the volume, whilst in a spheroid of the size of the earth only one-tenth is generated within 500 miles of the surface. In the paper on " Precession " the changes in the system of the sun, moon, and earth were traced backwards from the present lengths of day and month back to a common length of day and month of 5 hours 36 minutes, and it was found that in such a change heat enough must have been generated within the earth to raise its whole mass 3000 Fahr. if applied all at once, supposing the earth to have the specific heat of iron. It appeared to me at that time that, unless these changes took place at a time very long antecedent to geological history, then this enormous amount of in- ternal heat generated would serve in part to explain the increase of temperature in mines and borings. Sir WILLIAM THOMSON, however, pointed out to me that the distribution of heat-generation would probably be such as to prevent the realisation of my expectations. I accordingly made the further calculations, connected with the secular cooling of the earth, comprised in the latter portion of Part II. It is first shown that, taking certain average values for the increase of underground temperature and for the conductivity of the earth, then the earth (considered homo- geneous) must be losing by conduction outwards an amount of energy equal to its present kinetic energy of rotation in about 262 million years. It is next shown that in the passage of the system from a day of 5 hours 40 minutes to one of 24 hours, there is lost to the system an amount of energy equal to 13^ times the present kinetic energy of rotation of the earth. Thus it appears that, at the present rate of loss, the internal friction gives a supply of hea.t for 3,560 million years. So far it would seem that internal friction might be a powerful factor in the secular cooling of the earth, and the next investigation is directly concerned with that question. In the case of the tidally-distorted sphere the distribution of heat-generation depends on latitude as well as depth from the surface, but the average law of heat- generation, as dependent on depth alone, may easily be found. Suppose, then, that we imagine an infinite slab of rock 8,000 miles thick, and that we liken the medial plane to the earth's centre and suppose the heat to be generated uniformly in time, according to the average law above referred to. Then conceive the two faces of the slab to be always kept at the same constant temperature, and that initially, when the heat-generation begins, the whole slab is at this same temperature. The problem then is, to find the rate of increase of temperature going inwards from either face of the slab after any time. This problem is solved, and by certain considerations (for which the reader is referred back) is made to give results which must agree pretty closely with the temperature gradient at the surface of an earth in which 13^ times the present kinetic energy of earth's rotation, estimated as heat, is uniformly generated in time, with the average space distribution referred to. It appears that at the end of the heat-generation the WITH THE TIDES OF A VISCOUS SPHEROID. 593 temperature gradient at the surface is sensibly the same, at whatever rate the heat is generated, provided it is all generated within 1,000 million years ; but the temperature gradient can never be quite so steep as if the whole heat were generated instan- taneously. The gradient, if the changes take place within 1,000 million years, is found to be about 1 Fahr. in 2,600 feet. Now the actually observed increase of underground temperature is something like 1 Fahr. in 50 feet ; it therefore appears that perhaps one-fiftieth of the present increase of underground temperature may pos- sibly be referred to the effects of long past internal friction. It follows, therefore, that Sir WILLIAM THOMSON'S investigation of the secular cooling of the earth is not sensibly affected by these considerations. If at any time in the future we should attain to an accurate knowledge of the increase of underground temperature, it is just within the bounds of possibility that a smaller rate of increase of temperature may be observed in the equatorial regions than elsewhere, because the curve of equal heat generation, which at the equator is nearly 500 miles below the surface, actually reaches the surface at the pole. The last problem here treated is concerned with the effects of inertia on the tides of a viscous spheroid. As this part will be only valuable to those who are interested in the actual theory of tides, it may here be dismissed in a few words. The theory used in the two former papers, and in the first two parts of the present one, was founded on the neglect of inertia ; and although it was shown in the paper on " Tides" that the error in the results could not be important, in the case of a sphere disturbed by tides of a frequency equal to the present lunar and solar tides, yet this neglect left a defect in the theory which it was desirable to supply. Moreover it was possible that, when the frequency of the tides was much more rapid than at present (as was found to have been the case in the paper on "Precession"), the theory used might be seriously at fault. It is here shown (see (62) ) that for a given lag of tide the height of tide is a little greater, and that for a given frequency of tide the lag is a little greater than the approximate theory supposed. A rough correction is then applied to the numerical results given in the paper on " Precession" for the secular changes in the configuration of the system ; it appears that the time occupied by the changes in the first solution (Section 15) is overstated by about one-fortieth part, but that all the other results, both in this solution and the other, are left practically unaffected. To the general reader, therefore, the value of this part of the paper simply lies in its confirmation of previous work. From a mathematical point of view, a comparison of the methods employed with those for finding the forced oscillations of fluid spheres is instructive. Lastly, the analytical investigation of the effects of inertia on the forced oscillations of a viscous sphere is found to be applicable, almost verbatim, to the same problem concerning an elastic sphere. The results are complementary to those of Sir WILLIAM THOMSON'S statical theory of the tides of an elastic sphere. 4 G 2 ON THE / SECULAR CHANGES IN THE ELEMENTS OF THE DEBIT OF A SATELLITE REVOLVING ABOUT A TIDALLT DISTORTED PLANET. BY G. H. DAKWIN, F.RS. From the PHILOSOPHICAL TRANSACTIONS OP THE ROYAL SOCIETY. PART II. 1880. LONDON : HARRISON AND SONS, PRINTERS IN ORDINARY TO HER MAJESTY, ST. MARTIN'S LANE. [ 713 ] XX. On the Secular Changes in the Elements of the Orbit of a Satellite revolving about a Tidally distorted Planet. ERRATA. IN a paper " On the secular changes in the elements of the orbit of a satellite, i, we have by integration, 0),= -- : - .COS I --- COS Y --- T^Slll Y nsim\ d% (fy ] n di 1 / . dW dW\ 1 dW *).,= . ; COS I j -- - Sin Y --- COS x n sin i \ d% d-fy J n di Then substituting these values in the geometrical equations, di . ctt We have finally, = &>, cos x~r w 2 sm X .(Mr Sill I --:. = toj Sill X~ 0} 1 COS . di . fAV ,l\\ cos dW n sim = cos i -j -- dt dndW These are the equations which will be used for determining the perturbations of the planet's rotation. We now see that the same disturbing function W will serve for finding both sets of perturbations. It is clear that it is not necessary in the above investigation that cr should actually be a tide wave ; it may just as well refer to the permanent oblateness of the planet. Thus the ordinary precession and nutations may be determined from these formulas. 3. To find spherical liarmonic functions of Dianas coordinates with reference to axefl fixed in the earth. Let A, B, C be rectangular axes fixed in the earth, C being the pole and AB tlie equator. THE ELEMENTS OP THE ORBIT OF A SATELLITE. 723 Fig. 2. Let X, Y, Z be a second set of rectangular axes, XY being the plane of Diana's orbit. Let M be the projection of Diana in her orbit. Let i,=ZC, the obliquity of the equator to the plane of Diana's orbit. X/ =AX=BCY. ^=MX, Diana's longitude from the node X. Let M!= cos MAI M 2 = cos MB > Diana's direction-cosines referred to A, B, C. M 3 =cosMC J Then M L = cos l { cos x,+ sin l t sin x, cos i t "1 M.,= cos l t sinx,+ sin l t cos x/ cost, > (19) M 3 = sin Z 7 sin i / We may observe that M 2 is derivable from M t by writing x,~^~^ 7r i place of X ,- These expressions refer to the plane of Diana's orbit, but we must now refer to the ecliptic. Fig. 3. In fig. 3, let A be the autumnal equinox, B the ascending node of the orbit, C the intersection of the orbit with the equator, being the X of fig. 2, and let D be a point fixed in the equator, being the A of fig. 2. Then if we refer to the sides and angles of the spherical triangle A B C by the letters a, b, c, A, B, C as is usual in works on spherical trigonometry, we have A=i, the obliquity of the ecliptic. B=/, the inclination of the orbit. TT- C = i,=ZCof fig. 2. c =.2V, the longitude of the node measured from A, for at present we may suppose i/=0, without loss of generality. 724 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Then let x^DA, and we hav* Again, if M be Diana in her orbit, MB=/, and since MC=/ 7 , therefore Whence cos x,= cos x cos b+ sin x sin b sin x,= sin x cos b cosx sin b cos ^= cos I cos a sin I sin a sin l e = sin I cos a+ cos I sin a Substituting these values in the first of (19) we have M\=:cos x cos I (cos a cos b sin a sin b cos ij+sin x cos I (cos a sin b+sin a cos b cos i) cos x sin I (sin a cos b+cos a sin b cos i t ) sin x sin I (sin a sin b cos a cos b cos i t ) Now cos i = cos C, and cos a cos b+sin a sin b cos C=cos c=cos N cos a sin b sin a cos b cos C=sin a [cot a sin b cos b cos C]=sin a cot A sin C = cos i sin N sin a cos b cos a sin b cos C=sin b [cot b sin a cos a cos C]=sin b cot B sin C = cos^' sin N sin a sin b+cos a cos b cos C=sin a sin b+cos c cos C sin a sin b cos 2 C = sin a sin b siir C+cos c( cos A cos B+sin A sin B cos c) =sin A sin B sin 2 c+sin A sin B cos 2 c cos A cos B cos c =sin i sin j cos i cosj cos 2V Then substituting in the expression for M lf ^cos x cos I cos 2V+sin x cos / sin N cos icos x sin I sm N cos j sin x sin I (sin * sin^' cos i cosj cos N) Let P= Then = sin .',/ cos x cos I cos _2 cos gin i sin cos (x-l-N)+P*q z cos cos ) sin x cos I sin N _ ( - sin sin / cos N cos cos sin sin I -/)] . (20) THE ELEMENTS OF THE ORBIT OF A SATELLITE. 725 Since M. 2 is derivable from M x by writing X/+2 71 " f r X/> therefore it is also derivable by writing x+a 77 f r X- Hence M c is the same as M ls save that sines replace cosines Again M 3 =sin l f sin ^=3111 I cos a sin i^+cos I sin a sin i t But sin a sin i^sin i sin N=2PQ sin N And cos a sin {^sin i cot a sin c=sin i (cot A sin B+cos c cos B) =cos i sin _/+ sin i cos/ cos JV = 2pq(I K Q 2 ) + l 2PQ(p*q~) cos N Therefore * M 3 =2P[p 2 sin(/+^- ? 2 sin(Z-^)]+2^(P 2 -# 2 )sinZ . . . (21) For the sake of future developments it will be more convenient to replace the sines and cosines in the expressions for the M'-s by exponentials, and for brevity the \/l will be omitted in the indices. Then 2M 1 =e*-'-*[Pp Qqe K J+e x+l+N [Qp+Pqe- A J+ the same with the signs of the indices of the exponentials changed, 2M 2V /"-^l= the same with sign of second line changed, M 3 represent a function of the same form as XY, save that each simple time- harmonic term of XY is multiplied by some fraction expressive of reduction of height of tide, and that the argument of each such simple harmonic term is altered in phase ; the constants so introduced will be functions of the constitution of the spheroid, and of the speed of the harmonic terms. Also extend the same notation to the other functions of X, Y, Z which occur in V. Then it is clear that, if r=a+cr be the equation to the complete wave surface corresponding to the potential V, ' T a + ! u t/ This expression shows that cr is a surface harmonic of the second order. Then by (17) we have for the disturbing function for the moon, due to Diana's tides, where cr is the height of tide, at the point where the moon's radius vector pierces the wave surface. THE ELEMENTS OF THE ORBIT OF A SATELLITE. Hence in the expression (30) for cr, we must put =M/, ,=M S ', =M 3 ' Then by analogy with (29), let and we have 3 This is the required expression for the disturbing function on the moon, due to Diana's tides. So far the investigation is general, but we now have to develop this function so as to make it applicable to the several problems to be considered. II. SECULAR CHANGES IN THE INCLINATION OF THE ORBIT OF A SATELLITE. 5. The perturbed satellite moves in a circular orbit inclined to a fixed plane. Subdivision of the problem. In this case e=0, e'=0, r=c, r'=c, so that the functions X, Y, Z and X', Y', Z' are simply the direction cosines of Diana and the moon, referred to the axes A, B, C fixed in the earth. Hence X=M 1 , Y=M 2 , Z = M 3 , and the five formulas (24-8) give the functions X 2 -Y 2 , 2XY, 2YZ, 2ZX, Z 2 . In order to form the functions in gothic letters we must express these functions as simple time-harmonics. The formulas (24) to (28) are equivalent to the expression of the five functions as a series of terms of the type A cos (ax-}-ftO-\-yN+?>). Now x is the angle between a point fixed on the equator and the autumnal equinox, and therefore (neglecting alterations in the diurnal rotation and the precessional motion) increases uniformly with the time, being equal to nt+& constant, which constant may be treated as aero by a proper choice of axes A, B, C. Q is the true longitude measured from the autumnal equinox, and is equal to fl +/, since the orbit is circular ; also $ may for the present be put equal to zero, without any loss of generality. Then if in forming the expressions for the state of tidal distortion of the earth we neglect the motion of the node, the five functions are expressed as a series of simple time-harmonics of the type A cos (c 730 MR. G. H. DARWIN ON THE SECULAR CHANGES IN The corresponding term in the corresponding gothic-letter function will be KA cos (ant-\-f3flt-\- k), where K is the fraction by which the tide is reduced and k is the alteration of phase. It appears, from the inspection of the five formulas (24-8), that there are tides of seven speeds, viz. : 2( /2), 2n, 2(?i+/2), n 2/2, n, n+2/2, 2/2. The following schedule gives the symbols to be introduced for reduction of tide and alteration of phase or lag. Semi-diurnal. Diurnal. Fortnightly. A _ A. Slow. Sidereal. Fast. Slow. Sidereal. Fast. Speed ........ 2( /2), 2, 2(n+/2,) n 2/2, n, i+2/2, 2/2 Fraction of equilibrium tide . F x F F 2 G l G G 2 H Retardation of phase or lag. . 2fj 2f 2f 2 gj g g 2 2h The gothic-letter functions may now at once be written down from (24-8). Thus, ~ ( * . . . (32) 1 = the same, with second line of opposite sign . . . . (33) . . . (34) l=the same, with second line of opposite sign ...... (35) (36) The fact that there is no factor of the same kind as H in the first pair of (36) results from the assumption that the tides due to the motion of the nodes of the orbit are the equilibrium tides unaltered in phase. The formulas for 2(X' 2 -Y' 2 ), -4X'YV^T, 2Y'Z', 2X'ZV^1, i~Z' 2 are found by symmetry, by merely accenting all the symbols in the five formulas (24-8) for the M functions. In the use made of these formulas this accentuation will be deemed to be done. At present we shall not regard x as being accented, but in 12 and in Part III. we shall have to regard x as a l so accented. We now have to develop the several products of the X' functions multiplied by the $ functions. THE ELEMENTS OP THE ORBIT OF A SATELLITE. 731 Before making these multiplications, it must be considered what are the terms which are required for finding secular changes in the elements, since all others are superfluous for the problem in hand. Such terms are clearly those in which 6 and 0' are wanting, and also those where 06' occurs, for these will be wanting in when Diana is made identical with the moon. It follows therefore that we need only multiply together terms of the like speeds. In the following developments all superfluous terms are omitted. Semi-diurnal terms. These are 2XT' If we multiply (24) (with accented symbols) by (32), and (25) (with accented symbols) by (33), and subtract the latter from the former, we see that x disappears from the expression, and that, 8X'Y'9+2(X' 3 -Y' 2 )( 3 -l c ) = First line of (24) X second of (32) + Second of (25) X first of (33) Then as far as we are concerned . (37) If x had been accented in the X' functions, we should have had 2(% x') i n a indices of exponentials of the first line, and 2(x x) in all the indices of the second line. These three pairs of terms will be called W : , W n , W m . Diurnal terms. These are 2Y'Z'|$2 + 2X / Z'$& If the multiplications be performed as in the previous case, it will be found that x disappears in the sum of the two products, and, as far as concerns terms in ff and those independent of and & ' , we have V - . ( 38 ) If x had been accented in the X' functions we should have had x~X ' n a 732 MR. G. H. DARWIN ON THE SECULAR CHANGES IN indices of the exponentials of the first line, and (x~ x') m a ^ the indices of the second line. These three pairs of terms will be called W 1( W 2 , W 3 . Fortnightly term. This is f(i-Z'2)(i-^). Multiplying (36) by (28) when the symbols are accented, and only retaining desired terms, f(i-Z' 2 ) (%-&) =f (I- ZT^KK) (|- ZK'KK'K') +f H^VVV^-^'- 8 -->' + tH OT VyV J e 2(9 - (!)+ - h . (39) Even if ^ had been accented in the X' functions, neither ^ or x' would have entered in this expression. These terms will be called W . Then the sum of the three expressions (37), (38), and (39), when multiplied by rr'/g, is equal 'to W, the disturbing function. If Diana be a different body from the moon the terms in ff 6 are periodic, and the only part of W, from which secular changes in the moon's mean distance and inclination can arise, are the sidereal semi-diurnal and diurnal terms, viz. : those in F and G, and also the term independent of H in (39). These terms being independent of 6' are independent of e', the moon's epoch. Hence it follows that, as far as con- cerns the influence of Diana's tides upon the moon, c?W/de' is zero, and we conclude that the tides raised by any one satellite can produce directly no secular change in the mean distance of any other satellite* But Diana being still distinct from the moon, the F-, G-, and part of the fortnightly term, which are independent of 9, do involve N and N' ; for W contains terms of the forms e aN , e a *', e that #JV, )= The terms corresponding to the tides of the seven speeds will now be taken separately, the coefficients in CT, K will be developed, and the terms involving N'N selected, the operation (N, e) performed, and then N' put equal to N, and e to e'. For the sake of brevity the coefficient T 2 /g will be dropped and will be added in the final result. The component parts of W taken from the equations (37-9) will be indicated as W I} W n , W m for the slow, sidereal, and fast semi-diurnal parts ; as W x , W 2 , W 3 for the slow, sidereal, and fast diurnal parts ; and as W for the fortnightly part. Slow semi-diurnal terms (2/12/2). W I =iF 1 [crVV ( '- ) - 2f '+wV 4 e- 2( '- )+2f '] ..... (40) Let w. Since rs Therefore r'*= the same with N' in place of N Therefore Therefore where n=0, 1, 2, 3, 4. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 735 Then Therefore by addition - Now when n=0, A=J, (_p 2 +g 2 )-4(N) performed on the first term is zero, as it ought to be according to the general principles of energy for the system is a conservative one as far as regards these terms. Let - 2PQ(P 2 - nr' 3 /c /2 =the same with N' for N + 4P 2 @ 2 (P 2 g2) /> 2h r = - J==l + 1 2P 2 2 (P 2 - Q* Adding and arranging the terms Then let . . (59) and we have (60) This is the last of the seven sets of terms. Then collecting results from (42-5-8, 51-4-7, 60), we have Sin 2f i+ 2 ^ F sin 2f - 2 ^F 2 sin 2f s +2 1 G 1 sin gl 738 ME. G. H. DARWIN ON THE SECULAR CHANGES IN Let KK)TS' K (ia TS /c'/c') *-q*) +pq(P 2 e- N - K'(ts7a' K'K')= the same with N' instead of N Now Put therefore =(P 2 -Q 2 ){(p 2 -q 2 Y(Pi+Q*-GP 2 Q 2 ) + 8P 2 Qyq*} . . . (53) and we have , e)W s =2(ffifG sin gain.; ........ (54) diurnal terms (n+2/2). (55) By an analogy similar to that by which the fast semi-diurnal was derived from the slow, we have ] (56) and Fortnightly terms (2/2). V 2 e^^^ . (58) THE ELEMENTS OF THE ORBIT OF A SATELLITE. 739 It will be found that <(-ZV) performed on the first term is zero, as it ought to be according to the general principles of energy for the system is a conservative one as far as regards these terms. Let sr'V 2 =the same with N' for N + 4P 2 2 (P 2 - Q*) + 4P 2 $ 2 (P 2 Q + 1 GP 2 Af - Q ^J A f - 2 Af sm4f A f sm4f fl , . .=\, where \=~ sin 41 n And ct But by (62) and i-2)=i cos t and = cos THE ELEMENTS OF THE ORBIT OF A SATELLITE. 743 These results may of course be also obtained when the functions are expressed in terms of P, Q, p, q. Whence on this hypothesis ^, ; = sin ksmj dt g cos ^ (64) 8. Secular change in the mean distance of a satellite, ivhere there is a second disturbing body, and where the nodes revolve with sensible uniformity on the fixed plane of reference. By (11) the equation giving the rate of change of f is _ k'~dt~~~de 7 As before, we may drop the accents, except as regards e'. In 6 we wrote <(e) for the operation tan ^j ' -=-,; hence ^-7-=- <(e) W, and by reference to that section the result may be at once written down. We have i g= - {20^ sin 2fi 2 S F 8 sin 2f 2 +2r 1 G 1 sin g 1 2r 2 G 2 sin g 2 2AH sin 2h] . (65) 13 Where 2 = the same with Q and P interchanged T 2 = the same with Q and P interchanged A = . (66) These functions are reducible to the following forms 2(* 1 +<& 2 )=1 sin 2 y+| sin 4 ,/ sin 2 i(l 2 sin 2 y+| si +^ sin 4 i(l 5 sin 2 j 1 $ 2 ) = cos i cos ./[I J sin a y J sin 2 i(\ f sin 2 ^] !+r 2 ) = sin 2 ,/ 1 sin 4 y+ sin 2 t(l | sin 2 y+| sin 4 j) 1 sin 4 {(1 5 sin 2 y+^ sin 4 ^') A T 2 ) = cos i cosy[sin 2 y+ sin 2 i(l f sin 2 ^)] 2A =f sinV+ sin 2 i(| sin 2 /- V- sin 4 /) +| sin 4 t(l 5 sin s ./+- sin V) 5 c 2 (67) 744 MR, G. H. DARWIN ON THE SECULAR CHANGES IN 9. Application to the case where the planet is viscous. As in 7 1 /7fc fr& in2g 1 r 3 sin2g 3 Asin4h} . . (68) If j be put equal to zero this equation will be found to be the same as that used as the equation of tidal reaction in the previous paper on " Precession." If the viscosity be small, with the same notation as before ^f=^in4f[ an d similarly dw f Qq dK> Pq dHf Pi, , .., SN'= -7^1' W'= -T^L' dN'=^=l> When N = - Also after differentiation when ./V=0, cr=w=cos J (?+./), /c=/c=sin ^(i-\-j) In order to find dj/dt we must, as before, perform <(iV, e) on W. Then take the same notation as before for the Ws and w's with suffixes. Slow semi-diurnal term. d _ dN' and sV^--, also ^)e^-^=- Hence and CT 7 K F sin Sidereal semi-diurnal term. rf*' rfar' and since ^>(e)W n =0, therefore ,e) W n =2 CT 3 /c 3 Fsin2f semi-diurnal term. By symmetry Zoiv diurnal term, "- /g / 09 rfw' . , did or -- and sin gl 746 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Sidereal diurnal term. W'- K ' = ~ ^i<^ e + P )= - y^i and rflrfa'?' -*'*') = Therefore and -K sn g diurnal term. By symmetry <(JV, e)W 8 =wc 8 (3w s -ic 8 )G. e sin g, Fortnightly term. and Whence - *K sin 2h Then collecting terms we have, on applying the result to the case of viscosity, sin 4 f 1+CT 3 K 3 sin 4 f+l CTK 7 ^ 4f 2 +| CT 3 K 3 ( C T 2 -K 2 ) sin 4h -S/f 2 ) sin 2g 1 +^wK(w 2 -/f 2 ) 2 sin 2g+i CT K B (3w 2 -/c 2 ) sin 2g 2 ] . (71 ) In the particular case where the viscosity is small, this becomes ...... (72) The right hand side is necessarily positive, and therefore the inclination of the orbit to the invariable plane will always diminish with the time. The general equation (71) for any degree of viscosity is so complex as to present no idea to the mind, and it will accordingly be graphically illustrated. The case taken is where n/f2=15, which is the same relation as in the previous graphical illustration of 7. The general method of illustration is sufficiently explained in that section. Fig. 5 illustrates the various values which dj/dt (the rate of increase of inclination to the invariable plane) is capable of assuming for various viscosities of the planet, and THE ELEMENTS OF THE ORBIT OF A SATELLITE. 747 for various inclinations of the satellite's orbit to the planet's equator. Each curve corresponds to one degree of viscosity, the viscosity being determined by the lag of the slow semi-diurnal tide of speed 2n 2/2. The ordinates give dj/dt (not as before djjsmjdt] and the abscissae give i-\-j, the inclination of the orbit to the equator. Fig. 5. Diagram illustrating the rate of change of the inclination of a single satellite's orbit to the invariable plane, for various viscosities of the planet, and various inclinations of the orbit to the planet's equator ( ) We see from this figure that the inclination to the invariable plane will always decrease as the time increases, and the only noticeable point is the maximum rate of decrease for large viscosities, for inclinations of the orbit and equator ranging from 60 to 70. If n/n had been taken considerably smaller than 15, the inclination would have been found to increase with the time for large viscosity of the planet. 11. Secular change in the mean distance of the satellite, where there is no other disturbing body than the planet. Comparison with result of previous paper. To find the variation of we have to differentiate with respect to e', and the follow- ing result may be at once written down Q sin 4f 1 K 8 sin 4f 2 +4w r V 2 sin 2g l 4V sin 2g 8 6w*K* sin 4h]. (73) This agrees with the result of a previous paper (viz.: (57) or (79) of ' : Precession"), obtained by a different method ; but in that case the inclination of the orbit was zero, so that OT and K were the cosine and sine of half the obliquity, instead of the cosine and sine o 748 MR. G. H. DARWIN ON THE SECULAR CHANGES IN In the case where the viscosity is small this becomes ^ sin 4f [cos (;+./) -X] ........ (74) It will now be shown that the preceding result (71) for dj/dt may be obtained by means of the principle of conservation of moment of momentum, and by the use of the results of a previous paper. It is easily shown that the moment of momentum of orbital motion of the moon and earth round their common centre of inertia is C/k, and the moment of momentum of the earth's rotation is clearly Cn. Also j and i are the inclinations of the two axes of moment of momentum to the axis of resultant moment of momentum of the system. Hence smj=n sin i By differentiation of which dj . dn . . . .di 1 dt . . = - smt+n cost 1-~ sin j dt dt k dt ^ Cdn ..... . . . di~\ . [dn / , . . / , -\ di 1 d} . . - sin (i+j)+n cos (i+j) ^J cos;-^ cos (i+j)-n sm (i+j) -+ k rf *J smj Now from equation (52) of the paper on "Precession," the second term on the right- hand side is zero, and therefore dn . . . , ., , , . . .. di But by equations (21) and (16) and (29) of the paper on "Precession" (when CT and K are written for the p, q of that paper) ill! T- ~ -[^CT 8 sin 4f 1 +2rV sin 4f+i*c 8 sin 4f 2 +wV sin 2g x +OT 2 /c 2 ( B r 2 -/f 2 ) 2 sin 2g+wV sin 2g 2 ] -/f 2 ) sin 4f-i CTK 7 sin 4f 2 +^ K r 5 /c( C r 2 +3K 2 ) sin 2 gl /c 2 ) 3 sin 2g ^^(Sj^+K 2 ) sin 2g 2 fw 3 ^ sin 4h] Then if we multiply the former of these by sin (i+f) or 2orfc, and the latter by cos (i+j) or -at 1 /c 2 , and add, we get the equation (71), which has already been established by the method of the disturbing function. It seemed well to give this method, because it confirms the accuracy of the two long analytical investigations in the paper on " Precession " and in the present one. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 749 12. The method of the disturbing function applied to the motion of the planet. In the case where there are only two bodies, viz.: the planet and the satellite, the problem is already solved in the paper on " Precession," and it is only necessary to remember that the p and q of that paper are really cos \(i-\-j), sin \(i-\-j), instead of cos ^i, sin \i. This will not be reinvestigated, but we will now consider the case of two satellites, the nodes of whose orbits revolve with uniform angular velocity on the ecliptic. The results may be easily extended to the hypothesis of any number of satellites. In (18) we have the equations of variation of i, i/f, x m terms of W. But as the correction to the precession has not much interest, we will only take the two equations . .di .dW dW} n Bin 1 5 = coat ^7 -^ | dn_dW dt~d x ' which give the rate of change of obliquity and the tidal friction. In the development of W in 5, it was assumed that t|, i/' were zero, and x, x did not appear, because x was left unaccented in the X'-Y'-Z' functions. Longitudes were there measured from the autumnal equinox, but here we must conceive the N, N' of previous developments replaced by N$, N'ifi'; also ftt-\- e, fl't-\-t must be replaced by /2<+e i//, n't-\-e T/>'. It will not be necessary to redevelop W for the following reasons. n't-\-e \jj' occurs only in the exponentials, and N' \}i' does not occur there ; and N'\l/ only occurs in the functions of w and K, and n't-\-e */' does not occur there. Hence dW_dW dW d- de' ~*~' . Again, it will be seen by referring to the remarks made as to ^, x m the develop- ment of W in 5, that we have the following identities : For semi-diurnal terms, For diurnal terms, (77) For the fortnightly term, Also de" dtf-- dg' n _ s _ ~~ ' ~ MDCCCLXXX. 5 B 750 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Then making use of (7-6) and (77), and remembering that cos i=P 2 Q-, sin i=lPQ, we may write equations (75), thus . (78) (79) It is clear that by using these transformations we may put //= and x again disappear, and we may use the old development ofW. The case where Diana and the moon are distinct bodies will be taken first, and it will now be convenient to make Diana identical with the sun. In this case* after the differentiations are made we are not to put N=N' and e=e'. The only terms, out of which secular changes in i and n can arise, are those depend- ing on the sidereal semi-diurnal and diurnal tides, for all others are periodic with the longitudes of the two disturbing bodies. Hence the disturbing function is reduced to W n and W 2 . Also dW u /dN' and dW 2 /dN' can only contribute periodic terms, because NN' is not zero, and by hypothesis the nodes revolve uniformly on the ecliptic. Then if we consider that here p' is not equal p, nor q' to q, we see that, as far as is of present interest, W n =2F cos 2f W 2 = 2G cos g P^P 2 - Q*)l(p*-q*y-2p 2 q 2 ~][(p 2 -q' 2 ) 2 - 2/Y*] . Also the equations of variation of i and n are simply twfr iv II I " * * 2 ~dt = '"~dt "~dg Then if we put =irin*i(l-i sinV)(l-|sin 2 /) L ^ = i sin 2 i cos 2 t(l -f sin 2 ;) (1 -| sin 2 /) J THE ELEMENTS OF THE ORBIT OF A SATELLITK. 751 We have Sin i!l-t~yljr Sin gj n~= [2F sin 2f+yG sin g] cot i I (Tfi u It will be noticed that in (81) 2rr' has been introduced in the equations instead of TT' ; this is because in the complete solution of the problem these terms are repeated twice, once for the attraction of the moon on the solar tides, and again for that of the sun on the lunar tides. The case where Diana is identical with the moon must now be considered. This will enable us to find the effects of the moon's attraction on her own tides, and then by symmetry those of the sun's attraction on his tides, We will begin with the tidal friction, By comparison with (65) | / [W I -W III +1W 1 -1W 3 ]=2* 1 F 1 sin 2f 1 +2 and $ and simplification, we find that if 6 - 2 , F,, F 2 are given in equations (67), and and y in equations (80). The expressions for Fi an d Fa are found by symmetry with those for ^j and J? 2 , by interchanging i and j ; the first of equations (62) then corresponds with the second of (103), and vice versd. From (103) it follows that and Also 756 MR. G. H. DARWIN ON THE SECULAR CHANGES IN The complete solution of the problem may be collected from the equations (101) and (81). In the case of the viscosity of the earth, and when the viscosity is small, we easily find the complete solution to be di sin 4f . . . . f ,, 4 <\ > 4/4 < o -/\ 2/2 ., n= sm i cos i\ T 2 (l f sm-^+T '(1 f sm-^ ) --- r sec i cosj (it (\ 11 o/y 1 - "T' 2 secicos/-Tr'(l-f sinV)(l-f sin 2 /)[ (104) This result agrees with that given in (83) of "Precession," when the squares of j and/ are neglected, and when fl'/n is also neglected. The preceding method of finding the tidal friction and change of obliquity is no doubt somewhat artificial, but as the principal object of the present paper is to discuss the secular changes in the elements of the satellite's orbit, it did not seem worth while to develop the disturbing function in such a form as would make it applicable both to the satellite and the planet ; it seemed preferable to develop it for the satellite and then to adapt it for the case of the perturbation of the planet. In long analytical investigations it is difficult to avoid mistakes; it may therefore give the reader confidence in the correctness of the results and process if I state that I have worked out the preceding values of di/dt and dn/dt independently, by means of the determination of the disturbing couples 1L, ffil, jBt. That investigation separated itself from the present one at the point where the products of the X'-Y'-Z' functions and %-^j!}-%> functions are formed, for products of the form Y'Z' X ^^ had there to be found. From this early stage the two processes are quite independent, and the identity of the results is confirmatory of both. Moreover, the investigation here presented reposes on the values found for dj/dt and d/dt, hence the correctness of the result of the first problem here treated was also confirmed. III. THE PROPER PLANES OF THE SATELLITE, AND OF THE PLANET, AND THEIR SECULAR CHANGES. 13. On the motion of a satellite moving about a rigid oblate spheroidal planet, and perturbed by another satellite. The present problem is to determine the joint effects of the perturbing influence of the sun, and of the earth's oblateness upon the motion of the moon's nodes, and upon the inclination of the orbit to the ecliptic ; and also to determine the effects on the THE ELEMENTS OF THE ORBIT OF A SATELLITE. 757 obliquity of the ecliptic and on the earth's precession. In the present configuration of the three bodies the problem presents but little difficulty, because the influence of oblateness on the moon's motion is very small compared with the perturbation due to the sun ; on the other hand, in the case of Jupiter, the influence of oblateness is more important than that of solar perturbation. In each of these special cases there is an appropriate approximation which leads to the result. In the present problem we have, however, to obtain a solution, which shall be applicable to the preponderance of either perturbing cause, because we shall have to trace, in retrospect, the evanescence of the solar influence, and the increase of the influence of oblateness. The lunar orbit will be taken as circular, and the earth or planet as homogeneous and of ellipticity I, so that the equation to its surface is />=a{l + *a-cos 3 0)} The problem will be treated by the method of the disturbing /unction, and the method will be applied so as to give the perturbations both of the moon and earth. First consider only the influence of oblateness. Let p, 6 be the coordinates of the moou, so that p=c and cos 0=M 3 . Then in the formula (17) 2, r=c and - = t(^ M 3 2 ), so that the disturbing function This function, when suitably developed, will give the perturbation of the moon's motion due to oblateness, and the lunar precession and nutation of the earth. Then by (21) we have M 3 =sin i [p 2 sin (l-\-N)(f sin (lN)~\-\-&inj cos i sin /. Where I is the moon's longitude measured from the node, and JV is the longitude of the ascending node of the lunar orbit measured from the descending node of the equator. Then as we are only going to find secular inequalities, we may, in developing the disturbing function, drop out terms involving I ; also we must write N t/ for N, because we cannot now take the autumnal equinox as fixed. Then omitting all terms which involve I, M 3 2 =sin 2 i [(?*+?*) pV cos 2(N )]+ sin 8 ./ cos 2 i +sin/ sin i cos i [ p 2 ^ 2 ] cos (N \ Since p=cos ^j, q=sm ^j, we have p*+q*= 1 1 sin 2 ./, p 2 q 2 =% sin 2 ./, p 2 q 2 =cosj MDCCCLXXX. 5 E 2 = sin 2 tl -sin 2 sin 2 i sin 2/ cos (2V 1/) ^ sin 2 i &m*j cos 2(iV ^(sin 3 i+sin 2 ji') f sin 2 isin 2 ^' ^= ^-(1 f sin 2 i)(l f sin 2 ^) 758 ME. G. H. DARWIN ON THE SECULAR CHANGES IN and Now Wherefore W=it{(l | sin 2 0(1 f sin 2 y) sin 2* sin 2; cos (N\fi) + sin 2 i sin 2 ./ cos 2(N\jj) } . (105) This is the disturbing function. Before applying it, we will assume that i andj are sufficiently small to permit us to neglect sin 2 i sin 2 j compared with unity. Then |(l-f sin 2 0(1-1 anV^A+i i sin 2 t sinV+sin 2 i sin z j- sin 2 1 sin 3 ./ = T2+i cos 2t cos 2/ ^ sin 2 i sin 2 ^ Hence, when we neglect the terms in sin 2 i sin 2 ^' \p)} .... (10G) f . dj dW - sm = Then since this disturbing function does not involve the epoch or ^, we have by (13), (14), and (18) V_rfW . M_dV[ . .d^_dW dt dj ' dt dijr' dt di Thus as far as concerns the influence of the oblateness on the moon, and the reaction of the moon on the earth, j- Bmj-~ = \rt sin 2i sin 2j sin (N 1/) ! sin/ = ^rt (cos 2i sin 2/+sin 2i cos 2^' cos (2V rt sin 2i sin 2/ sin (2V ^/) n sin i-r~= ^T{{sin 2i cos 2/-j-cos 2i sin 2/ cos (2V *""V* .di n sin t-r- = < . . . (107) If there be no other disturbing body, and if we refer the motion to the invariable plane of the system, we must always have 2V=//. lu this case the first and third of (107) become dt dt and the second and fourth become THE ELEMENTS OP THE ORBIT OP A SATELLITE. 759 ' ~T=n sin i -r = ?t sin 2 But /k is proportional to the moment of momentum of the orbital motion, and n is proportional to the moment of momentum of the earth's rotation, and so by the defini- tion of the invariable plane j smj=n8in.i .......... (108) /c Wherefore - = ~, and it follows that the two nodes remain coincident. This at at result is obviously correct. In the present case, however, there is another disturbing body, and we must now consider The perturbing influence of the sun. Accented symbols will here refer to the elements of the solar orbit. We might of course form the disturbing function, but it is simpler to accept the known results of lunar theory; these are that the inclination of the lunar orbit to the ecliptic remains constant, whilst the nodes regrede with an angular velocity /ji'Y i T ' T ' Now 1(77 ^ i(i/2' 2 )X7r=it7: in our notation. Hence I shall write 1 for \Ji / it 11 II l(/f) ^~%f2 r 2 ' although if necessary (in Part IV.) I shall use the more accurate formula for numerical calculation. For the solar precession and nutation we may obtain the results from (107) by putting j= 0, and / for T. Thus for the solar effects we have dN "* n sin i ~= L T 't sin at ' The following seems worthy of remark. By the last of (109) we have d^jdt= r't cos i/n. In this formula e is the precessional constant, because the earth is treated as homogeneous. The full expression for the precessional constant is (2C A B)/2C, where A, B, C are the three prin- cipal moments of inertia. Now if we regard the earth and moon as being two particles rotating with an angular velocity fl about 5 E 2 760 MR. G. H. DARWDf OX THE SECULAR CHANGES DC Then when the system is perturbed both by the oblateness of the earth and by the sun, we have from (107) and (109), sinj = Tt sin Zi sin Zj sin (X f . ..2* their common centre of inertia, then the three principal moments of inertia of the system are jrM>/(ir+), Jf*e*/(Jf+), 0, and therefore the processional constant of the system is f Thus die formula for D?/A is preciselj analogous to that for W*. each of them being equal to T x prec. X cos inclin. - THE ELEMENTS OF THE ORBIT OF A SATELLITE. 761 or dy fl-rt . . . T' \ l-rt -=l cos 2 cosj+jf-cosj] z cosj cos 2;. Now let fat Tt T/t And we have (112) -f = (ofj cos 2i cosj-\-a* cos_/)y+o 1 cosj cos 2/.Tj -f = a cos 2i cos cos 2. (113) and by symmetry from the two latter of (110) it* = (&j cos 2; cos i + 60 cos 1)17 + 6j cos i cos 2t .y = (&! cos 2/ cos i+6 2 cos i) &! cos t cos 2i.z (114) These four simultaneous differential equations have to be solved. The as and b's are constant, and if it were not for the cosines on the right the equations would be linear and easily soluble. It has already been assumed that i and j are not very large, hence it would require large variations of t and j to make considerable variations in the coefficients, I shall therefore substitute for t and j, as they occur explicitly, mean values t' and y o ; and this procedure will be justifiable unless it be found subsequently that t and j vary largely. Then let =<*! cos 2i' cosy o +fo cos j /8=&i cos 2/ cos t' +6o cos *' a=a 2j b=& 1 cost' cos2j' *' 1 . (115) (Hereafter i andy will be treated as small and the cosines as unity.) Then (116) 762 MR. G. H. DARWIN ON THE SECULAR CHANGES IN These equations suggest the solutions =", cos ='L sin t=tL' cos (/rf+m) -=-*$,L' sin /c< + m Then substituting in (116), we must have Wherefore and -ab:=0 or K? This quadratic equation has two real roots (/q and /c 3 suppose), because (+/8) 2 4(/3 ab)=:(a /J) 2 +4ab is essentially positive. Then let (117) And the solution is r sin 2j cos N=z=L 1 cos ( r sin 2j sin N=y=L l sin ( r sin 2i cos i/ = =/ cos r sin 2i sin \lt =n=L/ sin cos ( -2 sin (*.,< +m 2 ) r*AGi I if f r-i m i V^vO 1 fVow | JJJ.1 ,' sin (W + m., where (H8) From these equations we have sin 2 2= sin 2 2i= cos [(KJ K 2 )+m 1 m 2 ] 1 'I 2 / cos [(*Cj K 2 )t+ra l m 2 ] -2) and From this we see that sin 2j oscillates between between 2(Z:/+Z 2 / ) and 2(L 1 '^Z/). Let us change the constants introduced by integration, and write and sin 2i '= sin 2i . Then our solution is THE ELEMENTS OF THE ORBIT OF A SATELLITE. . . -,-r . . , a sin 2j cos N=sm 2j cos (/c^+mj) sm 2i cos K 2 + a -\7 / \ a . . sin 2; sin ./V=sin 2? n sin (K,+m,) sin 2i n sin i! K 3 + a tC \ QL sin 2i cos i/> = - - sin 2/ cos (/c^+nij) + sin 2i Q cos (/c 2 <+m 2 ) *!+ . . . , \ i : ' - sin 2; sm (/< 1 <+m 1 )+ sin a sin 2i sin sin From this it follows that sin 2i sin 2j cos (N $) = i sin 3 2/ sin 2 2i' + Now Therefore Jj in 2i sin 2/ cos [( KI 763 . (119) i mj sin 2isin 2/ sin (N V) = (l l ) sin 2&' sin 2/' sin [(! *c 3 )^+m 1 mj sin 2i sin 2/ cos (iV , i cos /, 1. Now draw a sphere of unit radius, with the origin as centre ; draw a tangent plane * See the foot-note to 18 for a comparison of these results with those ordinarily given. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 765 to it at the point where the axis of z' meets the sphere, and project on this plane the poles of the lunar orbit and of the earth. We here in fact map the motion of the two poles on a tangent plane to the celestial sphere. Let x, y be a pair of axes in this plane parallel to our previous x, y' ; and let x', y' be the coordinates of the pole of the lunar orbit, and f , tj be the coordinates of the earth's pole. Then x'=j sin 2V, y'= j cos N ; f'= ism i/, ij'=i cosi// . . . . (123) Let x, y, g, 7) be the coordinates of these same points referred to another pair of rectangular axes in this plane, inclined at an angle to the axes x', y'. Then x= x' cos (f>-\-y' sin , = .' cos -\-v)' sin ^ y= x' sin <+/' cos (> , r = f ' sin <+ cos <> From (123) and (118) we have therefore = L l sin(/c 1 f + m 1 (f>)-\-L 2 sin (/ 2 i-(-m 2 ) I 1 <) L 2 cos (K 2 +m 2 $) \ 1 ) Z/sin ( cos Now suppose the new axes to rotate with an angular velocity K. 2 , and that s^t+in,. Then c=Z L sin [(KJ K^i+nij m 2 ] y+L. 2 = L! cos [(! K 2 f= Z/ sin [(KJ K^i+nij m 2 ] j m 2 ] (124) These four equations represent that each pole describes a circle, relatively to the rotating axes, with a negative angular velocity (because K I K 2 is negative). The centres of the circles are on the axis of y. The ratio distance of centra of terrestrial circle L 3 ' + _ b /ioc\ distance of centre of lunar circle ~~ L~ a ~~ the distances being measured from the pole of the ecliptic. And the ratio MDCCCLXXX. 5 F 766 MB. G. H. DARWIN ON THE SECULAR CHANGES IN radius of terrestrial circle Z,' *, + a b ; = L == ! radius of lunar circle Z t a i+/8 According to the definitions adopted in (117) of ^ and *c 2 , (/^-f )/a is negative and (/c 2 +)/a is positive ; hence L i has the same sign as L{, and Z 2 has the opposite sign from L. z '. When t=(m. 1 m 2 )/(K i K 2 ), we have *=0, y=(-L,)-L l , =Q, v= Fig. 6. In fig. 6 let Ox, Oy be the rotating axes, which -revolve with a negative rotation equal to K%, which is negative. Let M be the centre of the lunar circle, and Q of the terrestrial circle. Then we see that L and P must be simultaneous positions of the two poles, which revolve round their respective circles with an angular velocity K 2 K r , in the direction of the arrows. M and Q are the poles of two planes, which may be appropriately called the proper planes of the moon and the earth. These proper planes are inclined at a constant angle to one another and to the ecliptic, and have a common node on the ecliptic, and a uniform slow negative precession relatively to the ecliptic. The lunar orbit and the equator are inclined at constant angles to the lunar and terrestrial proper planes respectively, and the nodes of the orbit, and of the equator regrede uniformly on the respective proper planes. In the ' Me"canique Celeste' (livre vii., chap. 2, sec. 20) LAPLACE refers to the proper plane of the lunar orbit, but the corresponding inequality of the earth is ordinarily referred to as the 19-yearly nutation. It will be proved later, that the above results are identical with those ordinarily given. Suppose then that THE ELEMENTS OP THE ORBIT OP A SATELLITE. 767 Then I=the inclination of the earth's proper plane to the ecliptic J=the inclination of the lunar orbit to its proper plane ^3= the inclination of the equator to the earth's proper plane J 7 =the inclination of the moon's proper plane to the ecliptic J=-i, I=L. 2 f , !,=/, J x = L 2 and by (125-6) -J=- (127) Thus I and J are the two constants introduced in the integration of the simul- taneous differential equations (116). It is interesting to examine the physical meaning of these results, and to show how the solution degrades into the two limiting cases, viz. : where the planet is spherical, and where the sun's influence is non-existent. Let It be the speed of motion of the nodes, when the ellipticity of the planet is zero. Let I be the purely lunar precession, or the precession when the solar influence is nil. Let nt be the ratio of the moment of momentum of the earth's rotation to that of the orbital motion of the two bodies round their common centre of inertia. Then Jen Then by (121) and (115) we have /^ a=ml+n, a=ml, /3=l+ , b=l n first suppose that tt is large compared with I. This is the case at present with the earth and moon, because the speed of motion of the moon's nodes is very great compared with the speed of the purely lunar- precession. Then a, ft, b are small compared with . Therefore by (117) v ... rt-i-fi if I if n R K \ K -2 a 1 P> K i\ K -> a P and = a KO= 5 F 2 768 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Therefore 1. b II =- approximately TE n And by (127) a a I . = -- ~=nt- approximately , + /3 n j= --- 1, /G KjL=lt approximately l,=ij, J,=mli Now we have shown above that K. 2 is the common angular velocity of the pair of proper planes, and the above results show that it is in fact the luni-solar precession. * 2 K I is the angular velocity of the two nodes on their proper planes, and it is nearly equal to It. The ratio of the amplitude of the 19-yearly nutation to the inclination of the lunar orbit is l/lt. The ratio of the inclination of the lunar proper plane to the obliquity of the ecliptic is ml/it. In this case, therefore, the lunar proper plane is inclined at a small angle to the ecliptic, and if the earth were spherical would be identical with the ecliptic. Secondly, suppose that It is small compared with \. T'C Then d fortiori is small compared with I. Hence we may put /3=b. 7fr Therefore a-b . _ K 2 *!=/( ) i +4ab=a+b+ j^t, nearly jss (m+i)l n In if -i- 1 / "a ' r^ i ! ' ' - m + i I' a m\ m + l Therefore _ -- m+l l/ > -V m + i THE ELEMENTS OF THE ORBIT OF A SATELLITE. 769 From the last of these, I-J= " ' m+i I * 2 is the precession of the system of proper planes, and the above results show that the solar precession of the planet and satellite together, considered as one system, is one (llt+ l) th of the angular velocity which the nodes of the satellite would have, if the planet were spherical. K 2 KJ is the lunar precession of the earth which goes on within the system, and it is approximately the same as though the sun did not exist. (Compare the second and fourth of (107) with N=$, and use (108)). It also appears that the lunar proper plane is inclined to the planet's proper plane at a small angle the ratio of which to the inclination of the earth's proper plane to the ecliptic is equal to one (ttt+l) th part of tt/l. If It and I are of approximately equal speeds the proper plarje of the moon will neither be very near the ecliptic, nor very near the earth's proper plane. The results do not then appear to be reducible to very simple forms ; nor are the angular velocities K 2 and K. 2 KJ so easily intelligible, each of them being a sort of compound precession. If the solar influence were to wane, M and Q, the poles of the proper planes, would approach one another, and ultimately become identical. The two planes would have then become the invariable plane of the system ; and the two circles would be concentric and their radii would be inversely proportional to the two moments of momentum (whose ratio is JTl). Now in the problem which is to be here considered the solar influence will in effect wane, because the effect of tidal friction is, in retrospect, to bring the moon nearer and nearer to the earth, and to increase the ellipticity of the earth's figure ; hence the relative importance of the solar influence diminishes. We now see that the problem to be solved is to trace these proper planes, from their present condition when pne is nearly identical with the ecliptic and the other is the mean equator, backwards until they are both sensibly coincident with the equator. We also see that the present angular velocity of the moon's nodes on the ecliptic is analogous to and continupus with the purely lunar precession on the invariable plane of the moon-earth system ; and that the present luni-solar precession is analogous to and continuous with a slow precessional motion of the same invariable plane. Analytically the problem is to trace the secular changes in the constants of integra- tipn, when a, a, /3, b, instead of being constant, are slowly variable under the influence of tidal friction, and when certain other small terms, also due to tides, are added to the differential equations of motion. 770 ME. G. H. DARWIN ON THE SECULAR CHANGES IN 14. On the small terms in the equations of motion due directly to tidal friction. The first step is the formation of the disturbing function. As we shall want to apply the function both to the case of the earth and to that ot the moon, it will be necessary to measure longitudes from a fixed point in the ecliptic ; also we must distinguish between the longitude of the equinox and the angle x> as they enter in the two capacities (viz. : in the X'Y' and ^9 functions) ; thus the N and N' of previous developments must become N^J, N'\l>' ; e, e' must become e x/>, e' /'; and 2(x-~x') mus t be introduced in the arguments of the trigonometrical terms in the semi-diurnal terms, and x^~x' i the diurnal ones. The disturbing function must be developed so that it may be applicable to the cases either where Diana, the tide-raiser, is or is not identical with the moon ; but as we are only going to consider secular inequalities, all those terms which depend on the longitudes of Diana or the moon may be dropped. In the previous development of Part II. we had terms whose arguments involved e e'; in the present case this ought to be written (flt-\-e. ^) (f2't-\- i/'), for which it is, in fact, only an abbreviation. Now a term involving this expression can only give rise to secular inequalities, in the case where Diana is identical with the moon ; and as we shall never want to differentiate the disturbing function with regard to fl', we may in the present development drop the fit and fl't. Having made these preliminary explanations, we shall be able to use previous results for the development of the disturbing function. The work will be much abridged by the treatment of i, j, i', f as small. Unaccented symbols refer to the elements of the orbit of the tide-raiser Diana, or (in the case of i, x, ^) to the earth as a tidally distorted body ; accented symbols refer to the elements of the orbit of the perturbed satellite, or to the earth as a body whose rotation is perturbed. Then since i, i' &ndj,f are to be treated as small, (22) becomes The same quantities when accented are equal to the same quantities when i,j, N, i/ are accented. Then referring to the development in 5 of the disturbing function, we see that, for the same reasons as before, we need only consider products of terms of the same kind in the sets of products of the type XT' X ?. Hence the disturbing function W is the sum of the three expressions (37-9) multiplied by rr'/g. Now since we only wish THE ELEMENTS OF THE ORBIT OF A SATELLITE. 771 to develop the expression as far as the squares of i andj, we may at once drop out all those terms in these expressions, in which K occurs raised to a higher power than the second. This at once relieves us of the sidereal and fast semi-diurnal terms, the fast diurnal and the true fortnightly term. We are, however, left with one part of f (3 Z' 2 )(g Z? 2 ), which is independent of the moon's longitude and of the earth's rotation ; this part represents the permanent increase of ellipticity of the earth, due to Diana's attraction, and to that part of the tidal action which depends on the longitude of the nodes, in which the tides are assumed to have their equilibrium value. I shall refer to it as the permanent tide. Then as before, it will be convenient to consider the constituent parts of the dis- turbing function separately, and to indicate the several parts of W by suffixes as in 5 and elsewhere ; as above explained, we need only consider W I; W lt W 2 , and W . Semi-diurnal term. From (37) we have W,/ =i[F lCT VV (6 '- 9) - 2f >+F lW V 4 e- 2 <<'-'' )+2f '] / 8 To the indices of these exponentials we must add i2( x x')' anc ^ f r ^ write e 1/>, and for ff, e i//. Then by (128) ^ = 1 -l^-i/-;/^-*' w'*= l -\^-\j' z -i'j'cr'~ K '-^ Hence . . (129) Slow diurnal term. From (38) we have W = To the indices of the exponentials we must add (x~x') > ra3 > w ' 3 mav ^ e obviously put equal to unity, and by (128) 772 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Hence -A")-(V'-f)-g 1 ]} - (130) Sidereal diurnal term. From (38) we have ~W 3 / =G[CT/c(nn!T /fK^CT'/c^Er'n/ >c / /c / )tf~ ? -|-7*r/c(nrnr KX)CT 'K '(sr 'rs /c'/c')e g ] / 8 To the indices of the exponentials must be added ( X X ') OT > CT ma J ^ e treated as unity. Hence the expression becomes G [/c/c'e x " x '" g -j-KK'e~ (x ~ x ' )+! ''] and Permanent term. From (39) we have = KK *cV to our degree of approximation. Now x /c=i(t 2 +/+iy(e fl '-*-|-6-^-*>))=i(i 2 +/+2y cos (A T -^)) Hence ? cos (^-^))-i(t va +/ 2 +2ty cos (A"-f )) . (132) W 2 and W are the only terms in W which can contribute anything to the secular inequalities, unless Diana and the satellite are identical ; for all the other terms involve e e', and will therefore be periodic however differentiated, unless c=e'. We now have to differentiate W with respect to i', X ', /', /, e', 2V. The results will then have to be applied in the following cases. THE ELEMENTS OP THE ORBIT OP A SATELLITE. 773 For the moon: (i.) When the tide-raiser is the moon, (ii.) When the tide-raiser is the sun. For the earth: (iii.) When the tide-raiser is the moon, and the disturber the moon, (iv.) When the tide-raiser is the sun, and the disturber the sun. (v.) When the tide-raiser is the moon, and the disturber the sun. (vi.) When the tide-raiser is the sun, and the disturber the moon. The sum of the values derived from the differentiations, according to these several hypotheses, will be the complete values to be used in the differential equations (13), (14) and (18) for dj/dt, dN/dt, di/dt, d^/dt. A little preliminary consideration will show that the labour of making these differentiations may be considerably abridged. In the present case i and j are small, and the equations (110) which give the position of the two proper planes, and the inclinations of the orbit and equator thereto, become - Ttl cos n sin i~= (Tt-\-Tt)iTtj cos dv - We are now going to find certain additional terms, depending on frictional tides, to be added to these four equations. These terms will all involve r 2 , r' 2 , or rr' in their coefficients, and will therefore be small compared with those in (133). If these small terms are of the same types as the terms in (133), they may be dropped ; because the only effect of them would be to produce a very small and negligeable alteration in the position of the two proper planes.* In consequence of this principle, we may entirely drop W from our disturbing function, for W only gives rise to a small permanent alteration of oblateness, and therefore can only slightly modify the positions of the proper planes. Analytically the same result may be obtained, by observing that W in (132) has the same form as W in (105), when i and^' are treated as small. * For example, we should find the following terms in -sin^' , viz. : , Ctf (N--^) smSgTL + ifj + j-cos (-2V-YO) [sin 2 2^-8^ gl - sin 2 gjr 3 which may be all coupled up with those in the second of (133). If the viscosity be small, so that the angles of lagging are small, it will be found that all the terms of this kind vanish in all four equations, excepting the first of those just written down, viz. : %j-rr'/Q. MDCCCLXXX. 5 G 774 MR. G. H. DARWIN ON THE SECULAR CHANGES IN In each case, after differentiation, the transition will he made to the case of viscosity of the planet, and the proper terms will be dropped out, without further comment. First take the perturbations of the moon. For this purpose we have to find dW/df and dW/smf cLZV'+tan \j' dW/de or dW/fdN'+ifdW/de'. By the above principle, in finding dV?/df we may drop terms involving j and i cos (N\jj), and in finding dW/fdN'+^f dW/dc, we may drop terms involv- ing * sin (N\l>). We may now suppose x = x' 1 A =1 /''- Take the case (i.), where the tide-raiser is the moon. Then as the perturbed body is also the moon, after differentiation we may drop the accents to all the symbols. From (129) =Jt sin (TV |) sin 4fj .......... (134) From (130) ~ cos ~ = %ism(N$)Bm2g l ........ (135) From (131) and symmetry with (135) -t) sin 2g ....... (136) Adding these three (134-6) together, we have for the whole effect of the lunar tides on the moon >-=i sin (2V-t/r) [sin 4f 1 -sin2g 1 + sin 2g] .... <137) 8 Now take the case (ii.) where the tide-raiser is the sun. Here we need only consider W 2 , but although we may put X = x'> V' = V''' i=i/ > we must not puty==/', N=N', because the tide-raiser is distinct from the moon. From (131) '-f -g)+> cos (N- Here accented symbols refer to the moon (as perturbed), and unaccented to the sun (as tide-raiser). As we refer the motion to the ecliptic ^'=0, and the last term disappears. Also we want accented symbols to refer to the sun and unaccented to THE ELEMENTS OF THE ORBIT OF A SATELLITE. 775 refer to the moon, therefore make r and T' interchange their meanings, and drop the accents to N' and i/'. Thus as far as important ^/^ T =i*Bin(^-^)sm2g ....... (138) This gives the whole effect of the solar tides on the moon. Then collecting results from (137-8), we have by (14) ' . (139) This gives the required additional terms due to bodily tides in the equation for y/dt, viz. : the second of (133). If the viscosity be small sin 4^8^1 2g 1 +sin 2g= sin 4f . ' , . .- sin 2 =sm 41 Next take the secular change of inclination of the lunar orbit. For this purpose we have to find dW/fdN'+^fdW/de', and may drop terms in i sin (N 1/). First take the case (i.), where the tide-raiser is the moon. From (129) W /) sin 4f x . . . (141) fi (142 From (130) N ^))sin2g! (143) dW It 3 ^j' r l /= to present order of approximation ........ From (131) cos (&-*)) sin 2g . (145) /7=0 absolutely ................. (146) / y 5 G 2 776 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Collecting results from the six equations (141-6), we have for the whole perturbation of the moon by the lunar tides (1 ,rw ffw\ AT* J 3F+tf 77) / i =*tf+' COS (^-^))( sin 4f i~ sin 2 Si+ 8in 2g) ' ( 147 ) Next take the case (ii.), and suppose that the sun is the tide-raiser. Here we need only consider W 2 . Then noting that cfW 2 /c?e'=0 absolutely, we have from (131) Accented symbols here refer to the moon (as perturbed), unaccented to the sun (as tide-raiser). Therefore j= 0. Then reverting to the usual notation by shifting accents and dropping useless terms, this expression becomes i cos (JV" $ sin 2g ........ (148) Then collecting results from (147-8), we have by (13) & j o / -/= (./+* cos ( N ^}} - ( sin 4f i sin 2gi + sin 2 g) i* cos (^ /) s in 2 g (149) tc ctt This gives the additional terms due to bodily tides in the equation for dj/dt, viz. : the first of (133). If the viscosity be small ski 4fj sin 2gj+ sin 2g= sin 4f 1 sin 2g=^ sui 4f j Before proceeding further it may be remarked that to the present order of approxi- mation in case (i.) and in case (ii.) it is zero ; thus by (11) |^=l^sin4f 1 ...... (151) k dt 4 g We now turn to the perturbations of the earth's rotation. Here we have to find dW/di' and cot i dW/d^ cTW/sin i dty' or (1 ^i^dW/idx'dW/id^', and in the former may drop terms in i and /cos (N\}i), and in the latter terms in j sin (N\}i). THE ELEMENTS OF THE ORBIT OP A SATELLITE. 777 First take the case (iii.), where the moon is tide-raiser and disturber. Here we may take N=N r , e=^, ./=/ throughout, and after differentiation may drop the accents to all the symbols. From (129) /T 2 - = -P\ {i cos 2fi+y cos (2V-^r+2f,)} = iy sin (N-j) sin 4f\ . (152) From (130) r-,/,)sin2 gl . (153) From (131) W Ai-2 -= iG{tcosg +j C o S (N-t+g)} =-iJdr i (N-t)sin2g. (154) Therefore from (152-4) we have for the whole perturbation of the earth, due to attraction of the moon on the lunar tides, * ~=ij sin (tf-$[sin4fi+sin2 gl -- sin 2g] . . . (155) The result for case (iv.), where the sun is both tide-raiser and disturber, may be written down by symmetry; and since y=0 here, therefore (156) Next take the cases (v.) and (vi.), where the tide-raiser and disturber are distinct. Here we need only consider W 2 . From (131) = * cos + cos - When the moon is tide-raiser and sun disturber, this becomes -l/ sin (AT t/)sin2g ........ (157) When sun is tide-raiser and moon disturber it becomes zero. Then collecting results from (155-7), we have by (18) 778 MB. G. H. DARWIN ON THE SECULAR CHANGES IN n sin i~=^j sin (N\fj)\ ^Vsin 4f,+ sin 2g x sin 2g) sin 2g . (158) at \_n g This gives the additional terms due to bodily tides in the equation for dty/dt, viz. : the last of (133). If the viscosity be small sin 4fj + sin 2gj sin 2g= sin 4f(l 2X) sin 2g==i sin 4f where 1 Next consider the change in the obliquity of the ecliptic ; for this purpose we must find (l^i 2 )dW/idxdW/id\}i', and may drop terms involving/ sin (N\jj). First take the case (iiL), where the moon is both tide-raiser and disturber. Then from (129) -= -Fj{(l -t 8 -/) sin 2fi+# sin (N-^-2f 1 )-iJ8m (N-ijj+ZfJ} . (160) 8 -^{(l-i 2 -/) sin 2^+iysin (^-^-2^)-^)' sin (tf- Therefore =%(i+j cos (N-i}>)) sin 4fj ..... (161) From (130) sin (^-^+g 1 )+/Bin gl } . (162) -4 sn - Therefore n L. =t sm - sm "r ^ .... (163) THE ELEMENTS OF THE ORBIT OF A SATELLITE. 779 From (131) /I-IG^ sin g+ij sin (N-t+g)-ij sin (N-*-g)+f* sing} . (164) /7W /-i-2 * G( Therefore V V/))sin2g . . . . (165) Then collecting results from (161-3-5), we have for the whole perturbation of the earth due to the attraction of the moon on the lunar tides, The result for case (iv.), where the sun is both tide-raiser and disturber, may be written down by symmetry ; and since j=Q here, therefore =%i sin 4f . . v . . . (167) It is here assumed that the solar slow diurnal tide has the same lag as the sidereal diurnal tide, and that the solar slow semi-diurnal tide has the same lag as the sidereal semi-diurnal tide. This is very nearly true, because fi' is small compared with n. Next take the cases (v.) and (vi.), where the tide-raiser and disturber are distinct. Here we need only consider W 2 = - sn sn -< . (168) -if sin (^-^- g )+^ sin (N- 780 MB. G. H. DARWIN ON THE SECULAR CHANGES IN Therefore l/, ,wAVj 1 dW a ~] Irr' , 1 ~ l "~ ~ sm + sm When the moon is tide-raiser and the sun disturber, this becomes ~i( l ~h? 'cos (N 1/)) sin 2g (169) When the sun is tide-raiser and the moon disturber, this becomes (170) Then collecting results from (166-7-9, 170), we have by (18), n-j^=\(i-\-j cos (N 1/1)) -(sin 4f 1 -f- sin 2g 1 sin 2g) sin 2g + *1? sin 4f ~ Z g sin 2g ] (171) This gives the additional terms due to bodily tides in the equation for di/dt, viz. : the third of (133). If the viscosity be small where 8in 4fj-f sin 2g l sin 2g= sin 4f(l 2X) sin 2g =^ sin 4f n (172) Also we have from (160-2-4-8) to the present order of approximation, and by symmetry, Therefore by (18) /r . V~7 / ~ ==: ? sin 4t 8 (173) THE ELEMENTS OP THE ORBIT OP A SATELLITE. 781 Now let k ^ =i-r -{sin 4fi sin 2g : + sin 2g) & y k [~V 3 TT' "I G=i- I -(sin 4fi sin 2g x + sin 2g)+y sin 2gJ 1 pr 3 r' 3 TT' "1 A=27 -(sin 4f : + sin 2gj sin 2g)H sin 4f 2 sin 2g D==j-(sm 4fi+ sin 2 gl - sin 2g)- y sin 2gJ Then the four equations (139), (149), (158), and (171) may be written (174) .dN di (175) Also from (151) and (173) 1 d$* T^ =1 sin 4f, k dt "'g =i- sin 4f,+-i sin 4f rft fl 8 (176) These six equations (175-6) contain all the secular inequalities in the motions of the moon and earth, due to the bodily tides raised by the sun and moon, as far as is material for the present investigation. The terms which are omitted only represent a very small displacement of the proper planes and of the inclinations of the planes of motion of the two parts of the system to those proper planes. Then reverting to the earlier notation in which MDCCCLXXX. y=j sin N, r)=i sin t/> 2=7 cos N, t,=i cos \jj 5 H (177) 782 MR. G. H. DARWIN ON THE SECULAR CHANGES IN We easily find dt = ~ Tz ~ d dt=- T y dt' dr) (178) These equations contain the additional terms due to tides, which are to be added to the equations (116), in order to find the secular displacements of the proper planes. The first application, which will be made hereafter, will be to the case where the viscosity is small, and it will be more convenient to make the transition to that hypo- thesis at present, although the greater part of what follows in this part will be equally applicable whatever may be the viscosity. In the case of small viscosity the functions r, A, G, D will be indicated by the corresponding small letters y, 8, g, d. Then by (140), (150), (159), (172) we shall have ,k sin 4f k sin 4f where X=- n . (179) And in the present case where i andj are small, we have by (112) and (121) T' . u T a=-T*. b=- n n where e=^, the permanent ellipticity of the earth I . . (180) These equations (180) are the same whether the viscosity be supposed small or not. Then the complete equations are THE ELEMENTS OP THE ORBIT OF A SATELLITE. 783 J f (181) If the viscosity be not small we have T, G, A, D in place of y, g, 8, d. As it is more convenient to write small letters than capitals, in the whole of the next section the small letters will be employed, although the same investigation would be equally applicable with T, G, &c., in place of y, g, &c. The terms in y, g, 8, d are small compared with those in a, a, ft, b, and may be neglected as a first approximation. Also a, a, ft, b vary slowly in consequence of tidal reaction, tidal friction, and the consequent change of ellipticity of the earth, but as a first approximation they may be treated as constant. Then if we put L l cos L l sin A' cos L' sin 2 = Z 2 cos (/c 2 + m 2 ) , 2 = sn ' cos sin . . . (182) By (122) or (118) the first approximation is where z=z l +z. 2 , y= . . . (183) Before considering the secular changes in the constants L of integration, it will be convenient to take one other step. The equation of tidal friction (173) may be written approximately dn sm (184) because sin 4f will be nearly equal to sin 4fj as long as r /3 is not small compared with r 2 . (See however 22, Part IV.) 5 H 2 784 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Also the equation of tidal reaction (151) is ^=l-"sin4f 1 (185) Dividing one by the other and putting T 2 =r 2 ~ 12 , we have and integrating, This is the equation of conservation of moment of momentum of the moon-earth system, as modified by solar tidal friction. From it we obtain n in terms of 15. On the secular changes of the constants of integration. It is often found difficult on first reading a long analytical investigation to trace the general method amidst the mass of detail, and it is only at the end that the ruling idea is perceived ; in such circumstances it has often appeared to me that a preliminary sketch would be of great service to the reader. I shall act on this idea here, and consider some simple equations analogous to those to be treated. Let the equations be dz dy = ay, -f=a.z dt *' dt If a be constant, the solution is obviously 2=L cos (a+m), y= L sin (at-\-m) Now suppose a to be slowly varying ; put therefore <*. -\-aft for a, and treat a, a' as constants. Then dz , dy , -=ay+aty, - = -- to Differentiating THE ELEMENTS OF THE ORBIT OF A SATELLITE. 785 The terms on the right-hand side of these equations are small, because they involve a', and therefore we may substitute in them from the first approximation. Hence d?z + 2 z= a.'L sin (a.t-\-m) 2afa.tL cos (a(+m) Ctlr and a similar equation for y. The solution of this equation is / / / z=L cos (a+tn)+ Lt cos (ai+m) Lt cos (at+m) Lt 2 sin (a+m) 2tOL ''j~ Z The terms depending on t cut one another out, and z= L cos (a-|-m) -^Lt* sin (a^-f-m) 2 Similarly we should find y= L sin (a.t-\-m) -Lt z cos (at-\-m) * Zi The terms in t z are obviously equivalent to a change in m, the phase of the oscilla- tion ; but the amplitude L is unaffected. We might have arrived at this conclusion about the amplitude if, in solving the differential equations, we had neglected in the solutions the terms depending on t 2 , as will be done in considering our equations below. In those equations, however, we shall not find that the terms in t annihilate one another, and thus there will be a change of amplitude. That this conclusion concerning amplitude is correct, may be seen from the fact that the rigorous solution of the equations dz dy =oiy, = OLZ dt " dt is z=L cos = L cos (at + mo \aftdt], = L sin (at + m \a.'tdt) Whence L is unaffected, whilst m=m \a.'tdt So that dm da. ~ ' 786 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Next consider the equations dz dy ajry-n M=- az - Where a is constant, but y is a very small quantity compared with a, but which may vary slowly. Treat y as constant, and differentiate, and we have dz Then if we neglect y, we have the first approximation z=.L cos (a.t-\-m), y=.L sin (a-f m) Substituting these values for z, y on the right, we have cfiz . sn fit And a similar equation for y. The solutions are 2= L cos (af-(-m) yLt cos (a<+m) ?/= L sin (a<+m)+y- sin (ai+m) From this we see that, if we desire to retain the first approximation as the solution, we must have IdL L dt = -y (187) This will be true if y varies slowly ; hence and the solution is 2= L e~W cos (ai-j-ni) ij= LffTW* sin (a.t-\-m) It is easy to verify that these are the rigorous solutions of the equations, when a is constant but y varies. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 787 The equation (187) gives the rate of change of amplitude of oscillation. The cases which we have now considered, by the method of variation of parameters, are closely analogous to those to be treated below, and have been treated in the same way, so that the reader will be able to trace the process. They are in fact more than simply analogous, for they are what our equations (181) ^become if the obliquity of the ecliptic be zero and =0, 77=0. In this case Lj, and dj/dt= jy. This shows that the secular change of figure of the earth, and the secular changes in the rate of revolution of the moon's nodes do not affect the rate of alteration of the inclination of the lunar orbit to the ecliptic, so long as the obliquity is zero. This last result contains the implicit assumption that the perturbing influence of the moon on the earth is not so large, but that the obliquity of the equator may always remain small, however the lunar nodes vary. In an exactly similar manner we may show that, if the inclination of the lunar orbit be zero, di/dt=iS. This is the result of the previous paper " On the Precession of a Viscous Spheroid," when the obliquity is small. According to the method which has been sketched, the equations to be integrated are given in (181), when we write a.-\-a't for a, a+a' for a, /3-}-(?t for /3, b+b' for b, and then treat a, a, &c., a.', a', &c., y, g, &c., as constants. Before proceeding to consider the equations, it will be convenient to find certain relations between the quantities a, a, &c., and the two roots /q and * a of the quadratic We have supposed the two roots to be such that K! K 2 v(a p Then K 1 K 2 =( / 8 ab) (189) * } 788 ME. G. H. DARWIN ON THE SECULAR CHANGES IN (192) (193) Now suppose our equations (181) to be written as follows: dz ~\ -= ay+arj+s (194) = o.z f * Where s, M, er, u comprise all the terms involving a', a', &c., y, g, &c. Then if we write (z) as a type of z, ?/, , 77 ; (a) as a type of a, a, /3, b ; (a') as a type of a, a', ft, b'; (y) as a type of y, g, 8, d; and (s) as a type of s, u, cr, u; it is clear that (s) is (z)(a')+(y)(z). Differentiate each of the equations (194), and substitute for after differen- tiation. Then if we write ds S= TT U= 05 aar ar (195) The result is (196) From the first of these THE ELEMENTS OF THE ORBIT OF A SATELLITE. Therefore from the third 789 and by (190) Similarly a +) T* =( 2 +ab) -T| +K 1 z *c g a J7 T(a 2 . . (197) Differentiate the first of (196) twice, using the first of (197), and we have = _(s+ab) -G8N-ab) -^ Therefore by (190) Then writing (S) as a type of S, 2, U, T, (S) is of the type (z)()( '(OM-(y) Hence every term of (S) contains some small term, either (') or (y). Therefore on the right-hand side of the above equation we may substitute for (z) the first approximation, viz.: (zi)-\-(z 2 ) given in (182-3). When this substitution is carried out, let (Sj), (S 2 ) be the parts of (S) which contain all terms of the speeds iq and * 2 respectively. Then by (191) and (193) the right-hand side in the above equation may be written + the same with 2 and 1 interchanged. 71 75 Now let D 4 stand for the operation ^i+(: 1 2 +/c 2 2 )-^+K 1 2 K 2 2 , and we have MDCCCLXXX. dt* 5 I 790 MR. G. H. DARWIN ON THE SECULAR CHANGES IN the same with 2 and 1 reversed The last three of these equations are to be found by a parallel process, or else by symmetry. If the right hand sides of (198) be neglected, we clearly obtain, on integration, the first approximation (183) for z, y. , 77. This first approximation was originally obtained by mere inspection. We now have to consider the effects of the small terms on the right on the constants of integration L lt L 2 , L{, L 2 ' introduced in the first approximation. The small terms on the right are, by means of the first approximation, capable of being arranged in one of the alternative forms >f\o "] mti "1 [ K } t-\-t >K } t-\- the same with 2 for 1 inj cosj cos sin Now consider the differential equation a z b*x=A cos(at+7))+J3t cos(at+r,) . (199) First suppose that B is zero, so that the term in A exists alone. Assume x=Ct sin (at-^-rj) as the solution. Then = a^ sn cos (at-\-rj)} sn cos By substitution in (199), with jB=0, we have Therefore the solution is THE ELEMENTS OF THE ORBIT OF A SATELLITE. 791 By writing t] ^TT for 77, we see that a term A sin (at-\-rj) in the differential equation A would generate - rr.t cos (at-\-ri] in the solution. 2a(a 3 o-) From this theorem it follows that the solution of the equation JTJ1 2= - + the same with 2 and 1 interchanged ^K/C" K~ and the solution of s z= , , r+ the same with 2 and 1 interchanged 4h(*i * J Also (writing the two alternatives by means of an easily intelligible notation) the solutions of are y= ^ 2 -2\~ the same with 2 and 1 interchanged ^ K l( K l~~ K 2~) The similar equations for D 4 , DS? may be treated in the same way. The general rule is that y and 77 in the differential equations generate in the solution tz and t respectively ; and z and t, generate ty and trj respectively ; and the terms are to be divided by S/c^/q 3 K 2 2 ) or 2:.,(K 2 2 Kj 2 ) as the case may be. Next suppose that .4 = in the equation (199), and assume as the solution x=Ct 2 e,in. (at+r))+Dt cos (at+ rj) Then C{ aW sin (at+T)) + iat cos (at+t)) + 2 sin ( -\-D{a?t cos (at+r)) 2a sin (at-\-rf)} CaW sin at+r 8a 3 cos at+r 12cr sin cos o=0 Whence C- -^- D- 5a *- li 13 4a(a 2 -& 2 )' " 4a si ( s - &3 ) Hence the solution of (199), when A = Q, is If t be very small, the second of these terms may be neglected. By writing r) JTT for 77, we see that a term Bt sin (at-\-rj) in the differential equation, would have given rise in the solution to t being very small. By this theorem we see that the solutions of the two alternative differential equations are, when t is very small, ^o'F, \ * the same with 2 and 1 interchanged. J, cr=0, v= S 2 , S 3 have similar forms with 2 for 1 Clearly jj "i "6i7i (219) Therefore the equation for z is -D 4 z=2K 1 (/c 1 +K2)(/c 2 - r -a)?7 1 + the same with 2 for 1 Thence l Az r * + * i r *i + g b b b b ^ y .^^ y Qi Y\f*f* T * *? / - ft ^i ^^2 ' oiJiL-u ^i -ii j ^o "" "2 5 K 2 800 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Therefore \L dt I K fC \Jj (It I ~ K K ' Again and *! + _ b I" *! + v -fin "" Oi ^ _ Oi , <228> The last equation is approximate, for by writing it in this form we are neglecting r' 2 (sin 4f sin 4f 1 )/r 2 sin 4f x compared with unity. This is legitimate, because when (sin 4f sin 4f!)/sin 4fj is not very small, T^/V 8 is very small, and vice-versd ; see however 22. Hence (228) may be written THE ELEMENTS OF THE ORBIT OF A SATELLITE. 805 Let m=| l (230) HI is the ratio of the moment of momentum of the earth's rotation to that of the orbital motion of moon and earth round their common centre of inertia. (The /x of my paper on " Precession " is equal to the reciprocal of nt , where nt is the value of m when 2=0.) By (121) and (112) we have, k a=-r Now t=^n~/Q, the ellipticity of the earth due to rotation ; and as r=f/^m/c 3 and =\/c/c . therefore T=T O /^. Hence Differentiating logarithmically a'_2rfw_7Q B'-Y' r ' dn / rirf ^ T '/- - /T /VIV Therefore B'=- Lastly = T _'(l + ^\ (243) n\ T / By (174), (227), and (230), when the viscosity is not small, we have THE ELEMENTS OF THE ORBIT OF A SATELLITE. /I ........ (249) sin J = - sin I 3 + J These equations will give I 7 and J,, when J and I are found. Now suppose we divide the first and last of (224) by (?/nkdt, then their left-hand sides may be written log tan ^J and nk-rz log tan |I THE ELEMENTS OF THE ORBIT OF A SATELLITE. 809 In ths last section we have determined the functions a, a', &c., and have them in such a form that T, G, A, D (or y, g, S, d) have all a common factor d/nkdt. But this is the expression by which we have to divide the equations in order to change the variable. Therefore in computing T, G, &c. (or y, g, &c,), we may drop this common factor. Again a, a, ft, b were so written as all to have a common factor rtjn ; therefore K! and K 3 also have the same common factor. Also ' a', ft', b' all have a common factor (d^/kdt)(rtf-n z ). From this it follows that when the variable is changed, we may drop the factor rf/n from a, a, ft, b, /q, K. Z and the factor (d/kdt)(Tt/n*) from a', a', ft', b'. Hence the differential equations with the new variable become log tan AJ= Inlogtan ^1= b'a -, . (250) or similar equations with T, G, A, D in place of y, g, 8, d if the viscosity be not small. But we now have by (232-3-5-6-7-9, 242-3-4-5-6-7) a=ltt, =1+ -, b=l T ' / '\2 / _'\ T -+; + T T ) r _ i m sin 4f i - sin 1 a'" sin i T' /r'\ 2 (sin 4f, + sin 2gj sin 2g) 2 sin 2g + ( - j sin 4f 2 sin 4fj in 2(1 -X) 2(1 -X) 1(1+ -Jsin2g-2sin2g 1 sin 4fj . (251) 810 1N1R. G. H. DARWIN ON THE SECULAR CHANGES IN In these equations we have, recapitulating the notation tn=-* \= n . t=f . (252) f * 9 Also (253) ic, K 2 = \/(OL ft)*-^ 4ab | Lastly we have by (186) ...... (254) which gives parallel values of n and These equations will be solved by quadratures for the case of the moon and earth in Part IV. If T IT be so small as to be negligeable, and r'/2\tT small compared with unity, then the equations (250) admit of reduction to a simple form. With this hypothesis it is easy to find approximate values of K I and /o,, and then by some easy, but rather tedious analysis, it may be shown that (250) reduce to the following d , m + l T' 1 1 + llml *n- } (255) d , . 1T r' 1 l + llm These equations would give the secular changes of J and I, when the solar influence is very small compared with that of the moon. Of course if G be replaced by g, they are applicable to the case of small viscosity. It is remarkable that the changes of I are independent of the viscosity ; they depend in fact solely on the secular change in the permanent ellipticity of the earth. IV. INTEGRATION OF THE DIFFERENTIAL EQUATIONS FOR CHANGES IN THE INCLINATION OF THE ORBIT AND THE OBLIQUITY OF THE ECLIPTIC. 18. Integration in the case of small viscosity, where the nodes revolve uniformly. It is not, even at the present time, rigorously true that the nodes of the lunar orbit revolve uniformly on the ecliptic and that the inclination of the orbit is constant ; but it is very nearly true, and the integration may be carried backwards in time for a long way without an important departure from accuracy. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 811 The integrations will be carried out by the method of quadratures, and the process will be divided into a series of "periods of integration," as explained in 15 and 17 of the paper on "Precession." These periods will be the same as those in that paper, and the previous numerical work will be used as far as possible. It will be found, however, that it is not sufficiently accurate to assume the uniform revolu- tion of the nodes beyond the first two periods of integration. For these first two periods the equations of 7, Part II., will be used ; but for the further retrospect we shall have to make the transition to the methods of Part III. It is important to defer the transition as long as possible, because Part III. assumes the smallness of i and j, whilst Part II. does not do so. By (104) and (86) of Part II. we have, when/=0, and fi'/n is neglected, di sin 4f . . . . [ n/ . , ., ,, 2/2 - -- di sin 4f . . . . [ n/ . , ., ,, 2/2 . . ,, . ., ..") = - 5 sm i cos i{ r-(l f surj)-!-?-- -- r sec i COSJTT (1 f sm-j) y Sill , -. * o *\ / i> i /O\ l/i H * f> *\ 9 * ( * surz)(T-+r-) -|(l-f sin 2 i)r sm- j ' sin 2 i(l f sin~y) I o/2 T" COS I COS n If we put 1^ sin 2 i=cos i, 1 f sin 2 /= cos 3 /, and neglect sin 2 i sin 2 /, these may be written di sin4f . . . .[ ,., , 2/2 , .] 1 T: = * sm % cos cos d ?^ r-+r-sec' :S ; TT T sec t sec 2 ? \ \ dt ntt ' J n J [ J. (256) dn sm 4f . . I" .,.,., . M , i / , n -1 - cos i cos i\ T'-\-T - sec j T +TT sin i tan % cos- 7 } at c 1 -|-a) sin 2/ /a, where i and j Q are the mean values of i andy. With the numerical values corresponding to the initial condition (that is to say in the present configurations of earth, moon, and sun), it will be found on substituting in (fl'\Zf fl'\ T ' I 1 -^ }n instead of simply -, that J2 1 y 1L I SL a=-341251, ='000318, a='000059, b='000150, when the present tropical year is the unit of time. Since 4ab is very small compared with (a ft)' 2 , it follows that we have to a close degree of approximation Then since (/q-f a)/a=b/(/f 1 +/J), it follows that sin 2j oscillates between sin 2y o a sin 2i /(a/3), and sin 2i between sin Zi^b sin 2/ /(a /3). Let 8; and 8i be the oscillations of j and i on each side of the mean, then 8 sin 2y=a sin 2i/(aft) and 8 sin 2i=b sin 2j/(a.ft). Hence in seconds of arc 648000 ,a sin 2i TT * /3 cos2/ .. 648000 , b sin2/ (2GO) THE ELEMENTS OF THE ORBIT OF A SATELLITE. 815 Reducing these to numbers with j 5 9', i=23 28'. we have 8/=13"-13, 8i = ir-86.* Hence, if the earth were homogeneous, at the present time we should have S/ as the inclination of the proper plane of the lunar orbit to the ecliptic, and Sz as the amplitude of the 19-yearly nutation. These are very small" angles, and therefore initially the method of Part II. was applicable. * The formulas here used for the amplitude of the 19-yearly nutation and for the inclination of the lunar proper plane to the ecliptic differ so much from those given by other writers that it will be well to prove their identity. LAPLACE (' Mec. Gel.,' liv. vii., chap. 2) gives as the inclination of the proper plane to the ecliptic ap-^W . - ^ sin \ cos X g i a* Here a/> is the earth's ellipticity, and is my t; xi> is the ratio of equatorial centrifugal force to gravity, and is my n-ajrj, it is therefore ft when the earth is homogeneous. Thus his ap ^a0= my ft. His g 1 is the ratio of the angular velocity of the nodes to that of the moon, and is therefore my (a /3)//2. His D is the earth's mean radius, and is my a. His a is the moon's mean distance, and is my c. His \ is the obliquity, and is my i. Thus his formula is sin t cos i a. ft G in my notation. Now my T=3/.m/2c 3 , and | 2 =C/M. Therefore the formula becomes *-/* But by (5) Cnclfl.Mm=k. Therefore it becomes , Jirt Now by (115) and (112), when f =1, a=krt cosj cos 2j. Therefore in my notation LAPLACE'S result for the inclination of the lunar proper plane to the ecliptic is , a sin 2i * .. sec j 2 ft cos 2j J This agrees with the result (260) in the text, from which the amount of oscillation of the lunar orbit was computed, save as to the sec _;'. Since j is small the discrepancy is slight, and I believe my form to be the more accurate. LAPLACE states that the inclination is 20"'023 (centesimal) if the earth be heterogeneous, and 41"'470 (centesimal) if homogeneous. Since 41"'470 (centes.) = 13"'44, this result agrees very closely with mine. The difference of LAPLACE'S data explains the discrepancy. If it be desired to apply my formula to the heterogeneous earth we must take of my k, because the of the formula (6) for s will be replaced by i nearly. Also t, which is T 5 T , must be replaced by the processional constant, which is '003272. Hence my previous result in the text must be multiplied by -I of 232 x -003272 or -6326. This factor reduces the 13"'13 of the text to 8"'31. LAPLACE'S result (20"'023 centos.) is 6"'49. Hence there is a small discrepancy in the results; but it must be remembered that LAPLACE'S value of the actual ellipticity (1/334 instead of 1/295) of the earth was considerably in error. The more correct result is I think 8"'31. The amount of this inequality was found by BUKG and 5 M 2 816 ME. G. H. DARWIN ON THE SECULAR CHANGES IN Now consider the final condition. Since the integrations of the two periods have extended from = 1 to "88, and again from=l to 76, T =T (-88X76)- 6 , /2=/2 (-88x76)- 3 , i-=^(-88 X 76)- 1 , also the value of n which gives the day of 9 hrs. 55 m. is given by log n=374451, and log 2+10=1-21217, when the year is the unit of time. We now have t=17 4',j = 6 21'. Using these values in (115) and (112), I find a='10872, ='00627, a='00563, b='00510. ab is still small compared with (a ft), but not negligeable. Then by (117) *!-*,= -V(a-) 2 +4ab= -(-) , also KI + *=-(* a P Now 2ab/(a/3) = -00056. Hence we have Kl +K 2= -114991 whence ^= -10900 KI K 8 = -10301 J x-2= '00599 BUKCKHABDT from the combined observations of BRADLEY and MASKELYNE to be 8" (GKANT'S ' Hist. Phys. Astr.,' 1852, p. 65). For the amplitude of the 19-yearly nutation, AIRY gives (' Math. Tracts,' 1858, article " On Precession and Nutation," p. 214) T T o- -j- cos I sin 2i B is theprecess. const. = my t; his T'= my 2ir/ft; hisn=myi>; histt'=myw; hisl=myi; his 1= and his -t is the period of revolution of the nodes, and therefore = my 27r/( a /3). Then since my T=3 1 f2 2 /2(l + "), the above in my notation is i cos t sin 2j n a /3 Now by (115) and (112) b= - cos i cos 2i, when f =1. n Therefore his result in my notation is b sin 2j , * ' a ft COS 2f This is the result used above (in 260) for computing the nutations of the earth. If my formula is to be used for the heterogeneous earth, t must be replaced by the processional constant, and therefore the result in the text must be multiplied by 232 x '003272 or '759. Hence for the hetero- geneous earth the 11"'86 must be reduced to 9" '01. AIRY computes it as 10"'33, but says the observed amount is 9"'6, but he takes the precessional constant as '00317, and the moon's mass as l-70th of that of the earth. I believe that '00327 and l-82nd are more in accordance with the now accepted views of astronomers. THE ELEMENTS OP THE ORBIT OF A SATELLITE. 817 KJ and K. 2 have now come to differ a little from a and /3, but still not much. With these values I find + 10=876472, Substituting in the formulas a sin 2i ~. _ , b sin 2/ ' ~ 2 + cos 1 + y9 cos 2i I find 8/=57' 31", Si=:22' 42" Thus the oscillation of the lunar orbit has increased from 13" to nearly a degree, and that of the equator from 12" to 23'. It is clear therefore that we have carried out the integration by the method of Part II., as far back in retrospect as is proper, even for a speculative investigation like the present one. We shall here then make the transition to the method of Part III. Henceforth the formulas used regard the inclination and obliquity as small angles ; the obliquity is still however so large that this is not very satisfactory. 19. Secular changes in the proper planes of the earth and moon where the viscosity is small. We now take up the integration, at the point where it stops in the last section, by the method of Part III. The viscosity is still supposed to be small, so that y, 8, g, d (as defined in (251)) must be taken in place of F, A, G, D, which refer to any viscosity. The equations are ready for the application of the method of quadratures in (250), and the symbols are defined in (251-4). The method pursued is to assume a series of equidistant values of and then to compute all the functions (251-4), substitute them in (250), and combine the equi- distant values of the functions to be integrated by the rules of the calculus of finite differences. The preceding integration terminates where the day is 9 hrs. 55 m., and the moon's sidereal period is 8'17 m.s. days. If the present tropical year be the unit of time, we have, at the beginning of the present integration log =3'7445 I, log/2 =2'44836, and log k+ 10 = 6"20990, k being sf2 * of (7). The first step is to compute a series of values of n/n , by means of (254). As a fact, I had already computed n/n corresponding to =1, '92, '84, '76 for the paper on " Precession," by means of a formula, which took account of the obliquity of the ecliptic ; and accordingly I computed n/n , by the same formula, for the values of ='96, '88, '80, instead of doing the whole operation by means of (254), The difference between my results here used and those from (254) would be very small. 818 MR. G. H. DARWIN ON THE SECULAR CHANGES IN The following table exhibits some of the stages of the computation. The results are given just as they were found, but it is probable that the last place of decimals, and perhaps the last but one, are of no value. As however we really only require a solution in round numbers, this is of no importance. TABLE II. s 1- 96 92 88 84 80 76 M/JJ = 1-00000 1-04467 1-08931 1-13392 1-17852 1-22308 1-26763 logc + 10= 840016 8-43812 8-47446 8-50932 8-54284 8-57507 8-60614 10gT/T+10 = 8-61867 8-51230 8-40140 8-28557 8-16435 8-03721 7-90356 logX + 10= 8-70384 8-73805 8-77533 8-81581 8-85966 8-90712 8-95841 T'/2XtT = 16-3546 10-8418 7-0889 4-5647 2-8895 1-7947 1-0914 ni=a= 90035 97976 1-06603 1-16014 1-26320 1-37648 1-50172 log 7 +10= 9-67591 9-71452 9-75343 9-79287 9-83307 9-87430 9-91693 log + 10= 9-65551 9-65745 9-65824 9-65788 9-65631 9-65341 9-64900 log (gb- ad) + 10= 8-83030 8-86665 8-91307 8-96946 9-03549 9-11080 9-19510 *'= 36-696 23-186 12-583 4-144 - 2-747 - 8-605 -13873 a'= - 7-4782 - 8-6811 -10-0883 -11-7426 -13-6966 -16-0163 -18-7899 ft'= - 6-4455 - 6-9122 - 7-4220 - 7-9805 - 8-5940 - 9-2699 -10 0184 b'= - 6-4038 - 6-8796 - 7-3968 - 7-9612 - 8-5794 - 9-2590 -10-0104 log (. + ) + 10 = 8-74306 8-95453 9-16587 9-37077 9-55751 9-71146 9-82404 log (*2 + ) 1-21135 1-03659 86190 69374 54396 -42731 35255 log( 8 -i) 1-21283 1-04017 87056 71393 58660 -50372 46520 The further stage in the computation, when these values are used to compute the several terms of the expressions to be integrated, are given in the following table. TABLE III. f 1- 96 92 88 84 80 76 -(a'-/3'Xl + )/Mi=l/W, they become , d . , . 1 + m kn d log tan *? = -- m~ e b log tan #= (1 -f m)d (Compare with the first of equations (255) given in Part III., when r'=0.) These equations are not independent, because of the relationship which must always subsist between i &n a ir ~S are small, so that their cosines are approximately unity and their sines approximately equal to their tangents. Hence sln ^=-> sin 4f = sin 2 i= sin 2 ^ = Then introducing X=/2/n, we have f -l-X sin2 "'- sin4f X> sin 4- Introducing the transformations (262) into (251), we have 826 MR. G. H. DARWIN ON THE SECULAR CHANGES IN . (263) All the other expressions in (251) remain as they were. Then the terms in F, A, G, D in (250) are the only ones which have to be recomputed. And all the other arithmetical work of the last section will be applicable here. Also all the materials for calculating these new terms are ready to hand. The results of the computation are embodied in the following tables. TABLE IX. f r 96 92 88 84 80 76 iogr+io= 9-54901 9-57529 9-59914 9-61994 9-63663 9-64791 9-65092 logA = 52876 55517 58023 60484 63005 65708 68739 log(aD-bG) + 10= 9-08381 9-22356 9-34416 9-45433 9-55931 9-66259 976574 TABLE X. f = 1- 96 92 88 84 80 76 r( 1 +.)/fai(^-i)= -00133 -00328 -00800 -01853 -03818 -06513 -08961 r(,+)/&w(*2-'i)= 39185 39657 39712 38973 37003 33856 30260 A( 1 + )/toi(<2-*i)= -00199 - -00485 -01168 -02688 -05553 -09627 -13761 A(^ + *)/foi(* 2 - Kl )= 58529 58541 57994 56554 53826 50044 46468 (bG-aD)/to(, 2 - 1 )= -00825 -01622 -03034 -05388 -08850 -13092 -17504 Then combining these terms with those given in Table III., according to the formulas (250), (with T, &c., in place of y, &c.), we have the following equidistant values. TABLE XL f 1- 96 92 88 84 80 76 log tan -ij/df = -3477 -2925 -1587 + -1125 + -5036 + -7818 + -71'.'.-, lo g tanil/<*f= + 6168 + 6661 + 7796 + 1-0107 + 1-3406 + 1-5458 + 1-4103 By interpolation it appears that dJ/dg vanishes when ='8966. This value of corresponds with a period of 8 lire. 54 m. for the earth's rotation, and 5 '89 m. s. days for the moon's revolution. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 827 Integrating as in the last section, from =1 to '88, we have log, tan ^J= log, tan ^J +'0238 log, tan 1 1 = log, tan ^I '0895 Taking I =6, J =17, we have 1 = 15 34', J=6 9'. These values correspond to I/^l 15', ^=3 37'. Again integrating from =1 to "76, we have log, tan ij=log, tan |J "046 1 log, tan JI =log, tan ^I '2552 These give J=5 44', 1=13 13', which correspond to I, =2 33', J y =8 46'. The integration will now be continued over another period, as in the last section. The following are the results of the computations. TABLE XII. f 1- 96 92 88 log(r=G)+io= 9-65092 9-64491 9-62783 9-59299 log(A=D) + 10 = 9-84629 9-86040 9-87686 9-89622 TABLE XIII. 1- 96 92 88 G( Kl + a)/kn( K ,- Kl ) = -06781 -07617 -07802 - -07323 G(ic s +a)/-n(*z-Ki) = 23026 21018 19033 16832 D(*c 1 +)/An( K3 -K 1 )= 10634 . 12511 13843 14720 D(/OJ -\-ct)/kn(K. 2 K { ) = 36106 34521 33771 33835 (bG-aD)/^^-^)- 13815 16352 -19057 35054 Substituting these values in the differential equations (250), we have the following equidistant values : 828 MR. G. H. DARWIN ON THE SECULAR CHANGES IN TABLE XIV. t 1- 9G 92 88 d log tan ^J/dg= + -5547 + 3915 + 2088 + 1925 d log tan $I/d{= + T0746 + 8682 + 6391 + 3093 Then integrating from =1 to '88 we have log,, tan J=log, tan J '0382 log,, tan II =log, tan |T '0886 Then putting I =13 13' and J =5 44', from the previous integration, we have J=530', 1 = 12 6'. These values of J and I give J,= 10 49', I, =2 40'. The physical meaning of the results of the whole integration is embodied in the following table. TABLE XV. Results of integration in the case of large viscosity. Day i 11 in. s. hoars aud minutes. Moon's sidereal period in m. s. days. Inclination of earth's proper plane to ecliptic. Inclination of equator to earth's proper plane. Inclination of moon's proper plane to ecliptic. Inclination of lunar orbit to moon's proper plane. h. m. 9 55 Days. 8-17 O 1 17 22 6 57 o ' 6 8 45 5-57 15 34 1 15 3 37 6 9 7 49 3-59 13 13 2 33 8 46 5 44 7 15 2-45 12 6 2 40 10 49 5 30 If we compare these results with those^n Table VIII. for the case of small viscosity, we see that the inclinations of the two proper planes to one another and to the ecliptic are almost the same as before, but there is here this important distinction, viz. : that the inclinations of the two moving systems to their respective proper planes is less (compare 5 30' with 6 18', and 2 40' with 3 3'). And besides, if we had carried the integration, in the case of small viscosity, further back we should have found the inclination of the lunar orbit increasing. It will now be shown that, in the present case of large viscosity, the inclinations of THE ELEMENTS OF THE ORBIT OF A SATELLITE. 829 the equator and the orbit to their proper planes will continue to diminish, as the square root of the moon's distance diminishes, and at an increasing rate. Suppose that, in continuing the integration, the solar influence be entirely neglected, and the motion referred to the invariable plane of the system. This plane will be in some position intermediate between the two proper planes, but a little nearer to the earth's plane, and will therefore be inclined to the ecliptic at about 11 45'. The equations of motion are now those of 10, Part II., which may be written But since i/j=g/kn=l/m, they become i d i i 1 + nVi kn~ log tan A ; = Gr df m An log tan \i ( 1 + m)D (compare with the first of equations (255) given in Part III., when r'=0). These equations are not independent of one another, because of the relationship which must always subsist between i and j. Then substituting from (263) (in which T is put zero, and G, D written for T, A) we have for the case of large viscosity kn log tan 4/ 4 1 tan ii= When X=, 4X(1 X)/(l 2X) is infinite, and therefore both dj/dg and di/dg axe infinite. This result is physically absurd. The absurdity enters by supposing that an infinitely slow tide (viz. : that of speed n 2/2) can lag in such a way as to have its angle of lagging nearly equal to 90. The correct physical hypothesis, for values of X nearly equal to ^, is to suppose the lag small for the tide n 2/2, but large for the other tides. Hence when X is nearly =^, we ought to put M 2p 2( 2/2) sm 4f 1= n ^ n , sin 2g= -f, but sin 2 gl = -y~ MDCCCLXXX. 5 O 830 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Then we should have D= The last term in each of these expressions involves a small factor both in numerator and denominator, viz. : 1 2X because X=^ nearly, and J), because the viscosity is large. The evaluation of these terms depends on the actual degree of viscosity, but all that we are now concerned with is the fact that when X=^ the true physical result is that D changes sign by passing through zero and not infinity, and that G does the same for some value of X not far removed from ^. Now consider the function - 1. The following results are not stated retro- J. t\ spectively, and when it is said that i or j increase or decrease, it is meant increase or decrease as t or increases. (i.) From X=l to X='5 the function is negative. Hence for these values of X the inclination j decreases, or zero inclination is dynamically stable. When X='5 it is infinite ; but we have already remarked on this case. (ii.) From X='5 to X='191 it is positive. Therefore for these values of X the inclination j increases, or zero inclination is dynamically unstable. It vanishes when X='191. (iii.) From X='191toX=Oitis negative. Therefore for these values of X the inclination j decreases, or zero inclination is dynamically stable. Next consider the function 1 + jj J. ~ & (iv.) From X=l to X='809 it is positive. Therefore for these values of X the obliquity i increases, or zero obliquity is dynamically unstable. It vanishes when X= '809. (v.) From X='809 to X='5 it is negative. Therefore for these values of X the obliquity i decreases, or zero obliquity is dynamically stable. When X='5 it is infinite ; but we have already remarked on this case. (vi.) From X='5toX=Oitis positive. Therefore for these values of X the obliquity i increases, or zero obliquity is dynamically unstable. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 831 Therefore from X=l to '809 the inclination _; decreases and the obliquity i increases. From X='809 to '5 both inclination and obliquity decrease. From X='5 to '191 both inclination and obliquity increase. From X='191 to the inclination decreases and the obliquity increases. Now at the point where the above retrospective integration stopped, the moon's period was 2'45 days or 59 hours, and the day was 7 "25 hours ; hence at this point X='123, which falls between '191 and '5. Hence both inclination and obliquity decrease retrospectively at a rate which tends to become infinite when we approach X= - 5, if the viscosity be infinitely great. For large, but not infinite, viscosity the rates become large and then rapidly decrease in the neighbourhood of X='5. From this it follows that by supposing the viscosity large enough, the obliquity and inclination may be made as small as we please, when we arrive at the point where X='5. It was shown in 17 of "Precession" that X='5 corresponds to a month of 12 hours and a day of 6 hours. Between the values X='5 and '809 the solutions for both the cases of small and of large viscosity concur in showing zero obliquity and inclination as dynamically stable. But between X='809 and 1 the obliquity is dynamically unstable for infinitely large, stable for infinitely small viscosity ; for these values of X zero inclination is dynamically stable both for large and small viscosity. From this it seems probable that for some large but finite viscosity, both zero inclination and zero obliquity would be dynamically stable for values of X between '809 and unity. It appears to me therefore that we have only to accept the hypothesis that the viscosity of the earth has always been pretty large, as it certainly is at present, to obtain a satisfactory explanation of the obliquity of the ecliptic and of the inclination of the lunar orbit. This subject will be again discussed in the summary of Part VII. 21. Graphical illustration of the preceding integrations. A graphical illustration will much facilitate the comprehension of the numerical results of the last two sections. The integrations which have been carried out by quadratures are of course equivalent to finding the areas of certain curves, and these curves will afford a convenient illus- tration of the nature of those integrations. In 19, 20 two separate points of departure were taken, the first proceeding from =:! to 76, and the second from =1 to '88. It is obvious that was referred to different initial values c in the two integrations. 5 o 2 832 MR. G. H. DARWIN ON THE SECULAR CHANGES IN In order therefore to illustrate the rates of increase of log tan ^J and log tan il from the preceding numerical results, we must either refer the second sets of 's to the same initial value c as the first set, or (which will be simpler) we may take y/c as the independent variable. Then for the values between =1 and '76, the ordinates of our curves will be the numerical values given in Tables IV. and XL, each divided by ^/e y By the choice of a proper scale of length, c may be taken as unity. For the values in the second integration from f=l to '88, the ^/c is the final value of ^/c in the first integration. Hence in order to draw the ordinates in the second part of the curve to the same scale as those of the first, the numbers in Tables VII. and XIV. must be divided by 76. Also the second set of ordinates are not spaced out at the same intervals as the first set, for the d^/c of the second integration is '76 of the d 2 cos 2 X +#* cos 2( x +0) 2MjM 2 = The same with sines for cosines M 2 M 3 = -P S Q cos ( x -20)+PQ(P 2 -Q*) cos x+W cos MjM 3 = The same with sines for cosines cos 28 THE ELEMENTS OF THE DEBIT OF A SATELLITE. 837 By the definitions (29) Now let J cos (*+). *M= cos , B= . (264) Then X 2 -Y 2 = The same when X +TJ- is substituted for X . . (265) XZ= The same when x \TT is substituted for x Hence all the terms of the five X-Y-Z functions belong to one of the three types , , or R. The equation to the ellipse described by the satellite Diana is Hence R=14-fe 2 +3e(l+le 2 ) cos (0- CT )+f e 2 cos 2(0-)+^ cos 3(6 n) <&(a)=R COS (20+a) = (l + |e 2 ) COS (26+ a) iS/~iii9\r / o zi i \i / z) i i \n /o/*^r\ 1 iip/ I 1 ^k* 1 r* AQ I -{ H 1 ff 7^ 1 I f*J~I^I I 17 I tt. I TT I ^1^11/1 -|-|e 2 [cos (40+a 2zr)-f COS (a+2or)] + ^e 3 [cos (50+a 3sr)+ cos (5 a 3ra)] and ' v ! f (a)=R, cos a. Now by the theory of elliptic motion, the true longitude may be expressed in terms of flt+e and m, in a series of ascending powers of e the eccentricity. Hence <& (a), R, and "9 (a) may be expressed as the sum of a number of cosines of angles of the form l(flt+c)+1Mr+na,, and in using these functions we shall require to make a either a multiple of x or zero, or to differ from a multiple of X by a constant. Therefore the X-Y-Z functions are expressible as the sums of a number of sines or cosines of angles of the form l(ftt+t)+mar+ttx. Now x increases uniformly with the time (being equal to nt+a, constant) ; hence, if MDCCCLXXX. 5 P MR. G. H. DARWIN ON THE SECULAR CHANGES IN we regard the elements of the elliptic orbit as constant, the X-Y-Z functions are expressible as a number of simple time-harmonics. But in 4, where the state of tidal distortion due to Diana was found, they were assumed to be so expressible ; therefore that assumption was justifiable, and the remainder of that section concerning the formation of the disturbing function is applicable. The problem may now be simplified by the following considerations : The equation (12) for the rate of variation of the ellipticity of the orbit involves only differentials of the disturbing function with regard to epoch and perigee. It is obvious that in the disturbing function the epoch and perigee will only occur in the argument of trigonometrical functions, therefore after the required differentiations they only occur in the like forms. Now the epoch never occurs except in conjunction with the mean longitude, and the longitude of the perigee increases uniformly with the time (or nearly so), either from the action of other disturbing bodies or from the disturbing action of the permanent oblateness of the planet, which causes a progression of the apses. Hence it follows that the only way in which these differentials of the disturbing function can be non-periodic is when the tide-raiser Diana is identical with the moon. Whence we conclude that The tides raised by any one satellite can produce no secular change in the eccentricity of the orbit of any other satellite. The problem is thus simplified by the consideration that Diana and the moon need only be regarded as distinct as far as regards epoch and perigee, and that they are ultimately to be made identical. Before carrying out the procedure above sketched, it will be well to consider what sort of approximations are to be made, for the subsequent labour will be thus largely abridged. From the preceding sketch it is clear that all the terms of the X-Y-Z functions corresponding with Diana's tide-generating potential are of the form (a+6e+ce 2 +cZe 3 +/e*+&c.) cos [Z x +m(/2<-f e)+nw-f S]. From this it follows that all the terms of the %- ?-Z? functions are of the form cos Also by symmetry all the terms of the X'-Y'-Z' functions are of the form and in the present problem the accent to x mav be omitted. The products of the ^-^-2? functions multiplied by the X'-Y'-Z' functions occur in such a way that when they are added together in the required manner (as for example in Y'Z' J^Z:+ X'Z' ^j) only differences of arguments occur, and x disappears from the disturbing function. Also secular changes can only arise in the satellite's eccentricity and mean distance from such terms in the disturbing function as are independent of flt-{-f. and CT, when we put e'=e and ET'=ET. Hence we need only select from the THE ELEMENTS OF THE ORBIT OP A SATELLITE. 839 complete products the products of terms of the like argument in the two sets of functions. Whence it follows that all the part of the disturbing function, which is here important, consists of terms of the form or cos [m(e e')+n(sy ra-') /] Now it is intended to develop the disturbing function rigorously with respect to the obliquity of the ecliptic, and as far as the fourth power of the eccentricity. The question therefore arises, what terms will it be necessary to retain in developing the X-Y-Z functions, so as to obtain the disturbing function correct to e*. In the X-Y-Z functions (and in their constituent functions <(), ^(), R) those terms in which a is not zero will be said to be of the order zero ; those in which a is zero, but I) not zero, of the first order ; those in which a=6=0, but c not zero, of the second order, and so on. Then, by considering the typical term in the disturbing function, we have the following Rule of approximation for the development of the X-Y-Z functions and of <(), M'(a), R: develop terms of order zero to e*; terms of the first order to e 3 ; terms of the second order to e 2 ; and drop terms of the third and fourth orders. To obtain further rules of approximation, and for the subsequent developments, we now require the following theorem. Expansion of cos (k0-\-/3) in powers of the eccentricity. 6 is the true longitude of the satellite, /2i+e the mean longitude, and rs the longitude of the perigee. For the present I shall write simply fl in place of fit-\-e. By the theory of elliptic motion n=0 2e sin (6>- CT )+fe 2 (l + e 2 ) sin 2(6 r) |e 3 sin 3(0 t^+T&^sin 4(0 If this series be inverted, it will be found that* -e sn /2- CT +el- e sn 2/2-sr+e sn /-w +J- 9 ^e 4 sin 4(i2-r) * See TAIT and STEELE'S ' Dynamics,' art. 118, or any other work on elliptic motion. 5 P 2 840 MB. G. H. DARWIN ON THE SECULAR CHANGES IN By differentiation we find that, when e=0, =2 sin (fl- OT ), =-| sm 2(/2-r), =-f sin (/2- CT )+^ sin /- rf*$ /V//?\ 2 =-11 s i n 2(/2_ CT ) + ifl3 sin 4(/2- CT ), (-} = 2-2 cos (/2-sy) (dd\3 ftjfi\^ jjH =6 sin(/2 sr) 2 sin 3(/2 CT), f j =6 8 cos 2(/2 nr) + 2 cos 4(/2 dd d-8 /d6\ z d^0 rfe 5?~ * cos ( /2 CT ) I cos 3(fl CT), fj^ =5 sin 2(/2 w ) | sin 4(/2 = - cos 4 2-< =-+ 8 cos 2/J-- To expand cos (A;^-(-y8) by means of MACLAURIN'S theorem, we require the values of the following differentials when e=0 and 0=fl : cos (ke+/3)=-k sin cos (k0+/3)= -F cos (M+fl-k sin cos += sn +.-S CO s W+8 - sn d*0\* rfe 2 . -4 cos + _ sn Now when e=0, k0+fi=kf2-\-fi, and the values of the differentials and functions of differentials of e are given above. Then if we substitute for these functions their values, and express the products of sines and cosines as the sums of sines and cosines, and introduce the abridged notation in which &/2-|-/3+.s(/2 w) is written (&-|-s), we have THE ELEMENTS OF THE ORBIT OP A SATELLITE. 841 d = cos (M+$=.k cos (kl)+k cos (jfe+l) = cos -2-2 cos cos - 3 =^ cos (k8+/3)= -(P-i/F cos cos cos -l ^) cos (jfe+3) cos - V*) cos (k-2) + 3(2Jb* Y^+S* 2 ) cos (*) cos cos (268) where the 6's are merely introduced as an abbreviation. Then by MACLAURIN'S theorem cos v . . . (269) In order to obtain further rules of approximation we will now run through the future developments, merely paying attention to the order of the coefficients and to the factors by which /2i + e will be multiplied in the results. From this point of view we may write <&() = (e) cos (20) + (e)[cos (3(9)+ cos (0)]+ (e 2 )[cos (4(9)+ cos (0)] (e s )[cos(50)+cos(0)] (a)=R=(e) cos (0)+(e) cos (0)+(e 2 ) cos (20)+(e 3 ) cos (30) The cosines of the multiples of 9 have now to be found by the theorem (269) and substituted in the above equations. In making the developments the following abridged notation is adopted ; a term of the form cos [(&+s)/2+/3 sis] is written {&+}. Consider the series for 4>(a) first. We have by successive applications of (269) with k=l, 2, 3, 4, 5. 842 MR. G. H. DARWIN ON THE SECULAR CHANGES IN (e 2 )cos(0)=(e 2 ){0} (e 3 )cos(50)=(e 3 ){5}+(e4)[{4} + {6}] In these expressions we have no right, as yet, to assume that { 2} and { 1} are different from {2} and {1} ; and in fact we shall find that in the expansion for () they are different, but in that for R they are the same. Then adding up these, and rejecting terms of the third and fourth orders by the first rule of approximation, we have It will be observed that {5} and {6} are wanting, and might have been dropped from the expansions. Also < {0} and {4} are terms of the second order, therefore wherever they are multiplied by (e*) they might have been dropped. Hence (e 3 ) cos (50) need not have been expanded at all. A little further consideration is required to show that (e 3 ) cos (ff) need not have been expanded. (e 3 ) cos (0) is an abbreviation for e 3 cos (6 a 3ra-), and therefore in this case {!}= cos (/2 a SCT) and {2}= cos (2/2 a 4w) ; but in every other case {!}= cos (/2+a+Er) and {2}= cos (2/2 +). Hence the terms {1} and {2} in (e 3 ) cos (0) are of the third and fourth orders and may be dropped, and {0} may also be dropped. Thus the whole of (e 3 ) cos (6) may be dropped. With respect to { 2} and { 1), observe that {2} in the expansion of cos (kfl+P^ stands for cos[2/2+(& 1 2)ra-f-/3 1 ]; and { 2} in the expansion of cos (k.,0+p. 2 ) stands for cos [2/2 (& 2 -|-2)sr /3. z ~] ; and k lt & 2 are either 1, 2, 3, or 4 ; and /?j, /?., are multiples of x+ a constant. Hence {2} and { 2} are necessarily different, but if /3j and $., were multiples of XT they might be the same, and indeed in the expansion of II necessarily are the same. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 843 In the same way it may be shown that { 1} and {1} are necessarily different. Therefore { 1} and {2} being terms of the third and fourth orders may be dropped. It follows from this discussion that, as far as concerns the present problem, (e 2 )cos(0) = (e 2 ){0} And the sum of these expressions is equal to (a). We thus get the following rules for the use of the expansion (269) of cos for the determination of (a) : When k=2, omit in @ 3 terms in cos (k 3), cos (+3) in @ 4 terms in cos (& 4), cos (2), cos (&-f 2), cos (+4) When &=3, omit in @ 2 term in cos (+2) in 3 terms in cos (3), cos (+1), cos (+3) all of @ 4 When k=I, omit in 2 term in cos (k 2) in 3 term in cos (3), cos (kl), cos (+3) all of @ 4 . When k=4, omit in j term in cos (&+1) in 2 term in cos(&), cos (+2) aU of 6 3 , @ 4 Then following these rules we easily find, When k=2, /3=<* cos (20+) = (l 4e 2 +iie 4 ) cos (2/2+a) 2e(l |e 2 ) cos l-Ve s ) cos (3/2+a- CT )+|e 2 cos (a+2sr)+ J -/e 2 cos (4/2 + a-2sr) . (270) When k=3, /3=a CT cos (3^+a- w ) = (l-9e 2 ) cos (3^+a- CT )-3e(l- J ^e 2 ) cos (2/2+a) + 3e cos (4/2 + a 2w)+ a 8 i e 2 COS (/2 + a+sr) . (271) 844 MR. G. H. DARWIN ON THE SECULAR CHANGES IN When k=l, /3=a+nr cos (0+a+ OT ):=(l-e 2 ) cos (/2 + a+ CT )+e(l-fe 2 ) cos (2/2+a)-e cos (a +|e 2 cos (3/2+a CT ) . (272) When &=4, yS=a 2cr cos (40+a 2cr)=co6 (4/2+a 2w) 4e cos (3/2+a ^-f^e 2 cos (2/2+a) . (273) These are all the series required for the expression of <&(), since cos (a-f-2sr) does not involve 6, and by what has been shown above cos (50+ a 3 cr) and cos (6 a. 3sr) need not be expanded. We now return again to the series for R or (), and consider the nature of the approximations to be adopted there. With the same notation (e)cos(0)=(e){0} Since R is a function of 0m, therefore after expansion it must be a function of flvs, and hence {1} must be necessarily identical with { 1], and {2} with { 2}. Adding these up, and dropping terms of the third and fourth orders, Here {0} is a term of the order zero, {1} of the first order, and {2} of the second. Therefore by the first rule of approximation {2} and { 2} may be dropped when multiplied by (e 4 ). Also {3} and {4} may be dropped. Hence as far as concerns the present problem (e)cos(0) = (e (e 2 ) cos (2^ = (e 2 ){2] +(e s Ml] +(e*) and (e 3 ) cos (30) need not be expanded. And the sum of these expressions is equal to R. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 845 We thus get the following rules for the use of the expansion of cos (k6-\-t) for the determination of "R. When Jt=l, omit in 6 2 term in cos in 0, terms in cos (k 3), cos (*+!), cos (t+3) allofB t "When k=-2, omit in Bj term in cos in B e terms in cos (i), cos (t-f-2) all of 03, B 4 . Then following these rules, we find When*=l, )S=-=r cos (0 )=(! e 2 ) cos (/Jw)e+e cos 2(/2w) . . . . (274) ) 2ecos(/J w)-|- .... (275) These are the only series required for the expansion of R or f(a), since by what is shown above, cos 3(0 vr) need not be expanded. Now multiply (270) by 1+fe 2 ; (271) by f^l+^e 2 ); (272) by |e(l+^); and (273) by %er; add the four products together, and add fer cos (-f-2w), and we find from (267) after reduction -el -* cos . (276) Next multiply (274) by 3e(l+ie*); (275) by fe*; add the two products, and add , and we find from (267) after reduction, 2 wj-fle 2 cos 2(/2 w) . . (277) MDCOCLXXX. 5 Q 846 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Now let (278) And we have = cos whence cos (2/2+a) + E 3 cos (3/2 + a CT) + E 4 cos (4/2 + a cos 2(/2 CT) R=J +2J 1 cos (/2 (a) = J cos a-j-Jj[ cos (/2 + a cr)+ cos (/2 a w)] +Jo[ COS (2/2 + a 2ar)+ cos (2/2 a 2nr)]_ (279) These three expressions are parts of infinite series which only go as far as terms in e 2 , but the terms of the orders e and e have their coefficients developed as far as e* and e 3 respectively. Then substituting from (279) for <, ^, and E, their values in the expressions (265), we find X 2 -Y 2 =P*[E 1 cos (2 x -/2- CT )+E 2 cos (2 x -2/2)+E 3 cos (2 x - -f E. t cos (2 X 4/2 + 2 cos 2x+J!{cos (2 X /2+w)+ cos (2 x -f/2 w )} +J 2 {cos (2 X 2/24-2ar)4- cos (2^+ (2 x +/2+w)+E 2 cos (2 x +2/2)+E s cos (2 x +3/2 2XY=The same, with sines for cosines YZ=The same as X 2 -Y 2 , but with -P*Q for P 4 , PQ(P--Q-} for 2P 2 2 , Pg 3 for Q* and with x for 2 X XZ=The same as the last, but with sines for cosines cos /2-t (280) 8 ^ 2 [E 1 cos (!2+isr) + E 3 cos 2/2+E 3 cos (3/2 CT) +E i cos(4/2 Then if we regard rs as constant, and remember that x =n, and that /2 stands for /2i+e, and if we look through the above functions we see that there are trigonometrical THE ELEMENTS OF THE ORBIT OF A SATELLITE. 847 terms of 22 different speeds, viz. : 9 in the first pair all involving 2nt, 9 in the second pair all involving nt, and 4 in the last. Then since these five functions correspond to Diana's tide-generating potential, therefore we are going to consider the effects of 22 different tides, nine being semi- diurnal, nine diurnal, and the last four may be conveniently called monthly, since their periods are ^, -5, ^ of a month and one month. We next have to form the %-$}-%> functions. We found that in the X-Y-Z functions there were terms of 22 different speeds ; hence we shall now have to introduce 44 symbols indicating the reduction in the height of tide below its equi- librium height, and the retardation of phase. The notation adopted is analogous to that used in the preceding problem, and the following schedule gives the symbols. Semi-diurnal tides. speed 2n 4/2 In 3/2 2n 2/2 In /2 2n 2n+fl 2n + 2fl 2re + 3/2 height F" F ui F ' F F F, F ;i F m lag 2f ! ' 2f i;i 2f" 2f 2f 2f ; 2f u 2f ffi F iv Diurnal tides. speed n 4/2 n 3/2 n 2/2 , /2 71 height G iT G ui G" G ; G lag g" g iu g u g' g M + /2 M+2/2 tt + 3/2 i ga giii w + 4/2 G iv giT Monthly tides.* speed /2 2/2 3/2 4/2 height H 1 H u H m H" lag tf 2h u Sh 1 " 4h" The %-^-%& functions might now be easily written out; for each term of the X-Y-Z functions is to be multiplied, according to its speed by the corresponding height, and the corresponding lag subtracted from the argument of the trigonometrical term. For example, the first term of 2 9 2 is F^P* cos (2x~ /2 nr 2f ! ). It will however be unnecessary to write out these long expressions. In order to form the disturbing function W, the ^-9-2; functions have now to be multiplied by the X'-Y'-Z' functions according to the formula (31). Now the X'-Y'-Z' functions only differ from the X-Y-Z functions in the accentuation of /2 and CT, because Diana is to be ultimately identical with the moon. Then in the %-%&-%> functions /2 is an abbreviation for /2+e, and in the X'-Y'-Z' functions /2' for flt-\-^; hence wherever in the products we find fll', we may replace it by e e'. * With periods of , ^, , and one month. 5 Q 2 848 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Again, since we are only seeking to find the secular changes in the ellipticity and mean distance, therefore (as before pointed out) we need only multiply together terms whose arguments only differ by the lag. Secular inequalities, in the sense in which the term is used in the planetary theory, will indeed arise from the cross-multipli- cation of certain terms of like speeds but of different arguments, for example, the product of the term F H P*E 2 cos (2 X 2/2 2f a ) in 2 gE>- multiplied by the term 2P 2 $ 2 J 2 cos(2x 2f2'+2rv) in X' 2 Y' 2 , when added to the similar cross-product in 4X'Y'^|9 (which only differs in having sines for cosines) will give a term 2F ii P 6 # 2 E ; jJ 2 cos [2(e' e) 2/ 2f"]. This term in the disturbing function will give a long inequality, but it is of no present interest. The products may now be written down without writing out in full either the $-%H-i& functions or the X'-Y'-Z' functions. In order that the results may form the constituent terms of W, the factor ^ is introduced in the first pair of products, the factor 2 in the second pair, and the factor f in the last. Then from (280) we have 2 cos [(e'-e) + (/- CT )-2f i ]-f F'E 2 2 cos [2(e'- e )-2f ii ] +F U E 3 2 cos [3( / -e)-( CT / - CT )-2f m ]+F iy E/ cos [4(e'-e)-2( w '-T ; r)-2f iT ]} +F i J 1 2 cos [(e'-e)-( ro / - CT )- +FJ 2 2 cos [2(e'-e)-2( CT / - w )-2f i ]+F ii J 2 2 cos [2 (e'-e)- 2( CT '- cos e-e + 7 : - B r iii3 cos - +F m E 3 2 cos[3( / -e)-( CT / -B7) + 2f m ]+F iT E 4 2 cos [4(e'-e)-2( CT / - (281) =the same, when 2 PQ* replaces IP 8 ; 2P 2 g 2 (P 2 -^ 2 ) 2 replaces ; 2P 2 # 6 replaces ^Q 8 ; and G's and g's replace F's and 2f's . (282) 1 cos e _ e _ w _ +2H H J 2 2 cos [2(e'-e)-2(t ! 7'- cos [( e '-c) + ( CT '- CT )+h l J+H ii E 2 2 cos [2( C / -e)-f2h ii ] THE ELEMENTS OF THE ORBIT OF A SATELLITE. 849 The sum of these three last expressions (281-3) when multiplied by ^ is U ( J. ~~ o ) equal to W the disturbing function. 24. Secular changes in eccentricity and mean distance. Before proceeding to the differentiation of W, it is well to note the following coin- cidences between the coefficients and arguments, viz. : E x 2 occurs with (e')-\-(ry'7ff), E 3 2 with 2(e'-e), E 3 2 with 3(e'-e)-(s/- CT ), E/ with 4(e'-e)-2(s/- CT ), Jj 2 with (e' e) (/ CT), J 2 2 with 2(e' e) 2(sr' cr), and the terms in J 2 do not involve 6, e', rs, m . In consequence of these coincidences it will be possible to arrange the results in a highly symmetrical form. By equations (11) and (12) d . (d d \ 1 1 - ' X Y when y ' and ^-^l L-v l"W" \vhpn v ( 7. jt I j./ i 7j_/ / Y * Wlleu 7 l Hence the single operation d/de'-\-yd/d-ar' will enable us by proper choice of the value of y to find either d log y/kdt or d/kdt. Perform this operation ; then putting e'=e, CT'=OT, and collecting the terms accord- ing to their respective E's and J's, we have dW\r* 1 dV '' 8 (1-e 2 ) 6 =E!-(l-f y){iP 8 F sin 2f i +2P 6 ^ 2 G i sin g i -2P 2 ^ 6 G i sin g, J^F, sin 2f, , +E 2 2 (2){the same with ii for i, and 2h i; for h 1 } +E 3 2 (3 y){the same with iii for i, and 3h iH for h ; ) +E 4 2 (4 2y){the same with iv for i, and 4h iv for h 1 } + J : 2(l _ y ) {2P 4 g 4 (F i sin 2f i -F i sin 2f s ) + 2P 2 g 2 (P 2 - -i(P 4 -4P 3 Q 2 +^ i ) 2 H i sin h 1 } +J 2 2 (2 2y){ the same with ii for i, and 2h" for h 1 ) ....... (284) The functions of P and Q, which appear here, will occur hereafter so frequently that it will be convenient to adopt an abridged notation for them. Let x then represent either i, ii, iii or iv, and let 850 MR. G. H. DARWIN ON THE SECULAR CHANGES IN sin 2f*+2P 6 2 G* sin g x -2P 2 Q 6 G 3; sin g, J^F, sin 2f, * ' " x "" lX) } (285) - Q*~)"(G* sin g* G* sin g,) | -H x sin (xh x ) J And the generalised definition of the F's, G's, H's, &c., is contained in the following schedule speed 2n x/2, n xf2, x/2, n+x/2, 2n+x/2 1 height F* G* H 1 G, F x L. . . (286) g x 2f x We must now substitute for the E's and J's their values, and as the ellipticity is chosen as the variable they must be expressed in terms of t) instead of e. Also each of the E 2 's and J 2 's must be divided by (1 e 2 )". Then since \/i e 3 =l t), therefore B =2- 2 and l-e 2 - fl =l-- 12 = Then by (278) , and & =4 ,n- w a-'?) , and _ 4 --j 2 =289^ 3 } . (287) Jn 2 J 2=l-3e 2 + 3e 4 =l-67 ? +15T ? 2 , and -ffy 3 =] +6i ? +2l7 ? 2 J 2 2 2 =fi e *=^- 2 When y is put equal to - we shall also require the following . . . (288) THE ELEMENTS OF THE ORBIT OF A SATELLITE. 851 Therefore by putting y==- in equation (284) we have - log i7=W)+2(l-10i^ - and by putting y=0 in (284) ^^ The equations may be also arranged in the following form : k dt (ui) 578<(iv)-V^(i)-VV(i), <(ii), <(iii), /(i) with values of {=0, 15, 30, 45, 60, 75, 90. The results were then combined so as to give a series of values of d logrjfdt or de/edt, and these values were set out graphically in the accompanying fig. 8. Fig. 8. Diagram showing the rate of change in the eccentricity of the orbit of the satellite for various obliquities and viscosities of the planet ( - T-> when e is small j. In the figure the ordinates are proportional to de/edt, and the abscissas to i the obliquity ; each curve corresponds to one degree of viscosity. From the figure we see that, unless the viscosity be so great as to approach rigidity (when f"=45), the eccentricity will increase for all values of the obliquity, except values approaching 90. The rate of increase is greatest for zero obliquity unless the viscosity be very large, and in that case it is a little greater for about 35 of obliquity. It appears from the paper on " Precession " that if the obliquity be very nearly 90, the satellite's distance from the planet decreases with the time. Hence it follows from this figure that in general the eccentricity of the orbit increases or diminishes with the mean distance ; this is however not true if the viscosity approaches very near rigidity, for then the eccentricity will diminish for zero obliquity, whilst the mean distance will increase. THE ELEMENTS OF THE ORBIT OP A SATELLITE. 853 If the viscosity be very small, the equations (289-90) admit of reduction to very simple forms. In this case the sines of twice the angles of lagging are proportional to the speeds of the several tides, and we have (as in previous cases) sin4f sin2g* sin2xh* sin2g sin4f ;I - Therefore = sn = sin 4f (cos i ^ -isin4f(ixX)(f) And (i) = sin 4f(ll cost 18X) (i)-^ 1 /.(ii)= ; 1 sin 4f(297 cos i-756 Whence from (289) or From this we see that, in the case of small viscosity, tidal reaction is in general competent to cause the eccentricity of the orbit of a satellite to increase. But if 18 sidereal days of the planet be greater than 11 sidereal months of the satellite the eccentricity will decrease. Wherefore a circular orbit for the satellite is only dynamically stable provided 18 such days is greater than 11 such months. Now if we treat the equation (290) for -* in the same way, we find MV The first line =^ sin 4f(cos i\}. The second =ITJ sin 4f(27 cos i 46X). The third =^ 3 sin 4f (273 cos i 697X) MDCCCLXXX. 5 R 854 MR. G. H. DARWIN ON THE SECULAR CHANGES IN Therefore = sn 41 + 27^ + 273 cos or j> . . (292) sin 4f[cos {-^(i + 19^-89^)] _j From this it follows that the rate of tidal reaction is greater if the orbit be eccentric than if it be circular. Also for zero obliquity the tidal reaction vanishes when - = 1 19 + 45 , , (x) (295) Where 2 means summation for i, ii, iii, iv. This result follows from the fact that in all the E-terms of W, e' and is' enter in the form If -{-mm, where l-\-tn-=-1. In the F x -terms x enters in the form 2^', and is of the opposite sign from l+m ; in the Fjt-terms it enters in the form 2^', and is of the same sign as l-\-m ; in the G x -terms it enters in the form x'> an ^ * s f the opposite sign from l-\-m ; in the Gx-terms it enters in the form x', and is of the same sign as l+m. Hence as far as regards the E-terms of W, we have dW idW , dW\ in the F x -terms-7-, = -77+7-7 <'X \ de d *> I dW dW in the Fx-terms = ~T7+;TT de BT . /dW , dW\ in the G x -terms = i -77 +T~ / ) *\de' dm'] in the G x -terms i in the H-terms = In the J-terms of W, x enters with coefficient 2 in the F x - and Fx-terms, and with the coefficient 1 in the G x - and G,-terms, and is always of the same sign as the corresponding lag. Hence for the J-terms dW VoTW dW ~iT~/~~ \ T/fiT ' Where S means summation for the cases where x is zero and both upper and lower i and ii. From this we have THE ELEMENTS OF THE ORBIT OP A SATELLITE. 857 dndW sin 2f+2PW sin g^P-^G, sin g.+ ^F, sin 2f x } +J 2 {4P^F sin 2f+2P 2 2 (P 2 -. 884 MR. G. H. DARWIN ON THE SECULAR CHANGES IN The periodic time of the satellite Deimos is 30 hrs. 18 m.,* and as the period of rota- tion of Mars is 24hrs. 37m.,t Deimos must be still receding from Mars, but very slowly. The periodic time of the satellite Phobos in 7 hrs. 39 m. ; therefore Phobos must be approaching Mars. It does not seem likely that it has ever been remote from the planet. The eccentricities of the orbits of both satellites are small, though somewhat uncertain. The eccentricity of the orbit of Phobos appears however to be the larger of the two. If the viscosity of the planet be small, or if oceanic tidal friction be the principal cause of change, both eccentricities are diminishing ; but if the viscosity be large, both are increasing. In any case the rate of change must be excessively slow. As we have no means of knowing whether the eccentricities are increasing or diminishing this larger eccentricity of the orbit of Phobos cannot be a fact of much importance either for or against the present views. But it must be admitted that it is a slightly unfavourable indication. The position of the proper plane of a satellite is determined by the periodic time of the satellite, the oblateness of the planet, and the sun's distance. The inclination of the orbit of a satellite to its proper plane is not determined by anything in the system. Hence it is only the inclination of the orbit which can afford any argument for or against the theory. The proper planes of both satellites are necessarily nearly coincident with the equator of the planet ; but it is in accordance with the theory that the inclinations of the orbits to their respective proper planes should be small.J Any change in the obliquity of the equator of Mars to the plane of his orbit must be entirely due to solar tides. The present obliquity is about 27, and this points also to an advanced stage of evolution at least if the axis of the planet was primitively at all nearly perpendicular to the ecliptic. We now come to the system of Jupiter. This enormous planet is still rotating in about 10 hours, its axis is nearly perpen- dicular to the ecliptic, and three of its satellites revolve in 7 days or less, whilst the fourth has a period of 16 days 16 hrs. This system is obviously far less advanced than our own. The inclinations of the proper planes to Jupiter's equator are necessarily small, but * 'Observations and Orbits of the Satellites of Mars,' by ASAPH HALL. Washington Government Printing Office, 1878. t According to KAISER, as quoted by SCHMIDT. 'Ast. Naeh.,' vol. 82, p. 333. J For the details of the Martian system, see the paper by Professor ASAPH HALL, above quoted. With regard to the proper planes, see a paper by Prof. J. C. ADAMS read before the R. Ast. Soc. on Nor. 14, 1879, R. A. S. Month. Not. There is also a paper by Mr. MAKTII, 'Ast. Nach.,' No. 2280, vol. 95, Oct., 1879. THE ELEMENTS OF THE ORBIT OF A SATELLITE. 885 the inclinations of the orbits to the proper planes appear to be very interesting from a theoretical point of view. They are as follows : * a , -M-i Inclination of orbit Satellite. to proper plane. First .... O ' // ... 000 Second .... ... 27 50 Third .... ... 12 20 Fourth . 14 58 Now we have shown above that the orbit of a satellite is at first coincident with its proper plane, that the inclination afterwards rises to a maximum, and finally declines. If then we may assume, as seems reasonable, that the satellites are in stages of evolution corresponding to their distances from the planet, these inclinations accord well with the theory. The eccentricities of the orbits of the two inner satellites are insensible, those of the outer two small. This does not tell strongly either for or against the theory, because the history of the eccentricity depends considerably on the degree of viscosity of the planet ; yet it on the whole agrees with the theory that the eccentricity should be greater in the more remote satellites. It appears that the satellites of Jupiter always present the same face to the planet, just as does our moon.t This was to be 'expected. The case of Saturn is not altogether so favourable to the theory. The extremely rapid rotation, the ring, and the short periodic time of the inner satellites point to an early stage of development ; whilst the longer periodic time of the three outer satel- lites, and the high obliquity of the equator indicate a later stage. Perhaps both views may be more or less correct, for successive shedding of satellites would impart a modern appearance to the system. It may be hoped that the investigation of the effects of tidal friction in a planet surrounded by a number of satellites may throw some light on the subject. This I have not yet undertaken, and it appears to have peculiar difficulties. It has probably been previously remarked, that the Saturnian system bears a strong analogy with the solar system, Titan being analgous to Jupiter, Hyperion and lapetus to Uranus and Neptune, and the inner satellites being analo- gous to the inner planets. Thus anything which aids us in forming a theory of the one system will throw light on the other.J The details of the Saturnian system seem more or less favourable to the theory. The proper planes of the orbits (except that of lapetus) are nearly in the plane of the ring, and the inclinations of all the orbits to their proper planes appear not to be large. * HEESCHEL'S ' Astron.' Synoptic Tables in appendix. t HERSCHEL'S ' Astron.' 9th ed., 546. J An investigation, now (September, 1880) almost completed, seems to show pretty conclusively that tidal friction cannot be in all cases the most important feature in the evolution of such systems as that of Saturn and his satellites, and the solar system itself. I am not however led to reject the views maintained in this paper. WK'CCLXXX. 5 X 886 MR. G. H. DARWIN ON THE SECULAR CHANGES IN HERSCHEL gives the following eccentricities of orbit : Tethys '04 (?), Dione '02 (?), Rhea "02 (?), Titan "029314, Hyperion "rather large;" and he says nothing of the eccentricities of the orbits of the remaining three satellites. If the dubious eccentricities for the first three of the above are of any value, we seem to have some indication of the early maximum of eccentricity to which the analysis points ; but perhaps this is pushing the argument too far. The satellite lapetus appears always to present the same face to the planet.* Concerning Uranus and Neptune there is not much to be said, as their systems are very little known ; but their masses are much larger than that of the earth, and their satellites revolve with a short periodic time. The retrogade motion and high incli- nation of the satellites of Uranus are, if thoroughly established, very remarkable. The above theory of the inclination of the orbit has been based on an assumed small- ness of inclination, and it is not very easy to see to what results investigation might lead, if the inclination were large. It must be admitted however that the Uranian system points to the possibility of the existence of a primitive planet, with either retrograde rotation, or at least with a very large obliquity of equator. It appears from this review that the other members of the solar system present some phenomena which are strikingly favourable to the tidal theory of evolution, and none which are absolutely condemnatory. Perhaps by further investigations some light may be thrown on points which remain obscure. APPENDIX. (Added July, 1880.) A graphical illustration of the effects of tidal friction when the orbit of the satellite is eccentric. In a previous paper (Proc. Roy. Soc., No. 197, 1879t) a graphical illustration of the effects of tidal friction was given for the case of a circular orbit. As this method makes the subject more easily intelligible than the purely analytical method of the present paper, I propose to add an illustration for the case of the eccentric orbit. Consider the case of a single satellite, treated as a particle, moving in an elliptic orbit, which is co-planar with the equator of the planet. Let C^ be the resultant moment of momentum of the system. Then with the notation of the present paper, by 27 the equation of conservation of moment of momentum * HERSCHEL'S ' Astron.' 9th ed., 547. f The last sentence of this paper contains an erroneous statement; the line of zero eccentricity on the energy surface is not a ridge as there stated. See the figure on p. 8'JG. THE ELEMENTS OP THE ORBIT OF A SATELLITE. 887 Here On is the moment of momentum of the planet's rotation, and Cf(l >?)/& is the moment of momentum of the orbital motion; and the whole moment of momentum is the sum of the two. By the definitions of and k in 2, C = 7= , *.\/c, where p. is the attraction between unit masses at unit distance. By a proper choice of units we may make fj.Mm/^fj.(M-\-m) and C equal to unity."' Then let x be equal to the square root of the satellite's mean distance c, and the equation of conservation of moment of momentum becomes n-\-x(l >?) h .......... (a) If in (a) 77, the ellipticity of the orbit, be zero, we have equation (3) of the previous paper, No. 197, 1879. It is well known that the sum of the potential and kinetic energies in elliptic motion is independent of the eccentricity of the orbit, and depends only on the mean distance. Hence if CE be the whole energy of the system, we have (as in equations (2) and (4) of the above paper, No. 197), with the present units Then if z be written for 2E, and if the value of n be substituted from (a), we have *={*-(l-'?)} 8 -^ ......... (ft) This is the equation of energy of the system. * In the paper above referred to, and in another, Proc. Roy. Soc., No. 202, of 1880, the physical meaning of the units adopted is scarcely adequately explained. The units are such that C, the planet's moment of inertia, is unity, that /i(M+ni) is unity, and that a quantity called * and defined in (6) of this paper is unity. From this it may be deduced that the unit length is such a distance that the moment of inertia of planet and satellite when at this distance apart about their common centre of inertia is equal to the moment of inertia of the planet about its own axis. If .__ . " - M+m By the theory of the tides of a viscous spheroid (Phil. Trans., Part I., 1879, p. 13) tan2f=^---^, Hence sin 4f = ^ ,_ also T 2 = (f ) 2 Hence (1) becomes rfc* /gOi C (fj.ni) 2 (ro-/2)/p ,. dt ~~^> 9 c 6 - ' ' ' Now let Ch be the angular momentum of the whole system, namely that due to the Phil. Trans., Part II., 1879, p. 459. ATTENDED BY SEVEKAL SATELLITES. 495 planet's rotation and to the orbital motion of all the satellites. And let CE be the whole energy, both kinetic and potential, of the system. Then h is the angular velocity with which the planet would have to rotate in order that the rotational momentum might be equal to that of the whole system ; and E is twice the square of the angular velocity with which the planet would have to rotate in order that the kinetic energy of planetary rotation might be equal to the whole energy of the system. By the principle of conservation of momentum h is constant, and since the system is non-conservative of energy E is variable, and must diminish with the time. The kinetic energy of the orbital motion of the satellite m is ^Mm/c, and the potential energy of position of the planet and satellite is pMm/c ; the kinetic energy of the planet's rotation is ^Cn 2 . Thus we have, Ch=Cn +2, (3) > v ' (4) In the equations (3) and (4) we may regard C as a constant, provided we neglect the change of ellipticity of the planet's figure as its rotation slackens. Let the symbol b indicate partial ditferentiation ; then from (3) and (4) bn _ 1 pi Mm ~ j. C Uf+my"'~C f P 1 fiMm _ 1 /jJ'Mm C ~7~"' C (M+m)*" and therefore From equations (2) and (5) we may express the rate of increase of the square root of any satellite's distance in terms of the energy of the whole system, in the general case where the planet has any degree of viscosity. A good many transformations, analogous to those below, may be made in this general case, but as I shall only examine in detail the special case in which the viscosity is small, it will be convenient to make the transition thereto at once. When the viscosity is small, J), which varies inversely as the viscosity, is large. Then, unless nfl be very large, (n /2)/p is small compared with unity. Thus in (2) we may neglect (n /2) 3 /j) 2 in the denominator compared with unity. Substituting from (5) in (2), and making this approximation, we have _ (M+m? dt~ g 496 ME. G. H. DARWIN ON A PLANET Now let (= YfY .......... (7) where a is any constant length, which it may be convenient to take either as equal to the mean radius of the planet, or as the distance of some one of the satellites at some fixed epoch, f is different for each satellite and is subject to the suffixes 1, 2, 3, &c. The equation (6) may be written ( M ~\\c^- m'x 49 V "CV^Y 1 ^ \M+ m)' C dt~ * ~M& X \ M )lE And we have = A- ........... (9) dt b% [In order to calculate A it may be convenient to develop its expression further. ?= -, so that --,= 5 a 3 a J/ 7 and ^ = (|)(|)49, where p= ....... (10) Since p is an angular velocity A is a period of time, and A is the same for all the satellites.] In (9) is the variable, but it will be convenient to introduce an auxiliary variable x, such that _ - M+m m ml Then r. N* C = x (If + ) (3/ 4- f) f Let t K is different for each satellite and is subject to suffixes 1, 2, 3, &c. Thus (3) may be written ........... (13) uMm u,M*m 1 A am -^ ATTENDED BY SEVERAL SATELLITES. 497 Let X= ltM ' m t (14) X is different for each satellite and is subject to suffixes 1, 2, 3, &c. On comparing (12) and (14) we see that ..... ....... (15) This is of course merely a form of writing the equation Then (4) may be written (16) [In order to compute K and \ we may pursue two different methods. First, suppose a=a, the planet's mean radius. m* Then ic=-f[i/ 4 (l+i/) 3 ]~*(-) , of same dimensions as an angular velocity. \a / X=f[V 3 (l+i')]~' [ Y of same dimensions as the square of an angular velocity. \ a I If v be large compared with unity, as is generally the case, the expressions become 5m Secondly, suppose M large compared with all the m's, and suppose for example that the solar system as a whole is the subject of investigation. Then take as the earth's present radius vector, and o> as its present mean motion, and m , , . m a., and \= ^ or K =m -- , C is here the sun's moment of inertia.] Then collecting results from (9), (13), (1G), the equations which determine the changes in the system are ctt~ ~ i>f and a similar equation for each satellite MR. G. H. DARWIN ON A PLANET where x 7 =g; A is a certain time to be computed as above shown in (10) ; K an angular velocity to be computed as above shown in (17) and (18); and X the square of an angular velocity to be computed as above in (17) and (IS). If v be large compared with unity, is very approximately proportional to the seventh power of the square root of the satellite's distance. The solution of this system of simultaneous differential equations would give each of the f s in terms of the time ; afterwards we might obtain n and E in terms of the time from the last two of (19). These differential equations possess a remarkable analogy with those which repre- sent HAMILTON'S principle of varying action (THOMSON and TAIT'S ' Nat. Phil.,' 1879, 330 (14)). The rate of loss of energy of the system may be put into a very simple form. This function has been called by Lord RAYLEIGH ('Theory of Sound,' vol. i., 81) the Dissipation Function,* and the name is useful, because this function plays an important part in non-conservative systems. dE In the present problem the Dissipation Function or Dissipativity is C-rr. From (19) the dissipativity is therefore either This quantity is of course essentially positive. It is easy to show that = (n fl) og 7 3? Then on substituting for the various symbols in the expression for the dissipativity their values in terms of the original notation, we have Or if N be the tidal frictional couple corresponding to the satellite m, This last result would be equally true whatever were the viscosity of the planetary spheroid. * Sir W. THOMSON prefers to modify the name by calling it Dissipativity. ATTENDED BY SEVERAL SATELLITES. 499 The dissipativity, converted into heat by JOULE'S equivalent, expresses the amount of heat generated per unit time within the planetary spheroid. This result has been already obtained in a different manner for the case of a single satellite in a previous paper (" Problems, &c.," Phil. Trans., Part II., 1879, p. 557). 3. Sketch of method for solution of the equations by series. It does not seem easy to obtain a rigorous analytical solution of the system (19) of differential equations. I have however solved the equations by series, so as to obtain analytical expressions for the 's, as far as the fourth power of the time. This solu- tion is not well adapted for the purposes of the present paper, because the series are not rapidly convergent, and therefore cannot express those large changes in the con- figuration of the system which it is the object of the present paper to trace. As no subsequent use is made of this solution, and as the analysis is rather long, I will only sketch the method pursued. 7 Tj1 /\ ET \ 2 If -rs A be taken as the unit of time = -AS ( J - } dt \b^ ) Differentiating again and again with regard to the time, and making continued use of this equation, we find d~Ejdt", d 3 E/dt 3 , &c., in terms of bE/bg. It is then necessary to develop these expressions by performing the differentiations with regard to g. An abridged notation was used in which represented I j or . With this notation the whole operation may be shown to depend on the performance of b/bg on expressions of the form where y is independent of but may be a function of the mass of each satellite. Having evaluated the successive differentials of E we have Where the suffix indicates that the value, corresponding to < = 0, is to be taken. It is also necessary to evaluate the successive differentials of bE/b with regard to the time, and then we have 1.2.3 The coefficient of t* was found to be very long even with the abridged notation, and involved squares and products of S's. MDCCOLXXXI. 3 T 500 MR. G. H. DARWIN ON A PLANET 4. Graphical solution in the case when there are not more than three satellite*. Although a general analytical solution does not seem attainable, yet the equations have a geometrical or quasi-geometrical meaning, which makes a complete graphical solution possible, at least in the case where there are not more than three satellites. To explain this I take the case of two satellites only, and to keep the geometrical method in view I change the notation, and write z for E, x for j, and y for ,, also I write fl x for /2 1( and fl y for fl. 2 . The unit of time is chosen so that A = l. Then the equations (19) become dx__bz ^__*i dt br' an< ^ tnese are ^ )ot ' 1 eqal to 1/J f J +U J ATTENDED BY SEVERAL SATELLITES. 501 Therefore the equation to the tangent to the steepest path is X-x Y-ij Z-z bx/bx bzjby (23) If this steepest path 011 the energy surface is the path actually pursued by the point which represents the configuration of the system, equation (23) must be satisfied by And therefore we must have bz_ bs dx _ bx dy _ by bx But these are the values already found in (22) for dx/dz and dy/dz. Therefore we conclude that the representative point always slides down a steepest path on the energy surface. Hence it only remains to draw the surface, and to mark out the lines of steepest slope in order to obtain a complete graphical solution of the problem. Since the lines of greatest slope cut the contours at right angles, if we project the contours orthogonally on to the plane of x y, and draw the system of ortho- gonal trajectories of the contours, we obtain a solution in two dimensions. This solution will be exhibited below, but for the present I pass on to more general considerations. A precisely similar argument might be applied to the case where there are any number of satellites, only as space has but three dimensions, a geometrical solution is not possible. If there be r satellites, then the problem to be solved may be stated in geometrical language thus : It is required to find the path which is inclined at the least angle to the axis of E on the locus This locus is described in space of r+ 1 dimensions. One axis is that of E, and the remaining r axes are the axes of the r different 's. The solution may be depressed so as to merely require space of r dimensions, for we may, in space of r dimensions, construct the orthogonal trajectories of the contour loci found by attributing various values to E. Thus we might actually solve geometrically the case of three satellites. The energy locus here involves space of four dimensions, but the contour loci are a family of surfaces in three dimensions. If such a system of surfaces were actually constructed, it would be possible to pass through them a number of wires or threads which should be a good approximation to the orthogonal trajectories. The trouble of execution 502 MR. G. H. DARWIN ON A PLANET would however be hardly repaid by the results, because most of the interesting general conclusions may be drawn from the case of two satellites, where we have only to deal with curves. If the case of a single satellite be considered, we see that the energy locus is a curve, and the transit along the steepest path degenerates merely into travelling down hill. Now as the slopes of the energy curve are not altered in direction, but merely in steepness, by taking the abscissas of points on the curve as any power of the solution may still be obtained if we take x (or >) as the abscissa instead of This reduces the solution to exactly that which was given in a previous paper, where the graphical method was applied to the case of a single satellite.* 5. The graphical method in the case of two satellites. I now return to the special case in which there are only two satellites. The equation to the surface of energy is given in (21). The maxima and minima values of z (if any) are given by equating bz/bx and bz/by to zero. This gives h _ Ka t_ K1 > = ^ n K, X* ^ ^) , . . X.j 1 J By (15) and (19) we see that these equations may be written "i =/ H (25) n=fl y \ They also lead to the equations . (26) = J Now an equation of the form F* aF 3 +/3=0 may be written (F/3"') 4 a/8~ i (F/8~ i ) 3 +l = 0. And I have proved in a previous paper t that an equation x*hx z -\-l = has two real roots, if h be greater than 4/3 1 , but has no real roots if h be less than 4/3*. Hence it follows that this equation in Y has two real roots, if a be greater than 4^/3*, but no real roots if it be less. Then if we consider the two equations (26) as biquadratics for x* and y } respectively, we see that the first has, or has not, a pair of real roots, according as * Proc. Roy. Soc., No. 197, 1879. t Ibid., No. 202, 1880, p. 260-263. ATTENDED BY SEVERAL SATELLITES, 503 /4\ hK.iy* is greater or less than l^j) X^iq*, and the second has, or has not, a pair of real roots, according as /4 hKjX* is greater or less than (TJ Now if we substitute for the \'s and /c's their values, we find that 11 ~ X 2 } K.j 4 the same with m a in place of m v Now let yj and y a be two lengths determined by the equations 3 _ Mm, *_ 1 2 M+m Or in words let y l be such a distance that the moment of inertia of the planet (concentrated at its centre) and the first satellite about their common centre of inertia may be equal to the planet's moment of inertia about its axis of rotation ; and let y. 2 be a similar distance involving the second satellite instead of the first. And let Q> I} &>., be two angular velocities determined by the equations Or in words let ooj be the angular velocity of the first satellite when revolving in a circular orbit at distance y : , and w 2 a similar angular velocity for the second satellite when revolving at distance y 2 . Vp,Mn h , Mm 1 [C c, ,1 bo that and similarly X 2 i /e s *=the same with the suffix 2 in place of 1. Hence the first of the two equations (26) has, or has not, a pair of real roots, according as A JlfiiJ C(h- K .$>) is greater or less than ^ J/+ ^ w l7l 2 , and the second has, or has not, a pair of real roots, according as C(h K^X } ) is greater or less than - } - w^y., 2 . It is obvious that Mm l (a 1 -y l 2 /(M-\-m l ) is the orbital momentum of the first satellite when revolving at distance y 1; and similarly MmaO}. 2 -Y//(M-^-m.. z ) is the orbital momentum of the second satellite when revolving at distance y 2 . 504 MR. G. H. DARWIN ON A PLANET If the second or y-satellite be larger than the first or x-satellite the latter of these momenta is larger than the first. Now Ck is the whole angular momentum of the system, and in order that there may be maxima and minima determined by the equations bz/bx=Q, bz/by=Q, the equations (26) must have real roots. Then on putting y equal to zero in the first of the above conditions, and x equal to zero in the second we get the following results: First, there are no maxima and minima points for sections of the energy surface either parallel to x or y, if the whole momentum of the system be less than 4/3* times the orbital momentum of the smaller or ce-satellite when moving at distance y x . Second, there are maxima and minima points for sections parallel to x, but not for sections parallel to y, if the whole momentum be greater than 4/3* times the orbital momentum of the smaller or x-satellite when moving at distance y lt but less than 4/3 1 times the orbital momentum of the larger or y-satellite when moving at distance y.,. Third, there are maxima and minima for both sections, if the whole momentum be greater than 4/3 1 times the orbital momentum of the larger or y-satellite when moving at distance y 2 . This third case now requires further subdivision, according as whether there are not or are absolute maximum or minimum points on the surface. If there are such points the two equations (24) or (25) must be simultaneously satisfied. Hence we must have n=flz=fl y , in order that there may be a maximum or minimum point on the surface. But in this case the two satellites revolve in the same periodic time, and may be deemed to be rigidly connected together, and also rigidly connected with the planet. Hence the configurations of maximum or minimum energy are such that all three bodies move as though rigidly connected together. The simultaneous satisfaction of (24) necessitates that t ^-1^2 1 ^J^l * x*=-~y r or y<=~x* K l^2 S*1 Hence the equations (24) become ft /v,\i i , x, i "~~ These equations may be written y= *-- k W+, .=o ATTENDED BY SEVERAL SATELLITES. 505 Then treating these biquadratics in the same way as before, we find that they have, or have not, two real roots, according as h is greater or less than No, (V , ! Therefore there is, or there is not, a pair of real solutions of the equations ii=-n. r =f2 l/ , according as the total momentum of the system is, or is not, greater than And this is also the criterion whether or not there is a maximum, or minimum, or maximum-minimum point on the energy surface. In the case where the masses of the satellites are small compared with the mass of the planet, we may express the critical value of the momentum of the system in the form 4 [J 3 ! f* " J A comparison of this critical value with the two previous ones shows that if the two satellites be fused together, and if y be such that and if ~ Ao=< Kn~ 5 Wj l S OT 2 Now if we choose the mass of the first satellite as unit of mass, then W] = l, and we have The unit of length has been already chosen as equal to the mean radius of the planet. Then substituting in (21) we have as the equation to the energy surface And since we suppose m l and m. 2 to be small compared with M, we have H On account of the abruptness of the curvatures, this surface is extremely difficult to illustrate unless the figure be of very large size, and it is therefore difficult to choose appropriate values of h, M, m 2 , so as to bring the figure within a moderate compass. In order to exhibit the influence of unequal masses in the satellites, I choose m 2 = 2, the mass of the first sateUite being unity. I take M=50, so that f M=2Q. Then with these values for M and m, the first critical value for h is 3711, the second is 6'241, and the third is 8 - 459. I accordingly take h=9, which is greater than the third critical value. The surface to be illustrated then has the equation ATTENDED BY SEVERAL SATELLITES. 507 There is also another surface to be considered, namely which gives the rotation of the planet corresponding to any values of x and y. The equations n=fl. r , 11 = fly have also to be exhibited. The computations requisite for the illustration were laborious, as I had to calculate values of z and n corresponding to a large number of values of x and y, and then by graphical interpolation to find the values of x and y, corresponding to exact values of z and n. The surface of energy will be considered first. Plate 61 shows the contour-lines (that is to say, lines of equal energy) in the positive quadrant, z being either positive or negative. I speak below as though the paper were held horizontally, and as though positive z were drawn vertically upwards. The numbers written along the axes give the numerical values of x and y. The numbers written along the curves are the corresponding values of 2z. Since the numbers happen to be all negative, smaller numbers indicate greater energy than larger ones ; and, accordingly, in going down hill we pass from smaller to greater numbers. The full-line contours are equidistant, and correspond to the values 9, 8^, 8, 7^, 7, 6^, and 6 of 2z ; but since the slopes of the surface are very gentle in the central part, dotted lines (....) are drawn for the contours 7f and 7|. The points marked 5'529 and 7'442 are equidistant from x and y, and therefore correspond to the case when the two satellites have the same distance from the planet, or, which amounts to the same thing, are fused together. The former is a maximum point on the surface, the latter a maximum-minimum. The dashed line ( --- ) through 7'442 is the contour corresponding to that value of 2z. The chain-dot lines ( ---- ) through the same point will be explained below. An inspection of these contours shows that along the axes of x and y the surface has infinitely deep ravines ; but the steepness of the cliffs diminishes as we recede from the origin. The maximum point 5 '5 29 is at the top of a hill bounded towards the ravines by very steep cliffs, but sloping more gradually in the other directions. The maximum-minimum point 7*442 is on a saddle-shaped part of the surface, for we go up hill, whether proceeding towards or away from O, and we go down hill in either direction perpendicular to the line towards O. If the total angular momentum of the system had been less than the smallest critical value, the contour lines would all have been something like rectangular MDCCCLXXXI. 3 u 508 MR. G. H. DARWIN ON A PLANET hyperbolas with the axes of x and y as asymptotes, like the outer curves marked 6, 6^, 7 in Plate 61. In this case the whole surface would have sloped towards the axes. If the momentum had been greater than the smallest, and less than the second critical value, the outer contours would have still been like rectangular hyperbolas, and the branches which run upwards, more or less parallel to y, would still have preserved that character nearer to the axes, whilst the branches more or less parallel to x would have had a curve of contrary reflexure, somewhat like that exhibited by the curve 7^ in Plate 61, but less pronounced. In this case all the lines of steepest slope would approach the axis of x, but some of them in some part of their course would recede from the axis of y. If the momentum had been greater than the second, but less than the third critical value, the contours would still all have been continuous curves, but for some of the inner ones there would have been contrary reflexure in both branches, somewhat like the curve marked 7^ in Plate 61. There would still have been no closed curves amongst the contours. Here some of the lines of greatest slope would in part of their course have receded from the axis of x, and some from the axis of y, but the same line of greatest slope would never have receded from both axes. Finally, if the momentum be greater than the third critical value, we have the case exhibited in Plate 61. Plate 62, fig. 1, exhibits the lines of greatest slope on the surface. It was constructed by making a tracing of Plate 61, and then drawing by eye the orthogonal trajectories of the contours of equal energy. The dashed line ( --- ) is the contour corresponding to the maximum-mini mum point 7'442 of Plate 61. The chain-dot line ( ----- ) will be explained later. One set of lines all radiate from the maximum point 5 '5 29 of Plate 61. The arrows on the curves indicate the downward direction. It is easy to see how these lines would have differed, had the momentum of the system had various smaller values. Plate 62, fig. 2, exhibits the contour lines of the surface n=9 x* It is drawn on the same scale as Plate 61 and Plate 62, fig. 1. The computations for the energy surface, together with graphical interpolation, gave values of x and y corresponding to exact values of n. The axis of n is perpendicular to the paper, and the numbers written on the curves indicate the various values of n. These curves are not asymptotic to the axes, for they all cut both axes. The angles, however, at which they cut the axes are so acute that it is impossible to exhibit the intersections. None of the curves meet the axis of x within the limits of the figure. The. curve n=3 meets the axis of y when y=2150, and that for n='3^ when y= 1200, but for values of n smaller than 3 the intersections with the axis of y do not fall within ATTENDED BY SEVERAL SATELLITES. 509 the figure. The thickness, which it is necessary to give to the lines in drawing, obviously prevents the possibility of showing these facts, except in a figure of very large size. On the side remote from the origin of the curve marked 0, n is negative, on the nearer side positive. Since M=50, n c =20/X' and n y =2Q/y* Hence the lines on the figure, for which Sl x is constant, are parallel to the axis of y, and those for which fl y is constant are parallel to the axis of x. The points are marked off along each axis for which fl x or fi y are equal to 3ij, 3, 2|, 2, 1^, 1. The points for which they are equal to \ fall outside the figure, Now, if we draw parallels to y through these points on the axis of x, and parallels to x through the points on the axis of y, these parallels will intersect the n curves of the same magnitude in a series of points. For example, 12^=1^, when x is about 420, and the parallel to y through this point intersects the curve n=l^, where y is about 740. Hence the first or x-satellite moves as a rigid body attached to the planet, when the first satellite has a distance (420) f , and the second a distance (740) ? . In this manner we obtain a curve shown as chain-dot ( ) and marked f2^=n for every point on which the first satellite moves as though rigidly connected with the planet; and similarly there is a second curve ( ) marked I2 y =n for every point on which the second satellite moves as though rigidly connected with the planet. This pair of curves divides space into four regions, which are marked out on the figure. The space comprised between the two, for which n, x and fl y are both less than n, is the part which has most interest for actual planets and satellites, because the satellites of the solar system in general revolve slower than their planets rotate. If the sun be left out of consideration, the Martian system is exemplified by the space fl x > n, ft y < n, because the smaller and inner satellite revolves quicker than the planet rotates, and the larger and outer one revolves slower. The little quadrilateral space near O is of the same character as the external space fl x ~>n, fl y >n, but there is not room to write this on the figure. These chain-dot curves are marked also on Plates 61 and 62, fig. 1. In Plate 61 the line fl x =n passes through all those points on the contours of -energy whose tangents are parallel to x, and the line /2 y =n passes through points whose tangents are parallel toy. The tangents to the lines of greatest slope are perpendicular to the tangents to the contours of energy; hence in Plate 62, fig. 1, fl x =n passes through points whose tangents are parallel to y, and l y =n through points whose tangents are parallel to x. Within each of the four regions into which space is thus divided the lines of slope preserve the same character ; so that if, for example, at any part of the region they are receding from x and y, they do so throughout. This is correct, because dxjdt changes sign with n fl x and dy/dt with n fl y ; also 3 u 2 510 MR. G. H. DARWIN ON A PLANET either nfl x or nfi y changes sign in passing from one region to another. In these figures a line drawn at 45 to the axes through the origin divides the space into two parts ; in the upper region y is greater than x, and in the lower x is greater than y. Hence configurations, for which the greater or ^/-satellite is exterior to the lesser or aj-satellite, are represented by points in the upper space and those in which the lesser satellite is exterior by the lower space. In the figures of which I have been speaking hitherto the abscissas and ordinates are the f power of the distances of the two satellites ; now this is an inconveniently high power, and it is not very easy to understand the physical meaning of the result. I have therefore prepared another figure in which the abscissas and ordinates are the actual distances. In Plate 63, fig. 3, the curves are no longer lines of steepest slope. The reduction from Plate 62, fig. 1, to Plate 63, fig. 3, involved the raising of all the ordinates and abscissas of the former one to the y power. This process was rather troublesome, and Plate 63, fig. 3, cannot claim to be drawn with rigorous accuracy ; it is, however, sufficiently exact for the hypothetical case under consideration. If we had to treat any actual case, it would only be necessary to travel along a single line of change, and for that purpose special methods of approximation might be found for giving more accurate results. In this figure the numbers written along the axes denote the distances of the satellites in mean radii of the planet the radius of the planet having been chosen as the unit of length. The chain-dot curves, as before, enclose the region for which the orbital angular velocities of the satellites are less than that of the planet's rotation. The line at 45 to the axes marks out the regions for which the larger satellite is exterior or interior to the smaller one. Let us consider the closed space, within which fl x and fl y are less than n. The corner of this space is the point of maximum energy, from which all the curves radiate. Those curves which have tangents inclined at more than 45 to the axis of x denote that, during part of the changes, the larger satellite recedes more rapidly from the planet than the smaller one. If the curve cuts the 45 line, it means that the larger satellite catches up the smaller one. Since these curves all pass from the lower to the upper part of the space, it follows that this will only take place when the larger satellite is initially interior. According to the figure, after catching up the smaller satellite, the larger satellite becomes exterior. In reality there would probably either be a collision or the pair of satellites would form a double system like the earth and moon. After this the smaller satellite becomes almost stationary, revolves for an instant as though rigidly connected with the planet, and then slower than the planet revolves (when the curve passes out of the closed space) ; the smaller satellite then falls into the planet, whilst the larger satellite maintains a sensibly constant distance from the planet. ATTENDED BY SEVERAL SATELLITES. 511 If we take one of the other curves corresponding to the case of the larger satellite being interior, we see that the smaller satellite may at first recede more rapidly than the larger, and then the larger more rapidly than the smaller, but not so as to catch it up. The larger one then becomes nearly stationary, whilst the smaller one still recedes. The larger one then falls in, whilst the smaller one is nearly stationary. If we now consider those curves which are from the beginning in the upper half of the closed space, we see that if the larger satellite is initially exterior, it recedes at first rapidly, whilst the smaller one recedes slowly. The smaller and inner satellite then comes to revolve as though rigidly connected with the planet, and afterwards falls into the planet, whilst the distance of the larger one remains nearly unaltered. Either satellite comes into collision with the planet when its distance therefrom is unity. When this takes place the colliding satellite becomes fused with the planet, and the system becomes one where there is only a single satellite ; this case might then be treated as in previous papers. The divergences of the curves from the point of maximum energy shows that a very small difference of initial configuration in a pair of satellites may in time lead to very wide differences of configuration. Accordingly tidal friction alone will not tend to arrange satellites in any determinate order. It cannot, therefore, be definitely asserted that tidal friction has not operated to arrange satellites in any order which may be observed. I have hitherto only considered the positive quadrant of the energy surface, in which both satellites revolve positively about the planet. There are, however, three other cases, viz. : where both revolve negatively (in which case the planet necessarily revolves positively, so as to make up the positive angular momentum), or where one revolves negatively and the other positively. These cases will not be discussed at length, since they do not possess much interest. Plate 63, fig. 4, exhibits the contours of energy for that quadrant in which the smaller or a;-satellite revolves positively and the larger or y-satellite negatively. This figure may be conceived as joined on to Plate 61, so that the x-axes coincide. The numbers written on the contours are the values of 2z ; they are positive and pretty large. Whence it follows that these contours are enormously higher than those shown in Plate 61, where all the numbers on the contours were negative. The contours explain the nature of the surface. It may, however, be well to remark that, although the contours appear to recede back from the x-axis for ever, this is not the case ; for, after receding from the axis for a long way, they ultimately approach it again, and the axis is asymptotic to each of them. The point, at which the tangent to each contour is parallel to the axis of x, becomes more and more remote the higher the contour. The lines of steepest slope on this surface give, as before, the solution of the problem. If we hold this figure upside down, and read x for y and y for x, we get a figure 512 MR. G. H. DARWIN OX THE which represents the general nature of the surface for the case where the z-satellite revolves negatively and the y-satellite positively. But of course the figure would not be drawn correctly to scale. The contours for the remaining quadrant, in which both satellites revolve nega- tively would somewhat resemble a family of rectangular hyperbolas with the axes as asymptotes. I have not thought it worth while to construct them, but the physical interpretation is obviously that both satellites always must approach the planet. II. A DISCUSSION OF THE EFFECTS OF TIDAL FRICTION WITH REFERENCE TO THE EVOLUTION OF THE SOLAR SYSTEM. 6. General consideration of the problem presented by the solar system. In a series of previous papers I have traced out the changes in the manner of motion of the earth and moon which must have been caused by tidal friction. By adopting the hypothesis that tidal friction has been the most important element in the history of those bodies, we are led to coordinate together all the elements in their motions in a manner so remarkable, that the conclusion can hardly be avoided that the hypothesis contains a great amount of truth. Under these circumstances it is natural to inquire whether the same agency may not have been equally important in the evolution of the other planetary sub-systems, and of the solar system as a whole. This inquiry necessarily leads on to wide speculations, but I shall endeavour to derive as much guidance as possible from numerical data. In the first part of the present paper the theory of the tidal friction of a planet, attended by several satellites, has been treated. It would, at first sight, seem natural to replace this planet by the sun, and the satellites by the planets, and to obtain an approximate numerical solution. We might suppose that such a solution would afford indications as to whether tidal friction has or has not been a largely efficient cause in modifying the solar system. The problem here suggested for solution differs, however, in certain points from that actually presented by the solar system, and it will now be shown that these differences are such as would render the solution of no avail. The planets are not particles, as the suggested problem would suppose them to be, but they are rotating spheroids in which tides are being raised both by their own satellites and by the sun. They are, therefore, subject to a complicated tidal friction ; the reaction of the tides raised by the satellites goes to expand the orbits of the satellites, but the reaction of the tide raised in the planet by the sun, and that raised EVOLUTION" OF THE SOLAR SYSTEM. 513 in the sun by the planet both go towards expanding the orbit of the planet. It is this latter effect with which we are at present concerned. I propose then to consider the probable relative importance of these two causes of change in the planetary orbits. But before doing so it will be well as a preliminary to consider another point. In considering the effects of tidal friction the theory has been throughout adopted that the tidally-disturbed body is homogeneous and viscous. Now we know that the planets are not homogeneous, and it seems not improbable that the tidally-disturbed parts will be principally more or less superficial as indeed we know that they are in the case of terrestrial oceans. The question then arises as to the extent of error introduced by the hypothesis of homogeneity. For a homogeneous viscous planet we have shown that the tidal frictional couple is approximately equal * to ~r 2 nfl , qaw C , where =' 3 9 "* Now how will this expression be modified, if the tidally-disturbed parts are more or less superficial, and of less than the mean density of the planet ? To answer this query we must refer back to the manner in which the expression was built up. By reference to my paper " On the Tides of a Viscous Spheroid " (Phil. Trans., Part I., 1879, pp. 8-10, especially the middle of p. 8), it will be seen that J) is really (%gciw ^gaiv}/l9v, and that in both of these terms w represents the density of the tidally-disturbed matter, but that in the former g represents the gravitation of the planet and in the second it is equal to farfiCtw, where w is the density of the tidally- disturbed matter. Now let f be the ratio of the mean density of the spheroid to the density of the tidally-disturbed matter. Then in the former term gaw=- 3 ' 3 ' l And in the latter Hence if the planet be heterogeneous and the tidally-disturbed matter superficial, must be a coefficient of the form _3 __ //5_3\ l9iA2 2/7 If /be unity this reduces to the form gaiv/lQv, as it ought; but if the tidally- * I leave out of account the case of " large " viscosity, becanse as shown in a previous paper that could only be true of a planet which in ordinary parlance would be called a solid of great rigidity. See "Precession," Phil. Trans., 1879, Part II., p. 531. 514 MR, G. H. DARWIN ON THE disturbed matter be superficial and of less than the mean density, then J) must be a coefficient which varies as ^-JlY The exact form of the coefficient will of course depend upon the exact nature of the tides. Iff be large the term 3/5/ will be negligeable compared with unity. Again, if we refer to the following paper ("Precession, &c.," Phil. Trans., Part II., 1879, p. 456), it will be seen that the C in the expression for the tidal frictional couple represents |-(|7ra 3 M?)a 2 , where w is the density of the" tidally-disturbed matter ; hence C should be replaced by CJf. Then if we reconstruct the expression for the tidal frictional couple, we see that it is to be divided by /, because of the true meaning to be assigned to C, but is to be multiplied by/ on account of the true meaning to be assigned to J). From this it follows that for a given viscosity it is, roughly speaking, probable that the tidal frictional couple will be nearly the same as though the planet were homo- geneous. The above has been stated in an analytical form, but in physical language the reason is because the lagging of the tide will be augmented by the deficiency of density of the tidally-disturbed matter in about the same proportion as the frictional couple is diminished by the deficiency of density of the tide-wave upon which the disturbing satellite has to act. This discussion appeared necessary in order to show that the tidal frictional couple is of the same order of magnitude whether the planet be homogeneous or heterogeneous, and that we shall not be led into grave errors by discussing the theory of tidal friction on the hypothesis of the homogeneity of the tidally-disturbed bodies. We may now proceed to consider the double tidal action of a planet and the sun. Let us consider the particular homogeneous planet whose mass, distance from sun, and orbital angular velocity are m, c, fl. For this planet, let C'= moment of inertia ; a'= mean radius; w'= density; g'= gravity ; $'=%g'/a ' ', v'= the viscosity; an d n'= angular velocity of diurnal rotation. The same symbols when unaccented are to represent the parallel quantities for the sun. Now suppose the sun to be either perfectly rigid, perfectly elastic, or perfectly fluid. Then mutatis mutandis, equation (2) gives the rate of increase of the planet's distance from the sun under the influence of the tidal friction in the planet. It becomes Mm dC (M+my> dt~* g' c 6 p' If the planet have no satellite the right-hand side is equal to C'dri/dt, because the equation was formed from the expression for the tidal frictional couple. Hence, if none of the planets had satellites we should have a series of equations of the form with different h's corresponding to each planet. EVOLUTION OF THE SOLAR SYSTEM. 515 We may here remark that the secular effects of tidal friction in the case of a rigid sun attended by tidally-disturbed planets, with no satellites, may easily be determined. For if we put c*=a;, and note that fi varies as or 3 , and that ri has the form (hkx)/C', we see that it would only be necessary to evaluate a series of integrals of the form fx fx^dx -. This integral is in fact merely the time which elapses whilst x changes from x to x, and the time scale is the same for all the planets. It is not at present worth while to pursue this hypothetical case further. Now if we suppose the planet to raise frictional tides in the sun, as well as the sun to raise tides in the planets, we easily see by a double application of (2) that ll$ / 23 h 56 m Mars ! 24 h 37 m 19 h Jupiter .... Saturn .... 9 h 55 m 10 h 29 5 h 3 h 20 m to 2 h 20 This seems to me to illustrate the arguments used above. For there should in general be a diminution of effective period as we recede from the sun. It will be noted that the earth, although ten times larger than Mars, has a longer effective period. The larger masses should proceed in their evolution slower than the smaller ones, and therefore the greater proximity of the earth to the sun does not seem sufficient to account for this, more especially as it is 530 MR. G. H. DARWIN ON THE 9. General discussion and summary. According to the nebular hypothesis the planets and the satellites are portions detached from contracting nebulous masses. In the following discussion I shall accept that hypothesis in its main outline, and shall examine what modifications are neces- sitated by the influence of tidal friction. In 7 it is shown that the reaction of the tides raised in the sun by the planets must have had a very small influence in changing the dimensions of the planetary orbits round the sun, compared with the influence of the tides raised in the planets by the sun. From a consideration of numerical data with regard to the solar system and the planetary subsystems, it appears improbable that the planetary orbits have been much enlarged by tidal friction, since the origin of the several planets. But it is possible that part of the eccentricities of the planetary orbits is due to this cause. We must therefore examine the several planetary subsystems for the effects of tidal friction. From arguments similar to those advanced with regard to the solar system as a whole, it appears unlikely that the satellites of Mars, Jupiter, and Saturn originated very much nearer the present surfaces of the planets than we now observe them. But the data being insufficient, we cannot feel sure that the alteration in the dimensions of the orbits of these satellites has not been considerable. It remains, however, nearly certain that they cannot have first originated almost in contact with the present surfaces of the planets, in the same way as, in previous papers, has been shown to be probable with regard to the moon and earth. The numerical data in Table II., 7, exhibit so striking a difference between the terrestrial system and those of the other planets, that, even apart from the considera- tions adduced in this and previous papers, we should have grounds for believing that the modes of evolution have been considerably different. This series of investigations shows that the difference lies in the genesis of the moon close to the present surface of the planet, and we shall see below that solar tidal shown above that the efficiency of solar tidal friction is of about the same magnitude for the two planets. It is explicable however by the considerations in the text, for it was there shown that a large satellite was destructive of planetary rotation. If we estimate how fast the earth must rotate in order that the whole internal momentum of moon and earth should exist in the form of rotational momentum, then we find an effective period for the earth of 4 h 12 m . This again illustrates what was stated above, viz.: that a large satellite is preservative of tho internal momentum of the planet's system. The orbital momentum of the satellites of the other planets is so small, that an effective period for the other planets, analogous to the 4 12 m of the earth, would scarcely differ sensibly from the periods given in the table. If Jupiter and Saturn will ultimately be as condensed as the earth, then it must be admitted as possible- or even probable that Saturn (and perhaps Jupiter) will at some future time shed another satellite; for the efficiency of solar tidal friction at the distance of Saturn is small, and a period of two or three hours gives a very rapid rotation. EVOLUTION OF THE SOLAR SYSTEM. 531 friction may be assigned as a reason to explain how it happened that the terrestrial planet had contracted to nearly its present dimensions before the genesis of a satellite, but that this was not the case with the exterior planets. The numbers given in Table III., 8, show that the efficiency of solar tidal friction is very much greater in its action on the nearer planets than on the further ones. But the total amount of rotation of the various planetary masses destroyed from the beginning cannot be at all nearly proportional to the numbers given in that table, for the more remote planets must be much older than the nearer ones, and the time occupied by the contraction of the solar nebula from the dimensions of the orbit of Saturn down to those of the orbit of Mercury must be very long. Hence the time during which solar tidal friction has been operating on the external planets must be very much longer than the period of its efficiency for the interior ones, and a series of numbers proportional to the total amount of rotation destroyed in the several planets would present a far less rapid decrease, as we recede from the sun, than do the numbers given in Table III. Nevertheless the disproportion between these numbers is so great that it must be admitted that the effect produced by solar tidal friction on Jupiter and Saturn has not been nearly so great as on the interior planets. In 8 it has been shown to be probable that, as a planetary mass contracts, the rate of tidal retardation of rotation, and of destruction of rotational momentum increases, rises to a maximum, and then diminishes. This at least is so, when the acceleration of rotation due to contraction exceeds the retardation due to tidal friction ; and this must in general have been the case. Thus we may suppose that the rate at which solar tidal friction has retarded the planetary rotations in past ages was greater than the present rate of retardation, and indeed there seems no reason why many times the present rotational momenta of the planets should not have been destroyed by solar tidal friction. But it remains very improbable that so large an amount of momentum should have been destroyed as to materially affect the orbits of the planets round the sun. I will now proceed to examine how the differences of distance from the sun would be likely to affect the histories of the several planetary masses. According to the nebular hypothesis a planetary nebula contracts, and rotates quicker as it contracts. The rapidity of the revolution causes its form to become unstable, or, perhaps a portion gradually detaches itself; it is immaterial which of these two really takes place. In either case the separation of that part of the mass, which before the change had the greatest angular momentum, permits the central portion to resume a planetary shape. The contraction and increase of rotation proceed continually until another portion is detached, and so on. There thus recur at intervals a series of epochs of instability or of abnormal change. Now tidal friction must diminish the rate of increase of rotation due to contraction, and therefore if tidal friction and contraction are at work together, the epochs of instability must recur more rarely than if contraction acted alone. MDCCCLXXXI. 3 Z 532 ME. G. H. DARWIN ON THE If the tidal retardation is sufficiently great, the increase of rotation due to con- traction will be so far counteracted as never to permit an epoch of instability to occur. Now the rate of solar tidal frictional retardation decreases rapidly as we recede from the sun, and therefore these considerations accord with what we observe in the solar system. For Mercury and Venus have no satellites, and there is a progressive increase in the number of satellites as we recede from the sun. Moreover, the number of satellites is not directly connected with the mass of the planet, for Venus has nearly the same mass as the earth and has no satellite, and the earth has relatively by far the largest satellite of the whole system. Whether this be the true cause of the observed distri- 'bution of satellites amongst the planets or not, it is remarkable that the same cause also affords an explanation, as I shall now show, of that difference between the earth with the moon, and the other planets with their satellites, which has caused tidal friction to be the principal agent of change with the former but not with the latter. In the case of the contracting terrestrial mass we may suppose that there was for a long time nearly a balance between the retardation due to solar tidal friction and the acceleration due to contraction, and that it was not until the planetary mass had contracted to nearly its present dimensions that an epoch of instability could occur. It may also be noted that if there be two equal planetary masses which generate satellites, but under very different conditions as to the degree of condensation of the masses, then the two satellites so generated would be likely to differ in mass ; we cannot of course tell which of the two planets would generate the larger satellite. Thus if the genesis of the moon was deferred until a late epoch in the history of the terrestrial mass, the mass of the moon relatively to the earth, would be likely to differ from the mass of other satellites relatively to their planets. If the contraction of the planetary mass be almost completed before the genesis of the satellite, tidal friction, due jointly to the satellite and to the sun, will thereafter be the great cause of change in the system, and thus the hypothesis that it is the sole cause of change will give an approximately accurate explanation of the motion of the planet and satellite at any subsequent time. That this condition is fulfilled in' the case of the earth and moon, I have endeavoured to show in the previous papers of this series. At the end of the last of those papers the systems of the several planets were reviewed from the point of view of the present theory. It will be well to recapitulate shortly what was there stated and to add a few remarks on the modifications and additions introduced by the present investigation. The previous papers were principally directed to the case of the earth and moon, and it was there found that the primitive condition of those bodies was as follows : the earth was rotating, with a period of from two to four hours, about an axis inclined at 11 or 12 to the normal to the ecliptic, and the moon was revolving, nearly in EVOLUTION OP THE SOLAR SYSTEM. 533 contact with the earth, in a circular orbit coincident with the earth's equator, and with a periodic time only slightly exceeding that of the earth's rotation. Then it was proved that lunar and solar tidal friction would reduce the system from this primitive condition down to the state which now exists by causing a retardation of terrestrial rotation, an increase of lunar period, an increase of obliquity of ecliptic, an increase of eccentricity of lunar orbit, and a modification in the plane of the lunar orbit too complex to admit of being stated shortly. It was also found that the friction of the tides raised by the earth in the moon would explain the present motion of the moon about her axis, both as regards the identity of the axial and orbital revolutions, and as regards the direction of her polar axis. Thus the theory that tidal friction has been the ruling power in the evolution of the earth and moon completely coordinates the present motions of the two bodies, and leads us back to an initial state when the moon first had a separate existence as a satellite. This initial configuration of the two bodies is such that we are almost compelled to believe that the moon is a portion of the primitive earth detached by rapid rotation or other causes. There may be some reason to suppose that the earliest form in which the moon had a separate existence was in the shape of a ring, but this annular condition precedes the condition to which the dynamical investigation leads back. The present investigation shows; in confirmation of preceding ones/' that at this origin of the moon the earth had a period of revolution about the sun shorter than at present by perhaps only a minute or two, and it also shows that since the terrestrial planet itself first had a separate existence the length of the year can have increased but very little almost certainly by not so much as an hour, and probably by not more than five minutes. t With regard to the 11 or 12 of obliquity which still remains when the moon and earth are in their primitive condition, it may undoubtedly be partly explained by the friction of the solar tides before the origin of the moon, and perhaps partly also by the simultaneous action of the ordinary precession and the contraction and change of ellipticity of the nebulous mass.J '' Precession," 19. t If the change has been as much as an hour the rotational momentum of the earth destroyed by solar tidal friction must have been 33 times the present total internal momentum of moon and earth. For the orbital momentum of a planet varies as the cube root of its periodic time, and if we differentiate logarith- mically we obtain the increment of periodic time in terms of the increment of orbital momentum. Then taking the numerical data from Tables I. and II. we see that this statement is proved by the fact that 3 x 33 times [216-4--01720 X 10 10 ] X 365'25 x 24 is very nearly equal to unity. J See a paper " On a Suggested Explanation of the Obliquity of Planets to their Orbits," ' Phil. Mag.,' March, 1877. See however 21 " Precession." 3 z 2 534 ME. G. H. DARWIN ON THE In the review referred to I examined the eccentricities and inclinations of the orbits of the several other satellites, and found them to present indications favourable to the theory. In the present paper I have given reasons for supposing that the tidal fric- tion arising from the action of the other satellites on their planets cannot have had so much effect as in the case of the earth. That those indications were not more marked, and yet seemed to exist, agrees well with this last conclusion. The various obliquities of the planets' equators to their orbits were also considered, and I was led to conclude that the axes of the planets from Jupiter inwards \\vrc primitively much more nearly perpendicular to their orbits than at present. But the case of Saturn and still more that of Uranus (as inferred from its satellites) seem to indicate that there was a primitive, obliquity at the time of the genesis of the planets, arising from causes other than those here considered. The satellites of the larger planets revolve with short periodic times ; this admits of a simple explanation, for the smallness of the masses of these satellites would have prevented tidal friction from being a very efficient cause of cha,nge in the dimensions of their orbits, and the largeness of the planets' masses would have caused them to proceed slowly in their evolution. If the planets be formed from chains of meteorites or of nebulous matter the rotation of the planets has arisen from the excess of orbital momentum of the exterior over that of the interior matter. As we have no means of knowing how broad the chain may have been in any case, nor how much it may have closed in on the sun in course of concentration, we have no means of computing the primitive angular momentum of a planet. A rigorous method of comparison of the primitive rotations of the several planets is thus wanting. If however the planets were formed under similar conditions, then, according to the present theory, we should expect to find the exterior planets now rotating more rapidly than the interior ones. It has been shown above (see Table IV., note to 8) that, on making allowance for the different degrees of concentration of the planets, this is the case. That the ulterior satellite of Mars revolves with a period of less than a third of its planet's rotation is perhaps the most remarkable fact in the solar system. The theory of tidal friction explains this perfectly,* and we find that this will be the ultimate * It is proper to remark that the rapid revolution of this satellite might pcrhnps bp referred to another cause, although the explanation appears very inadequate. It has been pointed out above that the formation of a satellite out of a chain or ring of matter must bo accompanied by a diminution of periodic time and of distance. Thus a satellite might after formation have a shorter periodic time than its planet. If this, however, were the explanation, we should expect to find other instances elsewhere, but the case of the Martian satellite stands quite alone. [I believe that I now (July, 1881) see some reason to suppose that the earliest form of a satellite may not be annular. The investigation necessary to test this idea seems likely to prove a difficult one.] INVOLUTION OF T11K SOLAR SYSTEM. 535 fate of all satellites, because the solar tidal friction retards the planetary rotation without directly affecting the satellite's orbital motion. The numerical comparison in Table III. shows that the efficiency of solar tidal friction in retarding the terrestrial and Martian rotations is of about the same degree of importance, notwithstanding the much greater distance of the planet Mars. From the discussion in this paper it will have been apparent that the earth and moon do actually differ from the other planets in such a way as to permit tidal friction to have been the most important factor in their history. By an examination of the probable effects of solar tidal friction on a contracting planetary mass, we have been led to assign a cause for the observed distribution of satellites in the solar system, and this again has itself afforded an explanation of how it happened that the moon so originated that the tidal friction of the lunar tides in the earth should have been able to exercise so large an influence. In this summary I have endeavoured not only to set forth the influence which tidal friction may, and probably has had in the history of the system, but also to point out what effects it cannot have produced. The present investigations afford no grounds for the rejection of the nebular hypo- thesis, but while they present evidence in favour of the main outlines of that theory, they introduce modifications of considerable importance. Tidal friction is a cause of change of which LAPLACE'S theory took no account/' and although the activity of that cause is to be regarded as mainly belonging to a later period than the events described in the nebular hypothesis, yet its influence has been of great, and in one instance of even paramount importance in determining the present condition of the planets and their satellites. * Note added on July 28, 1881. Dr. T. R. MAYER appears to have been amongst the first, if not quite the first, to draw attention to the effects of tidal friction. I have recently had my attention called to his paper on " Celestial Dynamics " [Translation, 'Phil. Mag.,' 1863, vol. 25, pp. 241, 387, 417], in which he has preceded me in some of the remarks made above. He points out that, as the joint result of contraction and tidal friction, " the whole life of the earth therefore may be divided into three periods youth with increasing, middle ago with uniform, and old age with decreasing velocity of rotation." Phil. Tnuis. 1881. Hate 61. 2500 2000 1500- 500- 500 (500 2500 3000 Contours of the surface, 2 z =(9 xf 2y^) 2 20(~"j + *~j ) -when, oc. Jt y are. both positive. 3f B. Th*. vajiu&s of 2z ittdicoLtids by t}\A nujtibtrs an. the. Contow 50 t/LO* #ut smjidLir ruxnikzrs indicate higher Contours. en > Contour Lines of the surface 2z-(9-xl-2yi)-20('~l ^ -S) wJun r io fn>:-:tiw o^ii y t. ON THE STRESSES CAUSED IN THE INTERIOR OF THE EARTH BY THE WEIGHT OF CONTINENTS AND MOUNTAINS. BY G. H. DARWIN, F.RS. From the PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY. PART I. 1882. I, N D X : HAKRISON AND SOXS, PlilSl'ERS IN ORUIXAKY TO Hl:R MAJESTY, ST. MARTIN'S LANE. PHILOSOPHICAL TRANSACTIONS OF THE EOYAL SOCIETY. Note to be inserted at p. 187, Part I. 1882. CORRECTION TO A PAPER ENTITLED " On the Stresses caused in the Interior of the Earth by the Weight of Continents and Mountains." Since this paper has left my hands I have discovered an error in the work. The error does not affect the physical conclusions, except in one unimportant respect ; had it done so I should probably have found it out long ago. Throughout the paper the normal stresses P, Q, R require an additional term W t . The only function of these stresses used for obtaining physical results is P R, and it remains unchanged when the correction is made. 10 must however be erased. The error takes its origin in (1). Thomson's solution (1), when reduced to the form applicable to the incompressible solid, is the solution of the equations ~ + iva 2 = -j , and two others. The solution required is that of j- + uya 2 = 0, and two others. The W ax involved in my solution is not the potential of a true bodily force, but only an "effective potential " producing the same strains as those 'due to the weight of the continents and mountains, but causing a different hydrostatic pressure. When therefore p is determined from Thomson's solution, that p is really equal to p + W t of the problem of the continents. (i \ i l+-f]Wi, instead of p = yTF,-. . The correction to (3) must be carried on through the rest of the paper, and obviously it merely adds W t to the stresses P, Q, R, leaving P R unchanged. The error would have been avoided had I, as suggested on p. 190, worked directly from the equations of equilibrium of the elastic incompressible solid, instead of from Thomson's solution. When the solid is compressible, this method of " effective potential " [see " The Tides of a Viscous Spheroid," Phil. Trans. Part I. 1879, pp. 7 9] for including all the effects of gravitation, is not applicable without certain additional terms in a, /3, y. Hence 10 is erroneous, inasmuch as the expressions for the strains and stresses are incomplete. The correction of 10 (which is not difficult) would require too much space to be carried out in this note. G. H. DARWIN. Aug. 1, 1882. IV. On the Stresses caused in the Interior of the Earth l)y the Weight of Continents and Mountains. By G. H. DARWIN. F.R.S. Received June 11, Read June 16, 1881. [PLATES 19, 20.] TABLE OF CONTENTS. Page. Introduction 187 Part I. THE MATHEMATICAL INVESTIGATION. 1. On the state of internal stress of a strained elastic spliere 188 2. The determination of the stresses when the disturbing potential is an even zonal harmonic 191 3. On the direction and magnitude of the principal stresses in a strained elastic solid 198 4. The application of the previous analysis to the determination of the stresses produced by the weight of superficial inequalities 200 5. The state of stress due to ellipticity of figure or to tide-generating forces . . 201 6. On the stresses due to a series of parallel mountain chains 205 7. On the stresses due to the even zonal harmonic inequalities 208 8. On the stresses due to the weight of an equatorial continent 211 9. On the strength of various substances 213 10. On the case when the elastic solid is compressible 215 Part II. SUMMARY AND DISCUSSION 218 IN this paper I have considered the subject of the solidity and strength of the materials of which the earth is formed, from a point of view from which it does not seem to have been hitherto discussed. The first part of the paper is entirely devoted to a mathematical investigation, based upon a well-known paper of Sir WILLIAM THOMSON'S. The second part consists of a summary and discussion of the preceding work. In this I have tried, as far as possible, to avoid mathematics, and I hope that a considerable part of it may prove intelligible to the non-mathematical reader. 2 B 2 188 MR. G. H. DARWIN ON THE STRESSES I. THE MATHEMATICAL INVESTIGATION. 1. On the state of internal stress of a strained clastic spJicrc. Let there be a homogeneous elastic sphere, for which CD ^v is the modulus of com- pressibility (or incompressibility, as I shall call it) and u the rigidity. * Take the centre of the sphere as the origin for a set of rectangular axes x, y, z. Let the sphere be subjected to no surface stresses, let it be devoid of gravitation, but subject to internal force such that the force acting on a unit volume of the elastic solid is expres- sible by a gravitation potential W it a solid spherical harmonic of the i th degree of the coordinates x, y, z. Let w be the density of the elastic solid, a the radius of the sphere, and r the radius vector of any point measured from the centre of the sphere. Sir WILLIAM THOMSON has investigated the state of internal strain produced under the conditions above described. If a, ft, y be the displacements his solution is as follows : where "^~^ I c\ / -i \ r m / i -i \o n /* i -i \ i (1) 2t + 3)-(2t+l)u and similar expressions for y8 and y. t Now let P, Q, R, S, T, U be the six stresses, across three planes mutually at right angles at the point x, y, z, estimated as is usual in works on the theory of elasticity. Let P, Q, R be tractions and not pressures, and let p be the hydrostatic pressure at the point x, y, z. Then P+Q+R being an invariant of the stress quadric, we have, if 8=-^-j - -\ r. so that 8 is the dilatation, then according to the usual formulas,! ax ay dz * The phraseology adopted by THOMSON and TAIT (first edition) and others seems a little unfortunate-. One might be inclined to suppose that compressibility and rigidity were things of the same nature ; but rigidity and the reciprocal of compressibility are of the same kind. If one may give exact meanings to old words of somewhat general meanings, then one may pair together compressibility and "pliancy," and call the moduli for the two sorts of elasticity the " incompressibility " and rigidity. t THOMSON and TAIT'S ' Nat. Phil.,' 834, (8) and (9) ; or Phil. Trans., 1863, p. 573. J THOMSON and TAIT'S 'Nat. Phil.,' 693. DUE TO THE WEIGHT OF CONTINENTS. 189 dct dy + ^, and the other four stresses are expressed from these by cyclic changes of P, Q, R ; S, T, U ; a, & y ; a, y, z. The first task is to find p. Now by adding P, Q, R together we have, p= (to Jv)8 We must now find 8 from (1). By differentiation * rta; and similar expressions for d/3/dy, d-y/dz. Now Wi, Wp~~'~ l are spherical harmonics of degrees i, il, and are also homo- geneous functions of the same degrees. If therefore we add the three expressions together, and note the properties of harmonics and of homogeneous functions, we have Omitting for brevity that part of the divisors in the expressions for F and G which is common to both, and we have, on introducing the omitted denominator, S= r -^- And -w, Throughout the rest of this paper (excepting in 10) the elastic sphere will be treated as incompressible, so that w is to be considered as infinitely large compared with v. Henceforth I write 7=2(i+l) 3 +l ' (2 and when w is infinite compared with v, we have, P=-jWi (3) Also we may put 190 MR. G. H. DARWIN ON THE STRESSES (4) ^J J And on putting co infinite in (1) we have _- ,_ Jd^ t ^^d. r , ~/ l /Ll2(t-l) a 2(2i-il) f fo 2i + l' rf.iA ; . . (5) and symmetrical expressions for /3 and y. The hydrostatic pressure might have been found from this general solution for the case of incompressibility, but in order to do so it would have been necessary to go back to the equations of equilibrium of the solid, and I prefer to deduce it from Sir WILLIAM THOMSON'S solution in the more general case. Since r +s ^( Wp-*- 1 ) = -(2i+l)x Wi+ r- &c. ax dz ax Differentiating (6) with regard to x, 27 * r<2) a ,_ (l . +3) ^iw dx [ t 1 ' J da? dx DuTerentiating with regard to z, and by symmetry Adding (8) and (9) together and dividing by 2, we have /*, = rfi Hence from (3) (4) (7) and (10) we have, DUE TO THE WEIGHT OF CONTINENTS. 191 where /= 2(i+ 1) 2 +L The expressions for Q, R, S, U may be writteii down from these by means of cyclic changes of the symbols. These are the required expressions for the stresses at any point in the interior of the sphere. In order to find the magnitude and direction of the principal stress-axes at any point it would be necessary to solve a cubic equation. The solution of this equation appears to be difficult, but the special case in which it reduces to a quadratic equation will fortunately give adequate results. It may be seen from considerations of symmetry that if "Wj be a zonal harmonic, two of the principal stress-axes lie in a meridional plane and the third is perpendicular thereto. Moreover the greatest and least stress- axes are those which lie in that plane, and the mean stress-axis is that which is perpendicular thereto. If this is not obvious to the reader at present, it will become so later. 1 shall therefore take Wj to be a zonal harmonic, and as the future developments will be by means of series (which though finite will be long for the higher orders of harmonics) I shall attend more especially to the equatorial regions of the sphere. 2. The determination of the stresses when the disturbing potential is an even zonal harmonic. If be colatitude the expression for a zonal surface harmonic or LEGEXDRE'S function of order i is cos _ or if we begin by the other end of the series, and take i as an even number, the expression is ( - ^r/sin 1 0- 1 sin'- 2 cos 2 sn This latter is the appropriate form when we wish to consider especially the equatorial regions, because cos Q is small for that part of the sphere. There is of course a similar formula when i is odd, but of this I shall make no use. Now let p-=y~ + --, so that sin 0=p/r, cos 0= l'.)2 MR, G. H. DARWIN ON THE STRESSES Then we may put Wj is a solid zonal harmonic of degree i ; but r~' Wi requires multiplication by a factor ( ) w i!/2'{^'!} 2 in order to make it a LEGENDRE'S function. The factors by which Wi must be deemed to be multiplied in order that it may be a potential, will be dropped for the present, to be inserted later. Or we may, if we like, suppose that the units of length or of time are so chosen as to make the factor equal to unity. Now let Then, dropping the suffix to W for brevity, we may write (14) I shall now find P, Q, R, T at any point in the meridional plane which is determined by y=0. In evaluating the first differential coefficients of W we must not put y=0, in as far as these coefficients are a first step towards the determination of the second differential coefficients. But in as far as these first coefficients are directly involved in the expressions for P, Q, R, and T, and in the second coefficients in the same expressions, we may put y=0, and thus write x in place of p. dp dp Then AW -=?/ same series dy In differentiating a second time we may treat p as identical with x, because y is to be put equal to zero. Thus . (15) (PW DUE TO THE WEIGHT OF CONTINENTS. 193 . . .] d'W = dxdy dydz Also treating p as identical with x, and putting y=0, . (16) dW =0 rlW z~ / dW , dW\ -r-+-r- 1 dx dy 0, I AW (2/-r \ y th dy n = -...} L . (18) These various results have now to be introduced into the expressions (11) for P, Q, E, S, T, U. In performing these operations it will be convenient to write J for i(i-\-2)/(i 1). Also r 3 =/3 2 +2 2 =x 3 +2 2 , when y=Q. From these formulas we see that S=0, U=0; which shows that a meridional plane is one of the three principal planes, a result already observed from principles of symmetry. Now (19) - (t !- 8) j8 4 =^'- (i- (20) MDCCCLXXXII, 194 MR. G. H. DARWIN ON THE STRESSES z 6 - } (21) Then multiplying (19) by (t+3), (20) by 3, and (15) by Jo 2 , and adding them each to each, we get the expressions for P, Q, R. Also multiplying (21) by (t+3), (18) by 3, and (16) by Jo? and adding, we get the expression for T. The results are 7P= - The general law of formation of the successive coefficients is obvious, and it is easy to write down the general term in each of the eight series involved in these four expressions ; the best way indeed of obtaining the formulas given below is to write down and transform the general term. The semi-polar coordinates used hitherto are not so convenient as true polar coordinates; I therefore substitute r, radius vector, and /, latitude, for the x, z system, and putting x=r cos I, z=r sin / write DUE TO THE WEIGHT OF CONTINENTS. 195 ' =?* cos '/(4,+4j tan 2 l+A tan *Z+ . . .) K=r''cos'7( +.D S tan 2 Z+D 4 tan *Z+ . . .) sin Z cos '- 3 Z(F +F 2 tan 2 Z+^ 4 tan Q=r"' cos { l(G +G. 2 tan ^+^ tan 4 Z+ . . .) +aV~ 2 cos '- 2 Z#"+ JI tan s ^+ JI tan Then introducing for J and for the fi's their values in terms of i, I find that the coefficients A, B, &c., are reducible to the forms given in the following equations : &c.=&c. ^'(t-2) . !" -(* 4 )( 5 ) &c.=&c. (23) &c.=&c. t-l 0! t ^- ' &c.=&c. (24) 2 c 2 196 MR. G. H. DARWIN ON THE STRESSES 1 i* 3) j I E a= *,(* 7! &c.=&c. IF a = -M. -*- O . -) (i-2)(i-4) i-1 3! i-1 5! i-1 &c.=&c. 7! (25) 4! &c.=&c. (26) &c.=&c. These sets of coefficients are all written down in such a form that the laws of their formation are obvious, and the general terms may easily be found. I have computed their values from these formulas for the even zonal harmonics of orders 2, 4, 6, 8, 10, 1 2 ; the results are given in the following tables both in the form of fractions and of decimals approximately equal to those fractions. The Cfs and H's were not computed because their values were not required for subsequent operations. DUE TO THE WEIGHT OF CONTINENTS. 197 TABLE I. The coefficients for expressing the stress P. i A A, A 4, -Bo B 2 5 4 -Be 2 -if -8421 - -1-1579 +H + 8421 4 -tf -1-8824 + + 5490 +tt + 2-8235 + + 1-8824 128 5 1 -2-5098 6 3 2 1T -2-9091 + 18 + 18-0000 +w + 167273 -w- -4-9455 + + 2-9091 J.1S2 6 -20-9455 +/ +4-6545 8 640 163 -3-9264 + -4H* + 63-3620 -w -12-9571 "~~ i tTs -74-6012 +m + 3-9264 _,^_ -67-3094 + - 9 T ? i ! A - + 80-7713 -H*i -7-1797 400 10 -4-9383 + A4i|0 + 148-6831 _Aj||a -284-7737 'Mmr 1 -230-4527 +w + 4-9383 -"iW 4 -153-6352 +-s-yyv La + 438-9561 gi aop -210-6996 12 -HI -5-9469 +*Hi* + 285-9823 JL3 8 OO 11? -1221-2389 +*MF + 212-3894 +tti + 5-9469 -HMF -291-9389 H-^ftW 4 + 1513-7570 -"iVW* -17300080 TABLE II. The coefficients for expressing the stress R i G Co &< ^6 -Do A ^ 6 2 + H + 1-3684 -Li2. I 10 + 1-6842 - -1-6842 4 +W + 2-4314 -VT 1 - -2-1961 -w -5-0196 -W -2-5098 +w + 5-0196 6 + + 3-4545 -24 -24-0000 -W -18-9091 +w + 9-3091 _1_9_1 & 5 -3-4909 +-4F + 27-9273 -w -9-3091 8 + 148 16 3 + 4-4663 - i !fl 4 -75-7791 +w + 25-9141 + i ttf 4 + 93-6049 5 1 2 O "1 "1 i T -4-4873 L g 2 1 6 Q. ^^ 1141 + 80-7713 -4F -107-6950 +W& + 14-3593 10 4. 13 3_0 < 243 + 5-4733 41200 2 t 3 -169-5473 +^MF + 348-9712 +-5^ + 247-5720 000 72 9 -5-4870 fj-vym* + 175-5830 3 B 4 OOP " 7 2 U -526-7490 +^vw ia + 280-9328 12 + HI + 6-4779 3 (5 8 5 n rT3~ -317-3097 + i r/ a + 1401-7700 -i^a -339-8230 806* Tlf3' -6-4875 + A 5fi-:-s a + 324-3765 a i B o 4 oo 1 243 -1730-0080 + i WW i + 2076-0097 198 MB. G. H. DARWIN ON THE STRESSES TABLE III. The coefficients for expressing the stress T. i *, ** ^ ^6 r 6 * F* 'a 2 +A + 3158 4 + W + 5-0980 +*? + 4-7059 -w - 5-0196 6 + 14 + 14-0000 - -5-0909 -*H* -18-3273 -w -13-9636 +-4F + 18-6182 8 +w + 26-9448 1 8 163 -81-2761 i o s 2_a. IDS -63-6074 + i MH A +42-8088 -v,v, n -26-9238 +-4HF + 107-6950 -*AW -43-0780 10 +-4P +43-9095 14800 -307-8189 +H^ + 72-4280 +myp + 339-3769 isoo.o_ -43-8958 +-MrV

+ 865-0040 -"tf&* -2076-0097 + 1 4 7 4 S 6 Ta~*T + 1186-2912 If W be a 2nd, 4th, or 6th harmonic these tables give the complete expressions for P, E, and T ; if W be an 8th harmonic the only further coefficients required are A a and (7 8 . For the cases of the 10th and 12th harmonics the values in the tables are sufficient to give the stresses approximately over a wide equatorial belt, because the series for P, R, T proceed by powers of the tangent of the latitude, and the omitted terms involve high powers of that tangent. It would hardly be safe however to apply the formula at least as regards the 12th harmonic for latitudes greater than 15, because the coefficients are large. 3. On the direction and magnitude of the principal stresses in a strained elastic solid. Let P, Q, R, S, T, U specify the stresses in a homogeneously stressed and strained elastic solid. Let ?, m, n be the direction cosines of a principal stress axis. The consideration, that at the extremity of a principal axis the normal to the stress quadric is coincident with the radius vector, gives the equations (P-X)Z+Um+Tn=0 TZ+Sm+(R X)n=0 These equations lead to the discriminating cubic for the determination of X, and the solution for I, m, n is then n- (Q-X)(R-X)-S a ~~(P-X)(R-\)-T 2 ~~(P-\)(Q-X)-U s DUE TO THE WEIGHT OF CONTINENTS. 199 In the case considered in the preceding sections S and U vanish, and the cubic reduces to the quadratic (P-X)(R-X)-T 2 =0 of which the solution is m is obviously zero and I, n are determinable from Let Then it is easily proved that (27) This equation gives the directions of the principal stress-axes. The two principal stresses N I} N 3 are the two values of X, so that and the third principal stress, which we suppose intermediate in value between N x and N 3 , is of course Q. When an elastic solid is in a state of stress it is supposed, in all probability with justice, that the tendency of the solid to rupture at any point is to be estimated by the form of the stress quadric. At any rate the hypothesis is here adopted that the tendency to break is to be estimated by the difference between the greatest and least principal stresses. For the sake of brevity I shall refer to the difference between the greatest and least principal stresses as "the stress-difference," This quantity I shall find it convenient to indicate by A . We may also look at the subject from another point of view: It is a well-known theorem in the theory of elastic solids that the greatest shearing stress at any point is equal to a half of the stress-difference. It is difficult to conceive any mode in which an elastic solid can rupture except by shearing, and hence it appears that the greatest shearing stress is a proper measure of the tendency to break. This measure of ten- dency to break is exactly one-half of the stress-difference, and it is therefore a matter of indifference whether we take greatest shearing stress or stress-difference. For the sake of comparison with experimental results as to the stresses under which wires and rods of various materials will break and crush, I have found it more convenient to use stress-difference throughout; but the results may all be reduced to shearing stresses by merely halving the numbers given. From (28) we have then A = v/(P-R) 2 +4T 3 ......... (29) and the greatest shearing stress at the same point is ^ A . 200 MR. G. H. JJARWIN ON THE STRESSES 4. The application of previous analysis to the determination of the stresses produced by the weight of superficial inequalities. I have in a previous paper shown how Sir WILLIAM THOMSON'S solution for the state of internal strain of an elastic sphere subject to bodily forces, but not acted on by any surface forces, is to be adapted to the case of a spheroid (whose small inequalities are expressed as surface harmonics) of homogeneous elastic matter, endued with the power of mutual gravitation.* THOMSON'S solution is of course directly applicable for finding the state of strain due to a true external force, such as the tide- generating influence of the moon, but this forms only a part of the complete solution when the sphere has the power of gravitation. He introduced the effects of gravita- tion synthetically, but for my own purposes I prefer the analytical method pursued in my paper above referred to. Suppose that r=a-\-(n be the equation to an harmonic spheroid of the i th order, forming inequalities on the surface of the sphere, whose density is w. Then the causes producing a state of stress and strain in the mean sphere of radius a are, first a normal traction per unit area of the surface of the sphere equal to gwcr,, when g is the value of gravity, and secondly the attraction of the inequalities cr,, acting throughout the whole sphere. The first of these causes (viz. : the weight of the mountains or continents) is shown in my paper to produce the same state of strain as would be produced in the sphei'e, now free from surface action, by bodily forces corresponding with a potential As regards the second of these causes (viz. : the attraction of the mountains or continents), the potential of the layer of matter cr, on any internal point, estimated per unit volume, is 3<7Ztf(>'/a)' respectively. To apply this to the case of the earth, take a=G37x 10 c.m., and w=5'66, and we find the surface and central stress-differences to be respectively 152e and 1214e metric tonnes per square centimeter. If these numbers be multiplied by 6 '34, we get the same result expressed in tons per square inch. Thus in British units these two stress-differences are 926e and 7698e. 2 D 2 204 MR, G. H. DARWIN ON THE STRESSES If then the ellipticity e be yuWth, the surface and central stress-differences will be nearly 1 ton and nearly 8 tons to the square inch respectively. From the Table VII. in 9 it will appear that cast brass ruptures with a stress - difference of about 8 tons to the square inch. Thus a spheroid, made of material as strong as brass, and of the same dimensions and density as the earth, would only just support an excess or deficiency of ellipticity equal to TFoT)th, above or below the equilibrium ellipticity adapted for its rotation. The following is a second example : If the homogeneous earth (with ellipticity j^) were to stop rotating, the stress-difference at the centre would be 33 tons per square inch. Now suppose the cause of internal stress to be the moon's tide-generating influence, and let m= moon's mass, and c= moon's distance. Then the potential under which the earth is stressed is f (i/t- 3 )(-j cos 2 0)wr, or according to the notation of 4 ^(m/c 3 )wr 2 <;. 2 . If we took into account the elastic yielding of the earth and the weight and attraction of the tidal protuberance, this potential would have to be diminished. To estimate the diminution we must of course know the amount of elastic yielding, but as there is no means of approximating thereto, it will be left out of account. Then it is obvious that the factor by which A, as given in (35), must be multiplied in order to give the stress-difference is fynw/c*. Thus the surface stress-difference is /] ; this expression may be written ^(m/c 3 )^^ cos 2 0) + |( w V c3 ) r ' 3sni2 ^ cos ty- The former of these terms produces a permanent increase of the earth's ellipticity, and is confused and lost in the ellipticity due to terrestrial rotation, and can produce no stress in the earth. The second term is the true tide-generating potential, but it is a sectorial harmonic, and I have failed to treat such cases. Now the first of these terms causes ellipticity in a homogeneous earth equal to (fa/#)(fw/c 3 ) according to the equilibrium tide-theory. This ellipticity is equal to '1039 X 10~ 6 , an excessively small quantity. If however this permanent ellipticity does not exist (and the above investigation in reality presumes it not to exist), then there will be a superficial stress-difference equal DUE TO THE WEIGHT OF CONTINENTS. 205 to 152 X '1039 X 10~ metric tonnes per square centimeter, and a central stress- difference of eight times as much. Since a metric tonne is a million grammes this surface stress-difference is 16 grammes, and the central 128 grammes per square centimeter. These stress -differences are exactly the halves of those which have been computed above. Thus the remaining stress-difference which is due to the moon's tide-generating influence is 16 grammes at the surface and 128 grammes at the centre per square centimeter. A flaw in this reasoning is that stress-difference is a non-linear function of the sti'esses, and therefore the stress-difference arising from the sum of two sets of bodily stresses is not the sum of their separate stress-differences. I conceive however that the above conclusion is not likely to be much wrong. These stresses are very small compared with those arising from the weights of mountains and continents as computed below, nevertheless they are so considerable that we can understand the enormous rigidity which Sir WILLIAM THOMSON has shown that the earth must possess in order to resist considerable tidal deformations of its mass. 6. On the stresses due to a series of parallel mountain chains. Having considered the case of the second harmonic, I now pass to the other extreme and suppose the order of harmonics i to be infinitely great, whilst the radius of the sphere is also infinitely great. The equatorial belt now becomes infinitely wide, and the surface inequalities consist of a number of parallel simple harmonic mountains and valleys. If i be infinitely large, we have from (12) Now let be the depth below the surface of the point indicated in the sphere (now infinitely large) by x, y, z. As the formulas given above apply to the meridional plane for which y=0, we have p=a Now let b=aji, then when both i and a are infinite p'=a' 1 =a' and since in the limit /o/i=a/t=6, cos* This expression for W involves the infinite factor a', and in order to get rid of it we 200 MR. G. H. DARWIN OX THE STRESSES must now consider the factor by which it is to be multiplied, in introducing the height of the mountains and gravity. This factor is computed in 4 ; it is there shown that if r=a-j-fo, be a harmonic spheroid, the factor is 2(i' l)gwh/(2i+l)a'. Now if the harmonic i be of an infinitely high order, 9, becomes simply cos z b, and the equation to the surface is =h cos 7 b % being measured downwards. Thus the harmonic spheroid /is,- now represents a series of parallel harmonic mountains and valleys of height and depth /;, and wave- length 2irb. The factor becomes givh/a', when i is infinite. Thus the effective disturbing potential W, which is competent to produce the same state of stress and strain as the weight of the mountains and valleys, is given by W= -gwht-t'" cos z - (36) Now revert to the expressions (11) for the stresses. When i is infinite 7=2i 2 , and they become, on changing x into ( ) Now as shown above a 2 r 2 =2a and a/i=b in the limit; making these substitu- tions, and dropping the terms which become infinitely small when i is infinite, we have and by a similar process */x M /' Q = Then from (36) and (37) we have P __ /TJ/Tn ~*' f*(Vi ^^ (7 Wlb, OUo , (37) R= gwh* as the two moduli, which define the nature of the elastic solid. In the denominators in E- H F,, GI of (1), the expression (2(i+l)~+l)w (2t occurs, this I shall call K, in analogy with /. Then K=2i(i l)co-f 3(2i If we develop the last differential coefficient in the expression (1) for a, we find 2vKa= 3k(^-a 2 - i*Y^+ la [(a 2 - r 2 )^+ 2x w] (44) \i 1 / ax ' aj; Also K8=iW ........... (45) and u|' '' ........ Differentiating (44) with regard to x, and substituting in (46) we find ' / 2 *\* W t w (a 2 -r 2 ) . . . (47) Again, differentiating (44) with regard to z, writing down d-y/dx by symmetry, and adding the two together, we have from (46) * dW dW\ . , 2 *W . z-. -- x \+iit would be just on the point of rupture. Again if the homogeneous earth, with ellipticity -g-g-g-, were to stop rotating, the central stress-difference would be 33 tons per square inch, and it would rupture if made of any material excepting the finest steel. DUE TO THE WEIGHT OF CONTINENTS. 223 A rough calculation""' will show that if the planet Mars has ellipticity -- Q - (about twice the ellipticity on the hypothesis of homogeneity) the central stress-difference must be 6 tons per square inch. It was formerly supposed that the ellipticity of the planet was even greater than -g^, and even if the latest telescopic evidence had not been adverse to such a conclusion, we should feel bound to regard such supposed ellipticity with the greatest suspicion, in the face of the result just stated. The state of internal stress of an elastic sphere under tide-generating forces is identical with that caused by ellipticity of figure.t Hence the investigation of 5 gives the distribution of stress-difference caused in the earth by the moon's attraction. In Plate 19, fig. 1, the point called " the pole" is the point where the moon is in the zenith. Computation shows that the stress-difference at the surface, due to the lunar tide- generating forces, is 16 grammes per square centimeter, and at the centre eight times as much. These stresses are considerable, although very small compared with those due to terrestrial inequalities, as will appear below. In 6 the stresses produced by harmonic inequalities of high orders are considered. This is in effect the case of a series of parallel mountains and valleys, corrugating a mean level surface with an infinite series of parallel ridges and furrows. In this case compressibility makes absolutely no difference in the result, as shown in 1 0. It is found that the stress-difference depends only on the depth below the mean surface, and is independent of the position of the point considered with regard to ridge and furrow ; the direction of the stresses does however depend on this latter consideration. In Plate 19, fig. 2, is shown the law by which the stress-difference increases and then diminishes as we go below the surface. The vertical ordinates of the curve indicate the relative magnitude of the stress-difference, and the horizontal ones the depth below the surface. The depth OL on the figure is equal to the distance between contiguous ridges, and the figure shows that the stress-difference is greatest at a depth equal to -g^rs- of OL. The greatest stress-difference depends merely on the height and density of the mountains, and the depth at which it is reached merely on the distance from ridge to ridge. Numerical calculation shows that if we suppose a series of mountains, whose crests are 4000 meters or about 13,000 feet above the intermediate valley- bottoms, formed of rock of specific gravity 2 '8, then the maximum stress-difference is 2 '6 tons per square inch (about the tenacity of cast tin) ; also if the mountain chains are 314 miles apart the maximum stress-difference is reached at 50 miles below the mean surface. * The data for the calculation are : Ratio of terrestrial radius to Martian radius 1'878. Ratio of Martian mass to terrestrial mass '1020. Whence ratio of Martian gravity to terrestrial gravity '3590. Central stress-difference, due to ellipticity e, 996e tons per square inch. "Homogeneous" ellipticity of Mars r b ; and ||-f equal to C. t This is subject to certain qualiKcatious noticed iu 5. 224 MR. G. H. DARWJN ON THE STRESSES It may be necessary to warn the geologist that this investigation is approximate in a certain sense, for the results do not give the state of stress actually within the mountain prominences or near the surface in the valley -bottoms. The solution will however be very nearly accurate at some five or six miles below the valley-bottoms. The solution shows that the stress-difference is nil at the mean surface, but it is obvious that both the mountain masses and the valley-bottoms are in some state of stress. The mathematician will easily see that this imperfection arises, because the problem really treated is that of an infinite elastic plane, subjected to simple harmonic tractions and pressures. To find the state of stress actually within the mountain masses would probably be difficult. The maximum stress-difference just found for the mountains and valleys obviously cannot be so great as that at the base of a vertical column of this rock, which has a section of a square inch and is 4000 meters high. The weight of such a column is 7'1 tons, and therefore the stress- difference at the base would be 7'1 tons per square inch. The maximum, stress-difference computed above is 2'G, which is about three-eighths of 7'1 tons per square inch. Thus the support of the contiguous masses of rock, in the case just considered, serves as a relief to the rock to the extent of about five-eighths of the greatest possible stress-difference. This computation also gives a rough estimate of the stress-differences which must exist if the crust of the earth be thin. It is shown below that there is reason to suppose that the height from the crest to the bottom of the depression in such large undulations as those formed by Africa and America is about 6000 meters. The weight of a similar column 6000 meters high is nearly 11 tons. In 7 I take the cases of the even zonal harmonics from the 2nd to the 12th, but for all except the 2nd harmonic only the equatorial region of the sphere is considered. Plate 19, fig. 3, shows an exaggerated outline of the equatorial portion of the inequali- ties ; it only extends far enough to show half of the most southerly depression, even for the 12th harmonic. It did not seem worth while to trace the surfaces of equal stress- difference throughout the spheroid, but the laborious computations are carried far enough to show that these surfacs must be approximately parallel to the surface of the mean sphere. It is accordingly sufficient to find the law for the variation of stress- difference immediately underneath the equatorial belt of elevation. It requires com- paratively little computation to obtain the results numerically, and the results of the computation are exhibited graphically in Plate 20, fig. 4. Table V. (b), 7, gives the maximum stress-differences, resulting from these several inequalities, computed under conditions adequately noted in the table itself. It will be convenient to postpone the discussion of the results. In 8 I build up out of these six harmonics an isolated equatorial continent. The nature of the elevation is exhibited in Plate 20, fig. 5, in the curve marked " represen- tation ;" no notice need be now taken of the dotted curve. This curve exhibits a belt of elevation of about 15 of latitude in semi-breadth, and the rest of the spheroid is DUE TO THE WEIGHT OF CONTINENTS. 225 approximately spherical. This kind of elevation requires the 2nd as one of its harmonic constituents, and this harmonic means ellipticity of the whole globe. Now it may perhaps be fairly contended that on the earth we have no such continent as would require a perceptible 2nd harmonic constituent. I therefore give in Plate 20, fig. 5, a second curve which represents an equatorial belt of elevation counterbalanced by a pair of polar continents in such a manner that there is no second harmonic constituent. I have not attempted to trace the curves of equal stress-difference arising from these two kinds of elevation, but I believe that they will consist of a series of much elongated ovals, whose longer sides are approximately parallel with the surface of the globe, drawn about the maximum point in the interior of the sphere at the equator. The surfaces of equal stress-difference in the solid figure will thus be a number of flattened tubular surfaces one within the other. At the equator however the law of variation of stress-difference is easy to evaluate, and Plate 20, fig. 6, shows the results graphically, the vertical ordinates representing stress-difference and the horizontal the depths below the surface. The upper curve in Plate 20, fig. 6, corresponds with the "representation curve" of Plate 20, fig. 5, and the lower curve with the case where there is no 2nd harmonic constituent. The central stress-difference, which may be observed in the upper curve, results entirely from the presence of the 2nd harmonic constituent in the corresponding equatorial belt of elevation. The maximum stress-differences in these two cases occur at about 660 and 590 miles from the surface respectively. We now come to perhaps the most difficult question with regard to the whole subject namely, how to apply these results most justly to the case of the earth. The question to a great extent turns on the magnitude and extent of the superficial inequalities in the earth. As the investigation deals with the larger inequalities, it will be proper to suppose the more accentuated features of ridges, peaks, and holes to be smoothed out. The stresses caused in the earth by deficiency of matter over the sea beds are the same as though the seas were replaced by a layer of rock, having everywhere a thickness of about ~? * or nearly ~ of the actual depths of sea. The surface being partially smoothed and dried in this manner, we require to find an ellipsoid of revolution which shall intersect the corrugations in such a manner that the total volume above it shall be equal to the total volume below it. Such a spheroid may be assumed to be the figure of equilibrium appropriate to the earth's diurnal rotation ; if it departs from the equilibrium form by even a little, then we shall much underestimate the stress in the earth's interior by supposing it to be a form of equilibrium. Professor BRUNS has introduced the term " geoid " to express any one of the " level " surfaces in the neighbourhood of the earth's surface, and he endeavours to form an estimate of the departure of the continental masses and sea-bottoms from some mean MDCCCLXXXII. 2 G 226 MR. G. H. DARWIN" ON THE STRESSES geoid.* From the geodesic point of view the conception is valuable, but such an estimate is scarcely what we require in the present case. The mean geoid itself will necessarily partake of the contortions of the solid earth's surface, even apart from disturbances caused by local incqxialities of density, and thus it cannot be a figure of equilibrium. Thus, even if we were to suppose that the solid earth were everywhere coincident with a geoid which is far from being the case a state of stress would still be produced in the interior of the earth. An example of this sort of consideration is afforded by the geodesic results arrived at by Colonel CLARKE, R.E.,t who finds that the ellipsoid which best satisfies geodesic measurement, has three unequal axes, and that one equatorial semi-axis is 1524 feet longer than the other. Now such an ellipsoid as this, although not exactly one of BRUNS' geoids, must be more nearly so than any spheroid of revolution ; and yet this inequality (if really existent, and Colonel CLARKE'S own words do not express any very great confidence) must produce stress in the earth. Colonel CLARKE'S results show an ellipticity of the equator equal to yg-ysr, and this in the homogeneous elastic earth will be about equivalent to ellipticity 37000 ; such ellipticity would produce a central stress-difference of - 2 ^nfo% or nearly one-third of a British ton per square inch. From this discussion it may, I think, be fairly concluded that if we assume the sea- level as being the figure of equilibrium and estimate the departures therefrom, we shall be well within the mark. The average height of the continents is about 350 meters (1150 feet), and the average depth of the great oceans is in round numbers 5000 meters (16,000 feet) ; but the latter datum is open to much uncertainty.! When the sea is solidified into rock the 5000 meters of depth is reduced to 3200 meters below the actual sea-level. Thus the average effective depression of sea-bed is about nine times as great as the average height of the land. I shall take it as exactly nine times as great, and put the depth as 3150 meters; but it is of course to be admitted that perhaps eight and perhaps ten might be more correct factors. In the analytical investigation of this paper the outlines of the vertical section of the continents and depressions are always sweeping curves of the harmonic type, and the magnitude of the elevations and depressions are estimated by the greatest heights and depths, measured from a mean surface which equally divides the two. We have already supposed the outlines of continents and sea-beds to have been smoothed down into sweeping curves, which we may take as being, roughly speaking, of the harmonic type. The smoothing will have left the averages unaffected. * ' Die Figur der Erde.' Von Dr. H. BRUNS. Berlin : STANKIEWICZ, 1878. t Phil. Mag., Aug., 1878. J In a previous paper, " Geological Changes, Ac.," Phil. Trans., Vol. 1C7, Part I., p. 295, I have endeavoured to discuss this subject, and references to a few authorities will be found there. DUE TO THE WEIGHT OP CONTINENTS. 227 The averages are not however estimated from a mean spheroidal surface, but from one which is far distant from the mean. The questions now to be determined are as follows : What is the proper greatest height and depression, estimated from a mean spheroid, which will bring out the above averages estimated from present sea-level, and what is the position of the mean spheroid with reference to the sea-level. From the solution of the problem considered in the note below,* it appears that, if * Conceive a series of straight harmonic undulations corrugating a mean horizontal surface, and suppose them to be flooded with water. This will represent fairly well the undulations on the dried earth, and the water-level will represent the sea-level. Suppose that the average heights and depths of the parts above and below water are known, and that it is required to find the position of the mean horizontal surface with reference to the water-level, and the height of the undulations measured from that mean surface. . Take an origin of coordinates in the water-level, the axis of x in the water-level and perpendicular to the undulations, and the axis of ?/ measured upwards. Let ?/=7((cos :r cos a) be the equation to the undulations. 1 r +a h The average height of the dry parts is clearly ^~\ ydx or - (sin a iicosa). Similarly the average &(l'J d ft depth below water is - [sin (TT a) (ir a) cos (if )"| or sin a+ (TT a) cos a 7T L ' J 7T a If the latter average be p times as great as the former pJt cos o( - tan a 1 )= h cos I tan a + I } \a } V a / This is an equation for determining a. Now I find that a = 34 30' gives p=8'98'3, which corresponds rery nearly with^c=9 of the text above. This value of a corresponds with an average equal to 'H6oh for the height above water, and T0469A for the depth below water. Now if we put 1-0469 A =31 50 meters which gives '1165/f = 350 meters very nearly, we have ^=3009 meters. The depth below water-level of the mean level is 7* cos 34" 30' or 2480 meters. The greatest height of the dry part above the water-level is 30092480 or 429 meters, and the greatest depth of the submerged part below water-level is 3009 + 2480 or 5489 meters. [After the proof-sheets of this paper had been corrected, Professor STOKES pointed out to me that, according to RIGAUD (Cam. Phil. Soc., vol. 6), the area of land is about four-fifteenths of the whole area of the earth's surface. Now, in the ideal undulations we are here considering the area above water is about one-tenth of the whole area ; hence in this respect the analogy is not satisfactory between these undulations and the terrestrial continents. If I have not considerably over-estimated the average depth of the sea (and I do not think that I have done so), the discrepancy must arise from the fact that actual continents and sea-beds do not present in section curves which conform to the harmonic type; there must also be a difference between corrugated spherical and plane surfaces. The geological denudation of the land must, to some extent, render our continents flat-topped. Add^d May 4/A, 1882.] 2 u 2 228 MR. G. H. DARWIN ON THE STRESSES the continents and sea-beds have sections which are harmonic curves, then if we take, The mean level bisecting elevations and depressions as 2480 meters (8150 feet) below the sea-level, and the greatest elevation and depression from that mean level as 3009 meters (9840 feet), it results that the average height of the land above sea-level is 350 meters and the average depression of dried sea-bed is 3150 meters. It thus appears that 3000 meters would be a proper greatest elevation and de- pression to assume for the harmonic analysis of this paper, if the earth were homogeneous. But as the density of superficial rocks is only a half of the mean density of the earth, I shall take 1500 meters as the greatest elevation and depression from the mean equilibrium spheroid of revolution. It is proper here to note that the height of the undulations of elevation and depres- sion in the zonal harmonic inequalities is considerably greater towards the poles than it is about the equator f it might therefore be maintained that by making 1500 meters the equatorial height, we are taking too high an estimate. But the state of stress caused in the sphere at any point depends very much more on the height of the inequality in the neighbourhood of a superficial point immediately over the point considered, than it does on the inequalities in remote parts of the sphere. Now in all the inequalities, except the 2nd harmonic, I have considered the state of stress in the equatorial region, and it will therefore I think be proper to adhere to the 1 500 meters for the greatest height and depression. We have next to consider, what order of harmonic inequalities is most nearly analogous to the great terrestrial continents and oceans. The most obvious case to take is that of the two Americas and Africa with Europe. The average longitude of the Americas is between 60 and 80 W., and the average longitude of Africa is about 25 E., hence there is a difference of longitude of about a right-angle between the two masses. These two great continents would be more nearly represented by an harmonic of the sectorial class,* rather than by a zonal harmonic, nevertheless I think the solution for the zonal harmonic will be adequate for the present purpose. Now it has been explained above that the harmonic of the fourth order represents an equatorial continent and a pair of polar continents. In the case of the 4th harmonic therefore there is a right angle of a great circle between contiguous con- tinents. We may conclude from this that the large terrestrial inequalities are about equivalent to the harmonic of the fourth order. Table V. (b), 7, gives the maximum stress- differences under the centre of the equa- torial elevation of the several zonal harmonics, the height of each being 1500 meters. * The sectorial harmonic of the fourth order sin 4 cos 40 would well represent these two great con. tinents. It would fairly represent China and Australia; but would annihilate the Himalayan plateau, and place another great continent in mid-Pacific. It is not at all difficult to find the stress-dill, nim under the centre of a sectorial inequality, but to find it generally involves the solution of a cubic equation. DUE TO THE WEIGHT OF CONTINENTS. 229 The point at which this maximum is reached is given in each case, and Plate 20, fig. 4, illustrates graphically the law of variation of stress-difference. The second harmonic cannot be said to represent a continent, and the table shows that in each of the other cases the maximum stress-difference is very nearly 4 tons per square inch. The depths of the maximum point are of course very different in each case. We have concluded above that Africa and America are about equivalent to an harmonic of the fourth order, hence it may be concluded that the stress-difference under those continents is at a maximum at more than 1100 miles from the earth's surface, and there amounts to about 4 tons per square inch. A comparison with Table VII. shows that marble would break under this stress, but that strong granite would stand. The case of the isolated continent investigated in 8 appeared likely to prove the most interesting one, for the purpose of application to the case of the earth. But unfortunately I have found it difficult to arrive at a satisfactory conclusion as to the proper height to attribute to the continent. The average height of the American continent is about 1100 feet above the sea, and the average depth of the Pacific Ocean about 15,000 feet. If the water of the Pacific be congealed into rock, it will have an effective depth of 1 0,000 feet. The greatest height of the American continent above the bed of the dried Pacific when smoothed down must be fully 12,000 feet or 3700 meters. The height of the great central Asian plateau above the average bed of the southern ocean (after drying) must be considerably more than this. Now in the application to the homogeneous planet the heights are to be halved to allow for the smaller density of surface rock. I therefore take 2000 meters as the height of the top of the equatorial table-land above the remaining approximately spherical portion of the sphere. The investigation of 8 then shows that the equatorial table-land will give rise to a stress-difference of 4 - l tons per square inch at a depth of 660 miles ; and that the equatorial table-land counterbalanced by the pair of polar continents (the second harmonic constituent being absent) gives a stress difference of about 3 '8 tons per square inch at a depth of 590 miles. This estimate of stress-difference agrees in amount, with singular exactness, with that just found from the case of the 4th zonal harmonic, but the maximum is reached 400 or 500 miles nearer to the earth's surface. I think there can be no doubt but that there are terrestrial inequalities of much greater breadth than that of my isolated continent ; thus this investigation for the isolated continent will give a position for the maximum stress-difference too near the surface to correspond with the largest continents. On the other hand, I do not feel at all sure that I have not considerably underestimated the height of such a compara- tively narrow plateau. 230 ON THE STRESSES DUE TO THE WEIGHT OF CONTINENTS. In the present paper it has been impossible to take any notice of the stresses pro- duced by the most fundamental inequality on the earth's surface, because it depends essentially on heterogeneity of density. It is well known that the earth may be divided into two hemispheres, one of which consists almost entirely of land, and the other of sea. If the south of England be taken as the pole of a hemisphere, it will be found that almost the whole of the land, excepting Australia, lies hi that hemisphere, whilst the antipodal hemisphere consists almost entirely of sea. This proves that the centre of gravity of the earth's mass is more remote from England, than the centre of figure of the solid globe. A deformation of this kind is expressed by a surface harmonic of the first order, for such an harmonic is equivalent to a small displacement of the sphere as a whole, with- out true deformation. Now if we consider the surface forces produced by such a deformation in a homogeneous sphei'e, we find of course that there is an unbalanced resultant force acting on the whole sphere in the direction diametrically opposed to that of the equivalent displacement of the whole sphere. The fact that in the homogeneous sphere such an unbalanced force exists shows that in this case the problem is meaningless ; it is in fact merely equivalent to a mischoice in the origin for the coordinates. But in the case of the earth such an inequality does exist, and the force referred to must of course be counterbalanced somehow. The balance can only be maintained by inequalities of density, which are necessarily unknown. The problem therefore apparently eludes mathematical treatment. It is certain that so wide-spreading an inequality, even if not great in amount, must produce great stress within the globe. And just as the 2nd harmonic produces a more even distribution of stress than the 4th, so it is likely that the first would produ.ce a more even distribution than the 2nd. It is difficult to avoid the conclusion that the whole of the solid portion of the earth is in a sensible state of stress. I would not however lay very much emphasis on this point, because we are in such complete ignorance as to the manner in which the equilibrium of the solid part of the earth is maintained. From this discussion it appears that if the earth be solid throughout, then at a thousand miles from the surface the material must be as strong as granite. If it be fluid or gaseous inside, and the crust a thousand miles thick that crust must be stronger than granite, and if only two or three hundred miles in thickness much stronger than granite. This conclusion is obviously strongly confirmatory of Sir WILLIAM THOMSON'S view that the earth is solid throughout. JM. 7/V///.S. 1882. Fiq. 1 Z) iA*q ram- sliowinof curves oC fa n-af' si re ss - difftf&ice du* to the. weiafhts of 2"*^ harmonic in-ea ti&lities or to ticie - gentratisuf force . cufn shoWLna the 'profile, of tlte eve,n- h the- 22 yuuxtar . RouLUjus of splLere 18 2 . f roun, s h o wi ti cj lh* difference between^ the. prLttcifyaul sires s e-& due, to vallies on, a. KorucontcLi i i Da Phil. 7>v///,s. 1882. Plate 2Q. Via. 4-. -Z-5 -2-0 -1-5 -1-0 I Cr.nirr. or aphtn Diagram tkayring the difference of the, principal stresses at in, ''('"''" ''" (o in-etfimiities represented* lay the even zonal harmonica. 0> . 1 p/ f ' / I/ ^ ?/i? 4 Ji e Is 1 i- - .; r : -3' H H ' *1 3; ? 8 ll) 1 - 1 E cf,u K/I.'S Fiq. 6. showing the difference of principal, stresses ctfc the due to isolated; ei^uaioriaJi conUjients. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. ASTRONOMY LIBRARY LD 21-100m-ll,'49(B7146sl6)47i 706318 3 UNIVERSITY OF CALIFORNIA LIBRARY - ! : !