QB 633 C4 UC-NRLF in THE EARTH'S MOTION OF ROTATION. Cambrttrge: PRINTER BY C. J. CLAY, M;A. AT THE UNIVERSITY PRESS. THE EARTH'S MOTION OF EOTATION INCLUDING THE THEORY OF PRECESSION AND NUTATION. BY C. H. H. CHEYNE, M.A. 8T JOHN'S COLLEGE, CAMBRIDGE, SECOND MATHEMATICAL MASTER IN WESTMINSTER SCHOOL, AUTHOR OF "AN ELEMENTARY TREATISE ON THE PLANETARY THEORY.' anfc MACMILLAN AND CO. 1867. [AU Rights reserved. ,] "-'' PKEFACE. IN offering to the student a treatment of the Problem of the Earth's rotation somewhat different from that which has been usually given in elementary text-books, a few words of explanation are necessary. The FIRST PART consists of an application of the method of the variation of elements to the general problem of rotation. That the formulae for calculating these elements are identical in the motions both of translation and rotation, appeared so remarkable, that it might be well to present the latter in a form easily accessible. As far as I am aware, an elementary investigation of these formulae has not yet been given : in attempting to supply this, I have adopted a method somewhat similar to that which I have given for the corresponding equations of the motion of translation, in an Elementary Treatise on the Planetary Theory. The striking analogy, thus developed, between the solutions of problems, in appearance so dissimilar, may, I hope, lead to a more complete study of Lagrange's beautiful theoiy of the varia- tion of arbitrary constants. In the SECOND PART the general rotation-formulae are applied to the particular case of the Earth. These formulae 382410 VI PREFACE. afford a simple and accurate proof of the important theorem of the Stability of the Earth's axis and of the motion about it, so far as these depend upon the attractions of distant bodies. In this I have followed M. de Ponte'coulant. The remaining pages are devoted to a consideration of the motion of the Earth's axis in space. In this I have obtained the formulae for calculating Precession and Nutation, first, by an application of the general method, and afterwards, by an independent process; but I have not carried the approxi- mation further than has been usual in elementary text- books. C. H. H. CHEYNE. i, DEAN'S YARD, WESTMINSTER, September, 1867. CONTENTS. PART I. GENERAL THEORY OP ROTATION. ART. PAGE 1. Object of this treatise 1 2. Reason of the smallness of the' effect upon the rotation of a planet of the attractions of other bodies ( '> Undisturbed Motion. 3. Integration of equations of motion 2 4. Signification of the constants of integration 3 5. Equations for determining the motion in space 4 6. The same when the invariable plane is the plane of reference 6 7. When the motion with reference to the invariable plane is known, to determine it with reference to any other plane 8 Disturbed Motion. 8. Method of treating the problem. Elements of the motion. Definition of plane of maximum areas 9 9. Expression for the sum of the moments of the disturbing forces about any line through the centre of gravity of the planet 10 10. Various ways of expressing the function V 11 11. Equation for calculating h 13 12. Analogy of this equation to that for calculating the mean distance in the Planetary Theory 14 13. Equations of motion 14. Comparison of these equations with those employed in Chapter II. of the Planetary Theory 16 15. Equations for calculating a, y and k 17 Vlll CONTENTS. ART. PAGE 16. Second method of obtaining the equation for calculating y ... 19 17. Relation between certain partial differential coefficients of l y ^ and0 l 20 1 8. Relation between partial differential coefficients of V 22 19. Equation for calculating I ... 24 20. Equation for calculating g 25 21. Recapitulation of formulae for calculating the elements 27 22. Comparison of these with the corresponding equations of the Planetary Theory PART II. APPLICATION OF PRECEDING RESULTS. PRECESSION AND NUTATION. 23. Application to the case of the Earth : method of treatment... 31 24. Stability of the Earth's axis, the disturbing force being neglected 32 25. The same, including the first power of the disturbing force... 33 26. Stability of the velocity 35 27. Effect of tidal friction 28. Poisson's equations for calculating 6 and ^ 37 29. Expression for V 30. Form assumed when BA 39 31. Moment of the disturbing couple due to the Sun's attraction 32. Expressions for 6 and ^ obtained from Poisson's formulae. 40 33. 34. The same obtained independently 42 35. Examination of these formulae. Solar Precession and Nuta- tion 45 36. Effect of the Moon's action on the motion of the Earth's axis with reference to the plane of the Moon's orbit 37. The same with reference to the ecliptic : values of 6 and -v/r... 47 38. Luni-solar Precession 50 39. Geometrical representation of the motion of the Earth's axis in consequence of Precession and Nutation 51 40. Annual Luni-solar Precession 52 41 . Effect of Planetary attraction inappreciable THE EARTH'S MOTION OF ROTATION, PART L GENERAL THEORY OF ROTATION* 1. IN that part of Physical Astronomy which usually goes by the name of the Planetary Theory we are con- cerned with the motions of translation only of the planets in space : we now propose to consider their motions of rota- tion. The principles of the conservation of the motions of translation and rotation permit us to consider these sepa- rately, and to treat the latter as if the centre of gravity were a fixed point. We shall adopt a method perfectly rigorous, and free from all assumptions, with the single exception of the hypothesis, already required in the Planetary Theory, that the attracting bodies are so distant that their action may be supposed the same as it would be if their whole masses were condensed into their centres of gravity. Thus we shall obtain, for the determination of the motion, for- mulae applicable to the case of any planet or other rigid body : an interesting application of these will then be afforded by the special circumstances which occur in the Earth's Motion of Rotation. 2. If the planets were exactly spherical in shape, it is clear that the attractions of the Sun, Moon, and of the other planets could produce no effect upon their rotation, since they would all pass through the centre of gravity. But c. 1 2 THE EARTH'S* MOTION OF ROTATION. although tins isTiiot ifce- .case, yet the deviation from exact sphericity being very small, the motion will differ only slightly from what it would be if these disturbing forces did not exist. We shall, therefore, by neglecting them obtain first an approximate solution of the problem, and then by the method of the variation of parameters deduce from it accurate results. Since, however, the motion of a rigid body about a fixed point under the action of no forces is dis- cussed in works on Rigid Dynamics, we shall here consider it only so far as is necessary for the purpose of obtaining re- sults which will be required in the sequel. Undisturbed Motion. 3. Let ojj, o) 2 , w 3 be the angular velocities of a planet about the principal axes at its centre of gravity ; A, B, C the moments of inertia about these axes : then Euler's equations give Multiplying these equations by e^, o> 2 , &> 3 respectively, adding, and integrating, we have where h is the constant of integration. GENERAL THEORY OF ROTATION. 3 Again, multiplying by ACO I} Ba) z , CCO B) adding, and inte- grating, we have where k* is the constant of integration. From these two equations we obtain (A-B}A 2 " (A-B)B Substituting these values, the third equation of motion becomes dl >J(AB) whence t + 1 ] _ where I is the constant of integration. This integral cannot in the general case be found; we may however approximate: thus t is known in terms of o> 3 , and consequently ew 3 in terms of t\ and then from above, o) lt a> a are also known. 4. With respect to the constants introduced by the inte- gration, we may remark that h represents the vis viva (Routh's Rigid Dynamics, Art. 194), and k the area conserved on the invariable plane. To prove the latter point, the areas conserved on the principal planes being AW I} Bco 2 , Co) 3 (Routh's R. D., Art 179), and the direction cosines of the invariable plane with reference to the principal axes A co Ba) Ca) 12 4 THE EARTH'S MOTION OF ROTATION. (Routh's R. D., Art. 125), the area conserved on the invari- able plane k k 5. When 2 , o> 3 are known at any time, the result- ant angular velocity of the planet is known, and also the position of the instantaneous axis of rotation with reference to the principal axes. It remains to shew how the position of these axes in space may be determined. Suppose a sphere described with its centre at the centre of gravity of the planet and its radius of any magnitude : take as a plane of reference any fixed plane passing through the centre of gravity, and let it cut the sphere in the great circle ON\ also let the principal plane of xy cut the sphere in the great circle NAB, N being the node of this plane upon the fixed plane, and A, B, G the points where the sphere is cut by the principal axes of x, y, z. Take P the pole of ON, and join PA, PB, GA, GB by arcs of great circles. C Let the angle ON A <= 0, ON= ^ , NA = $ : then if the GENERAL THEORY OF ROTATION. 5 angles 0, , ^ be known, the position of the planet will be determined with reference to the fixed plane. Now we may consider the planet to be moving with angu- lar .velocities a> lt o> 2 , < 8 about the principal axes; or with angular velocities -^ , -y- , -^ , the first about a normal to the fixed plane, the second about the line of nodes, the third about the principal axis of z. We shall adopt the usual convention with respect to signs, and consider positive those angular velocities which tend to turn the planet round the axes of x, y, z from y to z, z to oc, sc to y, respectively. Thus, resolving about the principal axes, we have M + -^- sin ^ sin ^>, . . w 2 = -- sin ^ + - sin ^ cos ^ THE EARTH'S MOTION OF ROTATION. Hence also, d-fr . n -~ sin 6 &)j sm + &> 2 cos fa d6 ~j- = Wj cos

2 , &> 3 obtained as above (Art 3), and then integrating these equations, 0, fa and ty would be determined, and thus the position of the principal axes at any time would be known. The integration, how- ever, earn* in general be effected; so that we are obliged to have recourse to a special hypothesis with regard to the position of the fixed plane of reference. If we take for this purpose the invariable plane, the process becomes much sim- plified. i 6. Let then lt fa, fa denote relatively to the inva- riabL plane the same angles which relatively to the original plane of reference have been denoted by 6, fa fa Then, the direction cosines of the invariable plane with reference to the principal axes being ~k~ y ~k~ y ~k~ respectively, we have (see figure of preceding Article) r- 1 = cos PA = sin X sin fa , j-* = cos PB = sin 6 l cos fa , /c * a -- 3 =cos0 i; * There is much disagreement between writers as to the measurement of the angles employed in these kinematical equations ; the above, however, agrees with La Place, Poisson, and Ponte*coulant. GENERAL THEORY OF ROTATION. therefore Co) These equations give 6 l and ^ : to obtain T^, substitute in the first of the equations (A) of Art. 5 ; thus - 1 sin 2 0j = ^ sin t sin ^ + o> 2 sin X cos ^; therefore ^ l (F - <7 2 a> 3 2 ) = - (Aa,? + J5o> 2 2 ) X; * ,, - 7 therefore -p = n 777^ fc ; dt k* C w* combining this with the result of Art. 3, and integrating, we have where g is the constant of integration. Since o> 3 is known in terms of t from Art. 3, these equa- tions give 0j, fa, fa; so that the position of the principal axes is known at any time with reference to the invariable plane. Since, however, when the disturbing forces are taken into account, this plane ceases to be absolutely invari- able, it will be convenient to be able to refer the motion to some other plane which does remain fixed, and which may be taken as a plane of reference : this we can now do by Spherical Trigonometry. 8 THE EAETH'S MOTION OF ROTATION. 7. Let the surface of a sphere of any radius, with its' centre at the centre of gravity of the planet, be cut by the fixed plane of reference, the invariable plane, and the princi- pal plane of xy, in the great circles OMN, MI, INA respec- tively. As before let ON= -^, NA = <, the angle ON A = 6 ; also take M as the origin from which ^ is measured, and let Jf/=^, IA= l} the angle MIN=6 l : let OM (the longi- tude of the node of the invariable plane) = a, and IMN (its inclination to the plane of reference) = 7. Then the sides of the spherical triangle IMN are and the angles respectively opposite to these, ft, IT- ft 7- Hence, by the formulae of Spherical Trigonometry, we have cos 6 = cos 7 cos 0j sin 7 sin t cos -fy^ sin ((f> l ) sin 6 = sin 7 sin fa, sin (>|r ct) sin 6 = sin X sin fa, which determine ft <, ^r, when ft, ^> 15 ^, a, 7 are known. GENERAL THEORY OF ROTATION. Disturbed Motion. 8. Having now shewn how to determine the position and velocity of rotation of the planet on the hypothesis that no forces act upon it, we proceed to a rigorous treatment of the problem. We shall employ the principle of the varia- tion of parameters, and suppose the results already obtained to represent the true solution, the arbitrary constants or elements being no longer constants, but variable quantities, which it will be our object to determine. We shall speak of the forces which produce this variation as disturbing forces. The elements which have been already employed in the undisturbed motion are six in number, viz. h, Jc, 7, g, a, y : in considering these as variable we shall arrive at the very remarkable result that the equations for calculating their variations are precisely the same as the corresponding equa- tions for the motion of translation in the Planetary Theory. DEF. The plane of which the direction cosines are "IT' ' T' the area conserved upon which has been shewn (Art. 4) to be equal to &, will in future be termed the plane of maximum areas, on account of the property which it possesses, that k is a maximum*'; since it can no longer be considered in- variable. * See Routh's Rigid Dynamics, Art. 174. 10 THE EARTH'S MOTION OF ROTATION. 9. To find an expression for the sum of the moments of the disturbing forces about any line through the centre of gravity of the planet. We shall suppose the disturbing body (which may be the Sun, Moon, or another planet), so distant that it may be considered to attract as if condensed into its centre of gravity. Let m be the mass of the disturbed, m of the disturbing body, p t the distance of the centre of gravity of the latter m from an element Sm^ of the former : also let V l . Then, if a- denote the length of the arc of any curve measured from some fixed point to the element Sm^ the disturbing force on this element in direction of the tangent to the curve, and tending to increase cr, will be If we suppose this arbitrary curve so drawn that its tangent at the point where the element is situated is perpen- dicular to the axis about which the moments are to be taken, and denote by p the distance of Sm^ from this axis, the moment of the force will be Let the small arc Bo- subtend an angle 8^ at the nearest point of the axis; then &r = p%, and the moment becomes * ' Similarly, if F 2 refer to an element Sra 2 , the moment of the disturbing force on this element will be x * cm* - , 2 GENERAL THEORY OF ROTATION. 11 S^ being the same as for the element Sn^ since these ele- ments are supposed rigidly connected. Hence the sum of the moments of the disturbing forces on all the elements of the planet , dV , dV- or, if we write V for % (Bm^Y^), the sum of the moments will be w where F'=w'2 . P COR. If there are several disturbing bodies m, m", &c. and V, V", &c. are the functions corresponding, the sum of the moments of the forces due to their action will still be rIV 5r, where F= V'+V" + ... d x The result of this Article may be thus enunciated : Sup- pose a small arbitrary rotation given to the planet about any axis through an angle 8%; then, supposing V expressed in terms of % and quantities which do not vary in this hypotheti- cal motion, the sum of the moments of the disturbing forces about this axis will be expressed by the partial differential coefficient -3 . d % The function V is thus clearly analogous to the disturbing function R o'f the Planetary Theory. 10. "We may express V in various ways which will be found useful : (i) As a function of 0, , <\fr. Let x, y, z be the co- ordinates of an element 8m of the disturbed planet, the fixed 12 THE EARTH'S MOTION OF ROTATION. plane of reference being that of xy, and the axis of x the line from which ^fr and a are measured ; let x, y , z be the co-ordinates of the centre of gravity of the disturbing body referred to the same axes; and let x lt y lt z l be the co-ordi- nates of Sm referred to the principal axes of the planet. Then F=m'2 , (Art. 9), 8m Now if X, p, v be the angles which the fixed axis of x makes with the principal axes of #, y, z, we have x = ajj cos X + y t cos p + z l cos v ; and by Spherical Trigonometry (see fig. to Art. 5), cos X = cos ^> cos ty 4- sin < sin i|r cos 0, cos fju = sin cos ty -f cos ^> sin ^ cos 0, cos v sin >Jr sin 0. Thus x may be expressed in terms of 6, , -Jr and of the co-ordinates of $m referred to the principal axes. Similarly, y and z may be expressed in terms of 6, , ty. If the values of x, y, z so obtained be now substituted in the expression for F, it will become a function of 6, , i|r ; and, in so far as it depends upon the disturbed planet, of quantities indepen- dent of the time. (ii) As a function of 1? ^, i/^, a, y. By Art. 7, we have, -=/(<>,. GENERAL THEORY OF ROTATION. 13 the symbol f denoting a different function in each case. If then we suppose. V to have been expressed as a function of 0, (j), ft by (i), these equations will enable us to express it as a function of 6 V v ft, a, y. (iii) As a function of t and the elements. Collecting together the results of Arts. 6 and 3, and making Ca> a s, we mav write ft += t + l=f(h,k, s). Supposing F" to have been expressed as a function of ^i $i> ft' a > % by (ii)> the ^ rs ^ three f these equations will enable us to express it as a function of s, h, k, g, a, 7 : then, if s be eliminated by means of the fourth equation, it will become a function of t + I, h, k, g, a, 7 ; that is, of t and the elements. 11. We now proceed to obtain equations for calculating the values of the elements at any time, commencing with h, the element of vis viva. Let T denote the vis viva due to the rotation of the dV planet, Sw l -- the resolved part of the disturbing force on an element Sm l of its mass in the direction of motion of the element : then the equation of vis viva gives dT ^( dV^ da\ j- = 22 Bm. -j- -j- dt \ r da- dtj where in - , the differential coefficient is taken only in cit 14 THE EARTH'S MOTION, OF ROTATION. so far as t is involved through the co-ordinates of the element Bm of the disturbed planet. This equation may be written if, as in Art. 9, we write F=S (Sm 1 .F 1 ). Now from the result of (iii) in the preceding Article we notice that t + l always occurs in V as one quantity ; therefore d(V) _dV dt = dt ' also -in the undisturbed motion T=h', hence our equation becomes dh_ dV dt~ dl> from which h may be calculated. 12. This equation may also be written dh_ d(V) dt~ dt ' under which form it is easily seen to be identical with the corresponding equation S\ a of the Planetary Theory. (See Planetary Theory, Art. 26.) In fact it appears that the element of vis viva will be given by a similar equation in all cases of motion, whether of translation or rotation, when the disturbing bodies attract according to any law expressed by a function of the distance. GENEKAL THEORY OF ROTATION. 15 Equations of Motion. 13. Let Jc t , & 2 , 3 denote the areas conserved about three rectangular axes of x, y, z, originating in the centre of gravity of the disturbed planet, and moving with angular velocities /3 19 2 , /3 3 about their instantaneous positions ; also let L } M t N be the moments of the disturbing forces about these axes : then (Routh's Rigid Dynamics, Art. 120) we have the equations of motion Now let & 3 denote the area conserved on the plane of maximum areas : then from the undisturbed motion (Art. 4) we have If If - A/3 ti, , and since the areas conserved upon all planes perpendicular to the plane of maximum areas are zero*, we must also have always ; and therefore -<" * This follows from the fact that the area conserved upon any plane is equal to the projection upon it of the area conserved upon the plane of maximum areas. (Routh's Rigid Dynamics, Art. 174.) 16 THE EARTH'S MOTION OF ROTATION. hence our equations become -- dt~ We may express /8, and /3 2 in terms of -5- and -^ , a and 6t& CLt y being respectively the longitude of the node and the inclination of the plane of maximum areas to the fixed plane of reference. Since no assumption has yet been made with regard to the position of the axis of x, it will be con- venient to suppose it to coincide with the line of nodes. Employing the equations of Art. 5, and making co l =/3 1 , . = ft = 7> = > ^=, we have da . - 81117 = ft, and the equations of motion become , da . dk ~r N. dt 14. With respect to the last of these equations, we may remark that it is of the same form as it would be if the plane of maximum areas were still invariable. Now on referring to Art. 18 of the Planetary Theory it will be seen that the equations for the motion of translation, when re- ferred to axes in the plane of the orbit, take the same forms GENERAL THEORY OF ROTATION. 17 as if this plane were at rest, and moreover, that the second equation there obtained is identical with the third of the preceding Article*. This coincidence is not accidental, but arises from the fact that in the motion of translation, the plane of maximum areas is the plane of the orbit. In fact; the equations obtained in the preceding Article for the motion of rotation are equally applicable to the motion of transla- tion, and we shall see that the resulting formulae for the cal- culation of the elements involved are identical in the two motions. 15. It now only remains to find expressions for L, M, N in terms of the differential coefficients of the function V, in order to obtain from the equations of Art. 13 formulas for calculating the elements a, 7 and k. We shall suppose V expressed as a function of lf fa, fa, a, 7 (Art. 10 (ii)). To find L : Suppose a small arbitrary rotation given to the planet about the axis of x, that is, about the line of nodes of the plane of maximum areas : then we may repre- sent the change in position of the planet by supposing 7 alone to vary, lt (j> l , fa, a remaining constant^. Thus, the ten- dency of L being to diminish 7, we have L = -^, the differential coefficient being partial. Hence by the first equation of motion, da_ 1 dV dt k sin 7 dy * * The quantity here denoted by k corresponds with what in the Planetary Theory is denoted by 7t. 1* This is equivalent to supposing the plane of maximum areas fijred in the body during the rotation, a supposition perfectly allowable, since the rotation given to the planet is hypothetical, and has nothing to do with its actual c. 2 18 THE EARTHS MOTION OF ROTATION. To find M : Suppose a small arbitrary rotation given to the planet through an angle &% about the axis of y, that is, about a line through L the centre of gravity of the planet, in the plane of maximum areas perpendicular to LM its line of nodes : let the effect of the rotation be to change the position of this plane from MI to mi: draw Mn perpen- dicular to mi. Then Mn fy, mn = S^, Mm = Sa, the angle Mmn 7 : and we have ^ = da. = ~~ a cos 7 > da. 1 JA|T. dot. therefore -y- = -. , -y- 1 = cos 7 -y- = cot 7. av sin 7 av ay And dV c?% o?a a 7 ^ G?^ d% ' since we may suppose a and ^ alone to vary with y^ *> there- 1 ore ,, Jf = sin 7 - COt 7 Now (Art. 10 (iii)) ^ + ff =f(h, k, s), and l , < x do not involve g ; therefore motion. Any other supposition might be adopted with regard to the motion of this plane, but this is the most convenient, and will be retained throughout this Article. GENERAL THEORY OF ROTATION. 19 1 dV ^ dV hence M = -- 7 ~ + cot 7 -*- : sin 7 da dg ' and the second equation of motion gives - Q 7 dt~~k sin y dot. h sin 7 dg * To find N: Suppose a small arbitrary rotation given to the planet about the axis of z, that is, about a normal through the centre of gravity of the planet to the plane of maximum areas : then we may suppose fa alone to vary, and since the tendency of Nis to diminish fa, we shall have AT- _dV -- as in the preceding Article ; then the third equation gives dkdV 16. The equation for calculating 7 may also be simply obtained as follows : if we refer the motion to the fixed plane of reference, and suppose V expressed as a function of 6, , fa since Jc cos 7 is the area conserved on this plane, we shall have d . dV Now (Art. 10 (ii)), ^ - a =/(0 lf fa, 7), and 6, do not involve a; therefore dV_dV lt fa, a, 7, or simply of t and the elements. Hence dV 22 20 THE EARTH'S MOTION OF -ROTATION. dy dV tilt therefore &sm 7 = + cos 7 dV dV ,, r dy 1 dF cos 7 therefore ~ = -j. -j \- 7 ^-*- . eft A; sin 7 da A; sin 7 a^ The method of this Article is identical with the second method given in Art. 34 of the Planetary Theory for the de- termination of -v- ; we reserve the comparison of the results to a subsequent Article (Art. 22). 17. The following proposition will be useful 'in finding -7- . To shew that TT - 2 -nr > an ^ j- = - 2 ^p , #ie variables dk dh ds dh connected together by the equations of Art. 10, (iii). B-G 9 Let u = Jc*-Bh c where 6- = <7o> 3 . Then from Arts. 3 and 6, we have t + l = J(AB) JL-.Zds (3), ta ^=^ w- GENERAL THEORY OF ROTATION. 21 From equation (2), by differentiating under the integral sign, dv du dl 1 f/A ~ [dk dk, -^=-^*J(AB r ds d/c I J ( uv y i [*(""*) ds, by equations (1). J (uv)* From equation (3), in like manner, we have 1 /, s*\ f dv . du i__ _& ~2\ ~~~C ^dh~~ uv s To simplify this expression, we find from equations (1), , s 2 u + v 9 _ ~~ A-B ' Substituting these values, and reducing, 1 dl f ' therefore =2 dk dh 22 THE EARTH'S MOTION OF ROTATION. Again, from equation (2), we have dl _ I(AB\ ~~ Differentiating equation (4), and substituting for -^ and ^7 from equations (1), 2 , di A Bv Au 2 . therefore -- = 2 r=l . 18. The following relations between partial differential coefficients of the function V, as well as the results of the preceding Article, will be required in finding -7- . In Art. 10 we have shewn that, in order to express V as a function of t and the elements, we may first express it as a function of lt $> v ^ , !, ^, by means of the equations in (iii). The first three of these equations are /(*, Tc, s), GENERAL THEORY OF ROTATION. 23 If, then, we substitute for 19 (f> i , ^ from these equations, V will become a function of h, k, s, g, a, 7 ; and we shall have fdV\ dV dG. ' dh d(p dh d^If dh Jfc ^Z. ^fc dcfri dh dty v dh ' since 6 l does not involve h. We have used the brackets to distinguish the partial differential coefficient with respect to h, of V expressed as here supposed, from its partial diffe- rential coefficient when expressed as a function of t and the elements. In order to pass from the expression of V as a function of h, k, s, g, a, 7 to its expression as a function of t and the elements, we must solve the fourth equation of Art. 10 (iii), with respect to s ; thus we shall have s=f(h, k t t + l), and then dV (dV\ dVds f dh ~\dh)ds dh' 1 or, writing for f -rj- j the value obtained above, dh ds dh dfa dh d^jr dh ' a relation which will be required in the following Article. Again, differentiating the equation s =f(h, k, t + l) on the hypothesis that h and I alone vary, we have ds ds dl 2* THE EARTH'S MOTION OF ROTATION. where -yr is the same as would be obtained by direct differ- entiation of the last equation of Art. 10 (iii). Hence ^ _ ds dh ds dl dh _dVdl dl dh' since I is introduced into V only through s. ' This relation also will be required in the following Article. 19 1 . We are now in a position to find -y , and shall for Cit this, purpose refer the motion to the principal plane of xy. We must first, however, obtain an expression for the sum of the moments of the disturbing forces about the axis of z. Consider, then, V as a function of 1? ^, ty lt a., 7, and let a small arbitrary displacement be given to the planet about the axis of z. We may express this by supposing lt fa, &, y to remain constant while ^ alone varies. Since, then, with the usual convention with respect to signs, the ten- dency of the moment we are considering is to increase $ v it will be expressed by -yy- . Thus, from Euler's equations of motion, d t ds dl _ 9 dfa dl_ _ dfa dk~ ~dh> ds~ ~dh dg and by Art. 18, ^ = ^ dl d/i " ds dh dl from which I may be calculated. 20. To obtain the formula for g, we might of course proceed in the same manner, differentiating the third equa- tion of Art. 10 (iii) ; but the process would be complicated by the fact that fa is measured from a moving point and on 26 -r a moving plane, and consequently that the form of ^- l is not the same in the disturbed as in the undisturbed motion. We shall therefore employ a different method : availing ourselves of the formulae already obtained for calcu- lating five out of the six elements, we shall obtain the sixth by direct substitution, in the same manner as that in which the epoch was obtained in Art. 36 of the Planetary Theory. By Art. 10 (iii), we may write V=f(t+l,g, h,k, a, 7). Differentiating, the elements being considered variable, dV _ _ o dt dk ' dk = dR ~di~ dg ' dg _ dR' cost dR' dt dh k sin i di ' occurs explicitly in 72 : if we suppose a to vary also as contained implicitly in n, this differential coefficient will include the term proportional to the time, which may therefore with this understanding be omitted. In Art. 37 of the Planetary Theory this term was removed by a different transformation. 30 THE EAKTH'S MOTION OF EOTATION. = ^ dt k sin i di * di _ 1 dIC cos i dR' dt~ Jc sin i dQ k sin i dg On comparing these with the rotation formulae, it will be seen that they are identical, with the exception only that the sign of a differs from that of H ; and this is accounted for by the fact that a and fl are measured in opposite direc- tions. Thus, by employing elements having a like significa- tion in the two motions of translation and rotation, we have arrived at the very remarkable result that the complete solu- tion of the problem of Planetary perturbation, whether in the motion of translation or of rotation, is expressed by the above simple formulae. PAR1 II. APPLICATION OF PRECEDING RESULTS. PRECESSION AND NUTATION. 23. THE formulae obtained in the first part are suf- ficient completely to determine the motion of a planet or other rigid body about its centre of gravity. They are per- fectly rigorous, subject only to the hypothesis that the dis- turbing bodies may be supposed to attract as if condensed into their respective centres of gravity ; a hypothesis admissible if these bodies are either very distant or nearly spherical in form. But though the formulas are exact, they can be in- tegrated only by approximation. We propose, therefore, in the second part to restrict ourselves to the particular case of the Earth, taking advantage of such of the results of obser- vation as may be required to enable us to approximate. In order to treat the problem fully we shall consider, first, the motion of the axis of rotation in the Earth itself, with the velocity of rotation about it ; secondly, the motion of this axis in space. The first is of special interest, since any change in the position of the axis in the Earth, were such change possible, would affect the permanence of terrestrial latitudes ; any change in the velocity of rotation would affect the length of the day. The second is of great importance to astronomers, since it establishes the fact that the first point of Aries, or vernal equinox, to which they are accustomed to refer celestial longitudes, is not a fixed point. 32 THE EARTH'S MOTION OF ROTATION. Stability of the axis of rotation in the Earth and of the velocity about it. 24. We proceed then, first, to consider the motion of the axis of rotation within the Earth, and shall be able to shew that, in so far as it depends upon the attractions of other bodies, this axis can never separate appreciably from the axis of figure, and that the velocity about it must always remain appreciably constant. We have from Art. 3 Eliminating &> 3 from these equations, Now if C be either the greatest or least principal mo- ment, both the terms of the left-hand member of this equation must always retain the same sign ; and if we take G about the Earth's axis of figure it will be the greatest and this sign will be positive. We may therefore write thus A(C-A)a>? + B(C-B}o>?=(C-A)(C-B}e\ It follows from this that I~C~ ~B G)^ can never be greater than e ,\J -. , co 2 can never be greater than e If we neglect the disturbing force, h, Jc and therefore e are constant, and. the value of e will be found by substituting STABILITY OF THE AXIS. 33 the values of co l) o> 2 as determined by observation at any given epoch. Now the most delicate observations have hitherto shewn no appreciable separation of the axis of rotation from the axis of figure : hence o^, o> 2 are at present inappreciable ; thus e is so, and we conclude that, indepen- dently of the disturbing force, o x , eo 2 must always remain inappreciable. If in the equations of Art. 6, we write o^ = 0, o> 2 = 0, we have also O l = ; so that, if the disturbing force be neglected, we may consider the plane of maximum areas as coincident with the Earth's equator. 25. Let us now examine the effect of the disturbing force upon the angular velocities o> 1 , o> 2 . By the preceding Article Differentiating, and substituting the expressions for -, dk and -j- obtained in the first part (Art. 21), , dV If the approximation be carried to the first power of the disturbing force, we may, in calculating the second member of this equation, suppose l = : thus the equations of Art. 7 give = % * = ^i-V r i ^ = . Also, if n be the mean angular velocity, we may to the same order of approximation suppose &> 3 = n : thus, from the equations (A) of Art. 5, i|r and 6 are constant and c. 3 34} THE EARTH'S MOTION OF ROTATION. whence by integration n dt Now (Art. 3) V = A^ = (7V to the same order of approximation : therefore . n Hence C -jj - k -y- = C j-^ - - - ^ = ; dl. dg dt n dt, de and therefore -j- : dt whence it follows that to the first order of the disturbing force e continues absolutely constant. When the approximation is carried to the order of the square of the disturbing force*, it is found that e z can con- tain no term proportional to the time, nor ..any inequality, either secular or periodic, raised to the first order in the process of integration. Hence we may consider e 2 as prac- tically constant, and thus conclude, as in the preceding Article, that o) l9 o> 2 will always remain inappreciable. * See Pente'coulant's Systeme du Monde, tome II. p. 218. STABILITY OF THE VELOCITY. 35 It follows, as in the preceding Article, that to the same degree of approximation the plane of maximum areas may be regarded as coincident with the Earth's equator. 26. We have now to examine the effect of the disturb- ing force upon the velocity of rotation. Since 2 are in- appreciable, this velocity will be equal to o> 3 ; thus, as in the preceding Article, we have k therefore rr = -7, -jr = 7? -j- (Art. 21) dt G dt C do ^ ' dco a 1 dk 1 dV dg _1_ d(V) Cn dt ' Now considering V as a function of 0, <, i^, if our ap- proximation be carried to the first power of the disturbing force, we have = r } < so that V will involve t only under the form nt + c. It fol- lows that 7 will consist of a series of terms of the form at P COS (pnt + q), sm ^ where p is some positive integer. And this term will give 2?r rise to inequalities whose period is -- , that is, a day or J}'Yl fraction of a day. Now the most delicate observations have hitherto failed to detect any such inequality : hence we must altogether reject such terms, and conclude that o> 3 will always remain sensibly constant. 27. The preceding Articles prove the stability of the axis of rotation in the Earth and of the velocity about it, so 32 36 THE EARTH'S MOTION OF ROTATION. far as the motion depends upon the attractions of distant bodies ; but there is another disturbing influence to be found in the friction produced by the tides. The effect of this (see Thomson's and Tait's Natural Philosophy) would be to retard the Earth's velocity ; but whether to an appreciable extent is yet uncertain. The fact, discovered by Professor Adams, that a portion of the acceleration of the Moon's mean motion is yet unaccounted for, led Delaunay to suggest that this por- tion might be apparent only, and really due to the retarding effect of tidal friction upon the Earth's velocity. But other possible explanations have been given, and to pursue the subject further would be beyond the scope of the present treatise. Precession and Nutation. 28. We have now to consider the motion of the Earth's axis in space, resulting in the phenomena of Precession and Nutation. We have seen that, to the first order of the dis- turbing force, the plane of maximum areas may be con- sidered as coincident with the Earth's equator, and that consequently to this order of approximation we may write B = ry, i|r = CC. Hence the motion of the equator may be calculated by formula (v) and (vi) of Art. 21. Now it may be shewn as in Art. 25 that dV = ld(V) dg n dt ' and may therefore be neglected, since (as has been proved in Art. 26) it can give rise only to terms whose period is a day PRECESSION AND NUTATION. 37 or fraction of a day, which are wholly insensible. If, then, we write ty and for a and 7 respectively, these formulae become dt ksmd d6 ' dt k sin 6 dty ' or, since to the same order of approximation k Co> 3 = Cn, cty_ 1 dV ~dt~ Cn sin dti ' d6 1 sn 29. These elegant equations, due to Poisson, give ty and 0, and thus determine the motion of the Earth's axis in space. Before we can integrate them it will be necessary to obtain an expression for F. Let then m be the mass of the Earth, m that of the dis- turbing body which we shall suppose to be the Sun ; r the distance between the centres of gravity of these bodies, r the distance from the Earth's centre of gravity of an element Sm of its mass, p the distance of the same element from the centre of gravity of the Sun : also let the inclination of r to / be denoted by v. Then by Art. 9, . __ ,, Bm P m r M 2r , ^V*! 1 - T cosv4--75 r- \ r r / ) 38 THE. EARTH'S MOTION OF ROTATION. Now since r is very large in comparison of r, we shall T neglect all powers of - above the second : thus jr m ' * f* /'i , r 1 r 2 3 r 2 2 \] V = 7 2 |Sm (I + -7 cos v - ^ 7* + ~ 2 ^co 8 " Jj , tfl -.., 2 2 \ -f ~^ 5 (Sm . r cos v) r 2 (Sm) + -15 2 (Sw . r cos v) + j 2 Of these terms 2(&?i) and -^S^m.r 2 ) may be omitted, since they cannot contain the angles 6, $ or ty, and would not be affected by any arbitrary rotation such as is supposed in Art. 9 : also since r is measured from the Earth's centre of gravity, 2 (5m . r cos v) 0. Hence we may write 3m' ~ 2r 3 if Q denote the moment of inertia of the Earth about the line joining its centre of gravity with that of the Sun. We see, then, that the effective portion of the disturbing func- tion is proportional to this moment of inertia. 30. We have hitherto made no assumption respecting the Earth's figure. We learn from pendulum observations and geodetic measurements that it is approximately an PRECESSION AND NUTATION. 39 oblate spheroid; the moments of inertia about all axes in the plane of the equator are therefore nearly equal, and we may take B = A. If, then, 8 be the Sun's declination, also if n denote the Sun's mean motion, we have approxi- mately 2?r therefore or, neglecting the Earth's mass in comparison with that of the Sun, Hence, by substitution, 31. By Art. 9, the moment of the Sun's disturbing force about an equatorial diameter of the Earth perpendicular to the line joining the centres of gravity of the two bodies, and tending to increase B dV 3n'* . , It is clear that the moment about any axis perpendicular to this is zero, since an hypothetical rotation about any such axis could produce no change in 8. If, then, the Sun's dis- turbing force be reduced to a parallel force through the Earth's centre of gravity, and a couple, the tendency of the latter will be to give to the equator a rotation towards the ecliptic about an equatorial diameter perpendicular to the 40 THE EARTH'S MOTION OF ROTATION. line joining the centres of gravity of the two bodies. The moment of this disturbing couple is ~-0<7- A) sin 28. 32. We shall now return to the equations of Art. 28, and deduce from them the motion of the Earth's axis in space. The results obtained will be of sufficient accuracy for our present purpose if we suppose the ecliptic, to which plane the motion will be referred, to remain fixed. Let -^ denote the longitude of the first point of Aries measured from some fixed "point on the ecliptic, X the Sun's longitude measured from the same point, 6 the obliquity of the ecliptic. Then from the spherical triangle formed by the intersection of the equator, the ecliptic, and a declination circle through the Sun, we have sin 8 = sin (T|T X) sin ; 7 therefore cos 8 -^ = sin (>|r X) cos 6, at? 7 cos 8 -yy- = cos (^r X) sin 6. If T/T X = ?, then I will be the Sun's longitude measured from the first point of Aries in the opposite direction to that in which ^ and X are measured : thus fjft 7 cos 8 g = sin I cos 0, cos 8 ,-7 = cos I sin 0. dv dty Hence -^ = -r 37, = 3n' 2 ( G A} sin 8 sin I cos 0, du do do dV dV dS -j~r -^- ~T7 = ~ 3w (C A) sin 6 c< d-fr do aw PRECESSION" AND NUTATION. 41 Substituting these values in the equations of Art. 28, d^ 3n* C-A . . . 7 ~J- = - . ~ sm o sin I cot 6 dt n G 3/i' 2 C-A , = . ~ sin 2 1 cos 3?i G A /\ it ctj\ = - . 77 cos 0(1- cos 2Z). .Zw O J0 3?i" C-A . T- = -- - . 7=5 sin o cos 6 (ft n C Sri* C-A . 7 1 . a = sin ? cos? sin n 3n'* C-A . fi . = ?r . Ti sin u sm 21. 2/* 6 Let / denote the mean value of the obliquity, then since both and are very small quantities, we may in the second members of these equations write = 7, l= n 't: ( d^r 3n 2 C-A Tn M thus j^T'- n cos / (1 cos 2/i^), dt An, L> dO SH* C-A . T . _ . -j- = - ^ . ri sm /sin 2tt t. dt 2n U By integration, 3rc' 2 (7-^1 3^' C-A y = ^. Q . cos /J -j- . . cos 7. sin 2w , ^ = / + -: . 7= . sin /. cos 2nt, 4cn C ty being measured from the position of the first point of Aries at the epoch from which the time is reckoned. 42 THE EARTH'S MOTION OF ROTATION. 33. These formulae determine the motion of the Earth's axis in space. Although the method by which they have been obtained has the advantage of connecting the two prob- lems of translation and rotation, it may be worth while to arrive at them by an independent process. Since the Earth is nearly an oblate spheroid, we shall, as before, take C for the greatest principal moment, and sup- pose JB = A, so that all axes in the plane of the equator are principal axes. If, then, taking the plane of the equator for that of xy, we suppose the axes of x and y to revolve with an angular velocity 3 about the axis of e t we have the equations of motion* A --~ Atoflz -f (7w 2 o) 3 = L, . at if we take for the axis of x the projection of the Sun's radius vector on the plane of the equator, we have L = 0, JV= 0, M '2 Jlf=_^L((7-^)sm2S, (Art. 31); and since 3 is very small and occurs only in connexion with 2 which are also very small (Arts. 24 and 25), we may write 3 = ri its mean value. * See Routh's Rigid Dynamics, Art. 107: the equations there given reduce to the above if 63 is written for ws + -~ . at PRECESSION AND NUTATION. 43 Thus the third equation gives c S-*=o. whence a> 3 = const. = n suppose ; and the first two become Cn-An' or, denoting - - -- by p } dot. 3n" C-A . C A Now ^ i g a ver y small fraction ; ri, the Sun's mean ^cl motion, is very small in comparison of n ; and 8 varies very slowly; we shall, therefore, in integrating these equations, take no account of its variation. Thus, eliminating o^, the integral of which is o) 2 = A cos (pt J5), where A and B are constants of integration. This term is independent of the disturbing force ; but having for its period , which does not differ much from a day, it must be re- 44 THE EARTH'S MOTION OF ROTATION. jected, since, as we have already remarked in Art. 26, such terms are altogether insensible. Hence we conclude that and by substitution in the second equation, 3ri* C-A . o), = -TT . - sm 28 3/i' 2 C-A f, Ari\' 1 . -^ TV 1 77 sin 2/i C \ On] 28 G sin powers of n above the second being neglected. We see, then, that the effect of the Sun's attraction is to cause the Earth's axis to travel in a plane perpendicular to that of the disturbing couple which it produces, and with an angular velocity proportional to its moment. This velocity, though so small as to be insensible to the most delicate observations, yet leads to values of i|r and 6 which can by no means be neglected. 34. We proceed to determine the motion of the axes in space. Substituting in the equations of Art. 5, we have dty . a Sri* C-A . -y- sm 6 = - . . . sin o cos 8 sm , at n L> dO 3rc' 2 C-A ~j- . ^ . sin o cos o cos 9. Now from the spherical triangle formed by the intersec- tion of the equator, the ecliptic and a declination circle through the Sun, it is easily seen that sin $ = tan 8 cot 0, sin 8 = sin I sin #, cos I = cos (f> cos 8. PKECESSION AND NUTATION. 45 By means of these equations we obtain d 3n' 2 C-A . dt n C ae 3tt' 2 C-A Q . 27 . cos 6 sm 2 Z, whence, as in Art. 32, 3>i' 2 C-A e = 1+. . sin /cos 4/i 35. It appears from these formulae that the motion of the Earth's axis is of two kinds ; partly secular, partly periodic. It is convenient to consider these separately. The former affects the equinoxes alone, and, being proportional to the time, indicates uniform motion ; this motion is one of regression, since ty has been measured in a direction contrary to that of the apparent motion of the Sun. Considered with reference to the apparent diurnal motion of the stars, the effect is to place the equinoxes in advance of the position they would occupy if fixed : hence it obtained the name of the Solar Precession of the Equinoxes. The latter furnishes corrections both on ty and 0, which go through all their changes in half a year : it is called the Solar Nutation of the Earth's axis ; the correction on ty forming the Nutation in Longitude, that on the Nutation in Latitude. 36. We will now examine the effect produced by the Moon's action on the motion of the Earth's axis. Since the investigations which have been given of the effect of the Sun's disturbing force contain nothing to restrict their gene- rality except the special assumptions made with regard to 46 THE EARTH'S MOTION OF ROTATION. small quantities, we shall first consider what modifications are required in order that they may be applied in the case of the Moon. Let ty', ff, n", I' denote relatively to the plane of the Moon's orbit the same quantities which relatively to the ecliptic have been denoted by ^ 0, ri, /; also let m, m be the masses of the Earth and Moon : then, as in the case of the Sun, we have approximately, 2-7T n " Vm + m" ' but we cannot in this case neglect m, as it is in fact much larger than m" : retaining it, we have r' 3 m + m if m = Xm". Also n" is small, though not so small as ri in comparison of n. Hence to determine the motion of the Earth's axis with reference to the plane of the Moon's orbit, we have 3ri f * C-A . The periodical terms in these equations go through all their values in half a month, and are so small that they are usually neglected. Thus we may consider the incli- nation of the plane of the Earth's orbit to that of the Moon PRECESSION AND NUTATION. 47 as constant and equal to its mean value, and the precession as uniform and given by 3V' 2 C-A whence, the velocity of precession d 3n" C ' These results, deduced from the corresponding, formulae in the case of the Sun, suppose the plane of the Moon's orbit fixed. Now the line of nodes moves too rapidly to allow of this hypothesis, but the only effect of considering its motion would be to add to -r- a term depending upon its velocity and not upon the disturbing force of the Moon upon the Earth. Since our object is to trace the effects of this force only, such terms must be omitted. 37. It now remains to determine the motion of the Earth's axis with reference to the ecliptic. Let 6 be the obliquity of the ecliptic, -\/r the longitude of the equinox, cc the longitude of the node of the Moon's orbit measured from the same origin as ^, i the inclination of the Moon's orbit to the ecliptic, /' its inclination to the equator. Then supposing tf measured from the node of the Moon's orbit on the ecliptic, we have by Spherical Trigono- metry (as in Art. 7) sin (-v|r a) sin 6 = sin /' sin ifr (1), cos# = cos /cos/' sin /sin/' costy' (2), cos /' = cos 6 cos /+ sin 6 sin z'cos (ty a) (3). We must now differentiate these equations in order to d^ir , d6 . dilr T , . express -j- and -7- in terms of -~ . In so doing we .shall ut ut cit 48 THE EARTH'S MOTION OF ROTATION. omit the terms involving -j- and -j-, since these quantities depend not upon the disturbing force, but upon the motion of the Moon's orbit. Thus / \ a dty . f Q dO cos (ty a) sin -3- + sin (y a) cos 6 -j- at rt , i -T sm I cos tjr -Z- . a . . r . ., sin 6 -j- = sin i sm / sin -Jr I- . at at Writing O for i|r a, we obtain cos fl sin -~ = (sin /' cos tjr' 7 i / -f sin i sin /' sin ^r' cot 6 sin O) -i at (sin /' cos -tf + sin i cos 6 sin 2 H) -~ ...... by (1), /cos t cos /' cos sn )fj f ' -^- ...... by (2), /cos i cos /' cos cos 2 / . V sln^ . . - sin i cos 6 cos 7 . / = (cos e sin ^ cos O sin i cos ^ cos 2 II) *-|- , ...... by (3) ; ^>6 therefore -^ = (cos i sin z cot 6 cos fl) -^ . d6 sin t sin T sin -// J^' Also -j-. = -- : ^ -f- a^ sin ^ dt d^r sint sinfl J- .................. by (1). at We must now substitute the value of -- : thus at PRECESSION AND NUTATION. 4$ ^ = I^L-^ . ^^ . cos /' (cos i - sin i cot cos ft), - 5 sm 2 z (7 2 - cot / sin 2/ cos (vt + O )), The integrals of these equations are 1 , ' cos T {(cos z ~ sm -- cot 27 sin 2e sin (^ + H) J + const., n " 2 C A 38. These equations determine the motion of the Earth's axis due to the attraction of the Moon: They are similar to the equations expressing .the Sun's action, and the remarks made in Art. 35 might, mutatis mutandis, be re- peated here. The Lunar Precession is - Sri' 2 C-A '/ 1 2 s e- sm ^ t. 2/i (1 + X) ' If we add to this the Solar Precession (Art. 32), we find for the whole permanent effect of the Sun and Moon upon the equinoxes 3n C-A T'~7T~ This is called the Luni-solar Precession, to distinguish it from the Precession due to the secular motion of the ecliptic in consequence of the attractions of the Planets, and which is called Planetary Precession. . Adding together the effects of Lunar and Solar Nutation, we find for the whole Nutation in Longitude PRECESSION AND NUTATION. ! 51 i 3 C-A T ( n" 2 - . 7^ . cos I - -. -r cot 27 sm 2i sm fl 2 6 [nv (I + X) where fl and denote respectively the mean longitude of the ascending node of the Moon's orbit and of the Sun. Similarly, the whole Nutation in Latitude is 3 C-A T ( n m ri } -r . 7-. . cos / \ 7^ -r- sm 2^ cos II -\ --- cos 2 V. 4 (nv (1 + X) n } In each of these expressions the first term, due to the action of the Moon, is the most important, since n" is larger than both n' and v. 39. If we consider only the motion of the Earth's axis due to Precession, it appears from the preceding formula? that it maintains a constant inclination to the pole of the ecliptic, and describes a right circular cone about it with uniform velocity. An axis possessing this motion ,exactly we shall term the mean axis of the Earth. The motion of the axis due to Lunar Nutation can now be exhibited as follows : Let x = -r . , 7= . cos 27 sin 2i sin O, 4 nv (1 + X) 6 3 ri f * C-A y~ 7 TT - =r\ ri cos * sm 2t cos H : 4 nv (1 + X) C then, taking unity as the radius of the celestial sphere, x and y are the small linear spaces traversed by the intersec- tion of the Earth's axis with the circumference in two direc- tions at right angles. Eliminating II from these equations, we have x* /3 n" 2 C-A cos a 7~ nvl + \' C . V * Sia J ' 52 THE EARTH'S MOTION OF ROTATION. Hence, in consequence of Lunar Nutation, the extremity of the axis may be considered to move in an ellipse whose semi-axes are in the ratio of cos 21 to cos J. The centre of this ellipse is at the point of intersection of this mean axis with the celestial sphere, and its plane a tangent to the sphere at that point. By supposing the mean axis to de- scribe the cone uniformly, while the true axis describes this ellipse about it, the real motion will be represented. This conception is due to Bradley, who arrived at it by obser- vation. 40. The annual value of the Luni-solar precession (see Art. 38) on A = jr-; . COS 2n (J Observation gives about 50". 1 as the numerical value of this expression. Hence, by substituting the known values of G A n, ri, v, I, ^' we have a relation between -^ and X, by means of which either may be determined when the other is known. 41. We have taken no account in our calculations of the Precession caused by the attractions of the planets on the Earth, since it is too trifling to be appreciable. THE END. CAMBRIDGE: PRINTED AT THE UNIVERSITY TRESS. MATHEMATICAL WOEKS PUBLISHED BY MACMILLAN AND CO. An Elementary Treatise on the Planetary Theory. With a Collection of Problems. By C, H. H. CHEYNE, B.A. Crown 8vo. 6s. 6d. 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