OF 
 R. Tracy Crawford 
 
 A8TRUHUMI 
 
THE SECULAR VARIATIONS 
 
 OF THE 
 
 ELEMENTS OF THE CEBITS 
 
 OF THE 
 
 FOUR INNER PLANETS 
 
 COMPUTED FOR THE EPOCH 1850.0, G. M. T 
 
 BY 
 
 ERIC DOOLITTLE 
 \ 
 
 UNIVERSITY OF PENNSYLVANIA 
 
 Extracted from THE TRANSACTIONS OF THE AMERICAN PHILOSOPHICAL SOCIETY, 
 
 N.S., Vol. XXII, Part 2 
 
 PHILADELPHIA 
 1912 
 
To MY FATHER, 
 PROFESSOR CHARLES L. DOOLITTLE, 
 
 THIS WORK 
 
 is INSCRIBED. 
 
 ASTROUOVY 
 
[Extracted from the TRANSACTIONS OP THE AMERICAN PHILOSOPHICAL SOCIETY, N. S., Vol. XXII., Part 2.] 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS OF THE ORBITS OF THE FOUR 
 INNER PLANETS COMPUTED FOR THE EPOCH 1850.Q G. M. T. 
 
 BY ERIC DOOLITTLE. 
 
 (Read March 1, 1912.) 
 
 TABLE OF CONTENTS. 
 
 THE THEORY. 
 
 1. Introduction 39 
 
 2. The method of GAUSS 40 
 
 3. HILL'S first modification of GAUSS'S method 42 
 
 4. HILL'S second modification. The work of CALLANDREAU and INNES 47 
 
 5. The method of HALPHEN and its modifications by ARNDT and INNES 49 
 
 THE COMPUTATION. 
 
 6. The elements of the orbits and the adopted masses 52 
 
 7. The formulas employed in the computation 53 
 
 8. The values of the preliminary constants 56 
 
 9. The radii vectores and the true anomalies 59 
 
 10. The separate results: 
 
 Mercury by Venus 61 Earth by Mercury 123 
 
 " Earth 65 " " Venus 127 
 
 " Mars r." . . 70 " " Mars 132 
 
 " Jupiter 77 " " Jupiter 138 
 
 " Saturn 82 " " Saturn 142 
 
 " Uranus 86 " " Uranus 146 
 
 " " Neptune 89 " " Neptune 149 
 
 Venus by Mercury 93 Mars by Mercury 152 
 
 " Earth .100 " " Venus : 156 
 
 " " Mars 104 " " Earth 160 
 
 " " Jupiter 108 . " " Jupiter 164 
 
 " Saturn 112 " " Saturn 168 
 
 " Uranus 116 " " Uranus 173 
 
 " Neptune 120 " " Neptune 176 
 
 11. The final values of the perturbations 179 
 
 12. Comparison with the results of observation 
 
 13. Comparison with SEELIGER'S hypothesis on the constitution of the Zodiacal Light 185 
 
 . 37 
 
1. INTRODUCTION. 
 
 The usual method of determining the secular variations of the elements of any 
 planet is the well-known one based upon the development of the perturbing function 
 into an infinite series whose successive terms involve continually higher powers of 
 the eccentricities and the mutual inclination. This method possesses two advantages. 
 The first is that when an extreme degree of accuracy is not required, so that higher 
 terms of the development may be disregarded, it is the simplest method available; 
 and, in the second place, since the coefficients of all terms are general literal expres- 
 sions, the change produced in the value of any variation by a change in the assumed 
 values of one or more of the elements can readily be ascertained by a simple substi- 
 tution of the more accurate values. On the other hand, this method possesses the 
 disadvantage that the complexity of the expansion grows rapidly greater as the order 
 of the included terms is increased, so that a slight increase in the desired accuracy 
 greatly increases the labor of the computation. 
 
 The integral methods, founded upon the celebrated theorem of GAUSS (I) ,* are 
 wholly free from this latter disadvantage, for if it is desired to include all terms to 
 the twenty fourth order this can be done by a computation which is less than twice 
 as long as that required when the approximation is stopped at terms of the eleventh 
 order. But the integral method, though thus extremely accurate, leads only to the 
 numerical values of the variations dependent upon the values of the elements assumed ; 
 if they are desired for some other epoch at which the various elements possess different 
 values from those adopted, or if an improved value of any of the elements becomes 
 known, they can only be found by an entire repetition of the computation. 
 
 The only determinations of the secular perturbations of the four inner planets 
 which are in any sense modern ones are the classic investigation of LE VERRIER (T) 
 and the computation of NEWCOMB (15> . The latter furnishes the most accurate values 
 of these variations so far determined ; the series were extended to terms of the eighth 
 order, only those terms of this order being included, however, which seemed likely 
 to be most important, and in some cases terms of the tenth order were included, 
 though usually by induction merely. 
 
 In both of the above computations the usual expansion into an infinite series 
 was employed. As the GAUSSIAN method is so extremely accurate, and as its formulas 
 throughout are wholly different from those hitherto employed, it seemed that an 
 
 * These symbols wherever they occur refer to the list of titles at the end of the present paper. 
 
 39 
 
40 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 application of it to a re-determination of these variations based upon the most ac- 
 curate values of the several elements now obtainable would be of value. The results 
 of this work will be found in the following pages; the final comparison with the 
 earlier results is given in Article 11, and the comparison with the results of observa- 
 tion in Articles 12 and 13. The epoch throughout is 1850.0, G. M. T. 
 
 In the four following articles an attempt is made to state briefly the essential 
 features of the various methods of computing secular variations which are founded 
 on GAUSS'S theorem, but for a detailed account of the long and often complex trans- 
 formations which are involved, the original papers must be consulted. 
 
 2. THE METHOD OF GAUSS. 
 
 The equations which express the complete variations of the elements of the 
 orbit of any body revolving about the sun when it is disturbed in its motion by the 
 presence of a third body, may, as is well known, be put in a variety of different 
 forms; the form selected as the basis for all developments founded on GAUSS'S method ' 
 is that in which three rectangular components of the disturbing force enter into 
 the expressions for the differential coefficients. Thus, if R denote the component 
 lying in the direction of the radius vector of the disturbed body, positive outward 
 from the sun; S, the component lying in the plane of the orbit of the disturbed body 
 and perpendicular to the radius vector, positive in the direction of motion; and W, 
 the component perpendicular to this plane and positive northward, we will have 
 for the variation of the eccentricity of the orbit of the disturbed body, 
 
 de tfn cos<p 
 
 dt = fe'(l + m) ^ sm + ( - cos " + cos ^ J' 
 
 with similar expressions for the variations of the six remaining elements.* 
 
 In the original memoir of GAUSS the determination of the secular terms of 
 these expressions was given a geometrical aspect. Thus, since each variation may 
 obviously be expressed in terms of the two single variables M and M' , the secular 
 term in question will be that given by the equation, 
 
 [de-\ 1 r 2 ' ("'de ,..,.., 
 [dt 1=4^1 1 dt dMdM ' 
 
 * The usual notation is adopted throughout. Thus a, e = sin ip, i, 12, ir, n, and L are respectively the half major 
 axis, the eccentricity, the inclination, the longitude of the ascending node, the longitude of perihelion, the mean motion 
 and the longitude at the epoch of the disturbed body; M, E, v and r are respectively its mean, eccentric and true anomalies 
 and its radius vector, m is its mass, k* is the mass of the sun, and m = >nk 2 . The same letters with accents refer to 
 the disturbing body. 
 
 Watson, Theoretical Astronomy, pp. 516-523; Oppolzer, Lehrbuch zur Bahnbestimmung, Vol. II, p. 213; Tisserand, 
 Mecanique Celeste, Vol. I, pp. 431-433, etc. The final forms of the equations expressing the other variations may be 
 inferred from those stated at the end of Article 7. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 41 
 
 and this is the same as, 
 
 [de~\ a 2 ncos<f f 2 " f 1 f 2ir _ 1 f 2 ' , ,, , 1 , ,, 
 
 = o 70/1 sm t>- 5- /MM' + (cos v + cosE)- ^- SdM' dM, 
 
 \_dt !joo 27r 2 (l + m) J L 2irJ 2irJ J 
 
 since the variable of the first integration enters the expression only through R and S. 
 In the equation as thus written R and S are supposed to contain the mass, ra ', as a 
 factor so that if Ri and Si are the corresponding values produced by a unit mass, 
 R = m 'Ri and S = m 'Si. 
 
 If we now imagine an infinitely thin elliptic ring which coincides with the orbit 
 of m', whose total mass is equal to the mass ra</, and the density of any portion of 
 which is proportional to the time occupied by m' in describing that portion of its 
 orbit, we will have for the three components of the attraction exerted by any portion 
 dm f , 
 
 R.dmo', Sidm'o, and 
 
 and integrating about the entire ring, we find for the complete components, 
 
 flit /*2n- /2jr 
 
 Rdmo, I Sidmo', and I W4m<>'- 
 Jo Jo 
 
 But by the conditions, 
 
 dt dM' 
 
 OTo' " T 2;r ' 
 
 and hence the components are, 
 
 J rtn 1 r>2 1 /*2ir 
 
 m^'RidM', 5- m Q 'S t dM' and ~- mo'T^id/lf, 
 
 <i7T J ^TT J /7T J 
 
 which are identical with 
 
 ^ f '" fldM', ^ (^ SdM' and g. if" W^dM'. 
 
 Thus the expressions giving the secular variations are seen to be the same whether 
 these are derived from the moving planet or from the elliptic ring.* 
 
 The work of GAUSS contains no application to the determination of secular vari- 
 ations nor are all the formulas necessary for this purpose there developed; the first 
 integration alone is effected, and it is shown that by changing first to the variable 
 E' and afterward introducing a new variable, T, each of the complicated integrals 
 may be made to depend upon elliptic integrals whose values GAUSS obtained by the 
 introduction of a new algorithm called by him the Arithmetico-geometrical mean. 
 
 * Other interesting geometrical aspects of the problem are treated by Bour < 5) , Hill < s "' (38) , and Halphen < 28) , but 
 for brevity a detailed account of these is here omitted. 
 
42 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 The first application of GAUSS'S method was made by NicoLAi (2) , who determined 
 by it the secular variations of the Earth's orbit, but the results only were published.* 
 The first development of the method is by CLAUSEN (3) who also applied it to a determi- 
 nation of the perturbations of Tuttle's Comet produced by the action of Jupiter (4) , 
 dividing the disturbed orbit into 120 parts with reference to the true anomaly. It 
 was next, in 1867, applied by ADAMS ((!) to the orbit of the November meteors with a 
 special view to ascertaining the cause of the steady progression of the node of the 
 orbit, but in this investigation certain small terms were neglected by ADAMS and the 
 solution of a fundamental cubic equation which occurs in the original method was in 
 this manner avoided. 
 
 No further applications of GAUSS'S method seem to have been made until after 
 the publication of HILL'S extensive development (8) and modifications of it in 1882. 
 
 3. HILL'S FIRST MODIFICATION OF GAUSS'S METHOD. 
 
 Although the first of the above integrations may be rigorously effected, the 
 value of the second must be approximated to by a mechanical quadrature about the 
 orbit of m, a greater or less number of terms being employed in the quadrature 
 according as the disturbed orbit is more or less eccentric. Since either the true, 
 eccentric, or mean anomalies may be selected as the variables, it becomes of im- 
 portance to decide which of these must be chosen in order to render the quadrature 
 most accurate. It is readily proved! that the inequalities of distribution of a series 
 of points on an elliptic orbit corresponding to a series of equidistant values of the 
 eccentric anomaly are of the order of the square of the eccentricity while for the 
 other two anomalies they are of the order of the first power of this quantity, and 
 therefore HILL has employed the eccentric anomalies throughout his development, 
 although SEELiGER (9) showed that a still higher accuracy will be obtained if the true 
 anomalies are chosen. 
 
 If, therefore, we decide to make the integrations with reference to the eccentric 
 anomalies, we will obtain, since 
 
 dM = ^dE, AM' =-- r -,dE', and r' = a' (I - e' cos E'), 
 
 [de~\ ncos<p I f 2 " f 2ir . 
 j, = ,,,, r -r \ sin v Kar(I e cos E) 
 
 L^JOO & 2 (1 + TO) 47T 2 J J 
 
 + (cos v + cos E) Sar(l - e' cos E')]dEdE'dt, 
 
 *See Article 11. 
 
 t See Tisserand's Mecanique Celeste, Vol. I, page 442, 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 43 
 
 and writing, 
 
 1 r 2 " nr 1 C 2 " nr 
 
 R = ^ I ,R(l-e' cosE')dE', S = ^ ,S(l-e' cos E')dE', 
 
 JIT JQ TOO zir JQ mo 
 
 the expression for the secular variation will become, 
 
 [del m'n 1 T 2 " 
 
 '- cos <p JT- [sin v-Ro + (cos v + cos E)S u ]dE. 
 at Joo 1 -f- TO ZTT J 
 
 In order to find the values of R , S , and W , it is first necessary to express 
 R, S and W in terms of E' '. For this purpose that part of the disturbing force arising 
 from the action of the disturbing planet upon the sun need not be included, for it is 
 known that this has no secular term.| Considering therefore only the action of m' 
 upon m, it is evident from a figure that R, S and W will have the values, 
 
 _ m o' I r> cos # r 1 
 
 I / 
 
 _ TOO' r' sin & cos y 
 
 TT7 ' 
 
 : -^r sm y> 
 
 and also that 
 
 A-' = r 2 - 2rr' cos + r' 2 , 
 
 in which & is the angle included between the radii vectores, A is the distance between 
 the two bodies, and y is the inclination of the plane which includes r and r' to the 
 plane of the orbit of the disturbed body. 
 
 If n and n' denote the angular distances respectively of the perihelia of the two 
 orbits from the ascending node of the orbit of m' upon the orbit of m, and if / be their 
 mutual inclination, we will have, 
 
 cos = cos (v + n) cos (v r + II') + sin (v + U) sin (v' + n') cos /, 
 sin cos 7 = sin (v + n) cos (v' + II') + cos (v + II) sin (v' + II') cos I, 
 
 sin sin 7 = sin / sin (v' + n') 
 
 The values of n, n', and / are obtained from the original elements by a direct 
 solution of the spherical triangle whose sides are n and n', and in which the angle 
 included between these sides is I. (See Article 7.) 
 
 tSee Hill's "On Gauss's Method <8 >, , . ," page 321. 
 
44 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 If we now eliminate v' from the above expressions by the equations, 
 
 r' cos v' = a' (cos E' - e'), r' sin v' = a' cos <p' sin E', r' = a' (1 e' cos E'), 
 
 the resulting equations giving R, S, W, and A will be expressed wholly in terms of 
 the variable E' . In order to simplify these results, we assume certain new auxiliaries 
 defined by the equations, 
 
 k cos (.K-n) = cos IT, k sin (K-U) =- cos / sin IT, k' cos (K'-Il) = cos 7 cos II', 
 
 k' sin (K' - II) = - sin II', 
 A = r z + 2ka'e'r cos (v + K) + a'\ 
 B cos e = ka'r cos (v + K) + o'V, 
 B sin e = /r'a' cos <p' r sin (v + K'), 
 
 A c = ka' cos (v + K), A s = k'a' cos <?' sin (v + K') 
 
 B c = - ka' sin (v + K), B s = k'a' cos v ' cos (v + A'') 
 
 C c = a' sin n' sin /, C s = a' cos <p' cos II' sin /. 
 
 C = a'V 3 , 
 when the desired expressions become, 
 
 -*, R = 4, (cos E' - e') + A s sin E' - r 
 
 W?o 
 
 ^U = 5 c (cos ' - e') + 5 8 sin ' 
 
 W?0 
 
 ~ W = C c (cos ' - e') + C, sin #' 
 
 2 
 
 A 2 = A-2B cos (' - e) + C cos 
 
 In order to effect the integrations, GAUSS here introduced a new variable, T, 
 connected with E' by the relations, 
 
 N sin E' = a + a' sin T + a" cos T 
 N cos E' = /3 + 0' sin 7 1 + 0" cos 7 T 
 N = 7 + 7' sin T + j" cos 7 7 , 
 
 the quantities , a', a", 0, 0' . . . being subject to the conditions that 
 (N sin #') 2 + (N cos #') 2 - JV 2 and sin 2 T + cos 2 T - 1 
 
 shall be identically zero, and also being so chosen that the coefficients of sin T, cos T, 
 and sin T cos T shall vanish in the expression 2V 2 A 2 which therefore must take the 
 
 form, 
 
 G - G' sin 2 T + G" cos 2 T. 
 
 From these conditions it is derived that the coefficients G, G' and G" in the trans- 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 45 
 
 formed expression for JV 2 A 2 must severally satisfy the cubic equation, 
 
 x(x - A)(x + C) + B 2 x + B 2 C sin 2 e = 0, 
 
 and hence that they must be the roots of this equation. By substituting for x the 
 successive values, -- (7, 0, a' 2 cos 2 <p' and + A, the first member is seen to take in 
 succession the corresponding values, 
 
 - B 2 C cos 2 e 
 + B 2 C sin 2 e 
 
 - a' 4 cos 2 <p' r 2 sin 2 1 sin 2 (v + II) 
 + B*-(A + C sin 2 e). 
 
 Since, even when cos (v + K) has its maximum negative value, the value of A 
 exceeds that of (r a') 2 , it is evident that A is always positive, and therefore that 
 the above equation has one negative root which lies between C and 0, one positive 
 root lying between and a' 2 cos 2 <p', and that the third root lies between this value 
 and + A. The roots are represented by G", G', and G, respectively, and thus 
 G", G' and G are always positive quantities, the last being the largest and the first 
 the smallest except when <p' exceeds 45, a case not met with in any of the planetary 
 orbits. 
 
 Since a, j3, y, a', 0' . . . must retain the same values whatever the values of 
 E' and T, we may, by writing the equations arising from the three conditions above 
 stated and equating the coefficients of the like terms in the two members, obtain 
 a series of equations which are sufficient for the determination of these quantities 
 in terms of G, G', G" and the other known auxiliaries. Upon substituting the resulting 
 expressions for sin E' and cos E' in the equations defining R , S , and T^o, and noticing 
 that JV 2 A 2 may be written, 
 
 G - G' sin 2 T + G" cos 2 T = (G' + G"} 1 1 - ^-^'sin 2 T }, 
 
 I tr + tr J 
 
 we obtain each of the components in the form, 
 
 m s sin 2 T + m c cos 2 T 
 or 
 
 1 I'"" 
 e = - 
 
 If we now write, 
 
 G 
 
 /I I fin - 7, 
 and consider that from LANDEN'S well-known transformation, 
 
46 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 c ~ 
 
 and also notice that 
 
 r' 2 dr 
 
 Jo (l-c 2 sin* 
 
 * sin 2 TdT 
 
 cos* TdT 
 
 ![/ TT\ 
 
 = ?L v c>2 / 
 
 it is evident that each of the above three integrals becomes expressible wholly in 
 terms of the rapidly convergent series of LANDEN. 
 
 For the purposes of the present computation HILL (S) has computed to ten places 
 the logarithms of the quantities 
 
 K = sec 2 KL, L' = L ~ B , and N u = sec 2 (1 + '), 
 
 and these correct to eight places are tabulated at intervals of one tenth of a degree 
 for all values of from = to = 50. 
 
 From a direct substitution it is now seen that the final resulting values of Ro, 
 S and W are as follows, in which the symbols N, P, Q, etc., are written for abbrevi- 
 ation and have the meanings stated in Article 7 : 
 
 7? = - N - QG' + VJS, 
 S = PF* + VJ, 
 W = PF, + VJ 3 
 
 The integration with respect to E' having been thus entirely completed, that 
 in regard to E is effected by mechanical quadratures. Since each variation is a 
 function of E alone, it follows by the principles of quadratures that if any one of them 
 be expanded into a periodic series involving the sines and cosines of E and its multiples, 
 
 the secular term of the series, which is rigorously equal to |ir I f(E)dE, may be 
 
 I/O 
 
 also obtained by forming the values of f(E) for 2j equidistant values of E, from 
 E = to E = 360, and dividing the sum by 2j. The expression thus obtained, 
 
 will be subject only to the error involved in dropping those terms which contain a 
 multiple of E not lower than 2j. An inspection of the known forms of the series 
 which express the variations renders it evident that the error thus committed is of 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 47 
 
 the order 2j in terms of the eccentricities and mutual inclinations of the orbits except 
 in the one case of the variation of the Mean Longitude, in which, as this variation 
 depends wholly upon the expansion of 2(r/a)JRo> it is of the order 2j + 1. 
 
 The resulting equations giving the values of all the secular variations are those 
 stated in Article 7. 
 
 4. HILL'S SECOND MODIFICATION OF GAUSS'S METHOD. THE WORK 
 
 OF CALLANDREAU AND INNES. 
 
 In HILL'S second modification of GAUSS'S method (8> , the well-known expressions 
 for the roots of a cubic equation when this is solved by the trigonometric method are 
 introduced, and thus, throughout the integrals, the quantities p, q and 0' occur instead 
 of the roots G, G' and G", the equations connecting these quantities being, 
 
 G = 2 9 sin60 - + P, G' = 2gsin 
 G" = 2g sin (60+ I') -p. 
 
 It was shown in GAUSS'S original memoir (1) that 
 
 dT 
 
 f 
 
 (m 2 cos 2 T + n 2 sin 2 T) * J ( m ' 2 COS 2 
 
 if m' = \(m + w) and w' = V mn, and that by repeating this transformation by the 
 employment of the equations, 
 
 m " = i( m > + n '), n" = Jrnfri, 
 
 m'" = \(m" + n"), n'" = JriW, 
 etc. etc., 
 
 m ( *> and n (k) very rapidly approach a single limit, p., which GAUSS named the Arith- 
 metico-geometrical Mean. It thus follows that our first integral is equal to 7r/2^, 
 and that integrals of the form 
 
 p (sin 2 T - cos 2 T)dT 
 J (m 2 cos 2 T + n 2 sin 2 T)* 
 
 become equal to ir/2 w/ju in which w is a very rapidly converging series involving 
 m, n, m', n', etc., in its successive terms. 
 
 The integral expressions which actually enter into the equations for .R , S , and 
 W o are 
 
 (f\f \ -/o 
 
 60 - Q ) - r -r^ 
 3 / 4 (m 2 
 
 n 2 ) 
 
48 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 V3 w sin 6' 
 
 in which 
 
 tf 
 
 A 
 o 
 
 and 
 
 the values of # , , and T^ being connected by comparatively simple relations with 
 these quantities and with known auxiliaries. 
 
 HILL accordingly suggested that tables of these functions should be computed, 
 and this was first done by MONS. 0. CALLANDREAU (I:!> who however adopted as an 
 argument the quantity a defined by the relation 
 
 1 
 1 
 
 cos 
 
 1 + a 6' 
 
 V COS Q 
 
 * <5 
 
 and tabulated the logarithms of the functions r 4 n 4 x(0') and <K0') *- x(0') at intervals 
 of 0.001 from = 0.000 to a = 0.400; of 0.002 from = 0.400 to a = 0.600 and of 
 0.005 from this point to the extreme value, = 1.000. This paper repeats the 
 derivation of all formulas necessary when the second method alone is employed, 
 essentially as this was given by HILL, and also contains a direct proof that R , S , 
 and W can be expressed wholly in terms of the complete elliptic integrals, F and E. 
 
 Similar tables were also computed by MR. R. T. A. lNNES (22) , the functions 
 here tabulated being (1 -- )/(! + 4 ) \l/(6') and \l/(6') + *(0') to the argument 0' 
 at intervals of one degree, from 0' = 90 to 0' = + 90. 
 
 Whether the first or second methods be employed, the values of the integrals 
 involved may also, as was pointed out by HiLL (38) , be approximated to with great 
 rapidity by the use of JACOBI'S Nome, q (American Journal of Mathematics, Vol. 23, 
 page 321. In the Astronomical Journal, No. 511, a brief application is given to a case 
 in the action of Venus on the Earth). This function is defined by the equation, 
 
 
 q = e~'x, 
 
 in which K' is the complete elliptic integral of the first kind complementary to K, 
 from which there may be derived, 
 
 KE=(1+ 3V + 5V ) - U + <? 2 + <? 6 ' ) 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 49 
 
 The values of log [(//tan 2 0] computed to ten decimal places for each degree of 8 
 from 6 = to 6 = 45 are given by lNNES (39) . When exceeds 45, the values of K 
 and E are readily obtained from their expressions in terms of the complementary 
 complete integrals whose moduli are sin (ir/2 0), and to which the table is therefore 
 directly applicable. 
 
 Lastly, in the second method, HILL recommends that the quadratures be per- 
 formed upon the quantities a/r R , a/r S and a/r W directly, all constant and 
 evanescent factors which appear in the expressions for the variations being removed 
 from under the integral signs and reserved until the integration has been completed. 
 
 5. THE METHOD OF HALPHEN AND ITS MODIFICATIONS BY ARNDT 
 
 AND INNES. 
 
 It was first pointed out by BRUNS (29) that the periods of the elliptic functions 
 of the first and second integrals can be evaluated without a knowledge of the three 
 roots, but it was HALPHEN (28) who first applied this remarkably elegant method of 
 analysis to the present problem. It was shown by him that if o> and 17 are the two 
 periods in question, then R , So, and W may be obtained in the form aw + br\, in 
 which a and b are rational functions of the coefficients of the cubic equation and w 
 and TJ are expressible in terms of certain hyper-geometric series in which the common 
 variable is an absolute invariant of the elliptic functions. 
 
 The three integrals entering into the problem have the form, 
 
 /" 
 
 Jo 
 
 IdT 
 
 7 1 ) 3 ' 
 
 in which / has the values 1, sin 2 T and cos 2 T, respectively, in the three cases; by 
 introducing the new variable, s, defined by the relation, 
 
 G + G" s-G' 
 G' + G" ' s - G ' 
 these become, 
 
 G> + G " r-o" te G + G " r-o ds 
 
 ~Jl ~^s (s ' G}> ~ 2 ~^~i v! (s ' 
 
 and 
 
 n 
 respectively, in which 
 
 f/~< i r<u\(r< r\ir<t i r>\ 
 n = ((j -\- (j ; ((JT Cr ) (u -f- Cr ; 
 
 and 
 
 S = - 4( - G)(s - G')(s + (?") 
 
50 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 Introducing the WEIERSTRASSIAN r function through the relation 
 
 C ds 
 
 U = % 
 
 u being the elliptic integral of the first kind and 61, e 2 , and e 3 the roots of the cubic 
 equation increased by one third of the coefficient of x 2 , and considering that from the 
 theory of these functions, 
 
 s - G = r(u) - ei, T(w) = d, r(w + w') = e 2 , and T(w') = e 3 , 
 the first integral will become, 
 
 2 ^~ f" " f r () - e 'l d = 2 \ e ^ + ~ ( + ') - - ' w 'l 
 J u L " f J 
 
 ff and o-' being the second WEIERSTRASSIAN functions, which are connected with 
 the periods, w and rj, by the equations, 
 
 (<> + ') = 77 + ?;'; -co' = )?'. 
 
 The three integrals consequently take the final forms, 
 
 _G' + G" n G + G", n G'-G, 
 
 2 ~ - (eico + 77) ; 2 - (^co + ?;), and 2 - ~- (630) + 77). 
 
 fv ra 71 
 
 A direct substitution of these expressions for the integrals in the equations which 
 define .R > So and Wo leads, after some reduction, to forms which are seen to contain 
 only these integrals themselves, the coefficients of the cubic equation with other 
 known auxiliaries, and the quantity n. But if, for brevity, we write the original 
 cubic equation in the form, 
 
 x 3 - P,x 2 + P*x - P, = 0, 
 and let 
 
 X = P, 1 - 3P 2 and p = PjP 2 - 9P 3 , 
 
 then the invariants, g 2 and g 3 , and the absolute invariant, g, will have the values, 
 
 to = |X; 0i = A(2PiX - 3p), and g = 2 3 * 270 3 2 , 
 
 and w will be given by, 
 
 n ~ = leC^ 3 - 27^ 3 2 ), 
 
 in which the last factor is the discriminant. Thus, except for <o and TJ, our final 
 expressions are obtained wholly in terms of the coefficients of the cubic equation, 
 and a knowledge of the roots becomes unnecessary. 
 
 In the paper by BRUNS, before referred to, it is shown that w and i\ are directly 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 51 
 
 expressible in terms of a hyper-geometric series whose variable is the absolute invari- 
 ant, g. By a simple transformation the relations may be placed in the following 
 forms, which are more convenient in practical application. 
 
 A. , o- 
 
 ' 12' l > g 
 
 DR. Louis ARNDT i30) has fully developed this method, deriving all the formu- 
 las necessary for its application and stating tables for F(w) and F(T)) for values of 
 (g - l)/0 from (g - l)/g = 0.000 to (g - !),/</ = 0.980, the interval being 0.001. 
 
 In a recent paper by INNES (SI) the complete formulas for this method are derived 
 when the quadrature is applied directly to the expressions (a/r)R 0) (a/r)S Q and 
 (a}r)Wo, as suggested in the second method of HILL. The development is nearly 
 identical with that of ARNDT except that the forms of the hyper-geometric series 
 are slightly changed, the variable, 
 
 , * Vg - 1 
 sin 2 2 = *?=- 
 
 being preferred. The values of the logarithms of 
 
 F () = F(!, I 2, sin 2 -} and F (77) = F (- g, | 2, sin 2 * Y 
 
 were published by MR. FRANK RoBBiNS (32) for all values of i, at intervals of one 
 degree from i = 1 to i = 90, the computation having been made to ten places and 
 published to seven, and these tables, computed with seven place logarithms, have 
 been extended from i = 90 to i = 180 by MR. C. J. MERFiELD (33) . 
 
 Although the preceding methods are of great mathematical elegance, it is doubt- 
 ful whether their formulas lead to so accurate results as those of HILL'S first method 
 when seven place logarithms are employed. (See the computations of Jupiter on 
 Mars (24) and of Saturn on Mars (25> , Article 10.) Moreover, when the method is 
 applied which is explained in the computation of Jupiter on Mercury (Article 10), 
 the roots of the cubic equation are so readily obtained that the avoidance of its 
 solution becomes a matter of no practical importance. Accordingly HILL'S first 
 modification of GAUSS'S method has been employed throughout all of the following 
 computation. 
 
THE COMPUTATION. 
 
 6. THE ELEMENTS OF THE ORBITS AND THE ADOPTED MASSES. 
 
 The values adopted for the elements of the several orbits, to serve as the basis 
 for this computation, were taken in each case from HILL'S "New Theory of Jupiter 
 and Saturn." (16) Those of the four inner planets will be found on page 192; those 
 of Jupiter and Saturn on page 558; of Uranus on page 109, and of Neptune on page 
 161. The epoch throughout is 1850.0 G. M. T. 
 
 The values of the masses finally selected by HILL, and here adopted, will be 
 found on page 554 for Mercury, Venus and the Earth; on page 192 for Mars; on page 
 19 for Jupiter and Saturn, and on page 161 for Neptune. The mass of Uranus as 
 stated in the "New Theory " is 1 -r- 22640, but at DR. HILL'S suggestion this is here 
 diminished to 1 -4- 22800, (A. J., No. 316). The value assumed for the mass of 
 Mercury when the first of these computations were made was 1 -r- 5000000, but all 
 of 'the results are here changed to agree with the value 1 -r- 7500000 stated below. It 
 seems not improbable that even this latter fraction is too large, but the true value of 
 this element is still very uncertain. 
 
 
 X 
 
 i 
 
 S2 I e 
 
 n 
 
 Mercury 
 Venus 
 Earth 
 Mars 
 Jupiter 
 Saturn 
 Uranus 
 Neptune 
 
 o / // 
 
 75 7 13.62 
 129 27 42.83 
 100 21 39.73 
 333 17 51.74 
 11 54 31.67 
 90 6 41.37 
 168 15 6.70 
 43 17 30.30 
 
 o / // 
 
 7 7.71 
 3 23 35.01 
 0.00 
 1 51 2.24 
 1 18 42.10 
 2 29 40.19 
 46 20.54 
 1 47 1.68 
 
 46 33' 8^63 
 75 19 53.08 
 
 48 23 54.59 
 98 56 19.79 
 112 20 49.05 
 73 14 8.00 
 130 7 31.83 
 
 0.20560476 
 0.00684311 
 0.01677114 
 0.09326803 
 0.04825511 
 0.05606025 
 0.0469236 
 0.0084962 
 
 5381016^260 
 2106641.357 
 1295977.416 
 689050.784 
 109256.626 
 43996.21506 
 .15425.752 
 7864.935 
 
 
 logo 
 
 1-j-m 
 
 Mercury 
 
 9.5878217 
 
 7 500 000 
 
 Venus 
 
 9.8593378 
 
 408 134 
 
 Earth 
 
 0.0000000 
 
 327 000 
 
 Mars 
 
 0.1828971 
 
 3 093 500 
 
 Jupiter 
 
 0.7162374 
 
 1 047.879 
 
 Saturn 
 
 0.9794956 
 
 3 501.6 
 
 Uranus 
 
 1.2831044 
 
 22 800 
 
 Neptune 
 
 1.4781414 
 
 19 700 
 
 52 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 53 
 
 7. THE FORMULAS EMPLOYED IN THE COMPUTATION. 
 
 The following formulas are written in the order in which they were applied. 
 When the right hand member appears in two different forms, one of these was used 
 in the first computation and the other in the duplication, though sometimes other 
 obvious modifications were made use of in the several cases differing from those which 
 are here written. 
 
 The values of /, n, and n' were obtained from the general equations: 
 
 sin / sin (H w) = sin i' sin (ft' ft) , 
 
 sin I cos (II w) = sin i cos i' + cos i sin i' cos (ft' ft) 
 
 = cos i cos i' [ tan i + tan i' cos (ft' ft)], 
 sin / sin (n' w') = sin i sin (ft' ft), 
 sin/ cos (II' a/) = cos i sin i' sin i cos i' cos (ft' ft), 
 
 = cos i cos i' [tan it tan i cos (ft' ft)]. 
 
 When the Earth is the disturbing body, these become, 
 
 / = i; n = 180 + co; n' = 180 + *' - ft; 
 and when the Earth is the disturbed body, 
 
 / = i'- n = TT - ft'; II' = *-' - ft'. 
 
 As i, i' and / are always small, eight place logarithms were generally here used to 
 insure the accuracy of n and n'. 
 
 The auxiliaries k, k', K, K' and C were then found from the relations: 
 
 k sin (A' - n) = - cos I sin n'; k cos (K - n) = cos n'; k' sin (K' - n) = - sin n'; 
 
 A;' cos (K'-U) = cos / cos n'; C = o'V 2 , 
 
 and their values were tested by the equations, 
 
 tan .7 = p ; tan \(K - K' + 90) cot |(A' + K' - 90 - 211) = "!" ( ^, ~ "\ , 
 v sin (^11 -p G) 
 
 sin ( K- K') = sin I tan / sin (K' - n) sin (K - n) cot n'. 
 
 The orbit of the disturbed planet being then divided into 2j parts in regard to 
 the eccentric anomaly, the following equations were applied to each point of division, 
 of which those marked with an asterisk are test equations upon the sums of the 
 functions corresponding respectively to the odd and even points of division of the 
 orbit. The sums corresponding to the odd points are designated by Si, those to the 
 even points by S 2 , and 
 
 2 = S, + S 2 . 
 
54 THE SECULAR VARIATIONS OP THE ELEMENTS 
 
 r sin v = a cos ip sin E, 
 r cos v = a (cos E e), 
 
 r 2 = a 2 (1 - 2 e cos # + e 2 cos 2 .E), 
 
 (the last equation giving the value of r 2 for use in A, N, and J 3 . Since 
 
 i log r 2 = log T-, 
 this affords also an independent test of r). 
 
 *S,t; + 180 = S 2 *>; *Z^ = Z 2 r = ja. 
 A = r 2 + 2&aYr cos (w + X) + a' 2 = [r + ka'e' cos (t> + X)] 2 + a' 2 [l - fcV 2 cos 2 (v + 
 
 (the second form used with ZECH'S tables in the duplication). 
 
 *SiA = ^A = ja 2 + |jaV + j[a' 2 - 2kaa'ee' cos X] 
 J3 sin e = A;'o' cos ^>'r sin (v + K'} 
 B cos e = ka'r cos (t> + K) + a' 2 e' 
 *2iB sin e = S 2 B sin e = jk'aa' cos ^>' e sin ^' 
 *2iB cos = 2 2 B cos e = j[a'~e' kaa'e cos A'] 
 g = B 2 C sin 2 e 
 
 To effect the solution of the cubic equation, h and I were found from the equations, 
 
 the very convenient test equation, hi = B* -AC, being applied to each pair of values. 
 The first approximation to G was then obtained from 
 
 G = h ~ h(h -I)' 
 and further approximations by successive applications of 
 
 G(G - iy 
 
 (The number of trials required never exceeded three.) G' and G" then follow from 
 the equations, 
 
 G' = (A - C - G) + (A -C- GY + ; G" = 
 and we have for verification, 
 
 ft i rt/ _ rtii _ A ri . fir _ L _i_ _ a _ . fin _ 
 
 ^'-'' = " 
 
 . 
 
 G")' 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 55 
 
 (In some cases the first approximations to G were found by, 
 
 sin e' = - 3 ; G = 2q sin (60 - &') + p, 
 
 the solution being then finished as before). 
 
 The modulus, (c = sin 6), of the elliptic integrals employed in the computation 
 was separately found by the two equations, 
 
 C* T I /""' /"" I firr 
 
 sin 2 6 = Q , Q,, ; tan 2 6 = Q _ Q> , 
 
 and with 6 as an argument the values of log K , log L ', and log N were taken from 
 the tables of HILL'S memoir (8) , the interpolation being effected in both directions to 
 second differences by the well-known formulas, 
 
 
 in which n + n' = 1. 
 
 The logarithms of Af, P, Q, and F were then obtained from, 
 
 o . 
 
 "(G + G") 3 ' ~(G + (?") 2 ' y "(? + "' W 
 
 the first three being verified by similar operations performed upon the values of 2 t 
 and 2 2 formed from the respective logarithms, and the last by the use of ZECH'S 
 tables and also by the equation, 
 
 V = ar-(G + G'T l [GN + G"(N - L ')]K . 
 The following auxiliaries were next obtained : 
 
 Ji' = a' 2 cos 2 <p'[l - sin 2 / sin 2 (v + n)] + G" 
 
 = [a' cos >p' + a' cos <p' sin 7 sin (v + n)][a' cos <p' a' cos <?' sin / sin (v + II)] 4- G", 
 J 2 = ka'e'r sin (v + K) - |a' 2 cos 2 <p' sin 2 1 sin 2(i> + n) 
 
 = ka'e'r sin (t; + K) - a' 2 cos 2 ^' sin (v + n) cos (v + n) sin 2 /, 
 
 the second form being employed with ZECH'S tables in the duplication 
 Jz = cos 2 <p' sin I cos 7 r sin (t; + H) -- ' sin / sin II' r 2 , 
 
 tZ CL 
 
 a'e' 
 *Si/ 3 = SaJa = - ja- cos 2 ^' sin I cos / e sin n --- sin / sin n' S^ 2 . 
 
 (Z 
 
 = a' 2 sin <p' cos <p' cos / fi sin e, 
 
56 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 *2iF 2 = S 2 F 2 = jk'aa' 3 ee' cos 2 <p' cos 7 sin K', 
 
 F 3 = -- sin 49' cos y>' sin I r cos (v + II) .B sin e. 
 
 There were next obtained, 
 
 B = - N -QG' + 7JY; S = PF* + VJ,; W = PF, + VJ a ; 
 
 fl<"> = -R sin E; 5<-> = -S ; W = .Sfl<">; (c) = . 
 r r z? zj 
 
 and the very accurate test equation, 
 
 sin <p !-Ai (>) + cos 9? B (c) = 0, 
 was applied. 
 
 These values were then substituted in the following series of equations, and 
 the final values of the differential coefficients obtained: 
 
 [dc ~\ TH'TI 1 
 
 ~dt Joo = 1 + m ' COS *" ' 2?- 2 f sin " ' Ro + ^ cos " + cos 
 
 m'n cos ? 1 
 
 di 1 
 
 sJ o = 
 
 mn 
 
 [dftl m'n 
 ,. 
 a< Joo 1 + 
 
 sec <p 1 
 ^-s-.S 
 TO sin i 2j 
 
 TO r n If O r- P 1. , . 9 v[d x ~\ , .,iTd1 
 = ^r- - s-.ZI 2-J2p | + 2sin 1 7 +2sin 2 ^, -,. . 
 
 oo 1 + TO 2j a 2 L rf/ Joo 2 L rf/ Joo 
 
 <lt 
 
 When the Earth is the disturbed body, the third and fourth equations are re- 
 placed by, 
 
 " dp ~| m'n 1 
 
 - sec ip-- n -.2 sin ( + ) (r 
 
 L + TO 2j 
 
 m'n I _ 
 
 [da ~\ 
 ZJ l ,- 
 
 In this case 
 
 rdxl = 
 
 L dt Joo 
 
 and the last term of the expression for [dL/dt] 00 disappears, but the first two equations 
 remain unaltered. 
 
 8. THE VALUES OF THE PRELIMINARY CONSTANTS. 
 
 The values obtained for those constants which are direct functions of the ele- 
 ments of the orbits in the several cases are shown in the following tables. The last 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 57 
 
 columns of these tables contain the differences between the values of K K' formed 
 directly and the same angles obtained from the test formula of the preceding article. 
 The other test equations were also exactly satisfied. 
 
 An examination of the formulas of the preceding article renders it evident that 
 with any two planets / will have the same value whether the inner or the outer 
 planet is the disturbing one, while the value of II in the first case will differ 180 from 
 that of n' in the second, and that of n' in the first case will similarly differ 180 
 from that of II in the second. These conditions will be seen to be here satisfied 
 very exactly, the minute discrepancies which occur being due to the fact that in some 
 places eight place logarithms were employed, in others seven, and in still others the 
 attainment of a higher accuracy throughout the entire computation was sought by 
 the use of the dash, ( ), which was placed above the last figure of each logarithm for 
 which the interpolation led to a value coinciding more nearly with the mean of the 
 two adjacent figures than with either one of them. In combining such logarithms 
 the effect of the dash was taken into consideration by methods which are obvious. 
 
 Mercury by 
 
 Venus 
 
 Earth 
 
 Mars 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 Neptune 
 
 Mercury by 
 
 Venus 
 
 Earth 
 
 Mars 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 Neptune 
 
 Venus by 
 
 Mercury 
 
 Earth 
 
 Mars 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 Neptune 
 
 II 
 
 II' 
 
 K 
 
 4 
 
 7 
 5 
 6 
 6 
 6 
 7 
 
 20 
 
 9 
 17 
 23 
 19 
 1 
 
 42.982 
 7.710 
 10.165 
 15.310 
 44.130 
 17.399 
 42.654 
 
 230 
 208 
 209 
 218 
 229 
 211 
 223 
 
 39 
 34 
 13 
 5 
 26 
 43 
 12 
 
 31.39 
 4.99 
 54.31 
 54.72 
 43.69 
 10.39 
 39.15 
 
 284 
 233 
 107 
 154 
 244 
 304 
 191 
 
 54 
 48 
 24 
 49 
 17 
 49 
 16 
 
 1.27 
 31.10 
 19.31 
 24.01 
 50.53 
 47.06 
 25.42 
 
 305 
 334 
 101 
 63 
 345 
 266 
 324 
 
 43 
 57 
 45 
 8 
 17 
 43 
 34 
 
 2.40 
 50.59 
 36.16 
 32.52 
 17.36 
 32.95 
 3.91 
 
 K' 
 
 log k 
 
 logfc' 
 
 log<7 
 
 resid. 
 
 305 
 
 47 
 
 57.49 
 
 9.9988328 
 
 9.9999176 
 
 5.3891826 
 
 0.007 
 
 334 
 
 33 
 
 18.85 
 
 9.9978879 
 
 9.9988719 
 
 6.4491252 
 
 0.002 
 
 101 
 
 53 
 
 33.05 
 
 9.9983990 
 
 9.9998432 
 
 8.3052599 
 
 0.004 
 
 63 
 
 24 
 
 30.76 
 
 9.9995281 
 
 9.9978563 
 
 8.7995614 
 
 0.002 
 
 345 
 
 
 
 30.90 
 
 9.9978013 
 
 9.9994926 
 
 9.4563012 
 
 0.013 
 
 267 
 
 3 
 
 12.48 
 
 9.9982188 
 
 9.9991396 
 
 9.9089914 
 
 0.000 
 
 324 
 
 24 
 
 7.31 
 
 9.9968502 
 
 9.9998757 
 
 8.8147322 
 
 0.001 
 
 II 
 
 II' 
 
 K 
 
 4 
 
 20 
 
 42.980 
 
 104 
 
 54 
 
 1.27 
 
 50 
 
 39 
 
 31.37 
 
 54 
 
 19 
 
 21.08 
 
 3 
 
 23 
 
 35.010 
 
 234 
 
 7 
 
 49.75 
 
 205 
 
 1 
 
 46.65 
 
 29 
 
 8 
 
 21.75 
 
 1 
 
 56 
 
 2.460 
 
 208 
 
 26 
 
 43.81 
 
 52 
 
 18 
 
 22.07 
 
 156 
 
 9 
 
 18.63 
 
 2 
 
 15 
 
 11.352 
 
 247 
 
 36 
 
 52.56 
 
 130 
 
 2 
 
 45.43 
 
 117 
 
 32 
 
 48.56 
 
 2 
 
 3 
 
 12.046 
 
 281 
 
 7 
 
 33.71 
 
 241 
 
 43 
 
 52.16 
 
 39 
 
 24 
 
 36.81 
 
 2 
 
 37 
 
 16.883 
 
 233 
 
 30 
 
 46.37 
 
 272 
 
 18 
 
 13.25 
 
 321 
 
 12 
 
 24.44 
 
 2 
 
 46 
 
 38.369 
 
 265 
 
 47 
 
 34.23 
 
 179 
 
 34 
 
 46.34 
 
 86 
 
 12 
 
 46.11 
 
58 
 
 Venus by 
 
 Mercury 
 Earth 
 Mars 
 Jupiter 
 Saturn 
 Uranus 
 Neptune 
 
 Earth by 
 
 THE SECULAR VARIATIONS OF 
 K' log k 
 
 54 9 38.85 9.9992531 
 29 3 44.28 9.9998637 
 156 7 24.89 9.9998450 
 117 35 25.70 9.9998033 
 39 22 46.29 9.9997836 
 321 12 41.79 9.9995460 
 86 12 49.67 9.9999999 
 
 / II 
 
 THE ELEMENTS 
 log k' log C 
 
 9.9994984 7.8017097 
 9.9993746 6.4491252 
 9.9999075 8.3052599 
 9.9998610 8.7995614 
 9.9999375 9.4563012 
 9.9999992 9.9089914 
 9.9994896 8.8147322 
 
 II' 
 
 resid. 
 
 0.000 
 0.003 
 0.000 
 0.001 
 0.003 
 0.001 
 0.001 
 
 K 
 
 Mercury 
 Venus 
 
 o 
 
 7 
 3 
 
 / 
 
 
 23 
 
 7.710 
 35.010 
 
 
 
 53 
 
 25 
 
 48 
 1 
 
 31.10 
 46.65 
 
 O 
 
 28 
 54 
 
 34 
 
 7 
 
 n 
 
 4.99 
 49.75 
 
 
 
 25 
 330 
 
 i 
 25 
 56 
 
 13.33 
 
 48.79 
 
 Mars 
 
 1 
 
 51 
 
 2.240 
 
 51 
 
 57 
 
 45.14 
 
 284 
 
 53 
 
 57.15 
 
 127 
 
 3 
 
 21.25 
 
 Jupiter 
 Saturn 
 
 1 
 2 
 
 18 
 29 
 
 42.100 
 40.190 
 
 1 
 
 348 
 
 25 
 
 
 19.94 
 50.68 
 
 272 
 337 
 
 58 
 45 
 
 11.88 
 52.32 
 
 88 
 10 
 
 27 
 13 
 
 5.264 
 49.89 
 
 Uranus 
 
 
 
 46 
 
 20.540 
 
 27 
 
 7 
 
 31.73 
 
 95 
 
 
 
 58.70 
 
 292 
 
 6 
 
 31.40 
 
 Neptune 
 
 1 
 
 47 
 
 1.680 
 
 330 
 
 14 
 
 7.90 
 
 273 
 
 9 
 
 58.47 
 
 57 
 
 4 
 
 3.92 
 
 Earth by 
 
 K' 
 
 log A; 
 
 logfc' 
 
 logC 
 
 resid. 
 
 Mercury 
 Venus 
 
 25 
 330 
 
 3 
 51 
 
 36.28 
 5.12 
 
 9.9992608 
 9.9994999 
 
 9.9974965 
 9.9997387 
 
 7.8017097 
 5.3891826 
 
 0.002 
 0.008 
 
 Mars 
 
 127 
 
 4 
 
 14.72 
 
 9.9997885 
 
 9.9999850 
 
 8.3052599 
 
 0.009 
 
 Jupiter 
 Saturn 
 
 88 
 10 
 
 27 
 16 
 
 10.859 
 
 6.88 
 
 9.9998865 
 9.9999411 
 
 9.9999998 
 9.9996473 
 
 8.7995614 
 9.4563012 
 
 0.010 
 0.003 
 
 Uranus 
 
 292 
 
 6 
 
 34.66 
 
 9.9999609 
 
 9.9999997 
 
 9.9089914 
 
 0.005 
 
 Neptune 
 
 57 
 
 4 
 
 14.94 
 
 9.9997902 
 
 9.9999994 
 
 8.8147322 
 
 0.001 
 
 Mars by 
 
 n 
 
 n' 
 
 K 
 
 Mercury 
 
 5 
 
 9 
 
 10.165 
 
 287 
 
 24 
 
 19.31 
 
 29 
 
 13 
 
 54.31 
 
 258 
 
 16 
 
 20.56 
 
 Venus 
 
 1 
 
 56 
 
 2.460 
 
 232 
 
 18 
 
 22.07 
 
 28 
 
 26 
 
 43.81 
 
 203 
 
 52 
 
 27.49 
 
 Earth 
 
 1 
 
 51 
 
 2.240 
 
 104 
 
 53 
 
 57.15 
 
 231 
 
 57 
 
 45.14 
 
 232 
 
 57 
 
 4.23 
 
 Jupiter 
 
 1 
 
 26 
 
 6.381 
 
 149 
 
 47 
 
 4.35 
 
 188 
 
 22 
 
 45.31 
 
 321 
 
 24 
 
 28.37 
 
 Saturn 
 
 2 
 
 21 
 
 52.110 
 
 176 
 
 17 
 
 59.42 
 
 293 
 
 4 
 
 38.76 
 
 243 
 
 12 
 
 17.28 
 
 Uranus 
 
 1 
 
 11 
 
 40.460 
 
 120 
 
 39 
 
 30.31 
 
 315 
 
 36 
 
 26.40 
 
 165 
 
 2 
 
 41.49 
 
 Neptune 
 
 2 
 
 22 
 
 41.388 
 
 152 
 
 49 
 
 56.12 
 
 222 
 
 47 
 
 52.02 
 
 290 
 
 3 
 
 32.71 
 
 Mars by 
 
 K' 
 
 log k 
 
 log k' 
 
 logt' 
 
 resid. 
 
 Mercury 
 Venus 
 
 258 
 203 
 
 4 
 50 
 
 28.68 
 49.03 
 
 9.9995819 
 9.9999439 
 
 9.9986621 
 9.9998087 
 
 7.8017097 
 5.3891826 
 
 0.017 
 0.012 
 
 Earth 
 
 232 
 
 55 
 
 19.79 
 
 9.9998596 
 
 9.9999141 
 
 6.4491252 
 
 0.009 
 
 Jupiter 
 Saturn 
 
 321 
 243 
 
 24 
 
 14 
 
 9.72 
 23.99 
 
 9.9999971 
 9.9996870 
 
 9.9998667 
 9.9999432 
 
 8.7995614 
 9.4563012 
 
 0.007 
 0.007 
 
 Uranus 
 
 165 
 
 3 
 
 26.31 
 
 9.9999538 
 
 9.9999519 
 
 9.9089914 
 
 0.003 
 
 Neptune 
 
 290 
 
 
 
 35.50 
 
 9.9998274 
 
 9.9997986 
 
 8.8147322 
 
 0.006 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 59 
 
 9. THE RADII VECTORES AND THE TRUE ANOMALIES. 
 
 The values of log r and v for the points of division employed in the four different 
 cases are given in the following tables. In each case the equations, 
 
 2 ir = 2 2 r = ja, and 2 : y + 180 = S 2 t> 
 
 were exactly satisfied, and the values of r were also obtained from the equation stated 
 in Article 7 for obtaining the value of r 2 . 
 
 E 
 
 MEKCUBY. 
 
 log r 
 
 V 
 
 
 E 
 
 VENUS. 
 logr 
 
 V 
 
 
 
 
 
 O 
 
 i 
 
 // 
 
 O 
 
 
 O 
 
 1 
 
 // 
 
 
 
 9.4878584 
 
 
 
 
 
 0.00 
 
 
 
 9.8563557 
 
 
 
 
 
 0.00 
 
 15 
 
 9.4916716 
 
 18 
 
 25 
 
 28.96 
 
 15 
 
 9.8564576 
 
 15 
 
 6 
 
 6.54 
 
 22.5 
 
 9.4963313 
 
 27 
 
 32 
 
 14.93 
 
 30 
 
 9.8567564 
 
 30 
 
 11 
 
 47.87 
 
 30 
 
 9.5026623 
 
 36 
 
 32 
 
 7.50 
 
 45 
 
 9.8572313 
 
 45 
 
 16 
 
 40.52 
 
 45 
 
 9.5195925 
 
 54 
 
 4 
 
 7.02 
 
 60 
 
 9.8578493 
 
 60 
 
 20 
 
 24.50 
 
 60 
 
 9.5407098 
 
 70 
 
 50 
 
 41.41 
 
 75 
 
 9.8585680 
 
 75 
 
 22 
 
 44.64 
 
 67.5 
 
 9.5522314 
 
 78 
 
 55 
 
 7.36 
 
 90 
 
 9.8593378 
 
 90 
 
 23 
 
 31.50 
 
 75 
 
 9.5640735 
 
 86 
 
 46 
 
 40.73 
 
 105 
 
 9.8601064 
 
 105 
 
 22 
 
 42.20 
 
 90 
 
 9.5878217 
 
 101 
 
 51 
 
 53.65 
 
 120 
 
 9.8608213 
 
 120 
 
 20 
 
 20.31 
 
 105 
 
 9.6103385 
 
 116 
 
 9 
 
 54.15 
 
 135 
 
 9.8614342 
 
 135 
 
 16 
 
 35.65 
 
 112.5 
 
 9.6207149 
 
 123 
 
 3 
 
 1.59 
 
 150 
 
 9.8619040 
 
 150 
 
 11 
 
 43.65 
 
 120 
 
 9.6303194 
 
 129 
 
 46 
 
 44.60 
 
 165 
 
 9.8621990 
 
 165 
 
 6 
 
 4.12 
 
 135 
 
 9.6467730 
 
 142 
 
 49 
 
 52.77 
 
 180 
 
 9.8622996 
 
 180 
 
 
 
 0.00 
 
 150 
 
 9.6589887 
 
 155 
 
 27 
 
 29.02 
 
 195 
 
 9.8621990 
 
 194 
 
 53 
 
 55.88 
 
 157.5 
 
 9.6633518 
 
 161 
 
 39 
 
 20.97 
 
 210 
 
 9.8619040 
 
 209 
 
 48 
 
 16.35 
 
 165 
 
 9.6664956 
 
 167 
 
 48 
 
 0.75 
 
 225 
 
 9.8614342 
 
 224 
 
 43 
 
 24.35 
 
 180 
 
 9.6690267 
 
 180 
 
 
 
 0.00 
 
 240 
 
 9.8608213 
 
 239 
 
 39 
 
 39.69 
 
 195 
 
 9.6664956 
 
 192 
 
 11 
 
 59.25 
 
 255 
 
 9.8601064 
 
 254 
 
 37 
 
 17.80 
 
 202.5 
 
 9.6633518 
 
 198 
 
 20 
 
 39.03 
 
 270 
 
 9.8593378 
 
 269 
 
 36 
 
 28.50 
 
 210 
 
 9.6589887 
 
 204 
 
 32 
 
 30.98 
 
 285 
 
 9.8585680 
 
 284 
 
 37 
 
 15.36 
 
 225 
 
 9.6467730 
 
 217 
 
 10 
 
 7.23 
 
 300 
 
 9.8578493 
 
 299 
 
 39 
 
 35.50 
 
 240 
 
 9.6303194 
 
 230 
 
 13 
 
 15.40 
 
 315 
 
 9.8572313 
 
 314 
 
 43 
 
 19.48 
 
 247.5 
 
 9.6207149 
 
 236 
 
 56 
 
 58.41 
 
 330 
 
 9.8567564 
 
 329 
 
 48 
 
 12.13 
 
 255 
 
 9.6103385 
 
 243 
 
 50 
 
 5.85 
 
 345 
 
 9.8564576 
 
 344 
 
 53 
 
 53.46 
 
 270 
 
 9.5878217 
 
 258 
 
 8 
 
 6.35 
 
 
 
 
 
 
 285 
 
 9.5640735 
 
 273 
 
 13 
 
 19.27 
 
 
 
 
 
 
 292.5 
 
 9.5522314 
 
 281 
 
 4 
 
 52.64 
 
 
 
 
 
 
 300 
 
 9.5407098 
 
 289 
 
 9 
 
 18.59 
 
 
 
 
 
 
 315 
 
 9.5195925 
 
 305 
 
 55 
 
 52.98 
 
 
 
 
 
 
 330 
 
 9.5026623 
 
 323 
 
 27 
 
 52.50 
 
 
 
 
 
 
 337.5 
 
 9.4963313 
 
 332 
 
 27 
 
 45.08 
 
 
 
 
 
 
 345 
 
 9.4916716 
 
 341 
 
 34 
 
 31.04 
 
 
 
 
 
 
60 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 E 
 
 THE EARTH. 
 logr 
 
 V 
 
 
 E 
 
 MARS. 
 log r 
 
 V 
 
 
 O 
 
 O 
 
 1 
 
 // 
 
 O 
 
 
 
 O 
 
 i 
 
 II 
 
 
 
 
 9.9926546 
 
 
 
 
 
 0.00 
 
 
 
 
 
 .1403760 
 
 
 
 
 
 0.00 
 
 22. 
 
 5 
 
 9.9932181 
 
 22 
 
 52 
 
 14.25 
 
 30 
 
 
 
 .1463201 
 
 32 
 
 47 
 
 24.62 
 
 30 
 
 
 9.9936460 
 
 30 
 
 29 
 
 2.39 
 
 45 
 
 
 
 .1532670 
 
 48 
 
 54 
 
 53.41 
 
 45 
 
 
 9.9948189 
 
 45 
 
 41 
 
 0.84 
 
 60 
 
 
 
 .1621567 
 
 64 
 
 44 
 
 46.64 
 
 60 
 
 
 9.9963428 
 
 60 
 
 50 
 
 8.59 
 
 90 
 
 
 
 .1828971 
 
 95 
 
 21 
 
 5.913 
 
 67. 
 
 5 
 
 9.9972036 
 
 68 
 
 23 
 
 26.41 
 
 120 
 
 
 
 .2026920 
 
 124 
 
 31 
 
 47.15 
 
 90 
 
 
 0.0000000 
 
 90 
 
 57 
 
 39.46 
 
 135 
 
 
 
 .2106341 
 
 138 
 
 39 
 
 52.35 
 
 112. 
 
 5 
 
 0.0027784 
 
 113 
 
 23 
 
 5.92 
 
 150 
 
 
 
 .2166313 
 
 152 
 
 34 
 
 23.40 
 
 120 
 
 
 0.0036266 
 
 120 
 
 49 
 
 43.50 
 
 180 
 
 
 
 .2216237 
 
 180 
 
 
 
 0.00 
 
 135 
 
 
 0.0051200 
 
 135 
 
 40 
 
 31.82 
 
 210 
 
 
 
 .2166313 
 
 207 
 
 25 
 
 36.60 
 
 150 
 
 
 0.0062624 
 
 150 
 
 28 
 
 37.29 
 
 225 
 
 
 
 .2106341 
 
 221 
 
 20 
 
 7.65 
 
 157.5 
 
 0.0066776 
 
 157 
 
 51 
 
 53.72 
 
 240 
 
 
 
 .2026920 
 
 235 
 
 28 
 
 12.85 
 
 180 
 
 
 0.0072232 
 
 180 
 
 
 
 0.00 
 
 270 
 
 
 
 .1828971 
 
 264 
 
 38 
 
 54.087 
 
 202. 
 
 5 
 
 0.0066776 
 
 202 
 
 8 
 
 6.29 
 
 300 
 
 
 
 .1621567 
 
 295 
 
 15 
 
 13.36 
 
 210 
 
 
 0.0062624 
 
 209 
 
 31 
 
 22.71 
 
 315 
 
 0.1532670 
 
 311 
 
 5 
 
 6.59 
 
 225 
 
 
 0.0051200 
 
 224 
 
 19 
 
 28.18 
 
 330 
 
 
 
 .1463201 
 
 327 
 
 12 
 
 35.38 
 
 240 
 
 
 0.0036266 
 
 239 
 
 10 
 
 16.50 
 
 
 
 
 
 
 
 247. 
 
 5 
 
 0.0027784 
 
 246 
 
 36 
 
 54.08 
 
 
 
 
 
 
 
 270 
 
 
 0.0000000 
 
 269 
 
 2 
 
 20.54 
 
 
 
 
 
 
 
 292. 
 
 5 
 
 9.9972036 
 
 291 
 
 36 
 
 33.59 
 
 
 
 
 
 
 
 300 
 
 
 9.9963428 
 
 299 
 
 9 
 
 51.41 
 
 
 
 
 
 
 
 315 
 
 
 9.9948189 
 
 314 
 
 18 
 
 59.16 
 
 
 
 
 
 
 
 330 
 
 
 9.9936460 
 
 329 
 
 30 
 
 57.61 
 
 
 
 
 
 
 
 337. 
 
 5 
 
 9.9932181 
 
 337 
 
 7 
 
 45.75 
 
 
 
 
 
 
 
 10. THE SEPARATE RESULTS. 
 
 The values found for the intermediate auxiliary functions which depend upon ", 
 as well as the final perturbations of the four inner planets in each case are now stated 
 in the following tables. The results of the application of the more important test 
 equations are also shown, but all of the test equations of Article 7 were also applied, 
 and each computation (except the first), was, after its completion, duplicated from the 
 beginning, the forms of the equations being changed in the duplication when this 
 was possible. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 61 
 
 MERCURY. 
 
 ACTION OF VKNUS ON MERCURY. 
 
 E 
 
 A 
 
 B cos t 
 
 B sin e 
 
 1000000 Xff 
 
 h 
 
 
 
 0.619543952 
 
 + 0.13308441 
 
 - 0.18036925 
 
 0.7970904 
 
 0.52358614 
 
 30 
 
 0.627434998 
 
 + 0.22218381 
 
 - 0.06982371 
 
 0.1194506 
 
 0.52390836 
 
 60 
 
 0.647116316 
 
 + 0.24372756 
 
 + 0.07193966 
 
 0.1268000 
 
 0.52384406 
 
 90 
 
 0.675632886 
 
 + 0.19194286 
 
 + 0.20693555 
 
 0.0491867 
 
 0.52344851 
 
 120 
 
 0.706503003 
 
 + 0.08070542 
 
 + 0.29899200 
 
 2.1902889 
 
 0.52319742 
 
 150 
 
 0.730295757 
 
 - 0.06017874 
 
 + 0.32344233 
 
 1.5631633 
 
 0.52358280 
 
 180 
 
 0.738317327 
 
 - 0.19295989 
 
 + 0.27373528 
 
 1.8358797 
 
 0.52446104 
 
 210 
 
 0.727259050 
 
 - 0.28205939 
 
 + 0.16318979 
 
 0.6524819 
 
 0.52500778 
 
 240 
 
 0.701243272 
 
 - 0.30360314 
 
 + 0.02142638 
 
 0.0112481 
 
 0.52470755 
 
 270 
 
 0.669559472 
 
 - 0.25181838 
 
 - 0.11356958 
 
 0.3160138 
 
 0.52391075 
 
 300 
 
 0.641856586 
 
 - 0.14058090 
 
 - 0.20562585 
 
 1.0359483 
 
 0.52329644 
 
 330 
 
 0.624398293 
 
 + 0.00030325 
 
 - 0.23007624 
 
 1.2969588 
 
 0.52323374 
 
 z, 
 
 4.054580456* 
 
 - 0.17962654f 
 
 + 0.28009822J 
 
 5.9972554 
 
 3.14309264 
 
 2 2 
 
 4.054580456 
 
 - 0.17962659 
 
 + 0.28009814 
 
 5.9972551 
 
 3.14309193 
 
 E 
 
 G 
 
 G' 
 
 G" 
 
 o 
 
 
 
 0.09593332 
 
 0.52358258 
 
 0.09595274 
 
 0.000015866 
 
 O 
 
 25 
 
 / 
 
 20 
 
 n 
 53.90 
 
 30 
 
 0.10350215 
 
 0.52390782 
 
 0.10350489 
 
 0.000002203 
 
 26 
 
 23 
 
 25.33 
 
 60 
 
 0.12324776 
 
 0.52384346 
 
 0.12325032 
 
 0.000001964 
 
 29 
 
 
 
 59.15 
 
 90 
 
 0.15215988 
 
 0.52344311 
 
 0.15217844 
 
 0.000013171 
 
 32 
 
 37 
 
 46.70 
 
 120 
 
 0.18328109 
 
 0.52318510 
 
 0.18331625 
 
 0.000022837 
 
 36 
 
 17 
 
 45.71 
 
 150 
 
 0.20668846 
 
 0.52356735 
 
 0.20672760 
 
 0.000023681 
 
 38 
 
 55 
 
 52.70 
 
 180 
 
 0.21383179 
 
 0.52444977 
 
 0.21385942 
 
 0.000016369 
 
 39 
 
 41 
 
 12.31 
 
 210 
 
 0.20222678 
 
 0.52500393 
 
 0.20223677 
 
 0.000006145 
 
 38 
 
 21 
 
 51.31 
 
 240 
 
 0.17651123 
 
 0.52470749 
 
 0.17651140 
 
 0.000000121 
 
 35 
 
 27 
 
 1.91 
 
 270 
 
 0.14562423 
 
 0.52390915 
 
 0.14562996 
 
 0.000004142 
 
 31 
 
 49 
 
 7.06 
 
 300 
 
 0.11853565 
 
 0.52329155 
 
 0.11855723 
 
 0.000016698 
 
 28 
 
 25 
 
 30.42 
 
 330 
 
 0.10114005 
 
 0.52322787 
 
 0.10117042 
 
 0.000024501 
 
 26 
 
 5 
 
 20.70 
 
 Si 
 
 0.91134083 
 
 3.14305994 
 
 0.91144736 
 
 0.000073855 
 
 194 
 
 13 
 
 23.40 
 
 S 2 
 
 0.91134154 
 
 3.14305922 
 
 0.91144808 
 
 0.000073843 
 
 194 
 
 13 
 
 23.80 
 
 * 6a 2 + 3aV + 6[o' 2 - 2kaa'ee' cos K] = + 4.054580460. 
 t 6[a'V - kaa'e cos K] = - 0.17962650. 
 t - Qk'aa' cos <p'-e sin K' = + 0.28009816. 
 
62 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 
 ACTION 
 
 OF VENUS ON MERCUBY. 
 
 
 E 
 
 log*. 
 
 log Lo' 
 
 log N log N log P 
 
 logQ 
 
 
 
 0.06678154 
 
 0.36107029 
 
 0.27485672 9.0518226 9.9748963 
 
 9.6076810 
 
 30 
 
 0.07267844 
 
 0.36875602 
 
 0.28344481 9.0869397 0.0171823 
 
 9.6511278 
 
 60 
 
 0.08883727 
 
 0.38974368 
 
 0.30686976 9.1792738 0.1306112 
 
 9.7669404 
 
 90 
 
 .0.11429390 
 
 0.42259487 
 
 0.34345542 9.2994384 0.2842725 
 
 9.9240134 
 
 120 
 
 0.14429575 
 
 0.46098687 
 
 0.38608356 9.4147448 0.4383831 
 
 0.0821541 
 
 150 
 
 0.16872258 
 
 0.49199359 
 
 0.42040790 9.4960335 0.5500428 
 
 0.1974492 
 
 180 
 
 0.17620114 
 
 0.50144271 
 
 0.43084933 9.5225000 0.5845077 
 
 0.2336318 
 
 210 
 
 0.16325170 
 
 0.48506821 
 
 0.41274966 9.4887994 0.5335323 
 
 0.1813814 
 
 240 
 
 0.13698082 
 
 0.45165804 
 
 0.37573831 9.4055654 0.4173889 
 
 0.0613864 
 
 270 
 
 0.10823963 
 
 0.41480523 
 
 0.33478930 9.2928156 0.2691020 
 
 9.9083455 
 
 300 
 
 0.08503270 
 
 0.38481172 
 
 0.30136861 9.1761377 0.1234343 
 
 9.7587487 
 
 330 
 
 0.07094409 
 
 0.36649704 
 
 0.28092115 9.0860236 0.0150983 
 
 9.6482336 
 
 Si 
 
 0.69812922 
 
 2.54971331 
 
 2.07576629 5.7500443 1.6692215 
 
 9.5105424 
 
 2 2 
 
 0.69813034 
 
 2.54971496 
 
 2.07576824 5.7500501 1.6692301 
 
 9.5105508 
 
 E 
 
 logV 
 
 J/ 
 
 1000 X Ji J 3 
 
 1000 X Fj 
 
 
 
 9.6076650 
 
 0.521404654 
 
 - 2.7049984 - 0.024195167 
 
 + 0.6439191 
 
 30 
 
 9.6511256 
 
 0.520191194 
 
 - 0.6256166 - 0.032354328 
 
 + 0.2492710 
 
 60 
 
 9.7669384 
 
 0.521003862 
 
 + 1.8268277 - 0.030122813 
 
 - 0.2568249 
 
 90 
 
 9.9240003 
 
 0.522559008 
 
 + 2.6401451 - 0.018096342 
 
 - 0.7387609 
 
 120 
 
 0.0821316 
 
 0.523207843 
 
 + 2.0172777 + 0.000503667 
 
 - 1.0674025 
 
 150 
 
 0.1974260 
 
 0.522626872 
 
 + 1.0213969 + 0.020692271 
 
 - 1.1546906 
 
 180 
 
 0.2336159 
 
 0.521405157 
 
 + 0.3980040 + 0.037057786 
 
 - 0.9772364 
 
 210 
 
 0.1813754 
 
 0.520383901 
 
 + 0.3750436 + 0.045213990 
 
 - 0.5825883 
 
 240 
 
 0.0613863 
 
 0.520288911 
 
 + 0.7059590 + 0.042976535 
 
 - 0.0764923 
 
 270 
 
 9.9083414 
 
 0.521364846 
 
 + 0.6938449 + 0.030947096 
 
 + 0.4054437 
 
 300 
 
 9.7587319 
 
 0.522844254 
 
 - 0.4295092 + 0.012350057 
 
 + 0.7340853 
 
 330 
 
 9.6482088 
 
 0.523030858 
 
 - 2.2812182 - 0.007832614 
 
 + 0.8213734 
 
 Si 
 
 9.5104690 
 
 3.130154681* 
 
 + 1.8135608 + 0.038570065 
 
 - 0.9999517 
 
 S2 
 
 9.5104775 
 
 3.130156679 
 
 + 1.8235957 + 0.038570073 
 
 - 0.9999517 
 
 * 2i(J,' - G") = + 3.130080826. 
 2t(Ji - G") = + 3.130082836. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 63 
 
 E 
 
 30 
 60 
 90 
 120 
 150 
 180 
 210 
 240 
 270 
 300 
 330 
 
 2! 
 
 2 2 
 
 ACTION OF VENUS ON MERCUKY. 
 10000 X F 3 #o 1000 X -So W, R w 1000 X S<"> 
 - 0.24640136 0.05971623 - 0.4883004 - 0.009827036 0.00000000 - 1.5879204 
 - 0.00762098 0.06444673 - 0.0208471 - 0.014490447 + 0.10127648 - 0.0655216 
 - 0.09149188 0.08146579 + 0.7212240 - 0.017625381 + 0.20314033 + 2.0766362 
 - 0.49802851 0.11164771 + 0.7947010 - 0.015287000 + 0.28842180 + 2.0529668 
 - 0.89441990 0.15077545 - 0.4917115 + 0.000363093 + 0.30587345 - 1.1518366 
 - 0.92808606 0.18433784 - 2.4881611 + 0.032271915 + 0.20211372 - 5.4561938 
 - 0.56751805 0.19359808 - 3.0725539 + 0.063241297 0.00000000 - 6.5837414 
 - 0.13320998 0.17488253 - 1.4207354 + 0.068605698 - 0.19174661 - 3.1154764 
 + 0.01209827 0.14154301 + 0.6131517 + 0.049504580 - 0.28714394 + 1.4363109 
 - 0.19301611 0.10799082 + 1.3152406 + 0.025022994 - 0.27897487 + 3.3976883 
 - 0.46971306 0.08194590 + 0.7289596 + 0.007023584 - 0.20433749 + 2.0989094 
 - 0.49748432 0.06575439 - 0.1643583 - 0.003535808 - 0.10333144 - 0.5165700 
 
 - 2.25744598 0.70904446 - 1.9892305 + 0.092680137 + 0.01753235 - 3.7116419 
 - 2.25744596 0.70906002 - 1.9841603 + 0.092587352 + 0.01775908 - 3.7031067 
 
 
 
 Ro COS V 
 
 
 
 
 E 
 
 RO sin v 
 
 + (cos v + cos E)So 
 
 + ( -sec 2 <f + l ) sin v 
 \a ) 
 
 W cos u 
 
 Wo sin u 
 
 -2 -Bo 
 a 
 
 
 
 - 0.00097660 
 
 - 0.05971623 
 
 - 0.008630594 
 
 - 0.004699311 
 
 - 0.09487655 
 
 30 
 
 + 0.03833160 
 
 - 0.05180530 
 
 - 0.006100201 
 
 - 0.013143840 
 
 - 0.10594280 
 
 60 
 
 + 0.07755258 
 
 - 0.02541165 
 
 + 0.002882594 
 
 - 0.017388064 
 
 - 0.14618182 
 
 90 
 
 + 0.10909895 
 
 + 0.02454507 
 
 + 0.009914505 
 
 - 0.011635936 
 
 - 0.22329541 
 
 120 
 
 + 0.11643405 
 
 + 0.09565744 
 
 - 0.000337472 
 
 + 0.000133975 
 
 - 0.33255092 
 
 150 
 
 + 0.08098459 
 
 + 0.16537954 
 
 - 0.032192272 
 
 - 0.002265848 
 
 - 0.43432170 
 
 180 
 
 + 0.00614511 
 
 + 0.19359808 
 
 - 0.055541660 
 
 - 0.030242129 
 
 - 0.46680548 
 
 210 
 
 - 0.07011629 
 
 + 0.16039915 
 
 - 0.041182679 
 
 - 0.054870101 
 
 - 0.41204381 
 
 240 
 
 - 0.10947718 
 
 + 0.08954949 
 
 - 0.009624818 
 
 - 0.048559933 
 
 - 0.31218788 
 
 270 
 
 - 0.10595403 
 
 + 0.01957233 
 
 + 0.007191939 
 
 - 0.023967193 
 
 - 0.21598164 
 
 300 
 
 - 0.07680517 
 
 - 0.02822239 
 
 + 0.005196776 
 
 - 0.004724857 
 
 - 0.14704334 
 
 330 
 
 - 0.03941900 
 
 - 0.05265111 
 
 - 0.003501678 
 
 + 0.000490095 
 
 - 0.10809245 
 
 2! 
 
 + 0.01287279 
 
 + 0.26545474 
 
 - 0.066055174 
 
 - 0.105480319 
 
 - 1.49964599 
 
 2 2 
 
 + 0.01292582 
 
 + 0.26543968 
 
 - 0.065870386 
 
 - 0.105392823 
 
 - 1.49967781 
 
 
 sin (/> |4i (<) + cos <p-BiJ 
 
 c) = - 0.0000000083. 
 
 
 
 
64 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 [de/dt] w 
 [dx/dt] w 
 [di/dt]oQ 
 
 = + 11321.398 TO' 
 
 = +1133127.6 TO' 
 
 60449.278 m' 
 
 - 792605.00 TO' 
 
 = +1127216.0 TO' 
 
 = -1326653.0 TO' 
 
 log eoeff. 
 p 4.0539001 
 p 6.0542788 
 n 4.7813911 
 n 5.8990568 
 p 6.0520072 
 n 6. 1227573 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 [de/dt] m = +0.027739414 
 [dxAft]oo = +2.7763615 
 [di/dtlw = -0.14811133 
 [dfi/dfloo = -1.9420214 
 [dr/dt] m = +2.7618772 
 = -3.2505323 
 
 [dL/di] 
 
 w 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Hill. Method of Gauss. 
 
 [de/dt] m 
 e[dw/dt]oo 
 [di/dt] w 
 GITI i r/vo '/"//i 
 
 bill I [Ctlfcy CttJOO 
 
 [dx/dt] w 
 
 +0.02780 
 +0.56811 
 -0.14812 
 -0.23648 
 
 -3.2769 
 
 +0.02774 
 +0.57086 
 -0.14806 
 -0.23665 
 
 +0.0277391 
 +0.567852 
 -0.1481112 
 -0.2367447 
 +2.776347 
 -3.250522 
 
 +0.0277394 
 +0.567855 
 -0.1481113 
 -0.2367449 
 +2.776361 
 -3.250532 
 
 NOTES. 
 
 This is the only one of the twenty eight computations that was not duplicated, 
 but the values of 6 were computed by two different formulas and all known test 
 equations were applied. As an illustration of his first modification of GAUSS'S 
 method HILL published this complete computation from exactly the same elements 
 as here employed, and DR. Louis ARNDT states that he has verified the results and 
 found them correct. (Bulletin de la Societe des Sciences Naturelles de Neuchatel, 
 Vol. XXIV). INNES states however that the test arising from the constancy of the 
 major axis is not satisfied, the residual being 0.00075 (M. N., Vol. LII, page 87), 
 but this statement is an error, for the residual obtained from HILL'S figures is 
 0.0000000088, a practically exact agreement with that here obtained. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 65 
 
 Upon comparing the present computation with that of HILL, the following 
 slight discrepancies may be noticed : 
 
 n', K and K' differ by less than 0".l from HILL'S values, a difference doubtless 
 due to the fact that the preliminary computation was here effected with eight place 
 logarithms while HILL employed but seven. The value of I for 330 should be 
 0.10114009 instead of 0.11014009, and G" for 180 should be 0.00001637 instead of 
 0.00001617. These are misprints merely. The values of the logarithms of K , 
 LQ, and N Q in HILL seem to be slightly in error throughout, a double interpolation 
 to second differences from HILL'S values of 6 giving with the three functions most 
 in error, 
 
 Hill's Values. 
 
 For E = 60, log N 0.30686978 0.3068691 
 For E = 150, log L ' 0.49199342 0.4919942 
 For E = 180, log L ' 0.50144261 0.5014421 
 
 The effect of these differences upon the final coefficients is, however, almost in- 
 appreciable. 
 
 It is evident from an inspection of the final sums that a division into twelve parts 
 is necessary in this case, the terms from the sixth to the eleventh orders, inclusive, 
 amounting to l/600th of the whole for [di/dt] 00 and to l/1200th of the whole for 
 
 ACTION OF THE EARTH ON MERCURY. 
 
 Bcos 
 
 + 0.29403604 
 + 0.32704393 
 + 0.25768623 
 + 0.10454706 
 
 - 0.09134000 
 
 - 0.27748723 
 
 - 0.40401673 
 
 - 0.43702459 
 
 - 0.36766690 
 
 - 0.21452774 
 
 - 0.01864062 
 + 0.16750659 
 
 - 0.32994198f 
 
 - 0.32994198 
 
 * 6a 2 + 3aV + 6[a' 2 - 2kaa'ee' cos A] = + 6.90363352. 
 t 6[a'V - kaa'e cos A] = - 0.3299419S. 
 t - Qk'aa' cos <p'-e sin A' = + 0.20460788. 
 
 E 
 
 A 
 
 
 
 1.10386215 
 
 30 
 
 1.11164085 
 
 60 
 
 1.12870093 
 
 90 
 
 1.15278960 
 
 120 
 
 1.17861164 
 
 150 
 
 1.19808875 
 
 180 
 
 1.20368356 
 
 210 
 
 1.19273750 
 
 240 
 
 1.16934301 
 
 270 
 
 1.14208711 
 
 300 
 
 1.11943230 
 
 330 
 
 1.10628960 
 
 s, 
 
 6.90363359* 
 
 2 2 
 
 6.90363341 
 
 B sin 
 
 KXXXTxff 
 
 h 
 
 - 0.13175730 
 
 0.04882864 
 
 1.00008277 
 
 + 0.06103685 
 
 0.01047875 
 
 1.00111148 
 
 + 0.24661358 
 
 0.17106421 
 
 1.00173668 
 
 + 0.37524793 
 
 0.39606073 
 
 1.00128597 
 
 + 0.41247210 
 
 0.47853578 
 
 1.00022588 
 
 + 0.34831240 
 
 0.34124242 
 
 0.99978030 
 
 + 0.19995995 
 
 0.11246339 
 
 1.00066133 
 
 + 0.00716576 
 
 0.00014443 
 
 1.00216106 
 
 - 0.17841099 
 
 0.08952996 
 
 1.00285589 
 
 - 0.30704518 
 
 0.26517354 
 
 1.00212493 
 
 - 0.34426952 
 
 0.33336685 
 
 1.00067710 
 
 - 0.28010968 
 
 0.22068934 
 
 0.99977580 
 
 + 0.20460782t 
 
 1.23378883 
 
 6.00623965 
 
 + 0.20460808 
 
 1.23378921 
 
 6.00623954 
 
66 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 E 
 
 ACTION OF THE EARTH ON MERCURY. 
 G G' G" 
 
 
 
 
 
 0.10349811 
 
 1.00007732 
 
 0.10355071 0.00004715 
 
 O 
 
 18 
 
 46' 28.61 
 
 30 
 
 0.11024810 
 
 1.00111031 
 
 0.11025877 0.00000949 
 
 19 
 
 22 58.72 
 
 60 
 
 0.12668299 
 
 1.00171716 
 
 0.12683714 0.00013464 
 
 20 
 
 51 17.40 
 
 90 
 
 0.15122236 
 
 1.00123943 
 
 0.15152995 0.00026105 
 
 22 
 
 54 42.04 
 
 120 
 
 0.17810449 
 
 1.00016768 
 
 0.17843084 0.00026815 
 
 25 
 
 4.02 
 
 150 
 
 0.19802718 
 
 0.99973772 
 
 0.19824193 0.00017219 
 
 26 
 
 27 9.31 
 
 180 
 
 0.20274096 
 
 1.00064732 
 
 0.20281046 0.00005541 
 
 26 
 
 45 34.58 
 
 210 
 
 0.19029517 
 
 1.00216104 
 
 0.19029526 0.00000008 
 
 25 
 
 50 0.71 
 
 240 
 
 0.16620585 
 
 1.00284522 
 
 0.16627021 0.00005369 
 
 24 
 
 1 53.32 
 
 270 
 
 0.13968091 
 
 1.00209425 
 
 0.13990074 0.00018915 
 
 21 
 
 57 13.77 
 
 300 
 
 0.11847394 
 
 1.00063933 
 
 0.11879216 0.00028045 
 
 20 
 
 10 34.78 
 
 330 
 
 0.10623253 
 
 0.99975109 
 
 0.10646458 0.00020734 
 
 19 
 
 3 48.08 
 
 2, 
 
 0.89570634 
 
 6.00609403 
 
 0.89669152 0.00083949 
 
 135 
 
 35 52.71 
 
 2 2 
 
 0.89570625 
 
 6.00609384 
 
 0.89669123 0.00083930 
 
 135 
 
 35 52.63 
 
 E 
 
 
 logtfo 
 
 0.03586144 
 
 log Lo' 
 0.32053269 
 
 log Wo log N log P 
 0.22947519 8.5993188 8.9197434 
 
 
 logQ 
 8.8287400 
 
 30 
 
 0.03828768 
 
 0.32372814 
 
 0.23305754 8.6307049 8.9534609 
 
 
 8.8632763 
 
 60 
 
 0.04451494 
 
 0.33191845 
 
 0.24223558 8.7125510 9.0428624 
 
 
 8.9539831 
 
 90 
 
 0.05408514 
 
 0.34447345 
 
 0.25629369 8.8165734 9.1597444 
 
 
 9.0722159 
 
 120 
 
 0.06487989 
 
 0.35858872 
 
 0.27208270 8.9130565 9.2712667 
 
 
 9.1849500 
 
 150 
 
 0.07303960 
 
 0.36922627 
 
 0.28397010 8.9788974 9.3482020 
 
 
 9.2629066 
 
 180 
 
 0.07483693 
 
 0.37156571 
 
 0.28658308 8.0002544 9.3712099 
 
 
 9.2865324 
 
 210 
 
 0.06949181 
 
 0.36460451 
 
 0.27880654 8.9738846 9.3366140 
 
 
 9.2517535 
 
 240 
 
 0.05973313 
 
 0.35186484 
 
 0.26456369 8.9063079 9.2556584 
 
 
 9.1696145 
 
 270 
 
 0.04950056 
 
 0.33846383 
 
 0.24956627 8.8114799 9.1479627 
 
 
 9.0600557 
 
 300 
 
 0.04158092 
 
 0.32806159 
 
 0.23791429 8.7102233 9.0374863 
 
 
 8.9477383 
 
 330 
 
 0.03700237 
 
 0.32203558 
 
 0.23116016 8.6301758 8.9522476 
 
 
 8.8613541 
 
 2i 
 
 0.32140725 
 
 2.06253200 
 
 1.53285453 2.8417119 4.8982271 
 
 
 4.3715583 
 
 2 2 
 
 0.32140716 
 
 2.06253178 
 
 1.53285430 2.8417160 4.8982316 
 
 
 4.3715621 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 67 
 
 ACTION OF THE EARTH ON MERCURY. 
 
 E 
 
 30 
 60 
 90 
 120 
 150 
 180 
 210 
 240 
 270 
 300 
 330 
 
 Si 
 
 io g y 
 8.8287147 
 8.8632712 
 8.9539113 
 9.0720772 
 9.1848079 
 9.2628156 
 9.2865032 
 9.2517534 
 9.1695861 
 9.0599551 
 8.9475885 
 8.8612431 
 
 4.3711117 
 4.3711156 
 
 1000 X F, 
 
 J,' J 2 J, 
 
 0.9963685 - 0.0084115617 - 0.04554861 
 0.9875038 - 0.0046146363 - 0.08975347 
 0.9853935 + 0.0065532595 - 0.10655243 
 0.9913722 + 0.0136249408 - 0.09143424 
 0.9979641 + 0.0119853300 - 0.04844492 
 0.9998176 + 0.0047535770 + 0.01089164 
 0.9963768 - 0.0029434883 + 0.07066640 
 0.9902153 - 0.0070671940 + 0.11485779 
 0.9854773 - 0.0058653024 + 0.13162975 
 0.9862780 - 0.0010764519 + 0.11649805 
 0.9932756 + 0.0016290397 + 0.07352222 
 0.9996406 - 0.0026304325 + 0.01421265 
 
 5.9548558* + 0.0029472768 + 0.08527241 
 5.9548275 + 0.0029898031 + 0.08527242 
 
 Ko 1000 X So 1000 X TF <"> 
 
 1000 XPj 
 + 2.1929308 
 - 1.0158793 
 - 4.1045657 
 - 6.2455186 
 - 6.8650672 
 - 5.7972107 
 - 3.3280756 
 - 0.1192649 
 + 2.9694212 
 + 5.1103718 
 + 5.7299237 
 + 4.6620645 
 
 - 3.4054328 
 - 3.4054372 
 
 1000 X <S ( "> 
 
 
 
 - 0.18791333 
 
 0.020434768 
 
 - 0.3847186 
 
 - 3.085993 
 
 0.000000000 
 
 - 1.2510793 
 
 30 
 
 + 0.04317409 
 
 0.021303927 
 
 - 0.4280932 
 
 6.547345 
 
 + 0.033478638 
 
 - 1.3454762 
 
 60 
 
 - 0.07397385 
 
 0.025621038 
 
 + 0.1363110 
 
 9.590571 
 
 + 0.063887706 
 
 + 0.3925293 
 
 90 
 
 - 0.49750230 
 
 0.033590134 
 
 + 0.7062450 
 
 - 10.865960 
 
 + 0.086774000 
 
 + 1.8244563 
 
 120 
 
 - 0.86425140 
 
 0.043556201 
 
 + 0.5521780 
 
 7.575463 
 
 + 0.088361000 
 
 + 1.2934800 
 
 150 
 
 - 0.83674077 
 
 0.051546760 
 
 - 0.4218487 
 
 + 1.808294 
 
 + 0.056517506 
 
 - 0.9250562 
 
 180 
 
 - 0.43280690 
 
 0.053431000 
 
 - 1.3516860 
 
 + 13.566615 
 
 0.000000000 
 
 - 2.8963373 
 
 210 
 
 - 0.01035886 
 
 0.048659649 
 
 - 1.2877184 
 
 + 20.505307 
 
 - 0.053351926 
 
 - 2.8237889 
 
 240 
 
 - 0.07819780 
 
 0.040457621 
 
 - 0.3317444 
 
 + 19.465045 
 
 - 0.082075038 
 
 - 0.7771130 
 
 270 
 
 + 0.18040025 
 
 0.032377496 
 
 + 0.5949004 
 
 + 13.349022 
 
 - 0.083641596 
 
 + 1.5368184 
 
 300 
 
 - 0.46718624 
 
 0.026190705 
 
 + 0.7690320 
 
 + 6.465459 
 
 - 0.065308303 
 
 + 2.2142904 
 
 330 
 
 - 0.46610559 
 
 0.022213151 
 
 + 0.2265587 
 
 + 0.990810 
 
 - 0.034907444 
 
 + 0.7120630 
 
 2i 
 
 - 1.94793392 
 
 0.209691333 
 
 - 0.6106280 
 
 + 19.245092 
 
 + 0.004865365 
 
 - 1.0242299 
 
 2 2 
 
 - 1.94793368 
 
 0.209691117 
 
 - 0.6099562 
 
 + 19.240128 
 
 + 0.004869178 
 
 - 1.0209836 
 
 
 * 2,(J,' - G") = 
 
 5.9540163. 
 
 
 
 
 
 
 2, (j ' G") = 
 
 5.9539882. 
 
 
 
 
 
68 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF THE EARTH ON MERCURY. 
 
 COS V Ro 
 
 E 
 
 BUI v m> 
 + (cos v + cos E)S<> 
 
 . (r ,.\j 100 X TFo cos 100 X Wo sin u 
 + I - sec* if + 1 1 sin t'So 
 
 
 
 - 0.000769437 
 
 - 0.020434774 - 0.2710273 - 0.1475729 
 
 30 
 
 + 0.011967951 
 
 - 0.017591057 - 0.2756307 - 0.5938896 
 
 60 
 
 + 0.024315363 
 
 - 0.008157573 - 0.1568518 - 0.9461439 
 
 90 
 
 + 0.032727249 
 
 + 0.008319102 + 0.7047203 - 0.8270793 
 
 120 
 
 + 0.032844271 
 
 + 0.028781472 + 0.7040911 - 0.2795215 
 
 150 
 
 + 0.022159505 
 
 + 0.046499174 - 0.1803831 - 0.0126962 
 
 180 
 
 + 0.002703372 
 
 + 0.053431012 - 1.1914874 - 0.6487582 
 
 210 
 
 - 0.017924683 
 
 + 0.045456375 - 1.2308941 - 1.6399928 
 
 240 
 
 - 0.030714227 
 
 + 0.026434445 - 0.3784440 - 1.9093573 
 
 270 
 
 - 0.031808132 
 
 + 0.005466897 + 0.3836684 - 1.2785779 
 
 300 
 
 - 0.024103783 
 
 - 0.010000896 + 0.4783818 - 0.4349396 
 
 330 
 
 - 0.012845686 
 
 - 0.018098650 + 0.0981246 - 0.0137335 
 
 Si 
 
 + 0.004275559 
 
 + 0.070053686 - 0.5016340 - 4.3662934 
 
 2 2 
 
 + 0.004276204 
 
 + 0.070051841 - 0.5003946 - 4.3659693 
 
 i 
 
 in ,.M."> + COB *.' 
 
 > = - 0.00000000016. 
 
 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 
 
 log coeff. 
 // 
 
 
 
 [de/dt}^ = +3752.8345 TO' p 3.5743594 
 
 
 
 [dx/dt] M = +299037.72 m' p 5.4757260 
 
 
 
 [dildt] M = -4591.3713 m' n 3.6619424 
 
 
 
 [dB/<ft] M = -328217.95 TO' n 5.5161623 
 
 
 
 [d7r/d*]oo = +296589.74 TO' p 5.4721561 
 
 
 
 [dL/dt] m = -390282.17 TO' n 5.5913787 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO' 
 
 
 
 // 
 
 
 
 [d/dt] w = +0.011476557 
 
 
 
 [dx/dt] m = +0.91448833 
 
 
 
 [dildt]<n = -0.014040890 
 
 
 
 [dQ/dt]oo = -1.0037245 
 
 
 
 [d7r/di]oo = +0.90700208 
 
 
 
 [dLldtln = -1.1935233 
 
 0.03246657 
 0.03502118 
 0.04597425 
 0.06718020 
 0.09606764 
 0.12145034 
 0.12883338 
 0.11464781 
 0.08923343 
 0.06475510 
 0.04699652 
 0.03651581 
 
 0.43957179 
 0.43957044 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dtlw +0.01153 +0.01147 +0.0114766 
 
 e[dr/dt] M +0.18668 +0.18799 +0.186484 
 
 [di/dtlao -0.01414 -0.01404 -0.0140409 
 
 sin i [dfl/di]oo -0.12219 -0.12233 -0.122360 
 
 -1.1942 -1.19352 
 
 NOTES. 
 
 As a' and e' are both small in this case, the sums, up to and including .R , are in 
 very exact agreement. But as / and e are unusually large, the final sums differ 
 considerably, the greatest discrepancy being in W cos u, which shows that a neglect 
 of the terms from the 6th to the llth orders would produce an error in [di/dt] QO of 
 slightly more than 1 /1000th of the whole value of this coefficient. 
 
70 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MARS ON MERCURY. 
 
 E 
 
 A 
 
 B cos 
 
 B sin e 
 
 <7 
 
 h 
 
 
 
 2.3984504 
 
 + 0.12138918 
 
 + 0.45632916 
 
 0.0042055 
 
 2.3024514 
 
 15 
 
 2.3737047 
 
 - 0.02025060 
 
 + 0.40611421 
 
 0.0033309 
 
 2.3025214 
 
 30 
 
 2.3556032 
 
 - 0.14407564 
 
 + 0.32017434 
 
 0.0020703 
 
 2.3025321 
 
 45 
 
 2.3456111 
 
 - 0.24164776 
 
 + 0.20436670 
 
 0.0008435 
 
 2.3024889 
 
 60 
 
 2.3447258 
 
 - 0.30631712 
 
 + 0.06658267 
 
 0.0000896 
 
 2.3024184 
 
 75 
 
 2.3533241 
 
 - 0.33367707 
 
 - 0.08378742 
 
 0.0001418 
 
 2.3023625 
 
 90 
 
 2.3710514 
 
 - 0.32186246 
 
 - 0.23649650 
 
 0.0011296 
 
 2.3023658 
 
 105 
 
 2.3967846 
 
 - 0.27167903 
 
 - 0.38113729 
 
 0.0029337 
 
 2.3024640 
 
 120 
 
 2.4286852 
 
 - 0.18654648 
 
 - 0.50785318 
 
 0.0052088 
 
 2.3026710 
 
 135 
 
 2.4643476 
 
 - 0.07226635 
 
 - 0.60800831 
 
 0.0074658 
 
 2.3029751 
 
 150 
 
 2.5010251 
 
 + 0.06337314 
 
 - 0.67477750 
 
 0.0091956 
 
 2.3033343 
 
 165 
 
 2.5359018 
 
 + 0.21112855 
 
 - 0.70361065 
 
 0.0099983 
 
 2.3036852 
 
 180 
 
 2.5663693 
 
 + 0.36093056 
 
 - 0.69254258 
 
 0.0096862 
 
 2.3039570 
 
 195 
 
 2.5902662 
 
 + 0.50257031 
 
 - 0.64232765 
 
 0.0083325 
 
 2.3040870 
 
 210 
 
 2.6060491 
 
 + 0.62639549 
 
 - 0.55638795 
 
 0.0062519 
 
 2.3040409 
 
 225 
 
 2.6128739 
 
 + 0.72396747 
 
 - 0.44058010 
 
 0.0039202 
 
 2.3038234 
 
 240 
 
 2.6105920 
 
 + 0.78863682 
 
 - 0.30279611 
 
 0.0018517 
 
 2.3034773 
 
 255 
 
 2.5996753 
 
 + 0.81599664 
 
 - 0.15242602 
 
 0.0004692 
 
 2.3030742 
 
 270 
 
 2.5810993 
 
 + 0.80418226 
 
 + 0.00028298 
 
 0.0000000 
 
 2.3026229 
 
 285 
 
 2.5562149 
 
 + 0.76399902 
 
 + 0.14492387 
 
 0.0004242 
 
 2.3023962 
 
 300 
 
 2.5266327 
 
 + 0.66886612 
 
 + 0.27163969 
 
 0.0014902 
 
 2.3022243 
 
 315 
 
 2.4941374 
 
 + 0.55458616 
 
 + 0.37179487 
 
 0.0027917 
 
 2.3021798 
 
 330 
 
 2.4606272 
 
 + 0.41894655 
 
 + 0.43856414 
 
 0.0038844 
 
 2.3022350 
 
 345 
 
 2.4280691 
 
 + 0.27119119 
 
 + 0.46739720 
 
 0.0044120 
 
 2.3023425 
 
 2, 
 
 29.7509107* 
 
 + 2.89391842f 
 
 - 1.41728084J 
 
 0.0450638 
 
 27.6343997 
 
 2 2 
 
 29.7509107 
 
 + 2.89391853 
 
 - 1.41728059 
 
 0.0450638 
 
 27.6344002 
 
 * 12a 2 + 6aV + 12[a' 2 - 
 t 12[a'V - kaa'e COB K\ 
 J 12fc'aa' cos <p' e sin 
 
 - Zkaa'ee' cos A') = 29.7509106. 
 = + 2.89391844. 
 A"' = - 1.41728062. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 71 
 
 E 
 
 ACTION OF MARS ON MERCURY. 
 G G' 
 
 
 
 0.0758032 
 
 2.3016305 
 
 0.0957141 
 
 0.0190900 
 
 12 
 
 51 
 
 3.65 
 
 15 
 
 0.0509875 
 
 2.3018785 
 
 0.0717874 
 
 0.0201570 
 
 11 
 
 28 
 
 40.15 
 
 30 
 
 0.0328754 
 
 2.3021359 
 
 0.0509293 
 
 0.0176577 
 
 9 
 
 54 
 
 3.88 
 
 45 
 
 0.0229265 
 
 2.3023282 
 
 0.0338958 
 
 0.0108086 
 
 7 
 
 59 
 
 27.93 
 
 60 
 
 0.0221116 
 
 2.3024014 
 
 0.0237650 
 
 0.0016363 
 
 6 
 
 1 
 
 37.49 
 
 75 
 
 0.0307659 
 
 2.3023354 
 
 0.0326775 
 
 0.0018845 
 
 7 
 
 2 
 
 5.28 
 
 90 
 
 0.0484898 
 
 2.3021481 
 
 0.0572743 
 
 0.0085668 
 
 9 
 
 43 
 
 5.24 
 
 105 
 
 0.0741248 
 
 2.3018919 
 
 0.0890147 
 
 0.0143178 
 
 12 
 
 11 
 
 37.36 
 
 120 
 
 0.1058185 
 
 2.3016404 
 
 0.1249596 
 
 0.0181104 
 
 14 
 
 22 
 
 46.33 
 
 135 
 
 0.1411768 
 
 2.3014735 
 
 0.1626257 
 
 0.0199473 
 
 16 
 
 17 
 
 11.47 
 
 150 
 
 0.1774951 
 
 2.3014531 
 
 0.1994129 
 
 0.0200367 
 
 17 
 
 54 
 
 21.70 
 
 165 
 
 0.2120208 
 
 2.3016063 
 
 0.2327627 
 
 0.0186630 
 
 19 
 
 13 
 
 8.83 
 
 180 
 
 0.2422166 
 
 2.3019140 
 
 0.2604177 
 
 0.0161582 
 
 20 
 
 12 
 
 26.11 
 
 195 
 
 0.2659835 
 
 2.3023097 
 
 0.2806562 
 
 0.0128954 
 
 20 
 
 51 
 
 34.72 
 
 210 
 
 0.2818125 
 
 2.3026974 
 
 0.2924401 
 
 0.0092842 
 
 21 
 
 10 
 
 38.08 
 
 225 
 
 0.2888548 
 
 2.3029783 
 
 0.2954612 
 
 0.0057613 
 
 21 
 
 10 
 
 27.64 
 
 240 
 
 0.2869190 
 
 2.3030785 
 
 0.2900893 
 
 0.0027715 
 
 20 
 
 52 
 
 41.30 
 
 255 
 
 0.2764054 
 
 2.3029737 
 
 0.2772408 
 
 0.0007349 
 
 20 
 
 19 
 
 35.05 
 
 270 
 
 0.2582114 
 
 2.3026922 
 
 0.2582114 
 
 0.0000000 
 
 19 
 
 33 
 
 51.59 
 
 285 
 
 0.2336230 
 
 2.3023071 
 
 0.2344977 
 
 0.0007857 
 
 18 
 
 38 
 
 24.76 
 
 300 
 
 0.2042127 
 
 2.3019157 
 
 0.2076391 
 
 0.0031178 
 
 17 
 
 36 
 
 2.08 
 
 315 
 
 0.1717618 
 
 2.3016103 
 
 0.1791035 
 
 0.0067722 
 
 16 
 
 29 
 
 5.95 
 
 330 
 
 0.1381965 
 
 2.3014548 
 
 0.1502129 
 
 0.0112362 
 
 15 
 
 19 
 
 13.32 
 
 345 
 
 0.1055309 
 
 2.3014695 
 
 0.1221038 
 
 0.0156999 
 
 14 
 
 6 
 
 53.42 
 
 2, 
 
 1.8741623 
 
 27.6251620 
 
 2.0110657 
 
 0.1276658 
 
 185 
 
 31 
 
 50.77 
 
 2 2 
 
 1.8741617 
 
 27.6251624 
 
 2.0118270 
 
 0.1284276 
 
 185 
 
 48 
 
 12.56 
 
72 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MARS ON 
 
 MERCURY. 
 
 
 
 E 
 
 logtfo 
 
 log LJ 
 
 logtfo 
 
 log JV 
 
 logP 
 
 logQ 
 
 
 
 0.01657774 
 
 0.29504379 
 
 0.20087027 
 
 8.0316820 
 
 7.5954802 
 
 7.8669295 
 
 15 
 
 0.01319292 
 
 0.29055307 
 
 0.19582511 
 
 8.0355546 
 
 7.5943701 
 
 7.8655109 
 
 30- 
 
 0.00979383 
 
 0.28603832 
 
 0.19075139 
 
 8.0547661 
 
 7.6099058 
 
 7.8800682 
 
 45 
 
 0.00636436 
 
 0.28147805 
 
 0.18562486 
 
 8.0870691 
 
 7.6401445 
 
 7.9084927 
 
 60 
 
 0.00361341 
 
 0.27781623 
 
 0.18150720 
 
 8.1291203 
 
 7.6819573 
 
 7.9481379 
 
 75 
 
 0.00492729 
 
 0.27956556 
 
 0.18347444 
 
 8.1771100 
 
 7.7316276 
 
 7.9980604 
 
 90 
 
 0.00943284 
 
 0.28555854 
 
 0.19021211 
 
 8.2272783 
 
 7.7853440 
 
 8.0537440 
 
 105 
 
 0.01490785 
 
 0.29282894 
 
 0.19838217 
 
 8.2762397 
 
 7.8395128 
 
 8.1098440 
 
 120 
 
 0.02081742 
 
 0.30066160 
 
 0.20717937 
 
 8.3211159 
 
 7.8908949 
 
 8.1628540 
 
 135 
 
 0.02681707 
 
 0.30859802 
 
 0.21608807 
 
 8.3595540 
 
 7.9366442 
 
 8.2098882 
 
 150 
 
 0.03254501 
 
 0.31616047 
 
 0.22457224 
 
 8.3896939 
 
 7.9743208 
 
 8.2484993 
 
 165 
 
 0.03762579 
 
 0.32285663 
 
 0.23208058 
 
 8.4101312 
 
 8.0019112 
 
 8.2766735 
 
 180 
 
 0.04171228 
 
 0.32823434 
 
 0.23810787 
 
 8.4198969 
 
 8.0178772 
 
 8.2928778 
 
 195 
 
 0.04453613 
 
 0.33194630 
 
 0.24226678 
 
 8.4184648 
 
 8.0212321 
 
 8.2961421 
 
 210 
 
 0.04594748 
 
 0.33380024 
 
 0.24434354 
 
 8.4057702 
 
 8.0116018 
 
 8.2861294 
 
 225 
 
 0.04593448 
 
 0.33378318 
 
 0.24432441 
 
 8.3822399 
 
 7.9892733 
 
 8.2631894 
 
 240 
 
 0.04461766 
 
 0.33205341 
 
 0.24238678 
 
 8.3488317 
 
 7.9552231 
 
 8.228375 
 
 255 
 
 0.04222050 
 
 0.32890264 
 
 0.23885673 
 
 8.3070780 
 
 7.9111256 
 
 8.1835072 
 
 270 
 
 0.03902753 
 
 0.32470208 
 
 0.23414923 
 
 8.2591387 
 
 7.8593690 
 
 8.1310520 
 
 285 
 
 0.03533705 
 
 0.31984157 
 
 0.22870029 
 
 8.2078386 
 
 7.8030574 
 
 8.0742275 
 
 300 
 
 0.03141955 
 
 0.31467566 
 
 0.22290683 
 
 8.15CG450 
 
 7.7459663 
 
 8.0168746 
 
 315 
 
 0.02748746 
 
 0.30948385 
 
 0.21708212 
 
 8.1095325 
 
 7.6924008 
 
 7.9633068 
 
 330 
 
 0.02367787 
 
 0.30444739 
 
 0.21142959 
 
 8.0706478 
 
 7.6468600 
 
 7.9179598 
 
 345 
 
 0.02004728 
 
 0.29964170 
 
 0.20603415 
 
 8.0437755 
 
 7.6135016 
 
 7.8848519 
 
 Si 
 
 0.31918262 
 
 3.69919207 
 
 2.58841642 
 
 8.8145868 
 
 3.7748004 
 
 7.0335140 
 
 s s 
 
 0.31939818 
 
 3.69947951 
 
 2.58873971 
 
 8.8145879 
 
 3.7748012 
 
 7.0338946 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 73 
 
 ACTION OF MARS ON MERCURY. 
 
 E 
 
 logf 
 
 /' 
 
 Ji 
 
 
 
 7.8624692 
 
 2.3161036 
 
 + 0.034714816 
 
 15 
 
 7.8607965 
 
 2.3114562 
 
 + 0.028725995 
 
 30 
 
 7.8759318 
 
 2.3036617 
 
 + 0.023022893 
 
 45 
 
 7.9059548 
 
 2.2939379 
 
 + 0.017028648 
 
 60 
 
 7.9477527 
 
 2.2850812 
 
 + 0.009527241 
 
 75 
 
 7.9976170 
 
 2.2883310 
 
 - 0.000387926 
 
 90 
 
 8.0517340 
 
 2.2994643 
 
 - 0.012768514 
 
 105 
 
 8.1064943 
 
 2.3097716 
 
 - 0.026802369 
 
 120 
 
 8.1586286 
 
 2.3171693 
 
 - 0.041120811 
 
 135 
 
 8.2052458 
 
 2.3210340 
 
 - 0.054165965 
 
 150 
 
 8.2438461 
 
 2.3213532 
 
 - 0.064480743 
 
 165 
 
 8.2723467 
 
 2.3185110 
 
 - 0.070890916 
 
 180 
 
 8.2891363 
 
 2.3131717 
 
 - 0.072601961 
 
 195 
 
 8.2931581 
 
 2.3062073 
 
 - 0.069243148 
 
 210 
 
 8.2839812 
 
 2.2986444 
 
 - 0.060886015 
 
 225 
 
 8.2618557 
 
 2.2916136 
 
 - 0.048054879 
 
 240 
 
 8.2277453 
 
 2.2862701 
 
 - 0.031734816 
 
 255 
 
 8.1833367 
 
 2.2836646 
 
 - 0.013368882 
 
 270 
 
 8.1310520 
 
 2.2845306 
 
 + 0.005188260 
 
 285 
 
 8.0740447 
 
 2.2890080 
 
 + 0.021821056 
 
 300 
 
 8.0161484 
 
 2.2963713 
 
 + 0.034481515 
 
 315 
 
 7.9617277 
 
 2.3049376 
 
 + 0.041716329 
 
 330 
 
 7.9153379 
 
 2.3123762 
 
 + 0.043244020 
 
 345 
 
 7.8811861 
 
 2.3164879 
 
 + 0.040207969 
 
 S, 
 
 7.0037635 
 
 27.6341976* 
 
 - 0.133414115 
 
 2 2 
 
 7.0037641 
 
 27.6349607 
 
 - 0.133414088 
 
 
 *Si(Ji' -G") =27, 
 
 5065318. 
 
 
 
 2 2 (-/i' - G") = 27 
 
 ,5065331. 
 
 
 - 0.08283714 
 
 - 0.12496689 
 
 - 0.15748255 
 
 - 0.17817560 
 
 - 0.18564570 
 
 - 0.17939386 
 
 - 0.15985316 
 
 - 0.12835823 
 
 - 0.08705254 
 
 - 0.03874380 
 + 0.01328579 
 + 0.06550049 
 + 0.11434922 
 + 0.15650567 
 + 0.18909426 
 + 0.20988694 
 + 0.21745674 
 + 0.21127772 
 + 0.19176389 
 + 0.16024221 
 + 0.11886354 
 + 0.07045517 
 + 0.01832594 
 
 - 0.03396170 
 
 + 0.19026829 
 + 0.19026812 
 
 - 0.09798270 
 
 - 0.08720058 
 
 - 0.06874762 
 
 - 0.04388148 
 
 - 0.01429659 
 + 0.01799079 
 + 0.05078038 
 + 0.08183755 
 + 0.10904591 
 + 0.13055114 
 + 0.14488783 
 + 0.15107882 
 + 0.14870229 
 + 0.13792019 
 + 0.11946725 
 + 0.09460109 
 + 0.06501619 
 + 0.03272882 
 
 - 0.00006076 
 
 - 0.03111796 
 
 - 0.05832630 
 
 - 0.07983156 
 
 - 0.09416821 
 
 - 0.10035921 
 
 + 0.30431767 
 + 0.30431761 
 
74 
 
 THE SECULAE VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MARS ON MERCURY. 
 
 E 
 
 ft 
 
 #0 
 
 1000 X So 
 
 100 X Wv 
 
 fl<> 
 
 
 
 + 0.006125234 
 
 0.005413043 
 
 - 0.1331172 
 
 - 0.05793911 
 
 0.000000000 
 
 15 
 
 + 0.004244733 
 
 0.005395946 
 
 - 0.1341971 
 
 - 0.08902862 
 
 + 0.004501863 
 
 30 
 
 + 0.002091460 
 
 0.005581735 
 
 - 0.1069853 
 
 - 0.11749705 
 
 4- 0.008771561 
 
 45 
 
 + 0.000394535 
 
 0.005978460 
 
 - 0.0544826 
 
 - 0.14331159 
 
 + 0.012778556 
 
 60 
 
 - 0.000202379 
 
 0.006587462 
 
 + 0.0157366 
 
 - 0.16470025 
 
 4- 0.016426268 
 
 75 
 
 + 0.000673582 
 
 0.007397547 
 
 + 0.0931210 
 
 - 0.17804909 
 
 4- 0.019496561 
 
 90 
 
 + 0.003010065 
 
 0.008379102 
 
 + 0.1659322 
 
 - 0.17823953 
 
 4- 0.021645905 
 
 105 
 
 4- 0.006397644 
 
 0.009479745 
 
 + 0.2230373 
 
 - 0.15960687 
 
 + 0.022459598 
 
 120 
 
 + 0.010124788 
 
 0.010622853 
 
 + 0.2557091 
 
 - 0.11755697 
 
 4- 0.021550254 
 
 135 
 
 + 0.013355105 
 
 0.011710967 
 
 + 0.2593958 
 
 - 0.05060872 
 
 4- 0.018676901 
 
 150 
 
 + 0.015340435 
 
 0.012635664 
 
 + 0.2351775 
 
 + 0.03775311 
 
 4- 0.013854131 
 
 165 
 
 + 0.015613320 
 
 0.013293529 
 
 + 0.1902485 
 
 + 0.13831063 
 
 4- 0.007415508 
 
 180 
 
 + 0.014107795 
 
 0.013605711 
 
 4- 0.1366988 
 
 + 0.23722085 
 
 0.000000000 
 
 195 
 
 + 0.011176655 
 
 0.013535474 
 
 + 0.0883172 
 
 + 0.31912559 
 
 - 0.007550472 
 
 210 
 
 + 0.007500348 
 
 0.013096767 
 
 + 0.0561741 
 
 + 0.37133333 
 
 - 0.014359702 
 
 225 
 
 + 0.003911780 
 
 0.012350565 
 
 + 0.0447317 
 
 4- 0.38738325 
 
 - 0.019696946 
 
 240 
 
 + 0.001183526 
 
 0.011390146 
 
 + 0.0503246 
 
 + 0.36844987 
 
 - 0.023106835 
 
 255 
 
 - 0.000166297 
 
 0.010320547 
 
 + 0.0628133 
 
 + 0.32211263 
 
 - 0.024451640 
 
 270 
 
 + 0.000001635 
 
 0.009239635 
 
 + 0.0697179 
 
 + 0.25931088 
 
 - 0.023868989 
 
 285 
 
 + 0.001425688 
 
 0.008225494 
 
 + 0.0610457 
 
 + 0.19093564 
 
 - 0.021678655 
 
 300 
 
 + 0.003528129 
 
 0.007331827 
 
 + 0.0329172 
 
 4- 0.12533212 
 
 - 0.018282391 
 
 315 
 
 + 0.005583303 
 
 0.006590510 
 
 - 0.0111915 
 
 + 0.06726176 
 
 - 0.014086776 
 
 330 
 
 + 0.006923128 
 
 0.006018069 
 
 - 0.0617561 
 
 4- 0.01815027 
 
 - 0.009457250 
 
 345 
 
 + 0.007124120 
 
 0.005623237 
 
 - 0.1063107 
 
 - 0.02290733 
 
 - 0.004691492 
 
 Si 
 
 + 0.069734164 
 
 0.109902014 
 
 + 0.7165294 
 
 4- 0.78161752 
 
 - 0.006827048 
 
 s 2 
 
 4- 0.069734168 
 
 0.109902021 
 
 + 0.7165286 
 
 + 0.78161728 
 
 - 0.006826894 
 
 i"> + cos <f B <c) = + 0.000000000104. 
 
OF THE DEBITS OF THE FOUR INNER PLANETS. 
 
 75 
 
 ACTION OF MARS ON MERCURY. 
 
 1000x-.RoCosB+ 
 
 E 
 
 1000 X S<" 
 
 + B ' " - 10 X W " c s u 10 X W 8in 
 
 -2-#o 
 
 
 
 C ' 
 
 ( - secV + 1 1 sin vSf, 
 
 
 a 
 
 
 
 -0.43288822 
 
 -0.2662344 
 
 - 5.4130432 
 
 -0.5088501 - 0.2770661 
 
 -0.008600192 
 
 15 
 
 -0.43258505 
 
 + 1.4484906 
 
 - 5.1972533 
 
 -0.6072560 - 0.6510373 
 
 -0.008648634 
 
 30 
 
 -0.33624961 
 
 +3.1443027 
 
 - 4.6032128 
 
 -0.4946402 - 1.0657799 
 
 -0.009175713 
 
 45 
 
 -0.16468928 
 
 +4.7703837 
 
 - 3.5917373 
 
 -0.1836694 - 1.4212974 
 
 -0.010218569 
 
 60 
 
 +0.04531053 
 
 +6.2357682 
 
 - 2.1327368 
 
 +0.2693639 - 1.6248266 
 
 -0.011820509 
 
 75 
 
 +0.25408211 
 
 +7.4151862 
 
 - 0.2308954 
 
 +0.7622000 - 1.6090993 
 
 -0.014007787 
 
 90 
 
 +0.42865608 
 
 +8.1659686 
 
 + 2.0547246 
 
 + 1.1559864 - 1.3566975 
 
 -0.016758205 
 
 105 
 
 +0.54706506 
 
 + 8.3522568 
 
 + 4.6004959 
 
 + 1.3031443 - 0.9215481 
 
 -0.019968410 
 
 120 
 
 + 0.59900000 
 
 +7.8723845 
 
 + 7.2196098 
 
 + 1.0926174 - 0.4337649 
 
 -0.023429816 
 
 135 
 
 +0.58504581 
 
 +6.6852166 
 
 + 9.6761487 
 
 +0.5003962 - 0.0756834 
 
 -0.026827124 
 
 150 
 
 +0.51571179 
 
 +4.8307371 
 
 + 11.7119607 
 
 -0.3765995 - 0.0265069 
 
 -0.029771109 
 
 165 
 
 +0.41003953 
 
 +2.4394871 
 
 + 13.0838349 
 
 -1.3270492 - 0.3897732 
 
 -0.031867221 
 
 180 
 
 +0.29291257 
 
 -0.2733976 
 
 + 13.6057094 
 
 -2.0833919 - 1.1343956 
 
 -0.032806220 
 
 195 
 
 + 0.19034846 
 
 -3.0319641 
 
 + 13.1877783 
 
 -2.4169344 - 2.0838761 
 
 -0.032447216 
 
 210 
 
 +0.12318210 
 
 -5.5396099 
 
 + 11.8615372 
 
 -2.2290420 - 2.9698842 
 
 -0.030857525 
 
 225 
 
 +0.10088867 
 
 -7.5290371 
 
 + 9.7823314 
 
 -1.5918740 - 3.5316447 
 
 -0.028292299 
 
 240 
 
 + 0.11788553 
 
 -8.8108894 
 
 + 7.2045350 
 
 -0.7163503 - 3.6141908 
 
 -0.025122156 
 
 255 
 
 +0.15406833 
 
 -9.3069299 
 
 + 4.4325573 
 
 +0.1350559 - 3.2182941 
 
 -0.021739497 
 
 270 
 
 +0.18010328 
 
 -9.0565656 
 
 + 1.7602448 
 
 +0.7452938 - 2.4836977 
 
 -0.018479268 
 
 285 
 
 +0.16656410 
 
 -8.1932578 
 
 - 0.5835177 
 
 + 1.0058653 - 1.6229225 
 
 -0.015575563 
 
 300 
 
 +0.09477911 
 
 -6.8986293 
 
 - 2.4659985 
 
 +0.9273370 - 0.8431253 
 
 -0.013156194 
 
 315 
 
 -0.03382955 
 
 -5.3509513 
 
 - 3.8502686 
 
 +0.6070918 - 0.2895753 
 
 -0.011264707 
 
 330 
 
 -0.19409650 
 
 -3.6857779 
 
 - 4.7671354 
 
 +0.1797507 - 0.0251579 
 
 -0.009892996 
 
 345 
 
 -0.34269706 
 
 -1.9808199 
 
 - 5.2732726 
 
 -0.2254930 - 0.0403424 
 
 -0.009012936 
 
 2! 
 
 + 1.43430766 
 
 -4.2819430 
 
 +36.0361948 
 
 -2.0385248 -15.8550934 
 
 -0.229869903 
 
 2 2 
 
 + 1.43430113 
 
 -4.2819391 
 
 +36.0362016 
 
 - 2.0385225 - 15.8550938 
 
 -0.229869963 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 
 
 
 w 
 
 log coeff. 
 
 
 
 
 1 fif> //r/1 
 
 I U-C' / Ltt-lQQ 
 
 1879.077 
 
 TO' n 3.2739445 
 
 
 
 
 [dxMJoo 
 
 = +76914.75 
 
 TO' p 4.8860096 
 
 
 
 
 [dildt] m 
 
 934.0667 
 
 TO' n 2.9703779 
 
 
 
 
 [dn/dt] m 
 
 = -59594.26 
 
 TO' n 4.7752044 
 
 
 
 
 [dTr/dt} m 
 
 = +76470.27 
 
 TO' p 4.8834926 
 
 
 
 
 [dL/dt] w 
 
 = -101879.0 
 
 TO' n 5.0080846 
 
 
76 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUES OF m'. 
 
 = -0.00060742746 
 [dx/d4 = +0.024863343 
 [dt'/dfloo = -0.00030194497 
 [dQ/dfloo = -0.019264347 
 [drfdt] m = +0.024719659 
 
 = -0.032933242 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 
 [de/dtlw 
 
 -0.00060 
 
 -0.00061 
 
 -0.000607 
 
 < 
 
 >,[dir/dt]oo 
 
 +0.00508 
 
 +0.00511 
 
 +0.005082 
 
 
 [dt'/dflw 
 
 -0.00030 
 
 -0.00030 
 
 -0.000302 
 
 sin i 
 
 [dQ/d*]oo 
 
 -0.00234 
 
 -0.00235 
 
 -0.002348 
 
 
 [dL/d/]oo 
 
 -0.0331 
 
 
 -0.032933 
 
 NOTES. 
 
 On account of the very large values of the eccentricities of both orbits and their 
 high mutual inclination, the approximate test is here wholly inapplicable if but 
 twelve points of division are employed. Thus the two sums differ by 1 38' 46". 90 
 for 6 and by 40' 42". 47 for , while the sums of the functions immediately dependent 
 upon these quantities differ by proportionate amounts. When the number of points 
 of division is increased to twenty-four, the final sums are in almost exact agreement, 
 showing that the combined effect of all terms from the llth to the 23rd orders is 
 wholly inappreciable. The greatest difference which arises in the variations from 
 the employment of twenty-four points of division, instead of twelve, occurs in the 
 case of [di/dt] 00 and here produces a decrease of but three units in the seventh decimal 
 of the logarithm of the coefficient, 
 
OF THE ORBITS OF THE FOUK INNER PLANETS. 
 
 77 
 
 E 
 
 
 
 30 
 
 60 
 
 90 
 
 120 
 
 150 
 
 180 
 
 210 
 
 240 
 
 270 
 
 300 
 
 330 
 
 
 
 30 
 
 60 
 
 90 
 
 120 
 
 150 
 
 180 
 
 210 
 
 240 
 
 270 
 
 300 
 
 330 
 
 ACTION OF JUPITER ON MERCURY. 
 
 A 
 
 B cos e 
 
 Bam f 
 
 p 
 
 27.23340536 
 
 +2.0282403 
 
 + 1.4219711 
 
 9.05679111 
 
 27.14356714 
 
 + 1.0282450 
 
 + 1.6206004 
 
 9.02684503 
 
 27.06879996 
 
 +0.0526625 
 
 + 1.2863778 
 
 9.00192264 
 
 27.03145602 
 
 -0.6371012 
 
 +0.5088565 
 
 8.98947466 
 
 27.04270097 
 
 -0.8562240 
 
 -0.5036265 
 
 8.99322298 
 
 27.09836241 
 
 -0.5459922 
 
 -1.4797775 
 
 9.01177679 
 
 27.18120744 
 
 +0.2104677 
 
 -2.1580381 
 
 9.03939180 
 
 27.26787830 
 
 + 1.2104630 
 
 -2.3566674 
 
 9.06828209 
 
 27.33631108 
 
 +2.1860457 
 
 -2.0224444 
 
 9.09109301 
 
 27.37048776 
 
 +2.8758091 
 
 -1.2449232 
 
 9.10248524 
 
 27.36240999 
 
 +3.0949315 
 
 -0.2324396 
 
 9.09979265 
 
 27.31308297 
 
 +2.7846997 
 
 +0.7437107 
 
 9.08335031 
 
 163.22483480* 
 
 +6.7161237f 
 
 -2.2081997J 
 
 54.28221419 
 
 163.22483460 
 
 +6.7161234 
 
 -2.2082005 
 
 54.28221412 
 
 - 8' 
 
 9 
 
 h 
 
 I 
 
 0111 
 
 
 
 
 88 49 5 
 
 0.12745094 
 
 27.006742 
 
 +0.163630 
 
 89 2 45 
 
 0.16554389 
 
 27.007491 
 
 +0.073044 
 
 89 19 
 
 0.10430339 
 
 27.007569 
 
 -0.001801 
 
 89 39 30 
 
 0.01632120 
 
 27.006899 
 
 -0.038475 
 
 89 41 40 
 
 0.01598743 
 
 27.006246 
 
 -0.026577 
 
 89 11 43 
 
 0.13801114 
 
 27.006455 
 
 +0.028875 
 
 88 41 46 
 
 0.29354831 
 
 27.007536 
 
 +0.110638 
 
 88 20 11 
 
 0.35007250 
 
 27.008599 
 
 +0.196247 
 
 87 48 4 
 
 0.25781881 
 
 27.008698 
 
 +0.264581 
 
 88 19 32 
 
 0.09768920 
 
 27.007728 
 
 +0.299728 
 
 88 22 22 
 
 0.00340551 
 
 27.006562 
 
 +0.292816 
 
 88 35 32 
 
 0.03486337 
 
 27.006176 
 
 +0.243874 
 
 532 41' 57" 
 
 0.80250439 
 
 162.043353 
 
 +0.803287 
 
 533 9 13 
 
 0.80250130 
 
 162.043348 
 
 +0.803293 
 
 f 
 
 80.552426 
 80.826371 
 
 81.050843 
 81.156986 
 81.117340 
 80.952183 
 80.714558 
 80.466952 
 80.265977 
 80.156946 
 80.170254 
 80.311910 
 
 483.871398 
 483.871348 
 
 G 
 
 27.006566 
 27.007263 
 27.007426 
 27.006877 
 27.006224 
 27.006265 
 27.007132 
 27.008116 
 27.008341 
 27.007593 
 27.006557 
 27.006128 
 
 162.042246 
 162.042242 
 
 * 6o 2 + 3o% 2 + 6[a' 2 - 2Jtaa'ee' cos K] = + 163.22483477. 
 t 6[o' 2 e' - koa'e cos A'] = + 6.7161238. 
 J - Sk'aa' cos >' e sin K' = - 2.2082004. 
 
78 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 G' 
 
 ACTION OF JUPITER ON MERCURY. 
 
 G" 
 
 log ZV 
 
 
 
 0.188801 
 
 0.024995 
 
 5 
 
 6 
 
 8.07 
 
 0.0025877 
 
 0.2764500 
 
 30 
 
 0.123075 
 
 0.049804 
 
 4 
 
 35 
 
 5.15 
 
 0.0020887 
 
 0.2757852 
 
 60 
 
 0.061321 
 
 0.062980 
 
 3 
 
 53 
 
 7.75 
 
 0.0014995 
 
 0.2750001 
 
 90 
 
 0.011983 
 
 0.050435 
 
 2 
 
 45 
 
 10.71 
 
 0.0007524 
 
 0.2740043 
 
 120 
 
 0.014441 
 
 0.040995 
 
 2 
 
 35 
 
 41.32 
 
 0.0006684 
 
 0.2738923 
 
 150 
 
 0.087484 
 
 0.058419 
 
 4 
 
 12 
 
 38.17 
 
 0.0017613 
 
 0.2753489 
 
 180 
 
 0.173639 
 
 0.062597 
 
 5 
 
 21 
 
 37.00 
 
 0.0028566 
 
 0.2768082 
 
 210 
 
 0.248823 
 
 0.052092 
 
 6 
 
 3 
 
 11.63 
 
 0.0036449 
 
 0.2778582 
 
 240 
 
 0.297071 
 
 0.032133 
 
 6 
 
 20 
 
 5.30 
 
 0.0039929 
 
 0.2783216 
 
 270 
 
 0.311476 
 
 0.011613 
 
 6 
 
 16 
 
 40.56 
 
 0.0039213 
 
 0.2782263 
 
 300 
 
 0.293250 
 
 0.000430 
 
 5 
 
 59 
 
 8.38 
 
 0.0035638 
 
 0.2777502 
 
 330 
 
 0.249105 
 
 0.005182 
 
 5 
 
 34 
 
 4.63 
 
 0.0030827 
 
 0.2771094 
 
 
 
 
 o 
 
 / 
 
 // 
 
 
 
 Si 
 
 1.028523 
 
 0.224130 
 
 29 
 
 15 
 
 47.82 
 
 0.0151689 
 
 1.6582224 
 
 2 2 
 
 1.031946 
 
 0.227545 
 
 29 
 
 26 
 
 50.85 
 
 0.0152513 
 
 1.6583323 
 
 E 
 
 logtfo 
 
 log AT 
 
 logP 
 
 logQ 
 
 logV 
 
 
 
 0.1799707 
 
 6.4183196 
 
 3.8310274 
 
 5.1664192 
 
 5.1659174 
 
 30 
 
 0.1792229 
 
 6.4468140 
 
 3.8580378 
 
 5.1937562 
 
 5.1927566 
 
 60 
 
 0.1783398 
 
 6.5219986 
 
 3.9320091 
 
 5.2678436 
 
 5.2665795 
 
 90 
 
 0.1772196 
 
 6.6157906 
 
 4.0252257 
 
 5.3607256 
 
 5.3597128 
 
 120 
 
 0.1770937 
 
 6.7009450 
 
 4.1105921 
 
 5.4459161 
 
 5.4450927 
 
 150 
 
 0.1787322 
 
 6.7589559 
 
 4.1694988 
 
 5.5052851 
 
 5.5041125 
 
 180 
 
 0.1803735 
 
 6.7800058 
 
 4.1918462 
 
 5.5278954 
 
 5.5266396 
 
 210 
 
 0.1815544 
 
 6.7609474 
 
 4.1741434 
 
 5.5101707 
 
 5.5091259 
 
 240 
 
 0.1820755 
 
 6.7044320 
 
 4.1187250 
 
 5.4544932 
 
 5.4538485 
 
 270 
 
 0.1819683 
 
 6.6198775 
 
 4.0347585 
 
 5.3701732 
 
 5.3699402 
 
 300 
 
 0.1814329 
 
 6.5255910 
 
 3.9403890 
 
 5.2755478 
 
 5.2755392 
 
 330 
 
 0.1807123 
 
 6.4489104 
 
 3.8629284 
 
 5.1980770 
 
 5.1979730 
 
 Si 
 
 1.0792861 
 
 9.6512920 
 
 4.1245888 
 
 2.1381153 
 
 2.1336169 
 
 S 2 
 
 1.0794097 
 
 9.6512958 
 
 4.1245926 
 
 2.1381878 
 
 2.1336210 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 79 
 
 ACTION OF JUPITER ON MERCUKY. 
 
 E 
 
 JV 
 
 J. 
 
 J 3 
 
 F t 
 
 Vi 
 
 
 
 26.907806 
 
 -0.08848287 
 
 -1.4438584 
 
 -1.8440911 
 
 +0.12702123 
 
 30 
 
 26.754757 
 
 -0.00410757 
 
 -2.3330959 
 
 -2.1016844 
 
 +0.05043578 
 
 60 
 
 26.779325 
 
 +0.16212886 
 
 -2.4984534 
 
 -1.6682462 
 
 -0.05353664 
 
 90 
 
 26.922525 
 
 +0.18465069 
 
 -1.8956909 
 
 -0.6599133 
 
 -0.05566955 
 
 120 
 
 27.032841 
 
 +0.04256694 
 
 -0.6863554 
 
 +0.6531309 
 
 +0.07759141 
 
 150 
 
 27.046752 
 
 -0.14516517 
 
 +0.8055488 
 
 + 1.9190575 
 
 +0.24215469 
 
 180 
 
 26.945408 
 
 -0.26169947 
 
 +2.1803373 
 
 +2.7986639 
 
 +0.29255865 
 
 210 
 
 26.802715 
 
 -0.24648650 
 
 +3.0696706 
 
 +3.0562500 
 
 +0.18231424 
 
 240 
 
 26.714620 
 
 -0.10777763 
 
 +3.2352193 
 
 + 2.6228182 
 
 +0.00934620 
 
 270 
 
 26.757114 
 
 +0.06770763 
 
 +2.6325527 
 
 + 1.6144859 
 
 -0.07863175 
 
 300 
 
 26.911783 
 
 +0.13569580 
 
 + 1.4231210 
 
 +0.3014406 
 
 -0.02506303 
 
 330 
 
 27.011072 
 
 +0.02690450 
 
 -0.0689744 
 
 -0.9644853 
 
 +0.08731406 
 
 Si 
 
 161.291783* 
 
 -0.11756837 
 
 +2.2100104 
 
 +2.8637163 
 
 +0.42791782 
 
 2 2 
 
 161.294935 
 
 -0.11649642 
 
 +2.2100109 
 
 +2.8637104 
 
 +0.42791747 
 
 E 
 
 1000 X Ro 
 
 1000,000 X So 
 
 100,000 X W, 
 
 1000 X R (n > 
 
 100,000 X S<"> 
 
 
 
 0.12949124 
 
 -0.25462237 
 
 - 2.1070330 
 
 0.00000000 
 
 -0.82801500 
 
 30 
 
 0.13531974 
 
 -0.15796961 
 
 - 3.6329106 
 
 +0.21265173 
 
 -0.49649079 
 
 60 
 
 0.16094783 
 
 +0.15688051 
 
 - 4.6204172 
 
 +0.40133398 
 
 +0.45170948 
 
 90 
 
 0.20322837 
 
 +0.35279259 
 
 - 4.3458069 
 
 +0.52500398 
 
 +0.91137646 
 
 120 
 
 0.25064630 
 
 +0.20287634 
 
 - 1.9026678 
 
 +0.50847837 
 
 +0.47523891 
 
 150 
 
 0.28657259 
 
 -0.17989808 
 
 + 2.6073819 
 
 +0.31420703 
 
 -0.39449173 
 
 180 
 
 0.29756923 
 
 -0.44461029 
 
 + 7.3765067 
 
 0.00000000 
 
 -0.95269261 
 
 210 
 
 0.28082350 
 
 -0.33962404 
 
 + 9.9405132 
 
 -0.30790355 
 
 -0.74474862 
 
 240 
 
 0.24483424 
 
 +0.03827880 
 
 + 9.2004732 
 
 -0.49668759 
 
 +0.08966829 
 
 270 
 
 0.20310516 
 
 +0.33360150 
 
 + 6.1619379 
 
 -0.52468566 
 
 +0.86179960 
 
 300 
 
 0.16660105 
 
 +0.28219872 
 
 + 2.6818065 
 
 -0.41543058 
 
 +0.81254093 
 
 330 
 
 0.14104055 
 
 -0.02790130 
 
 - 0.1024400 
 
 -0.22164184 
 
 -0.08769243 
 
 Si 
 
 1.25008989 
 
 -0.01899829 
 
 + 10.6286684 
 
 -0.00230582 
 
 +0.04845000 
 
 S 2 
 
 1.25008991 
 
 -0.01899894 
 
 + 10.6286755 
 
 -0.00236831 
 
 +0.04975249 
 
 
 sin <f \ 4i ( "'+ cos if 
 
 Bo (c) = + 0.00000000000073. 
 
 *2,(Ji' -G") = 161.067653. 
 
 S 2 (Ji' - G") = 161.067390. 
 
80 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION or JUPITER ON MERCURY. 
 
 E 
 
 1000 X [ft. sin v + 
 (cos v + cos E)S(,] 
 
 1000 X Wi, cos u 1000 X W a sin u 
 
 1000 X 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log cqeff. 
 
 [dx/dt] w 
 [di/dt]oo 
 
 + 3.3470577 m' p 0.5246632 
 + 1613.8089 m' p 3.2078521 
 
 51.404941 m' 
 - 1550.4039 TO' 
 + 1602.2454 TO' 
 
 n 1.7110049 
 w 3.1904449 
 p 3.2047290 
 
 [dL/dt] w = -2312.2863 m' n 3.3640416 
 
 (-';*) 
 
 
 
 -0.00509245 
 
 -0.12949124 
 
 -0.018505013 
 
 -0.010075877 
 
 -0.20573445 
 
 30 
 
 +0.07792118 
 
 -0.11047546 
 
 -0.015293859 
 
 -0.032953015 
 
 -0.22244969 
 
 60 
 
 +0.15333585 
 
 -0.04994118 
 
 +0.007556597 
 
 -0.045582053 
 
 -0.28880400 
 
 90 
 
 +0.19816106 
 
 +0.04884221 
 
 +0.028185071 
 
 -0.033078773 
 
 -0.40645673 
 
 120 
 
 +0.19031365 
 
 +0.16372520 
 
 +0.017684089 
 
 -0.007020516 
 
 -0.55282671 
 
 150 
 
 +0.12222497 
 
 +0.25901646 
 
 -0.026009473 
 
 -0.001830673 
 
 -0.67519875 
 
 180 
 
 +0.00889221 
 
 +0.29756923 
 
 -0.064784149 
 
 -0.035274618 
 
 -0.71750180 
 
 210 
 
 -0.11061196 
 
 +0.25859897 
 
 -0.059670973 
 
 -0.079503145 
 
 -0.66165318 
 
 240 
 
 -0.18859570 
 
 +0.15601907 
 
 -0.017887815 
 
 -0.090249100 
 
 -0.54000750 
 
 270 
 
 -0.19945173 
 
 +0.03508579 
 
 +0.017710220 
 
 -0.059019459 
 
 -0.40621028 
 
 300 
 
 -0.15503995 
 
 -0.05982940 
 
 +0.019842790 
 
 -0.018040859 
 
 -0.29894808 
 
 330 
 
 -0.08443003 
 
 -0.11301587 
 
 -0.001014512 
 
 +0.000141991 
 
 -0.23185401 
 
 Zi 
 
 +0.00381361 
 
 +0.37805168 
 
 -0.056093501 
 
 -0,206243023 
 
 -2.60382254 
 
 22 
 
 +0.00381349 
 
 +0.37805210 
 
 -0.056093526 
 
 -0.206243074 
 
 -2.60382263 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'. 
 
 [de/dfloo = +0.00319413 
 
 [d x /dt] w = +1.540072 
 
 [di/d/]oo = -0.049056191 
 
 [dtt/dtlw = -1.4795642 
 
 [djr/dflw = +1.5290366 
 
 = -2.2066350 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 81 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt] m 
 
 +0^00320 
 
 +0.00320 
 
 +0.003194 
 
 e[dw/dt]M 
 
 +0.31437 
 
 +0.31664 
 
 +0.314377 
 
 [di/dt]oo 
 
 -0.04907 
 
 -0.04905 
 
 -0.049056 
 
 sin i [dfl/dt]oo 
 
 -0.18042 
 
 -0.18037 
 
 -0.180368 
 
 [dL/dt] M 
 
 -2.2078 
 
 
 -2.20663 
 
 NOTES. 
 
 The above results were published in 1896 in A. J., No. 386. In 1911, upon 
 applying to the various computations all of the test equations devised or learned of 
 by that time, a slight error was detected in the value of F z for 240. This rendered 
 the values of W , W cos u, W sin u, [di/dt] 00 and [dQ/dt} 00 incorrect. 
 
 In this computation the device was for the first time applied of finding the root 
 G by approximations and then depressing the cubic equation and solving the resulting 
 quadratic equation directly. When a' and hence g is large, some such device becomes 
 necessary as the solution by HILL'S formulas involves a great amount of labor. 
 Thus, while but three approximations were necessary with the Earth on Mercury, no 
 less than eleven were required in some cases with Mars on Mercury, and in the latter, 
 as well as in the present case, if the formulas of HILL'S second method are employed 
 the angle 6' will be found so nearly equal to 90 as to render the values of the roots 
 obtained from it but little better than first approximations. Accordingly all the 
 remaining computations have been effected by the method here outlined, a method 
 which, since the approximation to the value of G is always very rapid, leads so quickly 
 to the values of the roots that special devices for avoiding the solution of the cubic 
 seem unnecessary. 
 
 The final sums are here practically in exact agreement, showing that the effect 
 of all terms from the sixth to the eleventh orders inclusive is inappreciable. 
 
82 
 
 THE SECULAH VARIATIONS OF THE ELEMENTS 
 
 ACTION OF SATURN ON MERCURY. 
 
 E 
 
 A 
 
 B cos t 
 
 /.' sin . 
 
 9 
 
 h 
 
 
 
 91.40055452 
 
 7.9236831 
 
 -0.7566872 
 
 0.1637322 
 
 90.704247 
 
 22.5 
 
 91.41316673 
 
 8.0026121 
 
 +0.6479546 
 
 0.1200578 
 
 90.704730 
 
 45 
 
 91.37113281 
 
 7.5285556 
 
 + 1.9837667 
 
 1.1253368 
 
 90.704973 
 
 67.5 
 
 91.28182268 
 
 6.5736833 
 
 +3.0473832 
 
 2.6555561 
 
 90.704844 
 
 90 
 
 91.15980423 
 
 5.2833659 
 
 +3.6768805 
 
 3.8659866 
 
 90.704445 
 
 112.5 
 
 91.02405557 
 
 3.8540430 
 
 +3.7764200 
 
 4.0781377 
 
 90.704075 
 
 135 
 
 90.89484135 
 
 2.5033169 
 
 +3.3308508 
 
 3.1725737 
 
 90.704038 
 
 157.5 
 
 90.79086206 
 
 1.4368221 
 
 +2.4080044 
 
 1.6581198 
 
 90.704442 
 
 180 
 
 90.72697664 
 
 0.8169216 
 
 + 1.1483775 
 
 0.3771123 
 
 90.705145 
 
 202.5 
 
 90.71250894 
 
 0.7379931 
 
 -0.2562644 
 
 0.0187792 
 
 90.705802 
 
 225 
 
 90.75006384 
 
 1.2120495 
 
 -1.5920765 
 
 0.7248182 
 
 90.706059 
 
 247.5 
 
 90.83489465 
 
 2.1669211 
 
 -2.6556933 
 
 2.0167735 
 
 90.705781 
 
 270 
 
 90.95505790 
 
 3.4572383 
 
 -3.2851902 
 
 3.0861872 
 
 90.705088 
 
 292.5 
 
 91.09266180 
 
 4.8865615 
 
 -3.3847297 
 
 3.2760409 
 
 90.704321 
 
 315 
 
 91.22635509 
 
 6.2372881 
 
 -2.9391612 
 
 2.4702909 
 
 90.703855 
 
 337.5 
 
 91.33481374 
 
 7.3037834 
 
 -2.0163140 
 
 1.1625662 
 
 90.703855 
 
 2, 
 
 728.48478638* 
 
 34.9624190} 
 
 + 1.5667604t 
 
 14.9860379 
 
 725.637848 
 
 2 2 
 
 728.48478617 
 
 34.9624206 
 
 + 1.5667608 
 
 14.9860312 
 
 725.637850 
 
 *8a i + 4a s e* + 8[a' J - 2kaa'ee' cos A'] = + 728.48478640. 
 1 8(a'V - kaa'e cos K] = + 34.9624198. 
 t - Kk'aa' cos >' e sin K' = + 1.5667610. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 83 
 
 E 
 
 ACTION OF SATURN ON MERCURY. 
 G G' 
 
 G" 
 
 
 
 
 
 +0.410351 
 
 
 90.704227 
 
 
 0.4147226 
 
 0.0043526 
 
 
 
 3 
 
 53' 5078 
 
 22.5 
 
 +0.422480 
 
 
 90.704715 
 
 
 0.4256040 
 
 0.0031100 
 
 3 
 
 56 31.53 
 
 45 
 
 
 
 +0.380202 
 
 
 90.704836 
 
 
 0.4105578 
 
 0.0302188 
 
 3 
 
 59 47.96 
 
 67.5 
 
 +0.291021 
 
 
 90.704520 
 
 
 0.3703889 
 
 0.0790439 
 
 4 
 
 2 4.89 
 
 90 
 
 
 
 +0.169403 
 
 
 90.703974 
 
 
 0.3081768 
 
 0.1383038 
 
 4 
 
 1 12.34 
 
 112.5 
 
 +0.034023 
 
 
 90.703579 
 
 
 0.2300012 
 
 0.1954823 
 
 3 
 
 55 22.98 
 
 135 
 
 
 
 -0.095154 
 
 
 90.703652 
 
 
 0.1455476 
 
 0.2403156 
 
 3 
 
 44 5.05 
 
 157.5 
 
 -0.199537 
 
 
 90.704241 
 
 
 0.0683029 
 
 0.2676388 
 
 3 
 
 29 2.13 
 
 180 
 
 
 
 -0.264126 
 
 
 90.705099 
 
 
 0.0149026 
 
 0.2789825 
 
 3 
 
 15 29.12 
 
 202 
 
 .5 
 
 
 -0.279250 
 
 
 90.705800 
 
 
 0.0007394 
 
 0.2799874 
 
 3 
 
 11 3.15 
 
 225 
 
 
 
 -0.241953 
 
 
 90.705971 
 
 
 0.0294521 
 
 0.2713169 
 
 3 
 
 17 46.29 
 
 247.5 
 
 -0.156844 
 
 
 90.705536 
 
 
 0.0901199 
 
 0.2467190 
 
 3 
 
 29 20.24 
 
 270 
 
 
 
 -0.035987 
 
 
 90.704713 
 
 
 0.1675089 
 
 0.2031209 
 
 3 
 
 39 39.24 
 
 292.5 
 
 +0.102383 
 
 
 90.703922 
 
 
 0.2482641 
 
 0.1454820 
 
 3 
 
 46 28.97 
 
 315 
 
 
 
 +0.236543 
 
 
 90.703554 
 
 
 0.3215440 
 
 0.0847000 
 
 3 
 
 50 7.92 
 
 337.5 
 
 +0.345001 
 
 
 90.703713 
 
 
 0.3789646 
 
 0.0338216 
 
 3 
 
 52 2.72 
 
 
 
 
 
 
 
 
 
 
 O 
 
 / // 
 
 v 
 
 
 
 + 0.559279 
 
 
 725.636026 
 
 
 1.8124124 
 
 1.2513111 
 
 29 
 
 41 58.70 
 
 Zo 
 
 
 
 +0.559277 
 
 
 725.636026 
 
 
 1.8123850 
 
 1.2512850 
 
 29 
 
 41 56.61 
 
 
 
 
 * 
 
 
 ACTION OF 
 
 SATURN ON MERCURY. 
 
 E 
 
 
 
 log A'o 
 
 
 log LO' 
 
 
 log # 
 
 log N log P 
 
 
 logQ 
 
 
 
 0.00150875 
 
 0.27501245 
 
 0. 
 
 17835367 5.6285747 1.9882904 
 
 3.8492800 
 
 22.5 
 
 
 
 .00154356 
 
 0. 
 
 27505883 
 
 0. 
 
 17840585 5. 
 
 6455608 2.0053300 
 
 
 3.8663219 
 
 45 
 
 
 
 
 .00158664 
 
 0.27511623 
 
 0.17847042 5.6919307 2.0514967 
 
 3.912626(1 
 
 67.5 
 
 0.00161702 
 
 0. 
 
 27515671 
 
 0. 
 
 17851595 5. 
 
 7568907 2.1160330 
 
 
 3.9773995 
 
 90 
 
 0.00160533 
 
 0.27514113 
 
 0. 
 
 17849842 5. 
 
 8276384 2.1862035 
 
 
 4.0478488 
 
 112 
 
 .5 
 
 0.00152867 
 
 0.27503898 
 
 0. 
 
 17838353 5. 
 
 8929411 2.2508613 
 
 
 4.1127652 
 
 135 
 
 
 
 
 .00138529 
 
 0.27484791 
 
 0. 
 
 17816860 5. 
 
 9445922 2.3018923 
 
 
 4.1639869 
 
 157.5 
 
 
 
 .00120533 
 
 0.27460806 
 
 0. 
 
 17789879 5. 
 
 9773700 2.3341639 
 
 
 4.1963617 
 
 180 
 
 0.00105402 
 
 0.27440638 
 
 0.17767193 5. 
 
 9884810 2.3449566 
 
 
 4.2071875 
 
 202.5 
 
 0.00100673 
 
 0.27434335 
 
 0. 
 
 17760102 5.9770716 2.3334678 
 
 4.1956990 
 
 225 
 
 
 
 
 .00107884 
 
 0. 
 
 27443945 
 
 0. 
 
 17770914 5. 
 
 9440470 2.3006205 
 
 
 4.1628231 
 
 247.5 
 
 
 
 .00120882 
 
 0. 
 
 27461270 
 
 0.17790402 5. 
 
 8922401 2.2492258 
 
 
 4.1113306 
 
 270 
 
 
 
 
 .00133101 
 
 0. 
 
 27477556 
 
 0. 
 
 17808722 5. 
 
 8268942 2.1844672 
 
 
 4.0463801 
 
 292.5 
 
 
 
 .00141515 
 
 0. 
 
 27488768 
 
 0. 
 
 17821333 5. 
 
 7562165 2.1144600 
 
 
 3.9761077 
 
 315 
 
 0.00146116 
 
 0. 
 
 27494901 
 
 0. 
 
 17828232 5.6914235 2.0503133 
 
 3.9116762 
 
 337.5 
 
 0.00148559 
 
 0.27498156 
 
 0. 
 
 17831893 5. 
 
 6452895 2.0046971 
 
 
 3.8658214 
 
 2! 
 
 
 
 
 .01101104 
 
 1. 
 
 19868812 
 
 1. 
 
 42524172 6.5435816 7.4082403 
 
 2.3018085 
 
 2 2 0.01101087 
 
 1. 
 
 19868787 
 
 1. 
 
 42524142 6. 
 
 5435802 7.4082388 
 
 
 2.3018068 
 
84 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF SATURN ON MERCURY. 
 
 E 
 
 logy 
 
 Ji' J 2 
 
 J, 
 
 p t 
 
 
 
 3.8492539 
 
 90.058195 -0.59752575 
 
 - 6.047237 
 
 + 3.829727 
 
 22.5 
 
 3.8663033 
 
 89.638288 -0.20995430 
 
 - 7.910876 
 
 - 3.279413 
 
 45 
 
 3.9124452 
 
 89.669758 +0.36734212 
 
 - 8.328272 
 
 -10.040194 
 
 67.5 
 
 3.9769269 
 
 90.090673 +0.71855783 
 
 - 7.235732 
 
 -15.423345 
 
 90 
 
 4.0470220 
 
 90.582478 +0.67967125 
 
 - 4.799462 
 
 -18.609342 
 
 112.5 
 
 4.1115969 
 
 90.879839 +0.35660262 
 
 - 1.390299 
 
 -19.113128 
 
 135 
 
 4.1625507 
 
 90.892982 -0.04825504 
 
 + 2.472681 
 
 -16.858027 
 
 157.5 
 
 4.1947623 
 
 90.670885 -0.36411899 
 
 + 6.201244 
 
 -12.187337 
 
 180 
 
 4.2055204 
 
 90.332824 -0.49291773 
 
 + 9.227617 
 
 - 5.812141 
 
 202.5 
 
 4.1940258 
 
 90.018885 -0.40939877 
 
 + 11.090997 
 
 + 1.296998 
 
 225 
 
 4.1612018 
 
 89.853321 -0.15646464 
 
 + 11.507770 
 
 + 8.057780 
 
 247.5 
 
 4.1098562 
 
 89.914463 +0.15541932 
 
 +10.414614 
 
 + 13.440932 
 
 270 
 
 4.0451660 
 
 90.199819 +0.35977661 
 
 + 7.978084 
 
 + 16.626927 
 
 292.5 
 
 3.9752380 
 
 90.576579 +0.29271946 
 
 + 4.569179 
 
 + 17.130715 
 
 315 
 
 3.9111697 
 
 90.780941 -0.07365934 
 
 + 0.706820 
 
 + 14.875613 
 
 337.5 
 
 3.8656191 
 
 90.580690 -0.50178023 
 
 - 3.021123 
 
 + 10.204923 
 
 2, 
 
 2.2943297 
 
 722.370318* +0.03796748 
 
 + 12.718001 
 
 - 7.929657 
 
 22 
 
 2.2943284 
 
 722.370302 +0.03804694 
 
 + 12.718004 
 
 - 7.929655 
 
 
 
 ACTION OF SATURN ON MERCURY. 
 
 
 
 E 
 
 F t 
 
 10000 X fto 1000000 X So 1000000 X W, 
 
 1000 X ft'"' 
 
 1000000 XS'> 
 
 
 
 -0.2217173 
 
 0.20835581 -0.38501122 - 4.2759258 
 
 0.0000000 
 
 -1.2520309 
 
 22.5 
 
 +0.0670688 
 
 0.21359630 -0.18752118 - 5.8140290 
 
 +2.6067657 
 
 -0.5980245 
 
 45 
 
 -0.2247552 
 
 0.23766012 +0.18723103 - 6.8102258 
 
 +5.0798256 
 
 +0.5659595 
 
 67.5 
 
 -0.9885705 
 
 0.27944181 +0.47990507 - 6.8742610 
 
 +7.2389467 
 
 + 1.3456245 
 
 90 
 
 -1.8298399 
 
 0.33354940 +0.47167629 - 5.3763783 
 
 + 8.6166490 
 
 + 1.2184913 
 
 112.5 
 
 -2.2911663 
 
 0.39056810 +0.12052582 1.8384740 
 
 +8.6416600 
 
 -0.2886458 
 
 135 
 
 -2.1148434 
 
 0.43919690 -0.40799123 + 3.5527836 
 
 +7.0044048 
 
 -0.9201906 
 
 157.5 
 
 -1.3919042 
 
 0.46950930 -0.83324243 + 9.6804438 
 
 +3.9006063 
 
 -1.8089208 
 
 180 
 
 -0.5106652 
 
 0.47592790 -0.91983080 +14.8005740 
 
 0.0000000 
 
 -1.9709759 
 
 202.5 
 
 +0.0653940 
 
 0.45862525 -0.61203702 +17.3393050 
 
 -3.8101033 
 
 -1.3286969 
 
 225 
 
 +0.0610980 
 
 0.42282850 -0.06578321 +16.6811060 
 
 -6.7433594 
 
 -0.1483686 
 
 247.5 
 
 -0.4587079 
 
 0.37651300 +0.43874423 +13.4040390 
 
 -8.3306788 
 
 + 1.0507430 
 
 270 
 
 -1.1366346 
 
 0.32772690 +0.65346841 + 8.8350920 
 
 -8.4662365 
 
 + 1.6881183 
 
 292.5 
 
 -1.5401525 
 
 0.28276850 +0.49946088 + 4.2959022 
 
 -7.3251220 
 
 + 1.4004581 
 
 315 
 
 -1.4203565 
 
 0.24587472 +0.10699355 + 0.5601261 
 
 -5.2554073 
 
 +0.3234187 
 
 337.5 
 
 -0.8596732 
 
 0.22009701 -0.26508243 - 2.2258014 
 
 -2.6861019 
 
 -0.8453756 
 
 2, 
 
 -7.3977141 
 
 2.69112025 -0.35924718 +27.9671518 
 
 +0.2358762 
 
 -0.4955782 
 
 2 2 
 
 -7.3977118 
 
 2.69111927 -0.3592470? +27.9671246 
 
 +0.2365727 
 
 -0.4955464 
 
 sin 
 
 <f- j^li'" + COS <f-l 
 
 V c) = + 0.000000000034. 
 
 
 
 *2j 
 
 (Ji' - G") = 721. 
 
 119007. 2 2 (J/ - G") = 721.119017. 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 85 
 
 E 
 
 1000 X [ft) sin v 
 
 + (COS V + COS E)S ] 
 
 1000X 
 
 ACTION OF SATURN ON MERCURY. 
 
 [ 
 "COS" 
 
 o 
 
 1000000 XWt cos u 1000000 X Wo sin u 
 
 1000 X -2- Bo 
 a 
 
 
 
 -0.000770023 
 
 -0.0208355810 
 
 - 3.7553310 
 
 - 2.0447571 
 
 -0.033103367 
 
 22.5 
 
 +0.009535644 
 
 -0.0190997850 
 
 - 3.2422806 
 
 - 4.8260289 
 
 -0.034604548 
 
 45 
 
 +0.019486084 
 
 -0.0136593860 
 
 - 0.8728046 
 
 - 6.7540641 
 
 -0.040621607 
 
 67.5 
 
 +0.027699067 
 
 -0.0044469025 
 
 + 2.0656152 
 
 - 6.5565758 
 
 -0.051490988 
 
 90 
 
 +0.032545329 
 
 +0.0078015052 
 
 + 3.4868920 
 
 4.0923123 
 
 -0.066709877 
 
 112.5 
 
 +0.032625198 
 
 +0.0215155020 
 
 + 1.6174933 
 
 - 0.8739008 
 
 -0.084259712 
 
 135 
 
 +0.027148276 
 
 + 0.0344565700 
 
 - 3.5128323 
 
 + 0.5313050 
 
 -0.100609845 
 
 157.5 
 
 +0.016337313 
 
 +0.0439769600 
 
 - 9.5267356 
 
 - 1.7182294 
 
 -0.111738865 
 
 180 
 
 +0.001839662 
 
 +0.0475927900 
 
 -12.9986018 
 
 - 7.0776661 
 
 -0.114756176 
 
 202.5 
 
 -0.013287669 
 
 +0.0439638930 
 
 -11.8447875 
 
 -12.6630324 
 
 -0.109148595 
 
 225 
 
 -0.025446823 
 
 +0.0337808020 
 
 - 6.8547683 
 
 -15.2076154 
 
 -0.096860250 
 
 247.5 
 
 -0.031966152 
 
 +0.0197522140 
 
 1.0475614 
 
 -13.3630400 
 
 -0.081227519 
 
 270 
 
 -0.032206863 
 
 +0.0054309792 
 
 + 2.5393222 
 
 - 8.4623118 
 
 -0.065545379 
 
 292.5 
 
 -0.027462505 
 
 -0.0063965152 
 
 + 2.7411608 
 
 - 3.3076901 
 
 -0.052103952 
 
 315 
 
 -0.019770536 
 
 -0.0145922640 
 
 + 0.5055592 
 
 - 0.2411456 
 
 -0.042025673 
 
 337.5 
 
 -0.010655672 
 
 -'0.0192899750 
 
 - 2.2254410 
 
 - 0.0400332 
 
 -0.035657730 
 
 2i 
 
 +0.002825107 
 
 +0.0799753913 
 
 -21.4625646 
 
 -43.3484220 
 
 -0.560232174 
 
 22 
 
 +0.002825224 
 
 +0.0799754154 
 
 -21.4625365 
 
 -43.3485702 
 
 -0.560231909 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 // 
 
 log coeff. 
 
 
 
 
 [de/dJfe = + 1 
 
 .8596825 m' p 
 
 0.2694389 
 
 
 
 
 [d x /dt] M = +256 
 
 .04618 m' p 
 
 2.4083183 
 
 
 
 
 [di/dt] w = 14 
 
 .751452 m' n 
 
 1.1688348 
 
 
 
 
 [da/dfl M = -244 
 
 .39983 m' n 
 
 2.3881009 
 
 
 
 
 [dir/dt] m = +254 
 
 .22335 m' p 
 
 2.4052154 
 
 
 
 
 [dL/dfloo = -373 
 
 .17967 m' n 
 
 2.5719180 
 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'. 
 
 
 
 
 u 
 
 
 
 
 
 [de/df]) = 
 
 +0.00053109524 
 
 
 
 [dx/d*] M = 
 
 +0.073122627 
 
 
 
 
 
 [di/dt] M = 
 
 -0.0042127757 
 
 
 
 [dtt/dt] M = 
 
 -0.069796619 
 
 
 
 
 
 [dir/dt]o = 
 
 +0.072602050 
 
 
 
 
 
 [dL/dtlw = 
 
 -0.10657405 
 
 
 
86 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 [de/dt], 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss, 
 
 oo 
 
 sin i [dtt/dt] m 
 
 +o'.00053 
 
 +0.00053 
 
 +0.0005311 
 
 +0.01494 
 
 +0.01503 
 
 +0.0149273 
 
 -0.00421 
 
 -0.00421 
 
 -0.0042128 
 
 -0.00853 
 
 -0.00850 
 
 -0.0085087 
 
 -0.1070 
 
 
 -0.106574 
 
 NOTES. 
 
 The considerable disagreement of the first sums is caused by the rather large 
 value of e'. The very exact agreement toward the close of the computation shows, 
 however, that all terms above the 15th order are wholly inappreciable, the total 
 effect of all terms from the 8th to the 15th orders inclusive occurring with [de/dt] o 
 and amounting to but 1 /30000th of the value of this coefficient. 
 
 ACTION OF URANUS ON MERCURY. 
 
 E 
 
 A 
 
 B cos e 
 
 B sin . 
 
 a 
 
 h 
 
 
 
 368.36907643 
 
 16.94656315 
 
 -5.8755608 
 
 27.995613 
 
 367.49553 
 
 45 
 
 368.87526360 
 
 22.1817430 
 
 -3.9726882 
 
 12.798540 
 
 367.49652 
 
 90 
 
 369.14240318 
 
 24.5977597 
 
 + 1.1481399 
 
 1.069008 
 
 367.49606 
 
 135 
 
 369.01848670 
 
 22.7793354 
 
 +6.4872119 
 
 34.127748 
 
 367.49535 
 
 180 
 
 368.57162431 
 
 17.7916836 
 
 +8.9169755 
 
 64.480235 
 
 367.49625 
 
 225 
 
 368.05910248 
 
 12.5565017 
 
 +7.0141016 
 
 39.896574 
 
 367.49742 
 
 270 
 
 367.78562896 
 
 10.1404905 
 
 + 1.8932751 
 
 2.906824 
 
 367.49671 
 
 315 
 
 367.91587927 
 
 11.9589113 
 
 -3.4457976 
 
 9.628773 
 
 367.49535 
 
 2, 
 
 1473.86873228* 
 
 69.4764969t 
 
 +6.0828297J 
 
 96.451680 
 
 1469.98455 
 
 2 2 
 
 1473.86873205 
 
 69.4764914 
 
 +6.0828277 
 
 96.451635 
 
 1469.98464 
 
 * 4a 2 + 2aV + 4[a' 2 - Zkoa'ee' cos K] = + 1473.86873246. 
 t 4[a'V - kaa'e cos A'] = + 69.4764933. 
 t - 4k'aa' cos J e sin A" = + 6.0828300. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 87 
 
 ACTION OF URANUS ON MERCURY. 
 
 E 
 
 I 
 
 G G' G" 
 
 8 
 
 
 
 
 O / U 
 
 
 
 +0.06261 
 
 367.49532 0.30919 0.24638 
 
 2 13 39.23 
 
 45 
 
 +0.56779 
 
 367.49643 0.62373 0.05584 
 
 2 27 51.84 
 
 90 
 
 +0.83539 
 
 367.49605 0.83888 0.00347 
 
 2 44 38.85 
 
 135 
 
 +0.71220 
 
 367.49510 0.82501 0.11257 
 
 2 53 41.24 
 
 180 
 
 +0.26443 
 
 367.49577 0.57178 0.30687 
 
 2 48 5.47 
 
 225 
 
 -0.24927 
 
 367.49712 0.22774 0.47670 
 
 2 30 27.70 
 
 270 
 
 -0.52202 
 
 367.49669 0.01474 0.53674 
 
 2 13 6.46 
 
 315 
 
 -0.39041 
 
 367.49528 0.05839 0.44873 
 
 2 7 39.31 
 
 2, 
 
 +0.64041 
 
 1469.98383 1.73458 1.09346 
 
 9 59 30.01 
 
 2 2 
 
 +0.64031 
 
 1469.98393 1.73486 1.09383 
 
 9 59 40.09 
 
 E 
 
 log jr. 
 
 log L ' log No log N log P 
 
 logQ 
 
 
 
 0.00049251 
 
 0.27365789 0.17682994 4.7157168 9.8582891 
 
 2.3270039 
 
 45 
 
 0.00060285 
 
 0.27380499 0.17699541 4.7796309 9.9227977 
 
 2.3913072 
 
 90 
 
 0.00074754 
 
 0.27399787 0.17721240 4.9163276 0.0598121 
 
 2.5282833 
 
 135 
 
 0.00083193 
 
 0.27411036 0.17733893 5.0341227 0.1774639 
 
 2.6460770 
 
 180 
 
 0.00077916 
 
 0.27404002 0.17725980 5.0782321 0.2210425 
 
 2.6898771 
 
 225 
 
 0.00062422 
 
 0.27383348 0.17702747 5.0332669 0.1754670 
 
 2.6444777 
 
 270 
 
 0.00048849 
 
 0.27365254 0.17682391 4.9151227 0.0570006 
 
 2.5260593 
 
 315 
 
 0.00044928 
 
 0.27360025 0.17676511 4.7787835 9.9208203 
 
 2.3897669 
 
 Zj 
 
 0.00250770 
 
 1.09534833 0.70812605 9.6253991 0.1961442 
 
 0.0712235 
 
 22 
 
 0.00250828 
 
 1.09534908 0.70812692 9.6258039 0.1965489 
 
 0.0716217 
 
 E 
 
 logF 
 
 Ji' Ji J 3 
 
 Fi 
 
 
 
 2.3266400 
 
 366.509888 -2.2679092 -1.6777615 
 
 + 100.81429 
 
 45 
 
 2.3912247 
 
 363.119586 -0.5138777 -3.4254958 
 
 + 68.16435 
 
 90 
 
 2.5282782 
 
 365.160838 +2.2768225 -2.9100648 
 
 19.70006 
 
 135 
 
 2.6459108 
 
 367.567473 +0.7238013 -0.4333073 
 
 -111.30915 
 
 180 
 
 2.6894241 
 
 366.570373 -1.5747120 +2.5538323 
 
 -152.99965 
 
 225 
 
 2.6437740 
 
 364.094438 -1.1669936 +4.3014333 
 
 -120.34967 
 
 270 
 
 2.5252669 
 
 364.090394 +1.5139175 +3.7858697 
 
 - 32.48527 
 
 315 
 
 2.3891044 
 
 367.299619 +1.4067899 +1.3092445 
 
 + 59.12384 
 
 Si 
 
 0.0696092 
 
 1462.331493* -0.0518812 +1.7518757 
 
 -104.37069 
 
 2, 
 
 0.0700139 
 
 1462.082117 -0.4497199 +1.7518747 
 
 -104.37063 
 
 *Si(J,' - G") = 1461.238038. 
 
 Z 2 (./i' - G") = 1460.987287. 
 
THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 ACTION OF UKANUS ON MERCURY. 
 
 
 E 
 
 F 3 1000000 X Ro 1000000000 X <S 1000000 X TFo 100000 X ft"" 
 
 1000000 X S'"> 
 
 
 
 - 7.546840 2.572319 - 40.838728 -0.3564792 0.00000000 
 
 -0.13280483 
 
 45 
 
 - 0.473969 2.902866 6.943520 -0.8432739 +0.62046843 
 
 -0.02098878 
 
 90 
 
 - 1.504590 4.048391 + 74.582644 -0.9823294 +1.04582909 
 
 +0.19267091 
 
 135 
 
 -14.059822 5.410853 + 15.278801 -0.1938530 +0.86293520 
 
 +0.03446006 
 
 180 
 
 -17.382080 5.928252 -102.476485 +1.2462639 0.00000000 
 
 -0.21958237 
 
 225 
 
 - 5.500356 5.225873 - 69.412255 +1.8932075 -0.83343288 
 
 -0.15655363 
 
 270 
 
 + 1.222261 3.978026 + 47.038051 +1.2690546 -1.02765154 
 
 +0.12151438 
 
 315 
 
 - 5.177088 2.987389 + 39.388463 +0.3202880 -0.63853471 
 
 +0.11906293 
 
 , 
 
 -25.211249 16.526988 - 21.694518 +1.1765099 +0.01817755 
 
 -0.03820191 
 
 Ii 
 
 -25.211235 16.526981 - 21.688511 +1.1763686 +0.01143604 
 
 -0.02401942 
 
 JS 
 
 lOOOOOOXlffi sin4- 1000000 > < -#ocosv+ 
 OM. + WJW /' \. -, lOOOOOOXTF.coau 1000000 XTF sin 
 ( -sec 2 v> + ll sin vS \ 
 
 1000000 X- 2- ft 
 a 
 
 
 
 -0.0816775 -2.5723195 -0.31307783 -0.17046914 
 
 4.0868774 
 
 45 
 
 +2.3415248 -1.7140875 -0.10807474 -0.83631981 
 
 - 4.9616693 
 
 90 
 
 +3.9465628 +0.9815683 +0.63709735 -0.74771500 
 
 - 8.0967815 
 
 135 
 
 +3.2460626 +4.3319665 +0.19167313 -0.02898997 
 
 -12.3950199 
 
 180 
 
 +0.2049530 +5.9282527 -1.09453081 -0.59596603 
 
 - 14.2942. r )M> 
 
 225 
 
 -3.0528880 +4.2563802 -0.77797582 -1.72597460 
 
 -11.9712672 
 
 270 
 
 -3.9027069 +0.7238029 +0.36474303 -1.21550896 
 
 - 7.9560519 
 
 315 
 
 -2.3679844 -1.8134015 +0.28908589 -0.13789041 
 
 - 5.1061400 
 
 2! 
 
 +0.1671314 +5.0613044 -0.40576826 -2.72965913 
 
 -34.4339707 
 
 2 2 
 
 +0.1667150 +5.0608577 -0.40529154 -2.72917479 
 
 -34.4340964 
 
 
 sin p 1^,0 + cos v Bo (c > = - 0.00000000000024. 
 
 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 
 
 log coeff. 
 
 
 
 [de/dt] m = + 0.21975650 TO' p 9.3419417 
 
 
 
 [dxldt]m = +32.406731 TO' p 1.5106352 
 
 
 
 [dildt] M =-- 0.55745051 TO' 9.7462063 
 
 
 
 [rfn/<ft] M = -30.777028 TO' n 1.4882267 
 
 
 
 [dw/dt]^ = +32.177180 TO' p 1.5075480 
 
 
 
 [dLldt] M = -45.859693 TO' n 1.6614312 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 89 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'. 
 
 = +0.0000096384435 
 [d x /dt] m = +0.0014213479 
 [dtYdflw = -0.000024449584 
 [dQ/dt] m = -0.0013498699 
 [dT/dfloo = +0.0014112801 
 [dL/dt] m = -0.0020113907 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 // // // 
 
 {de/dt] m +0.00000 +0.00001 +0.0000096 
 
 e[dw/dt] w +0.00029 +0.00029 +0.0002902 
 
 [dt'/dfloo -0.00001 -0.00002 -0.0000244 
 
 sin i [dtt/dt] w -0.00016 -0.00016 -0.0001646 
 
 NOTES. 
 
 In the results of this computation, published in A. J., No. 398, the residual from 
 the test equation which arises from the constancy of the major axis was stated very 
 much too large. Its true value is as here given. 
 
 A comparison of the above figures with the corresponding tabulation for Saturn 
 on Mercury and Uranus on Venus shows 'that a division into eight parts is fully 
 sufficient. The effect of all terms of the 4th and higher orders may, however, in 
 some cases amount to l/1000th of the whole. 
 
 ACTION OF NEPTUNE ON MERCURY. 
 
 E 
 
 A 
 
 B COS e 
 
 Bsin e 
 
 g 
 
 h 
 
 
 
 904.45979027 
 
 + 15.1625927 
 
 - 5.380877 
 
 1.889897 
 
 904.17356 
 
 45 
 
 904.50658592 
 
 + 17.0408109 
 
 + 3.150684 
 
 0.647955 
 
 904.17446 
 
 90 
 
 904.46648308 
 
 + 12.3030460 
 
 + 10.652425 
 
 7.406776 
 
 904.17363 
 
 135 
 
 904.36745259 
 
 + 3.7246157 
 
 + 12.729927 
 
 10.577523 
 
 904.17288 
 
 180 
 
 904.26302599 
 
 - 3.6693551 
 
 + 8.166225 
 
 4.352861 
 
 904.17435 
 
 225 
 
 904.20989585 
 
 - 5.5475736 
 
 - 0.365336 
 
 0.008712 
 
 904.17570 
 
 270 
 
 904.24366422 
 
 - 0.8098086 
 
 - 7.867076 
 
 4.039717 
 
 904.17446 
 
 315 
 
 904.34902898 
 
 + 7.7686225 
 
 - 9.944584 
 
 6.455136 
 
 904.17292 
 
 Si 
 
 3617.43296356* 
 
 +22.9864685f 
 
 + 5.5706981 
 
 17.689251 
 
 3616.69599 
 
 2 2 
 
 3617.43296334 
 
 +22.9864755 
 
 + 5.570691 
 
 17.689326 
 
 3616.69596 
 
 * 4o 2 + 2aV + 4[a' 2 - 2kaa'ce' cos K] = + 3617.4329635, 
 t 4[a'V - kaa'e cos A'] = + 22.986469. 
 J 4fc'oo' cos <f e sin A'' = + 5.570695, 
 
90 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF NEPTUNE ON MERCURY. 
 
 E 
 
 I 
 
 G 
 
 G' 
 
 G" 
 
 e 
 
 
 
 
 
 
 1 II 
 
 
 
 0.22096 
 
 904.17356 
 
 0.2300460 
 
 0.0090860 
 
 55 54.56 
 
 45 
 
 0.26686 
 
 904.17446 
 
 0.2695190 
 
 0.0026589 
 
 59 38.88 
 
 90 
 
 0.22759 
 
 904.17362 
 
 0.2591945 
 
 0.0316047 
 
 1 1 39.23 
 
 135 
 
 0.12931 
 
 904.17286 
 
 0.1906740 
 
 0.0613537 
 
 57 23.73 
 
 180 
 
 0.02341 
 
 904.17434 
 
 0.0820755 
 
 0.0586556 
 
 42 53.31 
 
 225 
 
 0.03107 
 
 904.17570 
 
 0.0003070 
 
 0.0313770 
 
 20 20.99 
 
 270 
 
 0.00394 
 
 904.17445 
 
 0.0688415 
 
 0.0649018 
 
 41 48.59 
 
 315 
 
 0.11084 
 
 904.17291 
 
 0.1564755 
 
 0.0456255 
 
 51 23.82 
 
 s, 
 
 0.57589 
 
 3616.69597 
 
 0.6401575 
 
 0.1642481 
 
 3 22 15.69 
 
 22 
 
 0.57594 
 
 3616.69593 
 
 0.6169755 
 
 0.1410151 
 
 3 8 47.42 
 
 E 
 
 log A'o 
 
 log /,' 
 
 logtfo 
 
 log N log P 
 
 logQ 
 
 
 
 0.00008616 
 
 0.27311614 
 
 0.17622049 4 
 
 .1292404 8.4898441 
 
 1.3492047 
 
 45 
 
 0.00009807 
 
 0.27313202 
 
 0.17623836 4 
 
 .1927246 8.5533496 
 
 1.4127095 
 
 90 
 
 0.00010478 
 
 0.27314097 
 
 0.17624841 4 
 
 .3291694 8.6897764 
 
 1.5491508 
 
 135 
 
 0.00009080 
 
 0.27312233 
 
 0.17622746 4.4470372 8.8075977 
 
 1.6669838 
 
 180 
 
 0.00005070 
 
 0.27306887 
 
 0.17616731 4 
 
 .4915054 8.8520137 
 
 1.7113924 
 
 225 
 
 0.00001141 
 
 0.27301649 
 
 0.17610837 4 
 
 .4469773 8.8074580 
 
 1.6668178 
 
 270 
 
 0.00004818 
 
 0.27306552 
 
 0.17616353 4 
 
 .3290882 8.6895869 
 
 1.5489683 
 
 315 
 
 0.00007281 
 
 0.27309835 
 
 0.17620048 4, 
 
 .1926694 8.5532210 
 
 1.4125965 
 
 2, 
 
 0.00028982 
 
 1.09239150 
 
 0.70479974 7.2790034 4.7212211 
 
 6.1587162 
 
 2 2 
 
 0.00027309 
 
 1.09236919 
 
 0.70477467 7 
 
 .2794084 4.7216261 
 
 6.1591075 
 
 E 
 
 logF 
 
 J,' 
 
 J. 
 
 J3 
 
 F, 
 
 
 
 1.3491992 
 
 897.83542 
 
 -6.8010114 
 
 - 59.717292 
 
 +41.026915 
 
 45 
 
 1.4127079 
 
 890.85490 
 
 + 1.7283795 
 
 - 93.077013 
 
 -24.022644 
 
 90 
 
 1.5491318 
 
 899.76750 
 
 +6.4446186 
 
 - 62.851349 
 
 -81.220264 
 
 135 
 
 1.6669469 
 
 904.08417 
 
 -1.3097917 
 
 + 13.254268 
 
 -97.060333 
 
 180 
 
 1.7113571 
 
 897.88500 
 
 -6.6871696 
 
 + 90.657875 
 
 -62.264029 
 
 225 
 
 1.6667989 
 
 891.04449 
 
 -2.2340656 
 
 + 124.017084 
 
 + 2.785531 
 
 270 
 
 1.5489293 
 
 894.36354 
 
 +5.9480179 
 
 + 93.790933 
 
 +59.983153 
 
 315 
 
 1.4125691 
 
 903.73833 
 
 +2.4206062 
 
 + 17.685807 
 
 +75.823263 
 
 2! 
 
 6.1586174 
 
 3589.85146* 
 
 -1.0955445 
 
 + 61.880167 
 
 -42.474225 
 
 2 2 
 
 6.1590228 
 
 3589.72189 
 
 +0.6051284 
 
 + 61.880146 
 
 -42.474183 
 
 -G") = 3589.68721. 
 - G") = 3589.58087. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 91 
 
 
 
 ACTION OF NEPTUNE ON MERCURY. 
 
 
 E 
 
 F j 1000000 X Ko 1000000000 X So 1000000000 X W 1000000 X fi" 
 
 1000000 X <S (n) 
 
 
 
 - 2.928627 
 
 0.6591808 -15.070783 -133.45316 0.000000 
 
 -0.04900917 
 
 45 
 
 - 0.320722 
 
 0.7449111 + 4.384511 -240.74232 +1.592198 
 
 +0.01325344 
 
 90 
 
 - 8.201274 
 
 1.0513250 +22.423107 -222.59983 +2.715911 
 
 +0.05792607 
 
 135 
 
 -13.630154 
 
 1.3989891 - 6.706655 + 61.47306 +2.231138 
 
 -0.01512625 
 
 180 
 
 - 6.745208 
 
 1.5178690 -34.846090 +466.35639 0.000000 
 
 -0.07466679 
 
 225 
 
 + 0.065738 
 
 1.3382857 -10.354891 +575.81202 -2.134326 
 
 -0.02335460 
 
 270 
 
 - 3.847073 
 
 1.0317860 +21.345904 +331.94457 -2.665434 
 
 +0.05514330 
 
 315 
 
 - 7.846125 
 
 0.7779772 + 6.529864 + 45.70113 -1.662874 
 
 +0.01973839 
 
 r, 
 
 -21.722182 
 
 4.2601608 - 6.147862 +442.24897 +0.050477 
 
 -0.01060659 
 
 22 
 
 -21.731263 
 
 4.2601631 - 6.147171 +442.24389 +0.026136 
 
 -0.00548902 
 
 E 
 
 1000000000 Xtffo sm v lwooooooo ><L-tf<> c s <' 1000000000 1000000000 
 
 1000000000 
 
 7* 
 
 
 + (cosw+cos)<S 
 
 o] fr 2 \ . "! XWoCOsw XWosmu 
 
 x-afa 
 
 
 
 30.1416 
 
 - 659.1808 -117.20523 - 63.81760 
 
 - 1047.3000 
 
 45 
 
 + 608.8428 
 
 - 430.4075 - 30.85375 -238.75703 
 
 -1273.2255 
 
 90 
 
 + 1024.2533 
 
 + 261.0141 +144.36884 -169.43527 
 
 -2102.6500 
 
 135 
 
 + 855.3050 
 
 + 1105.9013 - 60.78178 + 9.19306 
 
 -3204.7615 
 
 180 
 
 + 69.6922 
 
 + 1517.8690 -409.57745 -223.01267 
 
 -3659.9008 
 
 225 
 
 - 792.9698 
 
 + 1080.1647 -236.61842 -524.94878 
 
 -3065.7028 
 
 270 
 
 -1014.1308 
 
 + 169.4383 + 95.40526 -317.93876 
 
 -2063.5719 
 
 315 
 
 - 621.4949 
 
 - 466.5348 + 41.24897 - 19.67525 
 
 -1329.7429 
 
 Si 
 
 + 49.6731 
 
 + 1289.1406 -287.00858 -774.20430 
 
 -8873.4227 
 
 S 2 
 
 + 49.6831 
 
 + 1289.1237 -287.00498 -774.18800 
 
 -8873.4327 
 
 
 sin <f \A i w + cos if L 
 
 t (c) = + 0.000000000000013. 
 
 
 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 
 
 
 log coeff. 
 
 
 
 
 [deldt]<n = + o'.065401848 TO' p 8.8155900 
 
 
 
 
 [dx/dt] m = + 8.2544736 TO' p 0.9166894 
 
 
 
 
 [di/dt] w = - 0.39452600 TO' n 9.5960756 
 
 
 [drfdt] 
 
 w 
 
 = - 8.7298700 TO' 
 = + 8.1893623 TO' 
 
 n 0.9410078 
 p 0.9132501 
 
 [dL/d*]oo = -11.826130 TO' n 1.0728427 
 
92 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'. 
 
 [de/dt} 00 = +0.00000331989 
 [dx/dtlw, +0.00041900885 
 
 [di/dt]oo = -0.00002002670 
 
 [dV/dt] w = -0.00044314061 
 
 [dw!dt] m = +0.00041570371 
 
 [dL/dt] w = -0.00060031125 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt] w +o'.()0000 
 
 +0.00000 
 
 +o'.0000033 
 
 c[d7r/d<]oo +0.00009 
 
 +0.00009 
 
 +0.0000855 
 
 [di/dt] m -0.00001 
 
 -0.00002 
 
 -0.0000200 
 
 sinz [dtt/dt] m -0.00005 
 
 -0.00005 
 
 -0.0000508 
 
 NOTES. 
 
 In the final results of this computation, published in A. J., No. 398, the value of 
 the residual arising from the equation [da/dt] 00 = is greatly overstated. Its true 
 value is that given above. 
 
 The very large disagreement in G', G", 6, etc., arises from the large values of e' 
 and I but the gradual lessening of the discrepancies as the end of the computation is 
 approached shows that terms of the 8th and higher orders are wholly inappreciable. 
 The greatest effect produced by all terms of the 4th and higher orders here occurs 
 with [de/dt] 00 and amounts to but 1 /10000th of the value of this coefficient. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 93 
 
 VENUS. 
 ACTION OF MEKCURY ON VENUS. 
 
 E 
 
 A 
 
 B cos 
 
 B gin e 
 
 1000 x g 
 
 
 
 0.73249627 
 
 +0.19271542 
 
 +0.22036223 
 
 0.30759837 
 
 15 
 
 0.70628935 
 
 +0.12839518 
 
 +0.25427819 
 
 0.40956972 
 
 30 
 
 0.67778527 
 
 +0.05734848 
 
 +0.27076206 
 
 0.46439247 
 
 45 
 
 0.64892716 
 
 -0.01558282 
 
 +0.26869056 
 
 0.45731389 
 
 60 
 
 0.62168257 
 
 -0.08542863 
 
 +0.24820463 
 
 0.39023784 
 
 75 
 
 0.59791027 
 
 -0.14742905 
 
 +0.21070058 
 
 0.28121662 
 
 90 
 
 0.57923027 
 
 -0.19735886 
 
 +0.15873412 
 
 0.15960636 
 
 105 
 
 0.56691636 
 
 -0.23181556 
 
 +0.09584678 
 
 0.05819218 
 
 120 
 
 0.56180731 
 
 -0.24845074 
 
 +0.02632410 
 
 0.00438952 
 
 135 
 
 0.56425026 
 
 -0.24613080 
 
 -0.04509594 
 
 0.01288203 
 
 150 
 
 0.57407787 
 
 -0.22501408 
 
 -0.11354628 
 
 0.08166866 
 
 165 
 
 0.59061890 
 
 -0.18653936 
 
 -0.17436209 
 
 0.19258116 
 
 180 
 
 0.61274521 
 
 -0.13332873 
 
 -0.22339892 
 
 0.31613445 
 
 195 
 
 0.63894853 
 
 -0.06900848 
 
 -0.25731488 
 
 0.41941058 
 
 210 
 
 0.66744400 
 
 +0.00203821 
 
 -0.27379880 
 
 0.47486769 
 
 225 
 
 0.69628968 
 
 +0.07496950 
 
 -0.27172719 
 
 0.46770903 
 
 240 
 
 0.72352217 
 
 +0.14481528 
 
 -0.25124139 
 
 0.39984532 
 
 255 
 
 0.74728586 
 
 +0.20681571 
 
 -0.21373734 
 
 0.28938127 
 
 270 
 
 0.76596211 
 
 +0.25674559 
 
 -0.16177082 
 
 0.16577156 
 
 285 
 
 0.77827946 
 
 +0.29120222 
 
 -0.09888350 
 
 0.06193800 
 
 300 
 
 0.78339732 
 
 +0.30783739 
 
 -0.02936080 
 
 0.00546066 
 
 315 
 
 0.78096661 
 
 +0.30551754 
 
 +0.04205926 
 
 0.01120554 
 
 330 
 
 0.77115125 
 
 +0.28440072 
 
 +0.11050957 
 
 0.07735875 
 
 345 
 
 0.75461912 
 
 +0.24592608 
 
 +0.17132542 
 
 0.18593159 
 
 2i 
 
 8.07130162* 
 
 0.356320051 
 
 -0.018220301 
 
 2.84733165 
 
 St 
 
 8.07130156 
 
 0.35632016 
 
 -0.01822015 
 
 2.84733161 
 
 
 * 12a 2 + 6aV + 12[a' 2 - 
 
 - 2kaa'ee' cos K] = 8.07130158. 
 
 
 1 12[a'V - kaa'e cos K] 
 
 = + 0.35632010. 
 
 
 
 
 t 12fc'aa' cos <p' e sin 
 
 K' = - 0.01822024. 
 
 
 
 0.58840054 
 0.56402284 
 0.53673295 
 0.50812387 
 0.48000449 
 0.45436878 
 0.43333042 
 0.41895978 
 0.41292353 
 0.41598961 
 0.42773114 
 0.44670373 
 0.47092914 
 0.49830819 
 0.52683118 
 0.55465379 
 0.58013161 
 0.60184004 
 0.61860745 
 0.62954599 
 0.63407808 
 0.63196045 
 0.62329194 
 0.60851491 
 
 6.33299247 
 6.33299197 
 
94 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 E 
 
 ACTION OF MERCUKY ON VENUS. 
 G G' 
 
 G" 
 
 
 
 0.13776127 
 
 0.58723527 
 
 0.14259981 
 
 0.00367327 
 
 29 
 
 50 
 
 12.97 
 
 15 
 
 0.13593205 
 
 0.56231468 
 
 0.14274284 
 
 0.00510263 
 
 30 
 
 41 
 
 37.56 
 
 30 
 
 0.13471786 
 
 0.53456025 
 
 0.14296704 
 
 0.00607649 
 
 31 
 
 40 
 
 18.70 
 
 45 
 
 0.13446883 
 
 0.50568767 
 
 0.14321940 
 
 0.00631437 
 
 32 
 
 42 
 
 44.77 
 
 60 
 
 0.13534362 
 
 0.47761736 
 
 0.14342737 
 
 0.00569662 
 
 33 
 
 44 
 
 34.65 
 
 75 
 
 0.13720703 
 
 0.45239646 
 
 0.14351084 
 
 0.00433149 
 
 34 
 
 40 
 
 36.18 
 
 90 
 
 0.13956539 
 
 0.43206750 
 
 0.14340426 
 
 0.00257595 
 
 35 
 
 25 
 
 4.49 
 
 105 
 
 0.14162212 
 
 0.41845750 
 
 0.14309623 
 
 0.00097183 
 
 35 
 
 52 
 
 44.73 
 
 120 
 
 0.14254931 
 
 0.41288415 
 
 0.14266320 
 
 0.00007451 
 
 36 
 
 
 
 33.62 
 
 135 
 
 0.14192619 
 
 0.41587654 
 
 0.14225701 
 
 0.00021775 
 
 35 
 
 48 
 
 50.91 
 
 150 
 
 0.14001227 
 
 0.42706500 
 
 0.14202490 
 
 0.00134648 
 
 35 
 
 20 
 
 41.47 
 
 165 
 
 0.13758072 
 
 0.44529837 
 
 0.14203102 
 
 0.00304495 
 
 34 
 
 40 
 
 10.49 
 
 180 
 
 0.13548161 
 
 0.46890720 
 
 0.14224327 
 
 0.00473972 
 
 33 
 
 51 
 
 11.13 
 
 195 
 
 0.13430588 
 
 0.49597000 
 
 0.14257523 
 
 0.00593116 
 
 32 
 
 57 
 
 11.97 
 
 210 
 
 0.13427836 
 
 0.52451120 
 
 0.14293246 
 
 0.00633413 
 
 32 
 
 1 
 
 25.39 
 
 225 
 
 0.13530143 
 
 0.55262582 
 
 0.14323803 
 
 0.00590863 
 
 31 
 
 6 
 
 52.56 
 
 240 
 
 0.13705610 
 
 0.57856627 
 
 0.14343948 
 
 0.00481804 
 
 30 
 
 16 
 
 21.83 
 
 255 
 
 0.13911137 
 
 0.60079677 
 
 0.14351100 
 
 0.00335627 
 
 29 
 
 32 
 
 27.98 
 
 270 
 
 0.14102020 
 
 0.61804514 
 
 0.14345226 
 
 0.00186975 
 
 28 
 
 57 
 
 29.65 
 
 285 
 
 0.14239902 
 
 0.62934391 
 
 0.14328793 
 
 0.00068684 
 
 28 
 
 33 
 
 26.40 
 
 300 
 
 0.14298478 
 
 0.63406059 
 
 0.14306246 
 
 0.00006020 
 
 28 
 
 21 
 
 52.79 
 
 315 
 
 0.14267170 
 
 0.63192420 
 
 0.14283210 
 
 0.00012415 
 
 28 
 
 23 
 
 50.37 
 
 330 
 
 0.14152485 
 
 0.62303406 
 
 0.14265313 
 
 0.00087040 
 
 28 
 
 39 
 
 39.46 
 
 345 
 
 0.13976975 
 
 0.60786141 
 
 0.14256873 
 
 0.00214548 
 
 29 
 
 8 
 
 52.62 
 
 Si 
 
 1.66229562 
 
 6.31855399 
 
 1.71486964 
 
 0.03813556 
 
 384 
 
 9 
 
 26.15 
 
 S 2 
 
 1.66229608 
 
 6.31855323 
 
 1.71487036 
 
 0.03813555 
 
 384 
 
 9 
 
 26.54 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 95 
 
 ACTION OF MEKCURY 
 
 ON VENUS. 
 
 
 
 E 
 
 log K, 
 
 log Lo' 
 
 logtfo 
 
 logtf 
 
 logP 
 
 logQ 
 
 
 
 0.09428087 
 
 0.39679023 
 
 0.31472571 
 
 0.0090498 
 
 0.8627996 
 
 0.5522553 
 
 15 
 
 0.10017341 
 
 0.40440450 
 
 0.32320948 
 
 0.0415726 
 
 0.9381720 
 
 0.6108796 
 
 30 
 
 0.10716555 
 
 0.41342180 
 
 0.33324961 
 
 0.0806578 
 
 1.0282684 
 
 0.6810018 
 
 45 
 
 0.11492344 
 
 0.42340404 
 
 0.34435531 
 
 0.1248163 
 
 1.1296769 
 
 0.7598999 
 
 60 
 
 0.12293987 
 
 0.43369427 
 
 0.35579389 
 
 0.1716324 
 
 1.2368681 
 
 0.8431970 
 
 75 
 
 0.13049856 
 
 0.44337414 
 
 0.36654485 
 
 0.2174860 
 
 1.3415449 
 
 0.9243732 
 
 90 
 
 0.13670353 
 
 0.45130400 
 
 0.37534554 
 
 0.2575171 
 
 1.4325547 
 
 0.9947294 
 
 105 
 
 0.14065788 
 
 0.45634998 
 
 0.38094249 
 
 0.2862203 
 
 1.4972527 
 
 1.0445040 
 
 120 
 
 0.14178785 
 
 0.45779080 
 
 0.38254019 
 
 0.2989083 
 
 1.5248859 
 
 1.0655419 
 
 135 
 
 0.14009658 
 
 0.45563407 
 
 0.38014861 
 
 0.2935151 
 
 1.5107656 
 
 1.0544719 
 
 150 
 
 0.13608370 
 
 0.45051253 
 
 0.37446742 
 
 0.2714379 
 
 1.4582282 
 
 1.0140442 
 
 165 
 
 0.13043971 
 
 0.44329885 
 
 0.36646127 
 
 0.2367595 
 
 1.3768369 
 
 0.9516100 
 
 180 
 
 0.12381666 
 
 0.43481823 
 
 0.35704266 
 
 0.1945715 
 
 1.2784801 
 
 0.8761594 
 
 195 
 
 0.11676724 
 
 0.42577304 
 
 0.34698954 
 
 0.1495758 
 
 1.1741125 
 
 0.7959472 
 
 210 
 
 0.10975153 
 
 0.41675185 
 
 0.33695547 
 
 0.1054452 
 
 1.0722608 
 
 0.7174326 
 
 225 
 
 0.10314673 
 
 0.40824135 
 
 0.32748247 
 
 0.0647779 
 
 0.9789193 
 
 0.6452104 
 
 240 
 
 0.09725105 
 
 0.40063000 
 
 0.31900464 
 
 0.0292994 
 
 0.8980200 
 
 0.5823493 
 
 255 
 
 0.09229597 
 
 0.39422222 
 
 0.31186325 
 
 0.0001263 
 
 0.8320547 
 
 0.5308426 
 
 270 
 
 0.08845847 
 
 0.38925289 
 
 0.30632243 
 
 9.9779738 
 
 0.7825625 
 
 0.4919641 
 
 285 
 
 0.08587486 
 
 0.38590395 
 
 0.30258708 
 
 9.9633062 
 
 0.7504867 
 
 0.4665316 
 
 300 
 
 0.08464926 
 
 0.38431433 
 
 0.30081370 
 
 9.9564277 
 
 0.7363980 
 
 0.4550694 
 
 315 
 
 0.08485629 
 
 0.38458290 
 
 0.30111334 
 
 9.9575313 
 
 0.7406137 
 
 0.4578944 
 
 330 
 
 0.08653835 
 
 0.38676424 
 
 0.30354673 
 
 9.9667119 
 
 0.7632399 
 
 0.475140o 
 
 345 
 
 0.08969689 
 
 0.39085723 
 
 0.30811150 
 
 9.9839477 
 
 0.8041353 
 
 0.5067244 
 
 Zi 
 
 1.32942669 
 
 5.01604517 
 
 4.05980799 
 
 1.3196325 
 
 13.0745661 
 
 8.7488847 
 
 Z 2 
 
 1.32942756 
 
 5.01604627 
 
 4.05980919 
 
 1.3196348 
 
 13.0745711 
 
 8.7488890 
 
96 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MERCURY ON VENUS. 
 
 E 
 
 log V 
 
 /' 
 
 Ji. 
 
 J 3 
 
 1000 X F, 
 
 
 
 0.5489818 
 
 0.14641488 
 
 +0.046569366 
 
 +0.007078444 
 
 -6.6249758 
 
 15 
 
 0.6061455 
 
 0.14799570 
 
 +0.053805593 
 
 +0.005998321 
 
 -7.6446250 
 
 30 
 
 0.6750909 
 
 0.14917691 
 
 +0.057280924 
 
 +0.004274648 
 
 -8.1401963 
 
 45 
 
 0.7534266 
 
 0.14962155 
 
 +0.056746961 
 
 + 0.002024896 
 
 -8.0779189 
 
 60 
 
 0.8370288 
 
 0.14915409 
 
 +0.052252409 
 
 -0.000597633 
 
 -7.4620305 
 
 75 
 
 0.9194294 
 
 0.14784239 
 
 +0.044134510 
 
 -0.003414224 
 
 -6.3345072 
 
 90 
 
 0.9916527 
 
 0.14602955 
 
 +0.032988273 
 
 -0.006232935 
 
 -4.7721868 
 
 105 
 
 1.0433053 
 
 0.14427332 
 
 +0.019614339 
 
 -0.008861684 
 
 -2.8815397 
 
 120 
 
 1.0654487 
 
 0.14317010 
 
 +0.004953440 
 
 -0.011121317 
 
 -0.7914084 
 
 135 
 
 1.0542014 
 
 0.14310862 
 
 -0.009985226 
 
 -0.012857840 
 
 + 1.3557653 
 
 150 
 
 1.0124149 
 
 0.14408811 
 
 -0.024195183 
 
 -0.013952904 
 
 +3.4136578 
 
 165 
 
 0.9480753 
 
 0.14573205 
 
 -0.036737898 
 
 -0.014331875 
 
 + 5.2420265 
 
 180 
 
 0.8709298 
 
 0.14748132 
 
 -0.046798849 
 
 -0.013968920 
 
 +6.7162708 
 
 195 
 
 0.7897501 
 
 0.14882171 
 
 -0.053732494 
 
 -0.012888777 
 
 +7.7359211 
 
 210 
 
 0.7111602 
 
 0.14942892 
 
 -0.057095855 
 
 -0.011165046 
 
 + 8.2314924 
 
 225 
 
 0.6396416 
 
 0.14920890 
 
 -0.056671026 
 
 -0.008915214 
 
 +8.1692132 
 
 240 
 
 0.5779992 
 
 0.14827058 
 
 -0.052477157 
 
 -0.006292610 
 
 +7.5533276 
 
 255 
 
 0.5279163 
 
 0.14686716 
 
 -0.044770588 
 
 -0.003475962 
 
 + 6.4258044 
 
 270 
 
 0.4903757 
 
 0.14532896 
 
 -0.034036602 
 
 -0.000657227 
 
 +4.8634820 
 
 285 
 
 0.4659576 
 
 0.14399775 
 
 -0.020965237 
 
 +0.001971497 
 
 +2.9728356 
 
 300 
 
 0.4550194 
 
 0.14316554 
 
 -0.006416142 
 
 +0.004231074 
 
 +0.8827039 
 
 315 
 
 0.4577910 
 
 0.14302191 
 
 +0.008631431 
 
 +0.005967519 
 
 -1.2644706 
 
 330 
 
 0.4744060 
 
 0.14361483 
 
 +0.023141851 
 
 +0.007062502 
 
 -3.3223626 
 
 345 
 
 0.5048724 
 
 0.14483260 
 
 +0.036096092 
 
 +0.007441419 
 
 -5.1507310 
 
 2, 
 
 8.7105081 
 
 1.75532379* 
 
 -0.003833505 
 
 -0.041341924 
 
 +0.5477741 
 
 2* 
 
 8.7105125 
 
 1.75532376 
 
 -0.003833543 
 
 -0.041341924 
 
 +0.5477737 
 
 
 * 2,(J,' - G") = 1 
 
 .71718823. 
 
 
 
 
 
 S 2 (j 2 ' _ G") = 1 
 
 .71718811. 
 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 97 
 
 ACTION OF MERCURY ON VENUS. 
 
 E 
 
 1000 X F 3 
 
 R 
 
 So 
 
 W 
 
 BW 
 
 <"> 
 
 
 
 +0.12855762 
 
 -1.0113711 
 
 +0.11654330 
 
 +0.02599380 
 
 0.0000000 
 
 +0.16223015 
 
 15 
 
 +0.28853773 
 
 -1.0855630 
 
 +0.15095431. 
 
 +0.02672256 
 
 -0.3910151 
 
 +0.21008155 
 
 30 
 
 +0.43551240 
 
 -1.1839703 
 
 +0.18420453 
 
 +0.02487781 
 
 -0.8232930 
 
 +0.25617923 
 
 45 
 
 +0.52994293 
 
 -1.3088599 
 
 +0.21275230 
 
 +0.01862044 
 
 -1.2857216 
 
 +0.29555816 
 
 60 
 
 +0.54641763 
 
 -1.4594255 
 
 +0.23029016 
 
 +0.00532100 
 
 -1.7533282 
 
 +0.31946706 
 
 75 
 
 +0.48046967 
 
 -1.6276742 
 
 +0.22753508 
 
 -0.01781201 
 
 -2.1774255 
 
 +0.31512321 
 
 90 
 
 +0.34977540 
 
 -1.7935752 
 
 +0.19439994 
 
 -0.05167293 
 
 -2.4796011 
 
 +0.26875615 
 
 105 
 
 +0.18941983 
 
 -1.9243133 
 
 +0.12616235 
 
 -0.09195653 
 
 -2.5651527 
 
 +0.17410988 
 
 120 
 
 +0.04248891 
 
 -1.9847155 
 
 +0.03108865 
 
 -0.12787906 
 
 -2.3681413 
 
 +0.04283321 
 
 135 
 
 -0.05148113 
 
 -1.9570464 
 
 -0.06917608 
 
 -0.14733858 
 
 -1.9039342 
 
 -0.09517474 
 
 150 
 
 -0.06711060 
 
 -1.8525243 
 
 -0.15091802 
 
 -0.14550282 
 
 -1.2730044 
 
 -0.20741354 
 
 165 
 
 +0.00001044 
 
 -1.7023416 
 
 -0.20114407 
 
 -0.12716787 
 
 -0.6051233 
 
 -0.27625376 
 
 180 
 
 +0.13212520 
 
 -1.5390947 
 
 -0.22013939 
 
 -0.10126618 
 
 0.0000000 
 
 -0.30227222 
 
 195 
 
 + 0.29405223 
 
 -1.3852923 
 
 -0.21561000 
 
 -0.07503510 
 
 +0.4924233 
 
 -0.29612150 
 
 210 
 
 +0.44259806 
 
 -1.2520945 
 
 -0.19638955 
 
 -0.05218717 
 
 +0.8604054 
 
 -0.26990714 
 
 225 
 
 +0.53811675 
 
 -1.1428753 
 
 -0.16935246 
 
 -0.03375802 
 
 + 1.1118587 
 
 -0.23300075 
 
 240 
 
 +0.55512295 
 
 -1.0569737 
 
 -0.13887026 
 
 -0.01942443 
 
 + 1.2611694 
 
 -0.19133211 
 
 255 
 
 +0.48911303 
 
 -0.9922445 
 
 -0.10732663 
 
 -0.00839923 
 
 + 1.3226839 
 
 -0.14811570 
 
 270 
 
 +0.35776744 
 
 -0.9463685 
 
 -0.07579537 
 
 +0.00013574 
 
 + 1.3083458 
 
 -0.10478640 
 
 285 
 
 +0.19621617 
 
 -0.9174587 
 
 -0.04456334 
 
 + 0.00686904 
 
 + 1.2273328 
 
 -0.06171770 
 
 300 
 
 +0.04762640 
 
 -0.9042927 
 
 -0.01348260 
 
 +0.01232297 
 
 + 1.0864013 
 
 -0.01870356 
 
 315 
 
 -0.04835263 
 
 -0.9063942 
 
 +0.01780841 
 
 +0.01685710 
 
 +0.8903706 
 
 +0.02473967 
 
 330 
 
 -0.06620421 
 
 -0.9240677 
 
 +0.04973149 
 
 +0.02067163 
 
 +0.6425655 
 
 +0.06916321 
 
 345 
 
 -0.00136711 
 
 -0.9584216 
 
 +0.08262385 
 
 +0.02378861 
 
 +0.3452194 
 
 +0.11498675 
 
 V 
 
 *-l 
 
 + 2.90467772 
 
 -5.9084737 
 
 +0.01066288 
 
 -0.40860954 
 
 -3.5384806 
 
 +0.02421404 
 
 2* 
 
 +2.90467759 
 
 -5.9084840 
 
 +0.01066372 
 
 -0.40860959 
 
 -3.5384837 
 
 +0.02421607 
 
 sin <p 
 
 ! ( "> + cos <p B (c) = + 0.000000033. 
 
98 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MERCURY ON VENCJS. 
 
 Ro COS V 
 
 E 
 
 [o sin v 
 
 L 
 
 ( \ 
 
 , Wo cos u 
 
 Wo sin u 
 
 -2-flo 
 
 
 + (cos v+cos E)S( 
 
 a 860 *" / 
 
 vSo 
 
 
 a 
 
 0* 
 
 +0.2330866 
 
 + 1.0113711 
 
 +0.015230839 
 
 +0.021064164 
 
 2.0089005 
 
 15 
 
 +0.0087244 
 
 + 1.1264720 
 
 +0.009475283 
 
 +0.024986282 
 
 2.1567748 
 
 30 
 
 -0.2767659 
 
 + 1.2080644 
 
 + 0.002459142 
 
 +0.024755972 
 
 2.3539076 
 
 45 
 
 -0.6298334 
 
 + 1.2226121 
 
 -0.003043897 
 
 +0.018369962 
 
 2.6050539 
 
 60 
 
 -1.0391050 
 
 + 1.1217558 
 
 -0.002204101 
 
 +0.004843036 
 
 2.9088638 
 
 75 
 
 -1.4586399 
 
 +0.8508163 
 
 +0.011332120 
 
 -0.013742297 
 
 3.2495836 
 
 90 
 
 -1.7948636 
 
 +0.3765263 
 
 +0.042079553 
 
 -0.029990049 
 
 3.5871504 
 
 105 
 
 -1.9215248 
 
 -0.2668018 
 
 +0.086138100 
 
 -0.032190511 
 
 3.8554443 
 
 120 
 
 -1.7441606 
 
 -0.9487539 
 
 +0.127283754 
 
 -0.012324508 
 
 3.9830138 
 
 135 
 
 -1.2790794 
 
 -1.4880990 
 
 +0.145357191 
 
 +0.024082111 
 
 3.9330324 
 
 150 
 
 -0.6591294 
 
 -1.7579567 
 
 +0.132584634 
 
 +0.059936528 
 
 3.7270060 
 
 165 
 
 -0.0490234 
 
 -1.7488888 
 
 +0.098503636 
 
 +0.080428204 
 
 3.4271881 
 
 180 
 
 +0.4402788 
 
 -1.5390947 
 
 +0.059336027 
 
 +0.082061396 
 
 3.0992536 
 
 195 
 
 +0.7728021 
 
 -1.2274791 
 
 +0.026854296 
 
 +0.070065081 
 
 2.7888981 
 
 210 
 
 +0.9628351 
 
 -0.8906639 
 
 +0.005513895 
 
 +0.051895060 
 
 2.5190296 
 
 225 
 
 + 1.0443017 
 
 -0.5731024 
 
 -0.005195888 
 
 +0.033355748 
 
 2.2968106 
 
 240 
 
 + 1.0518039 
 
 -0.2937729 
 
 -0.007836004 
 
 +0.017773731 
 
 2.1211810 
 
 255 
 
 + 1.0129579 
 
 -0.0559799 
 
 -0.005257515 
 
 -0.006550236 
 
 1.9880041 
 
 270 
 
 +0.9468650 
 
 +0.1451147 
 
 +0.000109448 
 
 -0.000080285 
 
 1.8927371 
 
 285 
 
 +0.8649660 
 
 +0.3177537 
 
 +0.006402057 
 
 -0.002489441 
 
 1.8316679 
 
 300 
 
 +0.7723975 
 
 +0.4708818 
 
 +0.012250672 
 
 -0.001332937 
 
 1.8023971 
 
 315 
 
 +0.6691423 
 
 +0.6125550 
 
 +0.016656295 
 
 +0.002594179 
 
 1.8040170 
 
 330 
 
 +0.5508294 
 
 +0.7487960 
 
 -f 0.018894170 
 
 +0.008386104 
 
 1.8371831 
 
 345 
 
 +0.4092814 
 
 +0.8824106 
 
 +0.018479728 
 
 +0.014979907 
 
 1.9041728 
 
 2, 
 
 -0.5559282 
 
 -0.3477320 
 
 +0.405702029 
 
 +0.226988212 
 
 31.8406236 
 
 2 2 
 
 -0.5559251 
 
 -0.3477313 
 
 +0.405701406 
 
 +0.226989461 
 
 31.8406376 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 
 
 [de/dt]oo = - 
 
 97592.111m' n4 
 
 .9894147 
 
 
 [dxldtiw = -8920493.9 TO' n 6 
 
 .9503889 
 
 
 
 
 [di/dt] M = + 
 
 71223.820 TO' p4 
 
 .8526253 
 
 
 
 
 [dQ/(ft]oo = + 
 
 673299.06 TO' p5 
 
 .8282080 
 
 
 [dirldt} w = -8919313.6 TO' 6 
 
 .9503315 
 
 
 [dL/dt]oo = +5590689.3 m' p 6 
 
 .7474654 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'. 
 
 [de/d/Joo = -0.013012279 
 
 = -1.1893992 
 
 = +0.0094965089 
 
 = +0.089773204 
 
 = -1.1892420 
 [dL/dt] w = +0.74542525 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt]^ -0.01304 
 
 -0.01301 
 
 -0.013012 
 
 e[dir/dt]oo -0.00810 
 
 -0.00814 
 
 -0.008138 
 
 [di/dt}^ +0.00950 
 
 +0.00949 
 
 +0.009497 
 
 sin i [dfl/d<]oo +0-00529 
 
 +0.00531 
 
 +0.005301 
 
 [dL/dfloo +0.747 
 
 
 +0.745425 
 
 NOTES. 
 
 This computation was originally made with but twelve points of division, but 
 it was found that, notwithstanding the small eccentricity of the orbit of Venus, the 
 values of e' and I are here so large that the tests which arise by comparing the sums 
 of the functions were, toward the close of the computation, entirely inapplicable. 
 The sums for [de/dt\ 00 agreed to but a single significant figure, while those for [dx/dt] o, 
 [di/dt] 00 , and [dQ/di\ 00 agreed to but two. It will be noticed that the increase of the 
 number of points of division almost wholly removes the discrepancy. 
 
 Notwithstanding the entire disagreement of the test equations when but twelve 
 points of division were employed, it is evident that this number would have been 
 sufficient. The greatest error would have occurred with [dLjdt] OQ , its amount being 
 0".00000016, showing that with this coefficient the sum of all terms of an order 
 higher than the 12th amounts to but 1 /4000000th of the remaining terms. 
 
100 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 .E 
 
 
 
 30 
 
 60 
 
 90 
 
 120 
 
 150 
 
 180 
 
 210 
 
 240 
 
 270 
 
 300 
 
 330 
 
 A 
 
 1.53711333 
 1.52932317 
 1.51985488 
 1.51125435 
 1.50583081 
 1.50503256 
 1.50906482 
 1.51684286 
 1.52628659 
 1.53487476 
 1.54031063 
 1.54113325 
 
 9.13846106* 
 9.13846095 
 
 ACTION OF THE 
 
 EARTH ON VENUS. 
 
 
 B cos t 
 
 B am t 
 
 1000 Xff 
 
 +0.64403672 
 
 +0.34841048 
 
 0.03414346 
 
 +0.38337105 
 
 +0.61703357 
 
 0.10708849 
 
 +0.02331740 
 
 +0.71967967 
 
 0.14568124 
 
 -0.33964844 
 
 +0.62884478 
 
 0.11122751 
 
 -0.60827014 
 
 +0.36886855 
 
 0.03827088 
 
 -0.71057033 
 
 +0.00941083 
 
 0.00002491 
 
 -0.61913831 
 
 -0.35321171 
 
 0.03509098 
 
 -0.35847281 
 
 -0.62183471 
 
 0.10876150 
 
 +0.00158081 
 
 -0.72448100 
 
 0.14763156 
 
 +0.36454672 
 
 -0.63364618 
 
 0.11293247 
 
 +0.63316838 
 
 -0.37366983 
 
 0.03927365 
 
 +0.73546881 
 
 -0.01421211 
 
 0.00005681 
 
 +0.07469486f 
 
 -0.01440384t 
 
 0.44009177 
 
 +0.07469500 
 
 -0.01440382 
 
 0.44009169 
 
 1.00234463 
 1.00372554 
 1.00314070 
 1.00120600 
 0.99983131 
 1.0003553T 
 1.00226402 
 1.00370935 
 1.00326747 
 1.00133527 
 0.99985008 
 1.00036791 
 
 6.01069819 
 6.01069936 
 
 E 
 
 G 
 
 G' 
 
 G" 
 
 
 
 0.53448743 
 
 1.00227181 
 
 0.53462398 
 
 0.00006372 
 
 46 
 
 55 
 
 3.02 
 
 30 
 
 0.52531637 
 
 1.00350237 
 
 0.52574251 
 
 0.00020298 
 
 46 
 
 22 
 
 33.07 
 
 60 
 
 0.51643292 
 
 1.00284204 
 
 0.51701255 
 
 0.00028098 
 
 45 
 
 53 
 
 55.40 
 
 90 
 
 0.50976708 
 
 1.00097979 
 
 0.51021108 
 
 0.00021779 
 
 45 
 
 33 
 
 45.30 
 
 120 
 
 0.50571824 
 
 0.99975382 
 
 0.50587139 
 
 0.00007567 
 
 45 
 
 20 
 
 44.51 
 
 150 
 
 0.50439599 
 
 1.00035525 
 
 0.50439608 
 
 0.00000005 
 
 45 
 
 14 
 
 29.83 
 
 180 
 
 0.50651954 
 
 1.00219338 
 
 0.50665928 
 
 0.00006911 
 
 45 
 
 19 
 
 11.90 
 
 210 
 
 0.51285225 
 
 1.00348844 
 
 0.51328431 
 
 0.00021116 
 
 45 
 
 39 
 
 53.44 
 
 240 
 
 0.52273786 
 
 1.00296095 
 
 0.52332564 
 
 0.00028127 
 
 46 
 
 15 
 
 21.69 
 
 270 
 
 0.53325823 
 
 1.00109414 
 
 0.53371072 
 
 0.00021137 
 
 46 
 
 54 
 
 18.41 
 
 300 
 
 0.54017928 
 
 0.99976461 
 
 0.54033745 
 
 0.00007270 
 
 47 
 
 19 
 
 22.49 
 
 330 
 
 0.54048407 
 
 1.00036779 
 
 0.54048430 
 
 0.00000011 
 
 47 
 
 18 
 
 38.53 
 
 2, 
 
 3.12607525 
 
 6.00978661 
 
 3.12783029 
 
 0.00084345 
 
 277 
 
 3 
 
 39.01 
 
 2 2 
 
 3.12607397 
 
 6.00978778 
 
 3.12782900 
 
 0.00084346 
 
 277 
 
 3 
 
 38.58 
 
 * 6a s + 3aV + 6[o' ! - 2fcaa'ee' cos A'] = 9.13846101. 
 t 6[a'V - koa'e cos A'] = + 0.07469471. 
 I - 6k'aa' cos <p' e sin A' = - 0.04440383. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 101 
 
 
 
 
 
 ACTION 
 
 OF 
 
 THE EARTH ON 
 
 VENUS. 
 
 
 
 
 E 
 
 
 logtfo 
 
 
 log Lo' 
 
 
 logJVo 
 
 
 \ogN 
 
 logP 
 
 
 logQ 
 
 
 
 q 
 
 .26000336 
 
 0.60598257 
 
 0.54577154 
 
 9 
 
 .8305330 
 
 0.4344894 
 
 
 0.3752914 
 
 30 
 
 o 
 
 .25288171 
 
 
 
 .59719127 
 
 0.53614961 
 
 9.8233230 
 
 0.4173018 
 
 
 0.3578664 
 
 60 
 
 
 
 .24673133 
 
 0.58958546 
 
 0. 
 
 52781896 
 
 9 
 
 .8197364 
 
 0.4066135 
 
 
 0.3462012 
 
 90 
 
 
 
 .24246622 
 
 0.58430371 
 
 0. 
 
 52203035 
 
 9.8196999 
 
 0.4029640 
 
 
 0.3412105 
 
 120 
 
 
 
 .23974361 
 
 0.58092896 
 
 0.51833025 
 
 9.8208350 
 
 0.4019120 
 
 
 0.3392393 
 
 150 
 
 
 
 .23844519 
 
 0.57931865 
 
 0. 
 
 51656434 
 
 9 
 
 .8213596 
 
 0.4003696 
 
 
 0.3377696 
 
 180 
 
 
 
 .23942219 
 
 0.58053038 
 
 0. 
 
 51789317 
 
 9.8218870 
 
 0.4004544 
 
 
 0.3387987 
 
 210 
 
 
 
 .24375790 
 
 0.58590391 
 
 0. 
 
 52378440 
 
 9.8244982 
 
 0.4071947 
 
 
 0.3466789 
 
 240 
 
 
 
 .25132635 
 
 0.59526901 
 
 0. 
 
 53404476 
 
 9.8301980 
 
 0.4226554 
 
 
 0.3628370 
 
 270 
 
 
 
 .25983876 
 
 0.60577958 
 
 0. 
 
 54554439 
 
 9.8370023 
 
 0.4416487 
 
 
 0.3819851 
 
 300 
 
 
 
 .26543283 
 
 0.61267386 
 
 0. 
 
 55308969 
 
 9 
 
 .8405753 
 
 0.4533906 
 
 
 0.3937357 
 
 330 
 
 
 
 .26526804 
 
 0.61247091 
 
 0. 
 
 55286781 
 
 !) 
 
 .8378789 
 
 0.4500302 
 
 
 0.3905869 
 
 2, . 
 
 1 
 
 .50265967 
 
 3.56497024 
 
 3. 
 
 19694837 
 
 8.9637646 
 
 2.5195152 
 
 
 2.1561032 
 
 2 2 
 
 1 
 
 .50265782 
 
 3.56496803 
 
 3. 
 
 19694594 
 
 8.9637618 
 
 2.5195089 
 
 
 2.1560973 
 
 E 
 
 
 logF 
 
 
 j. 
 
 
 J, 
 
 
 J 
 
 i 
 
 
 F, 
 
 
 
 
 0.3752596 
 
 
 0.9974825 
 
 
 +0.004202020 
 
 -0.047236424 
 
 -0.005832177 
 
 30 
 
 
 0.3577653 
 
 
 0.9964541 
 
 
 +0.010025286 
 
 -0.058127293 
 
 -0 
 
 .010328762 
 
 60 
 
 
 0.3460609 
 
 
 0.9970986 
 
 
 +0.013405599 
 
 -0.053273840 
 
 -0.012046994 
 
 90 
 
 
 0.3411015 
 
 
 0.9987570 
 
 
 +0.012206904 
 
 -0.033976547 
 
 -0.010526476 
 
 120 
 
 
 0.3392013 
 
 
 0.9997616 
 
 
 +0.006516023 
 
 -0.005406098 
 
 -0 
 
 .006174632 
 
 150 
 
 
 0.3377695 
 
 
 0.9991244 
 
 
 -0.001172845 
 
 +0.024782063 
 
 -0 
 
 .000157532 
 
 180 
 
 
 0.3387641 
 
 
 0.9974877 
 
 
 -0.007608410 
 
 +0.048499056 
 
 +0 
 
 .005912547 
 
 210 
 
 
 0.3465734 
 
 
 0.9964670 
 
 
 -0.010818504 
 
 +0.059389945 
 
 +0.010409132 
 
 240 
 
 
 0.3626968 
 
 
 0.9970681 
 
 
 -0.010873469 
 
 +0.054536468 
 
 +0.012127366 
 
 270 
 
 
 0.3818798 
 
 
 0.9987050 
 
 
 -0.008962465 
 
 +0.035239146 
 
 +0.010606846 
 
 300 
 
 
 0.3936994 
 
 
 0.9997505 
 
 
 -0.005884342 
 
 +0.006668715 
 
 +0.006255003 
 
 330 
 
 
 0.3905868 
 
 
 0.9991427 
 
 
 -0.001520954 
 
 -0.023519435 
 
 +0.000237902 
 
 2, 
 
 
 2.1556821 
 
 
 5.9886490* 
 
 
 -0.000242579 
 
 +0.003787877 
 
 +0 
 
 .000241113 
 
 2 2 
 
 
 2.1556763 
 
 
 5.9886502 
 
 
 -0.000242578 
 
 +0.003787879 
 
 +0 
 
 .000241110 
 
 * 
 
 - 1 (t) 
 
 r/ _ G") = 
 
 5.9878055. 
 
 
 - ^ (J 
 
 Y _ G") .= 
 
 5.9878067. 
 
102 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF THE EARTH ON VENUS. 
 
 E 
 
 1000 X F, 
 
 So 
 
 1000 X So 
 
 TF 
 
 B<" 
 
 S<"> 
 
 
 
 +0.20122394 
 
 0.42126107 
 
 -5.890097 
 
 -0.11153496 
 
 0.0000000 
 
 -0.008199111 
 
 30 
 
 +0.06017487 
 
 0.40675651 
 
 -4.150418 
 
 -0.13232119 
 
 +0.2828447 
 
 -0.005772122 
 
 60 
 
 -0.29485277 
 
 0.40441131 
 
 -0.984602 
 
 -0.11894044 
 
 +0.4858527 
 
 -0.001365876 
 
 90 
 
 -0.50823894 
 
 0.41101802 
 
 +0.151227 
 
 -0.07580674 
 
 +0.5682286 
 
 +0.000209069 
 
 120 
 
 -0.36563277 
 
 0.41646740 
 
 -1.349097 
 
 -0.01272800 
 
 +0.4969244 
 
 -0.001858754 
 
 150 
 
 -0.00856116 
 
 0.41403810 
 
 -2.948800 
 
 +0.05391799 
 
 +0.2845157 
 
 -0.004052670 
 
 180 
 
 +0.20680805 
 
 0.40710579 
 
 -1.730893 
 
 +0.10632289 
 
 0.0000000 
 
 -0.002376680 
 
 210 
 
 +0.06559223 
 
 0.40536093 
 
 +2.554045 
 
 +0.13208012 
 
 -0.2785530 
 
 +0.003510140 
 
 240 
 
 -0.29105383 
 
 0.41526071 
 
 +7.029019 
 
 +0.12494379 
 
 -0.4954847 
 
 +0.009684416 
 
 270 
 
 -0.50707628 
 
 0.43289840 
 
 +7.732079 
 
 +0.08349760 
 
 -0.5984781 
 
 +0.010689527 
 
 300 
 
 -0.36741788 
 
 0.44451245 
 
 +3.199250 
 
 +0.01546615 
 
 -0.5340294 
 
 +0.004438118 
 
 330 
 
 -0.01281582 
 
 0.43893384 
 
 -3.068000 
 
 -0.05784756 
 
 -0.3052198 
 
 -0.004266768 
 
 s, 
 
 -0.91092526 
 
 2.50901873 
 
 +0.273580 
 
 +0.00352943 
 
 -0.0467370 
 
 +0.000322113 
 
 S 2 
 
 -0.91092510 
 
 2.50900580 
 
 +0.270133 
 
 +0.00352022 
 
 -0.0466619 
 
 +0.000317176 
 
 _ [fl sin v 
 
 + (coav+coaE)S a ] 
 
 -0.01178019 
 
 30 +0.19740436 
 
 60 +0.35044516 
 
 90 +0.41100745 
 
 120 +0.36078900 
 
 150 +0.21090709 
 
 180 +0.00346179 
 
 210 -0.20590981 
 
 240 -0.36545661 
 
 270 -0.43294230 
 
 300 -0.38308894 
 
 330 -0.22607885 
 
 S, -0.04562979 
 2 2 -0.04561206 
 
 sin <f\Ai ( '' + coa <fB 
 
 
 
 - 
 
 Ro cos v 
 
 .('-sec? +\\smvsl 
 \o see *" / S " J 
 
 -0.42126107 
 -0.35572418 
 -0.20183130 
 +0.00311509 
 +0.20803111 
 +0.35633148 
 +0.40710579 
 +0.34919573 
 +0.19760050 
 -0.01250179 
 -0.22551772 
 -0.37629530 
 
 -0.03587269 
 -0.03587897 
 
 = + 0.00000000814. 
 
 TF.COSM 
 
 Wt, sin u 
 
 -2-flo 
 a 
 
 -0.065352939 
 
 -0.09038273 
 
 -0.83675673 
 
 -0.013079792 
 
 -0.13167315 
 
 -0.80869188 
 
 +0.049268295 
 
 -0.10825646 
 
 -0.80605518 
 
 +0.061732775 
 
 -0.04399687 
 
 -0.82203004 
 
 +0.012668751 
 
 -0.00122668 
 
 -0.83578481 
 
 -0.049130989 
 
 -0.02221027 
 
 -0.83298365 
 
 -0.062298971 
 
 -0.08615914 
 
 -0.81978340 
 
 -0.013955080 
 
 -0.13134085 
 
 -0.81552660 
 
 +0.050403535 
 
 -0.11432600 
 
 -0.83336328 
 
 +0.067326200 
 
 -0.04938656 
 
 -0.86579680 
 
 +0.015375410 
 
 -0.00167292 
 
 -0.88598306 
 
 -0.052873488 
 
 -0.02346769 
 
 -0.87266531 
 
 +0.000064081 
 
 -0.40202393 
 
 -5.01772646 
 
 +0.000019626 
 
 -0.40207539 
 
 -5.01770028 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 103 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 [de/dt] w = - 16017!410w' n 4.2045923 
 [dx/dtlw = -1840673.3 m' n 6.2649767 
 [di/dt] 00 = + 14.695 m' p 1.1671802 
 
 [dQ/<ft] M = -2385136.3 TO' n 6.3775132 
 [ArAftJoo = -1844854.1 TO' n 6.2659621 
 [dL/dt] m = -1765973.3 m' n 6.2469841 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 [de/dt] w = -0^04898290 
 
 [d x /dt}w = -5.6289701 
 
 [di/dt] m = +0.000044940 
 
 [dn/dt}^ = -7.293993 
 
 [dw/dt] m = -5.6417558 
 
 [dL/dt] m = -5.4005288 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt]oo 
 
 -0.04875 
 
 -0.04896 
 
 -0.048982 
 
 e[dTr/dt]oo 
 
 -0.03873 
 
 -0.03852 
 
 -0.038607 
 
 [dildt]oo 
 
 +0.00006 
 
 +0.00004 
 
 +0.0000449 
 
 sin i [dtt/dt]w 
 
 -0.43154 
 
 -0.43169 
 
 -0.431698 
 
 [dL/dt] m 
 
 -5.397 
 
 
 -5.4005 
 
 NOTES. 
 
 The close agreement of the sums of the functions toward the beginning of the 
 computation is here caused by the smallness of the term a'V; the ratio of the major 
 axes is, however, so large that the expansion of the perturbing function is not very 
 rapidly convergent. The greatest error arising from a division of the orbit into but 
 six parts would here occur with the coefficient [dttjdiloa, its amount being 0".0004, 
 which is l/16000th of the whole. 
 
104 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 
 ACTION OF 
 
 E 
 
 A 
 
 B cos e 
 
 
 
 2.6510232 
 
 -0.78427108 
 
 30 
 
 2.6356206 
 
 -0.87195528 
 
 60 
 
 2.6766222 
 
 -0.66613155 
 
 90 
 
 2.7630522 
 
 -0.22195070 
 
 120 
 
 2.8717556 
 
 +0.34156979 
 
 150 
 
 2.9736020 
 
 +0.87343460 
 
 180 
 
 3.0412895 
 
 + 1.23113141 
 
 210 
 
 3.0566810 
 
 + 1.31881567 
 
 240 
 
 3.0156549 
 
 + 1.11299237 
 
 270 
 
 2.9292114 
 
 +0.66881112 
 
 300 
 
 2.8205208 
 
 +0.10529076 
 
 330 
 
 2.7187000 
 
 -0.42657423 
 
 2, 
 
 17.0768662* 
 
 + 1.34058170f 
 
 2 
 
 17.0768672 
 
 + 1.34058118 
 
 E 
 
 I 
 
 G 
 
 O 
 
 
 
 
 
 0.32836475 
 
 2.30159781 
 
 30 
 
 0.31341820 
 
 2.30194311 
 
 60 
 
 0.35238060 
 
 2.30214618 
 
 90 
 
 0.43608420 
 
 2.30201186 
 
 120 
 
 0.54393270 
 
 2.30165423 
 
 150 
 
 0.64776765 
 
 2.30144287 
 
 180 
 
 0.71834405 
 
 2.30164190 
 
 210 
 
 0.73432570 
 
 2.30208690 
 
 240 
 
 0.69089570 
 
 2.30230785 
 
 270 
 
 0.60177725 
 
 2.30207736 
 
 300 
 
 0.49293070 
 
 2..30165418 
 
 330 
 
 0.39347570 
 
 2.30144156 
 
 2, 
 
 3.12684850 
 
 13.81100215 
 
 2, 
 
 3.12684870 
 
 13.81100366 
 
 
 * 6a 2 + 3aV + 6[a' 2 - 
 
 Zkaa'ee' cos K] = 17.076! 
 
 
 t 6[a'V - kaa'e cos K] 
 
 = + 1.34058156. 
 
 
 t 6fc'aa' cos <p' e sin 
 
 K' = - 0.0182328. 
 
 MARS ON VENUS. 
 
 /.' sin . 
 
 +0.4410282 
 -0.1200582 
 -0.6497891 
 -1.0062238 
 -1.0938564 
 -0.8892048 
 -0.4471057 
 +0.1139805 
 +0.6437115 
 + 1.0001462 
 + 1.0877783 
 +0.8831271 
 
 -0.0182332J 
 -0.0182329 
 
 G' 
 
 0.33433455 
 0.31388465 
 0.36444375 
 0.46014840 
 0.56837725 
 0.66243750 
 0.72188175 
 0.73455360 
 0.69835610 
 0.62106785 
 0.51868770 
 0.41360955 
 
 3.20608110 
 3.20570155 
 
 0.003928189 
 0.000291101 
 0.008527164 
 0.020447915 
 0.024164644 
 0.015968474 
 0.004037199 
 0.000262374 
 0.008368400 
 0.020201647 
 0.023896846 
 0.015750931 
 
 0.072922442 
 0.072922442 
 
 G" 
 
 0.00510484 
 0.00040288 
 0.01016345 
 0.01930383 
 0.01847156 
 0.01047414 
 0.00242983 
 0.00015516 
 0.00520478 
 0.01412953 
 0.02001680 
 0.01654686 
 
 0.06139126 
 0.06101240 
 
 2.30246275 
 2.30200670 
 2.30404590 
 2.30677230 
 2.30762720 
 2.30563865 
 2.30274975 
 2.30215960 
 2.30456350 
 2.30723845 
 2.30739440 
 2.30502860 
 
 13.82884350 
 13.82884430 
 
 o / 
 
 22 33 
 21 40 
 
 23 41 
 
 27 1 
 
 30 11 
 
 32 38 
 34 6 
 34 23 
 
 33 30 
 
 31 34 
 
 28 47 
 25 31 
 
 172 54 
 
 172 51 
 
 26.00 
 58.40 
 
 4.25 
 51.11 
 38.49 
 59.04 
 10.10 
 46.69 
 59.53 
 45.83 
 46.93 
 
 1.93 
 
 5.30 
 23.00 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 105 
 
 ACTION OF MARS ON VENUS. 
 
 E 
 
 log A!. 
 
 log ZV 
 
 log JVo log N log /' 
 
 logQ 
 
 
 
 0.05236235 
 
 0.34221622 
 
 0.25376720 9.0799241 8.6961571 
 
 8.9706997 
 
 30 
 
 0.04824440 
 
 0.36381569 
 
 0.24772073 9.0778391 8.6903136 
 
 8.9633892 
 
 60 
 
 0.05820469 
 
 0.34986592 
 
 0.26232763 9.0871721 8.7089460 
 
 8.9854537 
 
 90 
 
 0.07644597 
 
 0.37365893 
 
 0.28892064 9.1058581 8.7480486 
 
 9.0290445 
 
 120 
 
 0.09671054 
 
 0.39993150 
 
 0.31822634 9.1294237 8.7983322 
 
 9.0821385 
 
 150 
 
 0.11444649 
 
 0.42279102 
 
 0.34867356 9.1516340 9.8464806 
 
 9.1313354 
 
 180 
 
 0.12581914 
 
 0.43738414 
 
 0.35989311 9.1660121 8.8784042 
 
 9.1634092 
 
 210 
 
 0.12819869 
 
 0.44043118 
 
 0.36327720 9.1681182 8.8842476 
 
 9.1692445 
 
 240 
 
 0.12114959 
 
 0.43139835 
 
 0.35324262 9.1574139 8.8625240 
 
 9.1475124 
 
 270 
 
 0.10649224 
 
 0.41255431 
 
 0.33228403 9.1373395 8.8203390 
 
 9.1048461 
 
 300 
 
 0.08740987 
 
 0.38789402 
 
 0.30480686 9.1137453 8.7700379 
 
 9.0527515 
 
 330 
 
 0.06771795 
 
 0.36229172 
 
 0.27622192 9.0929018 8.7249711 
 
 9.0040125 
 
 2, 
 
 0.54165618 
 
 2.34869015 
 
 1.85226376 4.7336910 2.7144014 
 
 4.4019649 
 
 2 a 
 
 0.54154574 
 
 2.34854285 
 
 1.85209808 4.7336907 2.7144005 
 
 4.4018722 
 
 E 
 
 logF 
 
 Ji' 
 
 J 2 J, 
 
 F t 
 
 
 
 S.9695199 
 
 2.3059508 
 
 +0.040158678 -0.039429682 
 
 -0.09502735 
 
 30 
 
 8.9632959 
 
 2.2999322 
 
 -0.012465787 -0.068607310 
 
 +0.02586866 
 
 60 
 
 8.9831123 
 
 2.3089841 
 
 -0.060962061 -0.080069400 
 
 +0.14000859 
 
 90 
 
 9.0246325 
 
 2.3187331 
 
 -0.093160848 -0.070744754 
 
 +0.21680880 
 
 120 
 
 9.0779450 
 
 2.3192082 
 
 -0.101285499 -0.043131936 
 
 +0.23569076 
 
 150 
 
 9.1289682 
 
 2.3119133 
 
 -0.083189540 -0.004629729 
 
 +0.19159498 
 
 180 
 
 9.1628614 
 
 2.3032758 
 
 -0.042922865 +0.034445271 
 
 +0.09633687 
 
 210 
 
 9.1692095 
 
 2.2997005 
 
 +0.009559512 +0.063622976 
 
 -0.02455913 
 
 240 
 
 9.1463381 
 
 2.3040271 
 
 +0.060251438 +0.075085208 
 
 -0.13869908 
 
 270 
 
 9.1016472 
 
 2.3135287 
 
 +0.094787774 +0.065760616 
 
 -0.21549926 
 
 300 
 
 9.0481938 
 
 2.3207258 
 
 +0.103054621 +0.038147729 
 
 -0.23438118 
 
 330 
 
 9.0002162 
 
 2.3179852 
 
 +0.082763214 -0.000354612 
 
 -0.19028541 
 
 Si 
 
 4.3879705 
 
 13.8621718* 
 
 -0.001705675 -0.014952810 
 
 +0.00392861 
 
 2 2 
 
 4.3879695 
 
 13.8617930 
 
 -0.001705688 -0.014952813 
 
 +0.00392864 
 
 
 *2,(J,'-G") = 13 
 
 .800780G. 
 
 
 
 
 SzG/i' - G") = 13 
 
 .8007806. 
 
 
 
106 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MAKS ON VENUS. 
 
 E 
 
 1000 X F s 
 
 Ro 1000 X -So 
 
 1000 X Wo 
 
 fiw 
 
 1000 X <S<"> 
 
 
 
 + 2.8021516 
 
 0.06350844 -0.9770071 
 
 - 3.5365233 
 
 0.00000000 
 
 -1.3600097 
 
 30 
 
 - 0.4518739 
 
 0.06287361 +0.1223535 
 
 - 6.3268753 
 
 +0.04372018 
 
 +0.1701610 
 
 60 
 
 - 0.0998524 
 
 0.06461988 +1.2993923 
 
 - 7.7066718 
 
 +0.07763320 
 
 + 1.8025651 
 
 90 
 
 + 3.5312016 
 
 0.06860530 +2.2776450 
 
 - 7.2896438 
 
 +0.09484620 
 
 +3.1488232 
 
 120 
 
 + 6.8298095 
 
 0.07412522 +2.6943730 
 
 4.7318401 
 
 +0.08844541 
 
 +3.7122436 
 
 150 
 
 + 6.5062493 
 
 0.07970667 +2.2590820 
 
 - 0.1661617 
 
 +0.05477227 
 
 +3.1047600 
 
 180 
 
 + 2.8799133 
 
 0.08340064 +1.0358414 
 
 + 5.2294311 
 
 0.00000000 
 
 + 1.4223079 
 
 210 
 
 - 0.4389788 
 
 0.08380114 -0.4699266 
 
 + 9.3597858 
 
 -0.05758587 
 
 -0.6458416 
 
 240 
 
 - 0.1552801 
 
 0.08095204 -1.6671420 
 
 + 10.5057004 
 
 -0.09659109 
 
 -2.2969487 
 
 270 
 
 + 3.4223024 
 
 0.07610223 -2.2705920 
 
 + 8.5365166 
 
 -0.10521063 
 
 -3.1390719 
 
 300 
 
 + 6.6966141 
 
 0.07080065 -2.2876620 
 
 + 4.6568417 
 
 -0.08505867 
 
 -3.1735299 
 
 330 
 
 + 6.3844559 
 
 0.06631737 -1.8208410 
 
 + 0.3034394 
 
 -0.04611486 
 
 -2.5323023 
 
 Zi 
 
 + 18.9533560 
 
 0.43740687 -0.0977956 
 
 + 4.4169380 
 
 -0.01557115 
 
 +0.1066283 
 
 S 2 
 
 + 18.9533565 
 
 0.43740632 -0.0977209 
 
 + 4.4170612 
 
 -0.01557271 
 
 +0.1065284 
 
 
 \Ro sin v 
 
 1 Ro cos v 
 
 
 
 
 E 
 
 
 1 (T \ n 1000 
 
 X W cos u 1000 X W sin u 
 
 -2-flo 
 
 
 + ( COS v+ COS E)S ( 
 
 + 1 sec 2 <f + 1 1 sin vSo \ 
 \i / J 
 
 
 
 a 
 
 
 
 -0.001954014 
 
 -0.06350844 
 
 2.072195 
 
 - 2.865834 
 
 -0.12614768 
 
 30 
 
 +0.031835191 
 
 -0.05421922 
 
 0.625404 
 
 - 6.295890 
 
 -0.12500201 
 
 60 
 
 +0.057445987 
 
 -0.02972268 + 
 
 3.192309 
 
 - 7.014411 
 
 -0.12879747 
 
 90 
 
 +0.068588125 
 
 +0.00502476 + 
 
 5.936279 
 
 4.230780 
 
 -0.13721060 
 
 120 
 
 +0.061265775 
 
 +0.04210058 + 
 
 4.709813 
 
 - 0.456037 
 
 -0.14875771 
 
 150 
 
 +0.035700934 
 
 +0.07141598 + 
 
 0.151409 
 
 + 0.068446 
 
 -0.16035811 
 
 180 
 
 -0.002071682 
 
 +0.08340064 
 
 3.064140 
 
 - 4.237688 
 
 -0.16794274 
 
 210 
 
 -0.040838000 
 
 + 0.07318497 
 
 0.988919 
 
 - 9.307396 
 
 -0.16859554 
 
 240 
 
 -0.068190156 
 
 +0.04377273 + 
 
 4.238101 
 
 - 9.612920 
 
 -0.16245809 
 
 270 
 
 -0.076084930 
 
 +0.00506196 + 
 
 6.883205 
 
 - 5.049116 
 
 -0.15220446 
 
 300 
 
 -0.063800118 
 
 -0.03106656 + 
 
 4.629519 
 
 - 0.503716 
 
 -0.14111682 
 
 330 
 
 -0.036506252 
 
 -0.05549212 + 
 
 0.277348 
 
 + 0.123100 
 
 -0.13184872 
 
 Si 
 
 -0.017304208 
 
 +0.04497627 +11.633407 
 
 -24.690606 
 
 -0.87522051 
 
 2 2 
 
 -0.017304932 
 
 +0.04497633 +11.633918 
 
 -24.691636 
 
 -0.87521944 
 
 
 sin if J.AI'" + cos <f 
 
 B (c) = + 0.0000000018. 
 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 107 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 = - 6075.5972 m' n 3.7835890 
 = +2307588.8 m' p 6.3631584 
 [difdt] m = + 4084.7434m' p 3.6111648 
 [dQ/dfloo = - 146478.61 m' n 5.1657742 
 [dw/dt] w = +2307332.0 m' p 6.3631101 
 = - 307497.75 m' n 5.4878419 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'. 
 
 [de/dt] w = -0.0019639882 
 
 [d x /dt} 00 = +0.74594759 
 
 [dt'/dfloo = +0.0013204280 
 
 [dfl/dflw = -0.047350446 
 
 [d7r/d<]oo = +0.74586465 
 
 [dL/dt] w = -0.099401232 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 sn 
 
 [de/dt]w 
 
 -0^00195 
 
 -0.00196 
 
 -0.001964 
 
 [dw/dilw, 
 
 +0.00510 
 
 +0.00510 
 
 +0.005104 
 
 [di/dtlw 
 
 +0.00131 
 
 +0.00132 
 
 +0.001320 
 
 [dtt/dt] w 
 
 -0.00280 
 
 -0.00281 
 
 -0.002802 
 
 [dLfdt] w 
 
 -0.099 
 
 
 -0.099401 
 
 NOTES. 
 
 The close agreement of the final sums shows that, notwithstanding the high 
 eccentricity of the orbit of Mars, the expansion of the perturbing function is quite 
 rapidly convergent for this case. The greatest error arising from a division into but 
 six parts would here occur with the coefficient [dQJdt]^, and would amount to 
 1 /50000th of the whole value of this coefficient. 
 
108 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 E A 
 
 27.41848845 
 
 30 27.28099847 
 
 60 27.22724715 
 
 90 27.27164617 
 
 120 27.40230326 
 
 150 27.58420446 
 
 180 27.76860046 
 
 210 27.90607834 
 
 240 27.95980514 
 
 270 27.91539376 
 
 300 27.78474891 
 
 330 27.60287233 
 
 Zi 165.56119337* 
 
 2 2 165.56119353 
 
 E I 
 
 0.347678 
 
 30 0.210701 
 
 60 0.157583 
 
 90 0.202099 
 
 120 0.332232 
 
 150 0.513472 
 
 180 0.697718 
 
 210 0.835716 
 
 240 0.890108 
 
 270 0.845842 
 
 300 0.714692 
 
 330 0.532182 
 
 Zi 3.140010 
 
 2 2 3.140011 
 
 t 6[a'V - kaa'e cos K] = + 7.9087800. 
 | - Gfc'aa' cos <p' e sin K' = 0.1367433, 
 
 ACTION OF JUPITER ON VENUS. 
 
 B COS e 
 
 B sin t 
 
 o 
 
 
 h 
 
 -0.4215334 
 
 +3.3076466 
 
 0.68960365 
 
 
 27.007779 
 
 -1.8560261 
 
 + 1.9912702 
 
 0.24993195 
 
 
 27.007266 
 
 -2.4400062 
 
 +0.1352278 
 
 0.00115264 
 
 
 27.006632 
 
 -2.0169963 
 
 -1.7631551 
 
 0.19594869 
 
 
 27.006516 
 
 -0.7003432 
 
 -3.1952096 
 
 0.64351702 
 
 
 27.007039 
 
 + 1.1571588 
 
 -3.7772161 
 
 0.89930104 
 
 
 27.007701 
 
 +3.0577930 
 
 -3.3532271 
 
 0.70874081 
 
 
 27.007851 
 
 +4.4922856 
 
 -2.0368516 
 
 0.26150512 
 
 
 27.007330 
 
 +5.0762663 
 
 -0.1808092 
 
 0.00206064 
 
 
 27.006665 
 
 +4.6532564 
 
 + 1.7175743 
 
 0.18594838 
 
 
 27.006520 
 
 +3.3366031 
 
 +3.1496275 
 
 0.62528757 
 
 
 27.007025 
 
 + 1.4791014 
 
 +3.7316353 
 
 0.87772776 
 
 
 27.007658 
 
 +7.9087796f 
 
 -0.13674401 
 
 2.67036233 
 
 
 162.042990 
 
 +7.9087799 
 
 -0.1367430 
 
 2.67036294 
 
 
 162.042990 
 
 
 
 G' 
 
 G" 
 
 
 e 
 
 
 
 
 
 1 II 
 
 27.0068207 
 
 0.4107944 
 
 0.06215867 
 
 7 
 
 35 44.57 
 
 27.0069201 
 
 0.2483150 
 
 0.03726867 
 
 5 
 
 53 53.53 
 
 27.0066304 
 
 0.1578551 
 
 0.00027037 
 
 4 
 
 23 18.41 
 
 27.0062448 
 
 0.2334496 
 
 0.03108029 
 
 5 
 
 40 35.79 
 
 27.0061457 
 
 0.3936568 
 
 0.06053126 
 
 7 
 
 26 34.62 
 
 27.0064436 
 
 0.5728576 
 
 0.05812876 
 
 8 
 
 46 58.16 
 
 27.0068530 
 
 0.7344471 
 
 0.03573164 
 
 9 
 
 42 56.80 
 
 27.0069600 
 
 0.8475114 
 
 0.01142507 
 
 10 
 
 16 14.68 
 
 27.0066621 
 
 0.8901968 
 
 0.00008571 
 
 10 
 
 27 38.97 
 
 27.0062568 
 
 0.8541659 
 
 0.00806094 
 
 10 
 
 17 29.01 
 
 27.0061444 
 
 0.7465850 
 
 0.03101258 
 
 9 
 
 45 50.06 
 
 27.0064305 
 
 0.5886244 
 
 0.05521465 
 
 8 
 
 52 22.88 
 
 162.0392563 
 
 3.3335352 
 
 0.18979023 
 
 49 
 
 22 3.43 
 
 162.0392558 
 
 3.3449238 
 
 0.20117838 
 
 49 
 
 47 34.05 
 
 a'ee'cosK] = 165.5611934, 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 109 
 
 
 
 ACTION 
 
 OF JUPITER ON VENUS. 
 
 
 
 E 
 
 log/fo 
 
 log Lo' 
 
 log No log N 
 
 logP 
 
 logQ 
 
 
 
 0.00574768 
 
 0.28065745 
 
 0.18470221 7.4290891 
 
 4.8448027 
 
 6.1813194 
 
 30 
 
 0.00346017 
 
 0.27761215 
 
 0.18127767 7.4282000 
 
 4.8416643 
 
 6.1774038 
 
 60 
 
 0.00191340 
 
 0.27555166 
 
 0.17896019 7.4297378 
 
 4.8423401 
 
 6.1772233 
 
 90 
 
 0.00320448 
 
 0.27727161 
 
 0.18089469 7.4332724 
 
 4.8466166 
 
 6.1822034 
 
 120 
 
 0.00551791 
 
 0.28035168 
 
 0.18435840 7.4378460 
 
 4.8533279 
 
 6.1897695 
 
 150 
 
 0.00769513 
 
 0.28324824 
 
 0.18761506 7.4422392 
 
 4.8606851 
 
 6.1974532 
 
 180 
 
 0.00942826 
 
 0.28555246 
 
 0.19020526 7.4452930 
 
 4.8667491 
 
 6.2034501 
 
 210 
 
 0.01054435 
 
 0.28703562 
 
 0.19187230 7.4462011 
 
 4.8699179 
 
 6.2064140 
 
 240 
 
 0.01094130 
 
 0.28756298 
 
 0.19246500 7.4447133 
 
 4.8693318 
 
 6.205706T 
 
 270 
 
 0.01058711 
 
 0.28709242 
 
 0.19193614 7.4412095 
 
 4.8651139 
 
 6.2015516 
 
 300 
 
 0.00952253 
 
 0.28567775 
 
 0.19034610 7.4366174 
 
 4.8583732 
 
 6.1950025 
 
 330 
 
 0.00785489 
 
 0.28346069 
 
 0.18785389 7.4321742 
 
 4.8509265 
 
 6.1876739 
 
 2i 
 
 0.04307108 
 
 1.69535398 
 
 1.12103716 4.6232965 
 
 9.1349247 
 
 7.1524707 
 
 Zi 
 
 0.04334613 
 
 1.69572073 
 
 1.12144975 4.6232963 
 
 9.1349242 
 
 7.1526998 
 
 E 
 
 log V 
 
 Ji' 
 
 J* 
 
 J 3 
 
 Ft 
 
 
 
 6.1800737 
 
 27.032601604 
 
 +0.14514043 
 
 -0.9796507 
 
 -4.3121624 
 
 30 
 
 6.1766560 
 
 27.002428557 
 
 +0.10192507 
 
 -1.0502482 
 
 -2.5960090 
 
 60 
 
 6.1772178 
 
 26.980449138 
 
 +0.02691302 
 
 -0.8390983 
 
 -0.1762960 
 
 90 
 
 6.1815797 
 
 27.031356993 
 
 -0.07055224 
 
 -0.4027785 
 
 +2.2986164 
 
 120 
 
 6.1885563 
 
 27.065863046 
 
 -0.15999110 
 
 +0.1417995 
 
 +4.1655781 
 
 150 
 
 6.1962891 
 
 27.048572013 
 
 -0.20267009 
 
 +0.6487166 
 
 +4.9243381 
 
 180 
 
 6.2027348 
 
 27.006174576 
 
 -0.17674090 
 
 +0.9821448 
 
 +4.3715859 
 
 210 
 
 6.2061853 
 
 26.976509919 
 
 -0.09316228 
 
 + 1.0527424 
 
 + 2.6554331 
 
 240 
 
 6.2057043 
 
 26.979786076 
 
 +0.01124440 
 
 +0.8415929 
 
 +0.2357201 
 
 270 
 
 6.2013902 
 
 27.007935305 
 
 +0.09774464 
 
 +0.4052732 
 
 -2.2391928 
 
 300 
 
 6.1943816 
 
 27.036474318 
 
 +0.11682513 
 
 -0.1393049 
 
 -4.1061538 
 
 330 
 
 6.1865682 
 
 27.045934111 
 
 +0.16010585 
 
 -0.6462222 
 
 -4.8649146 
 
 Si 
 
 7.1486685 
 
 162.101348758* -0.00660902 
 
 +0.0074833 
 
 +0.1782719 
 
 2j 
 
 7.1486684 
 
 162.112736898 
 
 -0.00660905 
 
 +0.0074833 
 
 +0.1782712 
 
 
 * z,(J,' - G") = 
 
 161.911558528. 
 
 
 
 
 
 2 2 (Ji' - C") = 
 
 161.911558518. 
 
 
 
 
110 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 
 ACTION OF JUPITEK ON VKNUS. 
 
 E 
 
 Ft 
 
 1000 X flo 100000 X So 1000 X W 1000 X <"> 100000 X <"' 
 
 
 
 +0.06417143 
 
 1.3439836 -0.81929943 -0.14785238 0.0000000 -1.1404782 
 
 30 
 
 -0.01379964 
 
 1.3378723 -0.27202988 -0.15783804 +0.9303113 -0.3783208 
 
 60 
 
 -0.00425156 
 
 1.3439282 +0.28211779 -0.12622122 +1.6145721 +0.3913643 
 
 90 
 
 +0.08385809 
 
 1.3588647 +0.54292475 -0.06059607 +1.8786178 +0.7505883 
 
 120 
 
 +0.16287446 
 
 1.3765538 +0.50194299 +0.02305118 +1.6424891 +0.6915652 
 
 150 
 
 +0.15397825 
 
 1.3917093 +0.38822446 +0.10305711 +0.9563448 +0.5335546 
 
 180 
 
 +0.06595223 
 
 1.4018961 +0.39768404 +0.15712801 0.0000000 +0.5460578 
 
 210 
 
 -0.01357095 
 
 1.4066592 +0.47042587 +0.16914113 -0.9666180 +0.6465278 
 
 240 
 
 -0.00563629 
 
 1.4053074 +0.35504098 +0.13510528 -1.6767973 +0.4891671 
 
 270 
 
 +0.08123098 
 
 1.3964225 -0.08725072 +0.06503287 -1.9305409 -0.1206233 
 
 300 
 
 +0.15970886 
 
 1.3800824 -0.66641223 -0.02064190 -1.6580071 -0.9244718 
 
 330 
 
 +0.15112247 
 
 1.3602249 -0.99122256 -0.09822805 -0.9458544 -1.3785254 
 
 2i 
 
 +0.44281913 
 
 8.2517515 +0.05107414 +0.02056897 -0.0777432 +0.0532044 
 
 2 2 
 
 +0.44281920 
 
 8.2517529 +0.05107192 +0.02056895 -0.0777394 +0.0532012 
 
 E 
 
 1000 X[Ro sin v 
 
 + (cosv+cosE)S<>] 
 
 1000 x| -Ro cost) 
 , 1000 X Wo cos u 1000 X W sin u - 2 - 
 
 /; \ . 1 (j 
 
 ~\- 1 - SGC^ (f -\~\ isin v *oo I 
 
 
 
 -0.0163860 
 
 -1.3439836 -0.086632804 -0.11981268 -0.0026695736 
 
 30 
 
 +0.6682014 
 
 -1.1590573 -0.015602101 -0.15706504 -0.0026598877 
 
 60 
 
 + 1.1706512 
 
 -0.6601482 +0.052284193 -0.11488325 -0.0026786592 
 
 90 
 
 + 1.3587959 
 
 +0.0201574 +0.049346057 -0.03516889 -0.0027177294 
 
 120 
 
 + 1.1829926 
 
 +0.7039968 -0.022943878 +0.00222159 -0.0027625280 
 
 150 
 
 +0.6850081 
 
 + 1.2114930 -0.093907380 -0.04245200 -0.0027999142 
 
 180 
 
 -0.0079537 
 
 + 1.4018961 -0.092067787 -0.12732924 -0.0028229786 
 
 210 
 
 -0.7073256 
 
 + 1.2159042 -0.017870798 -0.16819438 -0.0028299915 
 
 240 
 
 -1.2164221 
 
 +0.7037025 +0.054502775 -0.12362395 -0.0028202318 
 
 270 
 
 -1.3963839 
 
 +0.0113009 +0.052437627 -0.03846518 -0.0027928449 
 
 300 
 
 -1.2058917 
 
 -0.6713715 -0.020520787 +0.00223277 -0.0027507212 
 
 330 
 
 -0.7013026 
 
 -1.1657067 -0.089781837 -0.03984932 -0.0027043278 
 
 2i 
 
 -0.0930097 
 
 +0.1340921 -0.115378288 -0.48119476 -0.0165046924 
 
 2 2 
 
 -0.0930067 
 
 +0.1340915 -0.115378432 -0.48119481 -0.0165046855 
 
 
 sin (p \A i (<) + cos (f 
 
 Bo (c) = + 0.0000000000028. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. Ill 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 u log coeff. 
 
 [de/dt] M = - 32.654970 m' n 1.5139493 
 
 [d x /dt] m = +6879.8159 TO' p 3.8375768 
 
 [di/dt] M = - 40.510972 TO' n 1.6075727 
 
 [<&/#], = -2854.6599 TO' n 3.4555544 
 
 [dir/eft]oo = +6874.8117 TO' p 3.8372608 
 
 [dL/dt] m = -5799.7390 TO' n 3.7634084 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 = -0.031162921 
 
 [dxldt]<n = +6.5654682 
 [di/dt] M = -0.038659982 
 [dQ/dt] w = -2.7242270 
 = +6.5606924 
 = -5.5347410 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 -0.03117 -0.0311629 
 
 +0.04482 +0.04491 +0.0448955 
 
 [di/dt] M -0.03865 -0.03865 -0.0386600 
 
 sin i [daAft] M -0.16114 -0.16122 -0.1612345 
 
 [dL/dt] w -5.535 -5.5347410 
 
 NOTES. 
 
 The term aV is here so large that the sums of the functions B, e, G', G", and 0, 
 as well as those of the functions immediately dependent upon these quantities are 
 in great disagreement; but, as the expansion of the perturbing function is here rapidly 
 convergent, the final sums agree almost exactly. The greatest effect of all terms from 
 the 6th to the llth orders is here produced with the coefficient [di/dt]^ and amounts 
 to 0".00000002. 
 
112 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF SATURN ON VENUS. 
 
 E 
 
 A 
 
 B cos t 
 
 B sin e 
 
 g 
 
 
 I 
 
 
 
 92.09886822 
 
 + 10.392665 
 
 +4.3401600 
 
 5.386574 
 
 
 90.704833 
 
 30 
 
 91.77434432 
 
 + 7.489741 
 
 +6.4166597 
 
 11.773870 
 
 
 90.705340 
 
 60 
 
 91.37861335 
 
 + 3.936959 
 
 +6.7658078 
 
 13.090030 
 
 
 90.705178 
 
 90 
 
 91.01772022 
 
 + 0.686282 
 
 +5.2940524 
 
 8.014522 
 
 
 90.704527 
 
 120 
 
 90.78837022 
 
 1.391274 
 
 +2.3957451 
 
 1.641279 
 
 
 90.704015 
 
 150 
 
 90.75201312 
 
 - 1.739027 
 
 -1.1525126 
 
 0.379833 
 
 
 90.704174 
 
 180 
 
 90.91838168 
 
 - 0.263800 
 
 -4.3999687 
 
 5.536055 
 
 
 90.704848 
 
 210 
 
 91.24289345 
 
 + 2.639122 
 
 -6.4764687 
 
 11.994380 
 
 
 90.705369 
 
 240 
 
 91.63859982 
 
 + 6.191904 
 
 -6.8256188 
 
 13.322485 
 
 
 90.705222 
 
 270 
 
 91.99948069 
 
 + 9.442580 
 
 -5.3538604 
 
 8.196630 
 
 
 90.704548 
 
 300 
 
 92.22884291 
 
 + 11.520135 
 
 -2.4555545 
 
 1.724250 
 
 
 90.704026 
 
 330 
 
 92.26522467 
 
 + 11.867891 
 
 + 1.0927030 
 
 0.341433 
 
 
 90.704165 
 
 S, 
 
 549.05167620* 
 
 +30.386589f 
 
 -0.179429U 
 
 40.700673 
 
 
 544.228122 
 
 *} 
 
 549.05167647 
 
 +30.386589 
 
 -0.1794266 
 
 40.700668 
 
 
 544.228123 
 
 E 
 
 I 
 
 G 
 
 G' 
 
 G" 
 
 
 e 
 
 
 
 
 
 
 
 
 
 i a 
 
 
 
 + 1.108078 
 
 90.7041702 
 
 1.1599384 
 
 0.0511977 
 
 6 
 
 38 1.200 
 
 30 
 
 +0.783048 
 
 90.7038960 
 
 0.9248448 
 
 0.1403539 
 
 6 
 
 12 59.177 
 
 60 
 
 +0.387477 
 
 90.7035801 
 
 0.6213419 
 
 0.2322660 
 
 5 
 
 33 35.601 
 
 90 
 
 +0.027235 
 
 90.7035526 
 
 0.3116929 
 
 0.2834826 
 
 4 
 
 38 20.618 
 
 120 
 
 -0.201603 
 
 90.7038159 
 
 0.0673334 
 
 0.2687363 
 
 3 
 
 29 4.472 
 
 150 
 
 -0.238118 
 
 90.7041279 
 
 0.0164527 
 
 0.2545245 
 
 3 
 
 7 43.816 
 
 180 
 
 -0.072423 
 
 90.7041756 
 
 0.2137666 
 
 0.2855179 
 
 4 
 
 14 53.293 
 
 210 
 
 +0.251567 
 
 90.7039071 
 
 0.5115373 
 
 0.2585083 
 
 5 
 
 16 44.963 
 
 240 
 
 +0.647420 
 
 90.7035910 
 
 0.8267173 
 
 0.1776657 
 
 6 
 
 2 4.051 
 
 270 
 
 + 1.008976 
 
 90.7035405 
 
 1.0926850 
 
 0.0827021 
 
 6 
 
 32 0.533 
 
 300 
 
 + 1.238859 
 
 90.7038135 
 
 1.2542286 
 
 0.0151565 
 
 6 
 
 47 36.301 
 
 330 
 
 + 1.275102 
 
 90.7041229 
 
 1.2780897 
 
 0.0029452 
 
 6 
 
 49 30.415 
 
 s, 
 
 +3.107808 
 
 . 544.2231463 
 
 4.1433262 
 
 1.0305401 
 
 32 
 
 45 14.918 
 
 2 2 
 
 +3.107809 
 
 544.2231470 
 
 4.1353023 
 
 1.0225166 
 
 32 
 
 37 19.522 
 
 
 * 6o 2 + 3aV + 6[a' 2 - 
 
 Zkaa'ee' cos K\ = 549.05167622. 
 
 
 t 6[a'V - kaa'e cos K\ 
 
 = + 30.386587. 
 
 
 
 
 
 
 J 6fc'aa' cos if' e sin 
 
 K' = - 0.1794290. 
 
 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 113 
 
 
 
 ACTION OF SATURN ON VENUS. 
 
 
 E 
 
 logtfo 
 
 log /V log No log N log P 
 
 logQ 
 
 
 
 0.00437971 
 
 0.27883659 0.18265468 6.6396204 3.0027123 
 
 4.8644028 
 
 30 
 
 0.00384466 
 
 0.27812417 0.18185352 6.6392492 3.0007786 
 
 4.8628053 
 
 60 
 
 0.00307377 
 
 0.27709752 0.18069890 6.6400076 2.9996349 
 
 4.8619714 
 
 90 
 
 0.00213852 
 
 0.27585161 0.17929758 6.6416827 2.9995754 
 
 4.8620009 
 
 120 
 
 0.00120578 
 
 0.27460866 0.17789947 6.6438207 3.0006087 
 
 4.862S099 
 
 150 
 
 0.00097200 
 
 0.27429705 0.17754895 6.6458518 3.0024609 
 
 4.8645568 
 
 180 
 
 0.00179285 
 
 0.27539102 0.17877951 6.6472415 3.0046480 
 
 4.8670287 
 
 210 
 
 0.00277061 
 
 0.27669370 0.18024473 6.6476233 3.0065929 
 
 4.8690060 
 
 240 
 
 0.00362228 
 
 0.27782805 0.18152050 6.6468912 3.0077706 
 
 4.8699374 
 
 270 
 
 0.00424801 
 
 0.27866126 0.18245753 6.6452313 3.0078525 
 
 4.8696688 
 
 300 
 
 0.00459387 
 
 0.27912172 0.18297533 6.6430830 3.0068083 
 
 4.8683601 
 
 330 
 
 0.00463699 
 
 0.27917912 0.18303987 6.6410259 3.0049227 
 
 4.8664247 
 
 Si 
 
 0.01866826 
 
 1.66288356 1.08452839 9.8606643 8.0221827 
 
 9.1945101 
 
 Y o 
 
 0.01861079 
 
 1.66280691 1.08444218 9.8606642 8.0221829 
 
 9.1944623 
 
 E 
 
 logF 
 
 J,' J 2 J 3 
 
 F 2 
 
 
 
 4.8640969 
 
 90.64262161 +0.26581387 -3.1529304 
 
 -22.089684 
 
 30 
 
 4.8619670 
 
 90.77821079 +0.41797337 -2.4126056 
 
 -32.658241 
 
 60 
 
 4.8605842 
 
 90.92403274 +0.41481456 -1.0167115 
 
 -34.435270 
 
 90 
 
 4.8603076 
 
 90.98237184 +0.27423234 +0.6607234 
 
 -26.944615 
 
 120 
 
 4.8612040 
 
 90.92121232 +0.07649557 +2.1702330 
 
 -12.193388 
 
 150 
 
 4.8630356 
 
 90.85355551 -0.10022240 +3.1073439 
 
 + 5.865829 
 
 180 
 
 4.8653231 
 
 90.87694181 -0.22507782 +3.2209585 
 
 +22.394088 
 
 210 
 
 4.8674621 
 
 90.89557561 -0.30595380 +2.4806343 
 
 + 32.962649 
 
 240 
 
 4.8688763 
 
 90.86858755 -0.34689500 +1.0847396 
 
 +34.739680 
 
 270 
 
 4.8691748 
 
 90.78219477 -0.32168268 -0.5926962 
 
 +27.249019 
 
 300 
 
 4.8682696 
 
 90.66900008 -0.19522740 -2.1022048 
 
 + 12.497792 
 
 330 
 
 4.8664071 
 
 90.60246403 +0.02557719 -3.0393154 
 
 - 5.561423 
 
 2j 
 
 9.1883540 
 
 544.90239611* -0.01007622 +0.2040844 
 
 + 0.913218 
 
 2 2 
 
 9.1883541 
 
 544.89437255 -0.01007598 +0.2040844 
 
 + 0.913218 
 
 
 * 2,(J,' - G") = 
 
 543.87185601. 
 
 
 
 2 2 (J,' - G") = 
 
 543.87185600. 
 
 
114 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF SATURN ON VENUS. 
 
 E 
 
 F 3 1000 X Ro 100000 X S 1000 X W 1000 X B (n) 
 
 100000 X <S<> 
 
 
 
 -0.15178108 0.21824849 -0.02789044 -0.023072725 0.00000000 
 
 -0.03882394 
 
 30 
 
 -0.76855930 0.21811010 -0.02299898 -0.017634117 +0.15166640 
 
 -0.03198543 
 
 60 
 
 -1.16657105 0.21852784 -0.04315220 -0.007491896 +0.26253554 
 
 -0.05986233 
 
 90 
 
 -0.94659087 0.21909707 -0.07037836 +0.004695343 +0.30289965 
 
 -0.09729742 
 
 120 
 
 -0.32871848 0.21963056 -0.06653491 +0.015732681 +0.26206084 
 
 -0.09167023 
 
 150 
 
 +0.06775179 0.22023472 -0.01412236 +0.022675427 +0.15133936 
 
 -0.01940900 
 
 180 
 
 -0.15599309 0.22103492 +0.06128516 +0.023605169 0.00000000 
 
 +0.08415031 
 
 210 
 
 -0.77884429 0.22186276 +0.10918347 +0.018202966 -0.15245807 
 
 +0.15005585 
 
 240 
 
 -1.18017310 0.22225231 +0.09717617 +0.007900358 -0.26518905 
 
 +0.13388702 
 
 270 
 
 -0.95986467 0.22180036 +0.03944793 -0.004483114 -0.30663694 
 
 +0.05453639 
 
 300 
 
 -0.33810816 0.22057741 -0.01719598 -0.015556236 -0.26499777 
 
 -0.02385490 
 
 330 
 
 +0.06476239 0.21916657 -0.03744378 -0.022338577 -0.15240100 
 
 -0.05207429 
 
 2, 
 
 -3.32134496 1.32027153 +0.00368780 +0.001117351 -0.00559044 
 
 +0.00382593 
 
 -"2 
 
 -3.32134495 1.32027158 +0.00368792 +0.001117928 -0.00559070 
 
 +0.00382610 
 
 E 
 
 m v [ff ,' 100 Xf-Socost; 
 J p T 1000 X TFo cos u 1000 X TFo sin u 
 
 1000X-2-Ro 
 
 
 1 + ^sec z Y,+ lJsin!%Sol 
 
 a 
 
 
 
 -0.00055781 -0.21824849 -0.013519262 -0.018697064 
 
 -0.43351010 
 
 30 
 
 +0.10930468 -0.18874417 -0.001743111 -0.017547753 
 
 -0.43363510 
 
 60 
 
 +0.18946670 -0.10888722 +0.003103342 -0.006818927 
 
 -0.43556030 
 
 90 
 
 +0.21909678 +0.00009174 -0.003823625 +0.002725091 
 
 -0.43819414 
 
 120 
 
 +0.19022138 +0.10978821 -0.015659442 +0.001516257 
 
 -0.44076418 
 
 150 
 
 +0.10971092 +0.19096268 -0.020662231 -0.009340619 
 
 -0.44307990 
 
 180 
 
 -0.00122570 +0.22103492 -0.013831594 -0.019129016 
 
 -0.44509495 
 
 210 
 
 -0.11216824 +0.19142744 -0.001923256 -0.018101079 
 
 -0.44635520 
 
 240 
 
 -0.19279204 +0.11058265 +0.003187081 -0.007228982 
 
 -0.44602556 
 
 270 
 
 -0.22179790 +0.00072885 -0.003614847 +0.002651640 
 
 -0.44360072 
 
 300 
 
 -0.19184805 -0.10885437 -0.015464964 +0.001682669 
 
 -0.43964535 
 
 330 
 
 -0.11088191 -0.18905102 -0.020417779 -0.009062352 
 
 -0.43573540 
 
 Si 
 
 -0.00673552 +0.00541570 -0.052184839 -0.048675063 
 
 -2.64060044 
 
 2 2 
 
 -0.00673567 +0.00541552 -0.052184849 -0.048675069 
 
 -2.64060046 
 
 
 sin <f Mi w + cos <p Bo w = + 0.00000000000029. 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 115 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 
 n 
 
 log coeff. 
 
 [de/dt]w 
 
 = - 2.3648522 TO' 
 
 n 0.3738040 
 
 [dx/dt] w 
 
 = +277.85744 TO' 
 
 p 2.4438220 
 
 [di/dt] w 
 
 18.322835 TO' 
 
 n 1.2629927 
 
 [dQ/dfloo 
 
 = -288.76199 TO' 
 
 n 2.4605400 
 
 [dir/dt]oo 
 
 = +277.35124 TO' 
 
 p 2.4430301 
 
 [dL/dt] w 
 
 = -927.63054 TO' 
 
 n 2.9673751 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 [de/dt] w = -0.00067536338 
 [dxldt] M = +0.079351564 
 
 = -0.0052327048 
 
 = -0.082465731 
 
 = +0.079207000 
 [dL/dt] w = -0.26491624 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 -0.00067 
 
 -0.00067 
 
 -0.00067536 
 
 +0.00055 
 
 +0.00054 
 
 +0.00054202 
 
 -0.00523 
 
 -0.00523 
 
 -0.00523270 
 
 -0.00489 
 
 -0.00488 
 
 -0.00488077 
 
 -0.265 
 
 
 -0.26491624 
 
 [di/dt] m 
 sin i [dQ/dt}oo 
 [dL/dt] 
 
 NOTES. 
 
 As in the previous case, the considerable disagreement of the sums of the functions 
 near the beginning of the computation nearly disappears as the work progresses, show- 
 ing that the convergence of the expansion of the perturbing function is here very 
 rapid. The greatest error which would have arisen from the neglect of all terms from 
 the 6th to the llth orders would have here occurred with the coefficient [dx/dt] 00 and 
 would have amounted to 1 /70000th part of the remaining terms. 
 
116 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF URANUS ON VENUS. 
 
 E 
 
 A 
 
 B cos ( 
 
 / >' sin i 
 
 9 
 
 i 
 
 
 
 369.8294733 
 
 28.016529 
 
 - 8.627090 
 
 60.35594 
 
 367.496220 
 
 30 
 
 370.1021929 
 
 30.912376 
 
 - 2.059265 
 
 3.43885 
 
 367.496155 
 
 60 
 
 370.0319613 
 
 30.136215 
 
 + 5.076263 
 
 20.89679 
 
 367.496140 
 
 90 
 
 369.6376057 
 
 25.896024 
 
 + 10.867539 
 
 95.77536 
 
 367.496165 
 
 120 
 
 369.0247981 
 
 19.327964 
 
 + 13.762797 
 
 153.60480 
 
 367.496245 
 
 150 
 
 368.3577351 
 
 12.191933 
 
 + 12.986248 
 
 136.75991 
 
 367.496230 
 
 180 
 
 367.8151465 
 
 6.400029 
 
 + 8.745974 
 
 62.03086 
 
 367.496275 
 
 210 
 
 367.5424142 
 
 3.504185 
 
 + 2.178150 
 
 3.84740 
 
 367.496245 
 
 240 
 
 367.6126214 
 
 4.280346 
 
 - 4.957378 
 
 19.92946 
 
 367.496135 
 
 270 
 
 368.0069649 
 
 8.520535 
 
 - 10.748653 
 
 93.69136 
 
 367.496165 
 
 300 
 
 368.6197847 
 
 15.088597 
 
 -13.643909 
 
 150.96247 
 
 367.496240 
 
 330 
 
 369.2868724 
 
 22.224622 
 
 -12.867361 
 
 134.26735 
 
 367.496240 
 
 ^ 
 
 2212.9337853* 
 
 103.2496801 
 
 + 0.356657} 
 
 467.78032 
 
 2204.977255 
 
 ^ 
 
 2212.9337852 
 
 103.249675 
 
 + 0.356658 
 
 467.78025 
 
 2204.977200 
 
 E 
 
 G 
 
 G' 
 
 G" 
 
 
 
 
 
 + 1.522310 
 
 367.495771 
 
 1.6238940 
 
 0.1011370 
 
 O 
 
 3 
 
 55 
 
 40^93 
 
 30 
 
 + 1.795095 
 
 367.496129 
 
 1.8003166 
 
 0.0051977 
 
 4 
 
 1 
 
 9.46 
 
 60 
 
 + 1.724880 
 
 367.495985 
 
 1.7573874 
 
 0.0323564 
 
 4 
 
 
 
 5.50 
 
 90 
 
 + 1.330495 
 
 367.495453 
 
 1.5044398 
 
 0.1732316 
 
 3 
 
 52 
 
 23.80 
 
 120 
 
 +0.717605 
 
 367.495105 
 
 1.0990547 
 
 0.3803068 
 
 3 
 
 38 
 
 8.91 
 
 150 
 
 +0.050560 
 
 367.495217 
 
 0.6363646 
 
 0.5847917 
 
 3 
 
 18 
 
 7.21 
 
 180 
 
 -0.492075 
 
 367.495816 
 
 0.2329563 
 
 0.7245710 
 
 2 
 
 55 
 
 22.89 
 
 210 
 
 -0.764775 
 
 367.496217 
 
 0.0134531 
 
 0.7782012 
 
 2 
 
 39 
 
 26.72 
 
 240 
 
 -0.694455 
 
 367.495988 
 
 0.0708724 
 
 0.7651843 
 
 2 
 
 43 
 
 51.72 
 
 270 
 
 -0.300145 
 
 367.495472 
 
 0.3769271 
 
 0.6763791 
 
 3 
 
 3 
 
 57.85 
 
 300 
 
 +0.312600 
 
 367.495121 
 
 0.8167025 
 
 0.5029833 
 
 3 
 
 25 
 
 59.41 
 
 330 
 
 +0.979685 
 
 367.495243 
 
 1.2686692 
 
 0.2879853 
 
 3 
 
 43 
 
 48.64 
 
 2, 
 
 +3.090865 
 
 2204.973786 
 
 5.6008673 
 
 2.5065388 
 
 20 
 
 39 
 
 9.36 
 
 2 2 
 
 +3.090915 
 
 2204.973731 
 
 5.6001704 
 
 2.5057866 
 
 20 
 
 38 
 
 53.68 
 
 * 6a 2 + 3aV + 6[a' 2 - 2kaa'ee' cos K] = 2212.9337852. 
 t 6[a'V - kaa'e cos K] = + 103.249685. 
 t 6fc'aa' cos <p' e sin A" = + 0.356657. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 117 
 
 
 
 ACTION OF UKANUS ON VENUS. 
 
 
 E 
 
 logtfo 
 
 log!/ logJV log AT logP 
 
 logQ 
 
 
 
 0.00153256 
 
 0.27504417 0.17838936 5.7255240 0.8698244 
 
 3.3385415 
 
 30 
 
 0.00160469 
 
 0.27514027 0.17849746 5.7265670 0.8711895 
 
 3.3398056 
 
 60 
 
 0.00159052 
 
 0.27512139 0.17847622 5.7286906 0.8732303 
 
 3.3418760 
 
 90 
 
 0.00149009 
 
 0.27498757 0.17832569 5.7313185 0.8753928 
 
 3.3441876 
 
 120 
 
 0.00131281 
 
 0.27475131 0.17805993 5.7337420 0.8770917 
 
 3.3461011 
 
 150 
 
 0.00108264 
 
 0.27444453 0.17771485 5.7353151 0.8778752 
 
 3.3470878 
 
 180 
 
 0.00084825 
 
 0.27413211 0.17736340 5.7356234 0.8775397 
 
 3.3468789 
 
 210 
 
 0.00070103 
 
 0.27393587 0.17714264 5.7345896 0.8761825 
 
 3.3455607 
 
 240 
 
 0.00074042 
 
 0.27398839 0.17720173 5.7324870 0.8741636 
 
 3.3435328 
 
 270 
 
 0.00093336 
 
 0.27424555 0.17749102 5.7298710 0.8720155 
 
 3.3413115 
 
 300 
 
 0.00117044 
 
 0.27456155 0.17784648 5.7274385 0.8703090 
 
 3.3394395 
 
 330 
 
 0.00138191 
 
 0.27484340 0.17816353 5.7258446 0.8695041 
 
 3.3384162 
 
 Si 
 
 0.00719500 
 
 1.64759892 1.06733712 4.3835054 5.2421585 
 
 0.0563696 
 
 2 2 
 
 0.00719372 
 
 1.64759719 1.06733519 4.3835056 5.2421594 
 
 0.0563691 
 
 E 
 
 logF 
 
 JY J 2 J, 
 
 ft 
 
 
 
 3.3383922 
 
 367.09942052 -0.77238411 -13.377206 
 
 + 148.77538 
 
 30 
 
 3.3397979 
 
 366.74089875 -0.18040606 -16.560521 
 
 + 35.51232 
 
 60 
 
 3.3418282 
 
 366.88453611 +0.52246333 -15.273725 
 
 - 87.54088 
 
 90 
 
 3.3439319 
 
 367.40160988 +0.87586900 - 9.861620 
 
 -187.41225 
 
 120 
 
 3.3455398 
 
 367.86665960 +0.72768333 - 1.774376 
 
 -237.34142 
 
 150 
 
 3.3462248 
 
 367.95568703 +0.32642368 . + 6.821041 
 
 -223.94969 
 
 180 
 
 3.3458097 
 
 367.72285452 +0.04295564 +13.621489 
 
 -150.82559 
 
 210 
 
 3.3444123 
 
 367.51508313 +0.01342241 +16.804806 
 
 - 37.56251 
 
 240 
 
 3.3424036 
 
 367.61069788 +0.04551264 +15.518012 
 
 + 85.49071 
 
 270 
 
 3.3403134 
 
 367.89469814 -0.13535069 +10.105905 
 
 + 185.36209 
 
 300 
 
 3.3386972 
 
 367.98729038 -0.54951786 + 2.018660 
 
 +235.29114 
 
 330 
 
 3.3379911 
 
 367.66272932 -0.88324571 - 6.576756 
 
 +221.89949 
 
 2i 
 
 0.0526706 
 
 2205.17145901* +0.01671297 + 0.732854 
 
 6.15066 
 
 S 2 
 
 0.0526714 
 
 2205.17070625 +0.01671263 + 0.732855 
 
 6.15055 
 
 
 *2,(J,'-G") = 
 
 2202.66492021. 
 
 
 
 Zi(Ji' - G") = 
 
 2202.66491965. 
 
 
118 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 ACTION OF URANUS ON VENUS. 
 
 
 
 E 
 
 F 3 1000 X Ro 1000000 X So 1000000 X W 
 
 1000 X fi <n) 
 
 1000000 X S ( "> 
 
 
 
 - 4.0226412 0.02650922 -0.05811066 -2.918780 
 
 0.00000000 
 
 -0.08089097 
 
 30 
 
 - 0.1770896 0.02652281 -0.01305234 -3.621493 
 
 +0.01844306 
 
 -0.01815231 
 
 60 
 
 1.6152186 0.02667654 +0.04940526 -3.356828 
 
 + 0.03204873 
 
 +0.06853681 
 
 90 
 
 - 6.9332675 0.02691093 +0.05269544 -2.182313 
 
 +0.03720412 
 
 +0.07285096 
 
 120 
 
 -10.8407275 0.02710189 -0.01759673 -0.401343 
 
 +0.03233767 
 
 -0.02424436 
 
 150 
 
 - 9.4434565 0.02715609 -0.09660941 +1.506695 
 
 +0.01866093 
 
 -0.13277471 
 
 180 
 
 - 4.1342719 0.02707758 -0.10424233 +3.017072 
 
 0.00000000 
 
 -0.14313458 
 
 210 
 
 - 0.2013093 0.02694790 -0.02527822 +3.713882 
 
 -0.01851786 
 
 -0.03474102 
 
 240 
 
 1.5455370 0.02684358 +0.07399797 +3.412656 
 
 -0.03202946 
 
 +0.10195266 
 
 270 
 
 - 6.7883563 0.02677475 +0.10841711 +2.207472 
 
 -0.03701585 
 
 +0.14988562 
 
 300 
 
 -10.6594103 0.02669990 +0.05468656 +0.432404 
 
 -0.03207678 
 
 +0.07586323 
 
 330 
 
 - 9.2743255 0.02659629 -0.02803268 -1.439065 
 
 -0.01849416 
 
 -0.03898596 
 
 2! 
 
 -32.8178065 0.16090871 -0.00185993 +0.185181 
 
 +0.00028016 
 
 -0.00191721 
 
 2, 
 
 -32.8178047 0.16090877 -0.00186010 +0.185178 
 
 +0.00028024 
 
 -0.00191742 
 
 
 1000 X f Bo sin D + 1000 X [- So cos v + 
 
 
 r 
 
 E 
 
 , \ 1 /r \ 1 100000 X ^o C08 " 1000000 X We sin u 
 (cos v + cos E] So 1 (~ secV + 1 1 sin vSo \ 
 
 1000 X -2- Ro 
 
 
 
 -0.00011622 -0.02650922 -1.710234 
 
 - 2.365243 
 
 -0.05265561 
 
 30 
 
 +0.01331757 -0.02293687 -0.357980 
 
 - 3.603757 
 
 -0.05273125 
 
 60 
 
 +0.02323048 -0.01311517 +1.390488 
 
 - 3.055297 
 
 -0.05317053 
 
 90 
 
 +0.02690995 +0.00028955 +1.777154 
 
 - 1.266575 
 
 -0.05382187 
 
 120 
 
 +0.02340803 +0.01365914 +0.399475 
 
 - 0.038680 
 
 -0.05438924 
 
 150 
 
 +0.01366523 +0.02346772 -1.372927 
 
 - 0.620648 
 
 -0.05463405 
 
 180 
 
 +0.00020848 +0.02707758 -1.767867 
 
 - 2.444894 
 
 -0.05452574 
 
 210 
 
 -0.01335043 +0.02340859 -0.392395 
 
 - 3.693094 
 
 -0.05421519 
 
 240 
 
 -0.02324178 +0.01343114 +1.376698 
 
 - 3.122646 
 
 -0.05387086 
 
 270 
 
 -0.02677488 -0.00003361 +1.779939 
 
 - 1.305659 
 
 -0.05354950 
 
 300 
 
 -0.02314722 -0.01330732 +0.429867 
 
 - 0.046772 
 
 -0.05321706 
 
 330 
 
 -0.01342562 -0.02295918 -1.315326 
 
 - 0.583802 
 
 -0.05287735 
 
 Zi 
 
 +0.00034177 +0.00123615 +0.118427 
 
 -11.073532 
 
 -0.32182904 
 
 2 2 
 
 +0.00034182 +0.00123620 +0.118465 
 
 -11.073535 
 
 -0.32182921 
 
 
 sin v Mi (t) + cos v B <c) = + 0.000000000000020. 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 119 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 n log coeff. 
 
 [defdl] M = + 0.12000343m' p 9.0791936 
 
 [dx/dt] M = + 63.424159 TO' p 1.8022547 
 
 [di/<ft]oo = + 0.04158807 TO' p 8.6189687 
 
 = -- 65.693091 TO' n 1.8175197 
 
 = + 63.308999 TO' p 1.8014655 
 
 [dL/dt] m = -113.109825 TO' n 2.0535003 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO' 
 
 [dejdt] m = +o'.0000052633084 
 [dx/dt}^ = +0.0027817616 
 [di/dtloo = +0.000001824038 
 
 = -0.0028812762 
 
 = +0.0027767109 
 
 = -0.0049609570 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt] w +0^00000 +o!o0001 +o!o00005263 
 
 e[dirldt] M +0.00002 +0.00002 +0.000019001 
 
 [dildt] m +0.00000 +0.00000 +0.000001824 
 
 sin i \daldi\m -0.000165 -0.00017 -0.000170530 
 
 NOTES. 
 
 That a division into eight parts is here fully sufficient is shown by the agreement 
 of the final sums. Thus the greatest effect produced by all terms from the 4th to the 
 7th order is seen to occur with the coefficient [dx/dt] o and to amount to but 
 0".00000004. 
 
120 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF NEPTUNE ON VENUS. 
 
 E 
 
 A B cos f B sin 
 
 h 
 
 
 
 904.77843877 + 9.109428 +21.528911 30.253556 
 
 904.17419 
 
 45 
 
 904.51260261 - 6.657698 +16.194152 17.117826 
 
 904.17356 
 
 90 
 
 904.39237664 -14.030230 + 1.286186 0.107979 
 
 904.17298 
 
 135 
 
 904.48820486 - 8.689452 -14.462107 13.651972 
 
 904.17345 
 
 180 
 
 904.74393521 + 6.236084 -21.825589 31.093114 
 
 904.17407 
 
 225 
 
 905.00974680 +22.003203 -16.490829 17.750770 
 
 904.17359 
 
 270 
 
 905.12994825 +29.375736 - 1.582864 0.163538 
 
 904.17281 
 
 315 
 
 905.03414463 +24.034972 +14.165428 13.097596 
 
 904.17334 
 
 2i 
 
 3619.04469887* +30.691018f - 0.593356f 61.618187 
 
 3616.69405 
 
 2, 
 
 3619.04469890 +30.691025 - 0.593356 61.618164 
 
 3616.69394 
 
 E 
 
 1 G G' G" 
 
 e 
 
 
 
 O / // 
 
 
 
 0.53898 904.17415 0.5952295 0.0562134 
 
 1 32 17.018 
 
 45 
 
 0.27377 904.17354 0.3309885 0.0571984 
 
 1 11 14.024 
 
 90 
 
 0.15413 904.17298 0.1548950 0.0007710 
 
 45 6.502 
 
 135 
 
 0.24949 904.17343 0.2998555 0.0503537 
 
 1 7 39.561 
 
 180 
 
 0.50459 904.17403 0.5654480 0.0608162 
 
 1 30 28.921 
 
 225 
 
 0.77088 904.17357 0.7955805 0.0246764 
 
 1 43 33.466 
 
 270 
 
 0.89187 904.17281 0.8920680 0.0002028 
 
 1 48 0.658 
 
 315 
 
 0.79553 904.17332 0.8133615 0.0178097 
 
 1 44 14.706 
 
 2, 
 
 2.08957 3616.69397 2.2076405 0.1180034 
 
 5 35 53.099 
 
 2 2 
 
 2.08967 3616.69386 2.2397860 0.1500382 
 
 5 46 41.757 
 
 
 ACTION OF NEPTUNE ON VENUS. 
 
 
 E 
 
 log K log La' log No log N log P 
 
 logQ 
 
 
 
 0.00023476 0.27331427 0.17644338 5.1378654 9.4986215 
 
 2.3580297 
 
 45 
 
 0.00013986 0.27318776 0.17630105 5.1395214 9.5001506 
 
 2.3595431 
 
 90 
 
 0.00005608 0.27307605 0.17617538 5.1436917 9.5042639 
 
 2.3636152 
 
 135 
 
 0.00012618 0.27316951 0.17628052 5.1479186 9.5085363 
 
 2.3679232 
 
 180 
 
 0.00022568 0.27330217 0.17642976 5.1497409 9.5104807 
 
 2.3698895 
 
 225 
 
 0.00029563 0.27339543 0.17653468 5.1481063 9.5089743 
 
 2.3683773 
 
 270 
 
 0.00032162 0.27343007 0.17657365 5.1439578 9.5048849 
 
 2.3642800 
 
 315 
 
 0.00029957 0.27340068 0.17654059 5.1397097 9.5005900 
 
 2.3599901 
 
 Si 
 
 0.00083814 1.09312256 0.70562217 0.5752556 8.0182509 
 
 9.4558143 
 
 2 2 
 
 0.00086124 1.09315338 0.70565684 0.5752558 8.0182511 
 
 9.4558336 
 
 * 4o 2 + 2aV + 4[a' J - Zkaa'ee' cos K] = 3619.04469884. 
 f 4[a'V - kaa'e cos A'] = + 30.691024. 
 t - 4k'aa' cos *' e sin A'' = - 0.593359. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 121 
 
 ACTION OF NEPTUNE ON VENUS. 
 
 E 
 
 logF 
 
 Jl' J2 
 
 J> 
 
 Ft 
 
 
 
 2.3579959 
 
 902.1174996 +0.0278169 
 
 -43.343250 
 
 -165.19752 
 
 45 
 
 2.3595087 
 
 903.0233587 +1.1892143 
 
 -32.830864 
 
 -124.26237 
 
 90 
 
 2.3636147 
 
 904.1640829 +0.1518779 
 
 - 2.911684 
 
 9.86927 
 
 135 
 
 2.3678929 
 
 903.3068137 -1.1744691 
 
 +28.888013 
 
 + 110.97189 
 
 180 
 
 2.3698530 
 
 902.1221024 -0.3409776 
 
 +43.940408 
 
 + 167.47400 
 
 225 
 
 2.3683624 
 
 902.9705135 +0.9081699 
 
 +33.428015 
 
 + 126.53886 
 
 270 
 
 2.3642798 
 
 904.1592634 +0.1562063 
 
 + 3.508843 
 
 + 12.14577 
 
 315 
 
 2.3599794 
 
 903.2945923 -0.9279474 
 
 -28.290864 
 
 -108.69539 
 
 Si 
 
 9.4557434 
 
 3612.5629483* -0.0050765 
 
 + 1.194317 
 
 + 4.55298 
 
 2s 
 
 9.4557434 
 
 3612.5952782 -0.0050323 
 
 + 1.194300 
 
 + 4.55299 
 
 E 
 
 Pi 
 
 100000 X Bo 100000000 X So 1000000 X Wo 
 
 100000 X B<> 
 
 1000000 X S<> 
 
 
 
 +0.5839042 
 
 0.6821434 -0.4573134 -0.9883466 
 
 0.0000000 
 
 -0.006365877 
 
 45 
 
 -3.9412736 
 
 0.6867470 +2.3281642 -0.7513860 
 
 +0.6746065 
 
 +0.032343148 
 
 90 
 
 -0.4777116 
 
 0.6961065 +0.3193229 -0.0672756 
 
 +0.9623607 
 
 +0.004414608 
 
 135 
 
 +4.0782664 
 
 0.7008255 -2.3820088 +0.6740551 
 
 +0.6818058 
 
 -0.032772474 
 
 180 
 
 +0.6001079 
 
 0.7010447 -0.2565237 +1.0297346 
 
 0.0000000 
 
 -0.003522313 
 
 225 
 
 -4.0073139 
 
 0.7005531 +2.5294517 +0.7805505 
 
 -0.6815408 
 
 +0.034801040 
 
 270 
 
 -0.5873116 
 
 0.6967370 +0.4002346 +0.0811602 
 
 -0.9632323 
 
 +0.005533205 
 
 315 
 
 +3.9893101 
 
 0.6879052 -2.4698966 -0.6479491 
 
 -0.6757442 
 
 -0.034312118 
 
 Si 
 
 +0.1189889 
 
 2.7760316 +0.0057204 +0.0552726 
 
 -0.0008716 
 
 +0.000059623 
 
 y 
 
 +0.1189890 
 
 2.7760308 +0.0057105 +0.0552705 
 
 -0.0008727 
 
 +0.000059596 
 
 E 
 
 inonm v r ; , 100000 X -Bo cos w 
 
 ItAAJlH) X l/VO Sin V innnnn ^, lir 
 
 , , . _, 100000 X Wo cos u 
 + (cos v+cos)S ] , IT 2 N . 1 
 
 \a sec *" r l " J 
 
 100000 X W sin u 
 
 1000 X - 2 - Bo 
 a 
 
 
 
 -0.00091463 
 
 -0.68214344 -0.057911307 
 
 -0.080091019 
 
 -0.013549511 
 
 45 
 
 + 0.49123773 
 
 -0.47994176 +0.012282960 
 
 -0.074127847 
 
 -0.013668483 
 
 90 
 
 +0.69608806 
 
 +0.00540218 +0.005478547 
 
 -0.003904555 
 
 -0.013922131 
 
 135 
 
 +0.49653747 
 
 +0.49458425 -0.066499046 
 
 -0.011017258 
 
 -0.014084333 
 
 180 
 
 + 0.00051305 
 
 +0.70104468 -0.060336403 
 
 -0.083444904 
 
 -0.014116841 
 
 225 
 
 -0.49655471 
 
 +0.49418261 +0.012013897 
 
 -0.077124930 
 
 -0.014078857 
 
 270 
 
 -0.69672347 
 
 +0.00396738 +0.006544151 
 
 -0.004800407 
 
 -0.013934740 
 
 315 
 
 -0.49226067 
 
 -0.48055578 -0.064023059 
 
 -0.009971443 
 
 -0.013691533 
 
 Si 
 
 -0.00103699 
 
 +0.02827080 -0.106225012 
 
 -0.172240885 
 
 -0.055523223 
 
 S 2 
 
 -0.00104018 
 
 +0.02826932 -0.106225248 
 
 -0.172241478 
 
 -0.055523206 
 
 
 sin <f ^A\ M + cos 
 
 v . BO M = - 0.0000000000000085. 
 
 
 
 
 *2,W,'-G"1 =3i 
 
 312.4449449. 
 
 
 
 i' - G") = 3612.4452400. 
 
122 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 [de/dt] w = - - 0.0054696734 m' n 7.7379614 
 [dx!dt] m = +21.756678 TO' p 1.3375926 
 [di/dt] m = -- 0.55945727 TO' n 9.7477669 
 [dB/(ft]oo =-15.327159 w' n 1.1854617 
 [drldt] m = +21.729810 TO' p 1.3370559 
 [dL/dt] M = -29.268164 TO' n 1.4663954 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO' 
 
 [defdt] m = -0 / .00000027764841 
 
 [dx/dt]^ = +0.0011044000 
 [difdt] M -0.000028398849 
 
 [dQ/dt] w = -0.00077802855 
 
 [dT/ft]oo = +0.0011030360 
 
 [dL/dt] w = -0.0014856935 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 // // // 
 
 oo -0.00000 -0.00000 -0.00000028 
 
 e[drldt] m +0.00001 +0.00001 +0.00000755 
 
 [di/dt] w -0.00004 -0.00003 -0.00002840 
 
 sin ?: [dQ/dt] M -0.00006 -0.00005 -0.00004605 
 
 NOTES. 
 
 The large disagreement of the sums of the functions near the beginning of the 
 computation is caused, as in previous cases, by the presence of the term a'V. The 
 greatest disagreement in the final sums occurs in the second column and shows that 
 the effect of all terms from the 4th to the 7th orders is to produce a change of 
 0".00000003 in the value of [d x /dt] 00 . 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 123 
 
 EARTH. 
 ACTION OF MERCUKY ON THE EARTH. 
 
 E 
 
 A 
 
 B COS e 
 
 It sin e 
 
 1000 xg 
 
 h 
 
 
 
 1.25770017 
 
 +0.37398164 
 
 +0.15686177 
 
 0.15586327 
 
 1.11042819 
 
 30 
 
 1.20879374 
 
 +0.24429166 
 
 +0.30606106 
 
 0.59337027 
 
 1.06575342 
 
 60 
 
 1.14343370 
 
 +0.05583076 
 
 +0.37253457 
 
 0.87910918 
 
 1.00282212 
 
 90 
 
 1.07923573 
 
 -0.14090326 
 
 +0.33847094 
 
 0.72569217 
 
 0.93669980 
 
 120 
 
 1.03345314 
 
 -0.29319554 
 
 +0.21299723 
 
 0.28738066 
 
 0.88633014 
 
 150 
 
 1.01830180 
 
 -0.36023958 
 
 +0.02973421 
 
 0.00560044 
 
 0.86904450 
 
 180 
 
 1.03773900 
 
 -0.32407097 
 
 -0.16221302 
 
 0.16667904 
 
 0.89145124 
 
 210 
 
 1.08650443 
 
 -0.19438096 
 
 -0.31141230 
 
 0.61430099 
 
 0.94482277 
 
 240 
 
 1.15158355 
 
 -0.00592005 
 
 -0.37788591 
 
 0.90454667 
 
 1.01121409 
 
 270 
 
 1.21564099 
 
 +0.19081399 
 
 -0.34382220 
 
 0.74882000 
 
 1.07228810 
 
 300 
 
 1.26156411 
 
 +0.34310630 
 
 -0.21834847 
 
 0.30200208 
 
 1.11392105 
 
 330 
 
 1.27699637 
 
 +0.41015028 
 
 -0.03508545 
 
 0.00779765 
 
 1.12755062 
 
 2, 
 
 6.88547367* 
 
 +0.14973214f 
 
 -0.01605383J 
 
 2.69558090 
 
 6.01616683 
 
 2 2 
 
 6.88547306 
 
 +0.14973213 
 
 -0.01605374 
 
 2.69558152 
 
 6.01615921 
 
 E 
 
 G 
 
 G' 
 
 G" 
 
 
 
 0.14093752 
 
 1.11028337 
 
 0.14207046 
 
 0.00098811 
 
 21 
 
 1 
 
 34.45 
 
 30 
 
 0.13670586 
 
 1.06515341 
 
 0.14124977 
 
 0.00394390 
 
 21 
 
 37 
 
 28.51 
 
 60 
 
 0.13427712 
 
 1.00181061 
 
 0.14149060 
 
 0.00620197 
 
 22 
 
 30 
 
 21.00 
 
 90 
 
 0.13620147 
 
 0.93572981 
 
 0.14260963 
 
 0.00543817 
 
 23 
 
 21 
 
 59.90 
 
 120 
 
 0.14078854 
 
 0.88589477 
 
 0.14348474 
 
 0.00226084 
 
 23 
 
 53 
 
 48.72 
 
 150 
 
 0.14292284 
 
 0.86903562 
 
 0.14297679 
 
 0.00004507 
 
 23 
 
 55 
 
 58.88 
 
 180 
 
 0.13995330 
 
 0.89120229 
 
 0.14152378 
 
 0.00132152 
 
 23 
 
 34 
 
 54.52 
 
 210 
 
 0.13534720 
 
 0.94401808 
 
 0.14077439 
 
 0.00462250 
 
 23 
 
 2 
 
 51.29 
 
 240 
 
 0.13403500 
 
 1.01019211 
 
 0.14138996 
 
 0.00633298 
 
 22 
 
 24 
 
 31.27 
 
 270 
 
 0.13701843 
 
 1.07154031 
 
 0.14266461 
 
 0.00489838 
 
 21 
 
 43 
 
 51.55 
 
 300 
 
 0.14130860 
 
 1.11364215 
 
 0.14347758 
 
 0.00189008 
 
 21 
 
 9 
 
 39.20 
 
 330 
 
 0.14311129 
 
 1.12754360 
 
 0.14316663 
 
 0.00004830 
 
 20 
 
 52 
 
 41.84 
 
 Si 
 
 0.83130008 
 
 6.01302530 
 
 0.85343712 
 
 0.01899550 
 
 134 
 
 34 
 
 49.16 
 
 2 2 
 
 0.83130709 
 
 6.01302083 
 
 0.85344182 
 
 0.01899632 
 
 134 
 
 34 
 
 51.97 
 
 * 6a 2 + 3aV + 6[a' 2 - 2kaa'ee' cos K] = 6.8854738. 
 t 6[a'V - kaa'e cos K] = + 0.14973211. 
 | - Gfc'aa' cos <p'-e sin K' = - 0.01605375, 
 
124 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MEKCUKY ON THE 
 
 EARTH. 
 
 
 E 
 
 
 logtfo 
 
 log LQ 
 
 
 logtfo 
 
 
 log N log P 
 
 logQ 
 
 
 
 
 
 .04527343 
 
 0.33291492 
 
 
 
 .24335185 
 
 9. 
 
 9618524 0.2031270 
 
 0.1593841 
 
 30 
 
 0.04797642 
 
 0.33646399 
 
 
 
 .24732687 
 
 9. 
 
 9917424 0.2701720 
 
 0.2100521 
 
 60 
 
 
 
 .05211516 
 
 0.34189221 
 
 
 
 .25340453 
 
 0. 
 
 0396020 0.3745624 
 
 0.2895406 
 
 90 
 
 0.05634203 
 
 0.34742859 
 
 0.25960067 
 
 0.0958412 0.4959354 
 
 0.3817747 
 
 120 
 
 
 
 .05903754 
 
 0.35095528 
 
 
 
 .26354624 
 
 0. 
 
 1435571 0.5975343 
 
 0.4586143 
 
 150 
 
 
 
 .05922392 
 
 0.35119901 
 
 
 
 .26381890 
 
 0.1631586 0.6362374 
 
 0.4879174 
 
 180 
 
 
 
 .05742739 
 
 0.34884899 
 
 
 
 .26118992 
 
 0. 
 
 1459441 0.5935535 
 
 0.4565142 
 
 210 
 
 0.05475392 
 
 0.34534936 
 
 
 
 .25727397 
 
 0. 
 
 1016262 0.4927722 
 
 0.3817985 
 
 240 
 
 
 
 .05164961 
 
 0.34128199 
 
 0.25272142 
 
 0.0482257 0.3752715 
 
 0.2938290 
 
 270 
 
 
 
 .04846610 
 
 0.33710662 
 
 0.24804652 
 
 0. 
 
 0004822 0.2736102 
 
 0.2165394 
 
 300 
 
 0.0-1587421 
 
 0.33370403 
 
 
 
 .24423576 
 
 9.9673367 0.2060765 
 
 0.1640904 
 
 330 
 
 
 
 .04461832 
 
 0.33205428 
 
 0.24238775 
 
 9. 
 
 9536821 0.1814324 
 
 0.1439179 
 
 2l 
 
 
 
 .31137734 
 
 2.04959742 
 
 1 
 
 .51844973 
 
 0.3065179 2.3501251 
 
 1.8219725 
 
 Sj 
 
 
 
 .31138070 
 
 2.04960185 
 
 1 
 
 .51845468 
 
 0. 
 
 3065326 2.3501595 
 
 1.8219999 
 
 E 
 
 
 logF 
 
 Ji' 
 
 
 J, 
 
 
 J, 
 
 F, 
 
 
 
 
 0.1589092 
 
 0.14310992 
 
 
 +0.032517759 
 
 
 +0.009293505 
 
 -0.004694216 
 
 30 
 
 
 0.2080805 
 
 0.14534318 
 
 
 +0.064629382 
 
 
 +0.012521451 
 
 -0.009159128 
 
 60 
 
 
 0.2862522 
 
 0.14795099 
 
 
 +0.079427870 
 
 
 +0.011088166 
 
 -0.011148401 
 
 90 
 
 
 0.3786919 
 
 0.14823938 
 
 
 +0.072184328 
 
 
 +0.005377209 
 
 -0.010129019 
 
 120 
 
 
 0.4572602 
 
 0.14575314 
 
 
 +0.044711897 
 
 
 +0.003081410 
 
 -0.006374115 
 
 150 
 
 
 0.4878898 
 
 0.14319523 
 
 
 +0.004967377 
 
 
 -0.012020967 
 
 -0.000889820 
 
 180 
 
 
 0.4557266 
 
 0.14344333 
 
 
 -0.035694279 
 
 
 -0.019045640 
 
 +0.004854357 
 
 210 
 
 
 0.3791987 
 
 0.14602945 
 
 
 -0.066230415 
 
 
 -0.022272938 
 
 +0.009319268 
 
 240 
 
 
 0.2904985 
 
 0.14803624 
 
 
 -0.078997000 
 
 
 -0.020838344 
 
 +0.011308544 
 
 270 
 
 
 0.2141065 
 
 0.14763142 
 
 
 -0.071296738 
 
 
 -0.015126732 
 
 +0.010289159 
 
 300 
 
 
 0.1631852 
 
 0.14536909 
 
 
 -0.045398479 
 
 
 -0.006668768 
 
 +0.006534256 
 
 330 
 
 
 0.1438950 
 
 0.14322488 
 
 
 -0.007686344 
 
 
 +0.002269483 
 
 +0.001049961 
 
 Si 
 
 
 1.8118319 
 
 0.87366271* 
 
 
 -0.003432232 
 
 
 -0.029252491 
 
 +0.000480425 
 
 2i 
 
 
 1.8118624 
 
 0.87366354 
 
 
 -0.003432410 
 
 
 -0.029252494 
 
 +0.000480421 
 
 
 *s,(J 
 
 V - G") = 
 
 0.85466721. 
 
 
 
 
 
 
 St(Ji - G") = 
 
 0.85466722. 
 
 
 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 125 
 
 
 ACTION OF MERCURY ON THE EARTH. 
 
 
 E 
 
 1000 X F 3 R a So Wo <"> 
 
 gto) 
 
 
 
 -0.3347370 -0.9146346 +0.03939096 +0.012865148 0.0000000 
 
 +0.04006286 
 
 30 
 
 -0.1102483 -0.9755942 +0.08729250 +0.020012484 -0.4949864 
 
 +0.08857904 
 
 60 
 
 +0.5661785 -1.0850638 +0.12713088 +0.022775691 -0.9476313 
 
 +0.12820596 
 
 90 
 
 + 1.0162060 -1.2358930 +0.14090497 +0.016043819 -1.2358930 
 
 +0.14090497 
 
 120 
 
 +0.7859945 -1.3865198 +0.10290744 -0.005719611 -1.1907761 
 
 +0.10205179 
 
 150 
 
 +0.1010653 -1.4553467 +0.01142556 -0.036530916 -0.7172557 
 
 +0.01126198 
 
 180 
 
 -0.3579653 -1.3946588 -0.08289463 -0.055794474 0.0000000 
 
 -0.08152732 
 
 210 
 
 -0.1348464 -1.2530893 -0.12959913 -0.053749951 +0.6175749 
 
 -0.12774376 
 
 240 
 
 +0.5468018 -1.1065943 -0.12737525 -0.039380718 +0.9503695 
 
 -0.12631602 
 
 270 
 
 + 1.0072426 -0.9942912 -0.09740907 -0.022874526 +0.9942912 
 
 -0.09740907 
 
 300 
 
 +0.7898462 -0.9252302 -0.05560180 -0.008440800 +0.8080489 
 
 -0.05607200 
 
 330 
 
 +0.1166998 -0.8987688 -0.00911126 +0.003338197 +0.4560076 
 
 -0.00924555 
 
 Zi 
 
 + 1.9961187 -6.8127015 +0.00355760 -0.073694764 -0.3799890 
 
 +0.00640527 
 
 * 
 
 + 1.9961190 -6.8129832 +0.00350357 -0.073760893 -0.3802614 
 
 +0.00634761 
 
 E 
 
 . Rocosv 
 
 Ho sm v ,,, , . , . . . 
 m o , l r i \ i\ o " o cos (v + JT) TV o sm (v + IT) 
 + (cosv+cosE)S +(-sec 2 ip + llsmv<So 
 
 -2-fl 
 o 
 
 
 
 +0.0787819 +0.9146346 -0.002313800 +0.012655370 
 
 1.7985901 
 
 30 
 
 -0.3440930 +0.9286742 -0.013088473 +0.015139066 
 
 1.9228487 
 
 60 
 
 -0.8219877 +0.7498962 -0.021560174 +0.007341046 
 
 2.1519297 
 
 90 
 
 -1.2380823 +0.2610826 -0.015731601 -0.003149764 
 
 2.4717860 
 
 120 
 
 -1.2947997 -0.5330552 +0.004304192 +0.003766681 
 
 2.7962929 
 
 150 
 
 -0.7369917 -1.2550382 +0.011990880 +0.034506913 
 
 2.9529690 
 
 180 
 
 +0.1657893 -1.3946588 -0.010034649 +0.054884684 
 
 2.8360980 
 
 210 
 
 +0.8424959 -0.9617143 -0.034466377 +0.041244714 
 
 2.5425788 
 
 240 
 
 + 1.0791999 -0.3473975 -0.036894585 +0.013770630 
 
 2.2317474 
 
 270 
 
 +0.9957850 +0.1781428 -0.022567368 -0.003736033 
 
 1.9885826 
 
 300 
 
 + 0.7530386 +0.5475911 -0.006510755 -0.005371886 
 
 1.8349430 
 
 330 
 
 +0.4402010 +0.7837111 +0.001148460 +0.003134422 
 
 1.7714297 
 
 Z, 
 
 -0.0399777 -0.0629896 -0.073009771 +0.087046525 
 
 13.6496011 
 
 2j 
 
 -0.0406851 -0.0651418 -0.072714479 +0.087139318 
 
 13.6501948 
 
 
 sin <p J.Ai ( '> + cos ? Bo M = + 0.00000006. 
 
 
126 THE SECULAR VARIATIONS OP THE ELEMENTS 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 [de!dt]w = - 8710J780TO' n 3.9400270 
 
 [d x /dt] M = [dTT/dt] m =--824986.23 TO' n 5.9164467 
 
 [dp/dt]*, = + 18814.333 TO' p 4.2744888 
 
 [dqfdt] m = - 15740.112 TO' n 4.1970078 
 
 [dL/dt] w = +2948201.7 w' p 6.4695572 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 = -o(X)l 1613570 
 
 [d x /dt] m = [drfdt] m = -0.10999815 
 [dpfdt] M = +0.0025085775 
 [dq/dt] w = -0.0020986812 
 [dL/dt] M = +0.39309355 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt] m 
 
 -0.00116 
 
 -0.00116 
 
 -0.001 16136 
 
 e[dir/dt]w 
 
 -0.00184 
 
 -0.00184 
 
 -0.00184479 
 
 [dp/dt] M 
 
 +0.00250 
 
 +0.00251 
 
 +0.00250858 
 
 [dq/dt]w 
 
 -0.00209 
 
 -0.00210 
 
 -0.00209868 
 
 [dL/dt] w 
 
 +0.3931 
 
 
 +0.39309355 
 
 NOTES. 
 
 Although / and e' are here very large, the error in the approximate test with 
 e, G, G', G" and 6 is small in consequence of the smallness of the factor a'. As we 
 approach the end of the computation, however, the difference of the sums steadily 
 increases, indicating the rather slow convergence of the perturbing function. The 
 greatest difference is in the coefficient [dTr/dt] OQ where terms from the fifth to the 
 eleventh orders inclusive amount to one sixtieth part of the remaining terms and 
 produce an effect of 0".0018 in the value of [dir/dt] 00 . A division into twelve parts 
 is thus necessary in this case, but a comparison with the computation of the action of 
 Mars on Mercury, and especially with the similar case of Mercury on Venus, where 
 twenty-four points of division are employed, renders it evident that more than twelve 
 points are in the present case unnecessary. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 127 
 
 The agreement with previous values is exact. The results obtained by HILL 
 in the "New Theory, 11 pages 511 and 512, are, 
 
 These are, however, 
 Venus on the Earth. 
 
 = +0.0025049 [dq/dt] m = -0.0020956 
 but provisional -values. (See the note to the computation of 
 
 E 
 
 A 
 
 ACTION 
 logB 
 
 OF VENUS ON 
 i 
 
 THE EARTH. 
 
 
 e' 
 
 
 
 
 
 
 o / 
 
 ,, 
 
 O 
 
 1 
 
 H 
 
 
 
 1.49844749 
 
 9.8537612 
 
 331 
 
 3.89 0.07950559 
 
 7 
 
 
 
 55.62 
 
 30 
 
 1.50411382 
 
 9.8546442 
 
 1 19 
 
 46.170 0.08070333 
 
 5 
 
 48 
 
 28.79 
 
 60 
 
 1.51484369 
 
 9.8567645 
 
 31 34 
 
 37.20 0.08262793 
 
 4 
 
 19 
 
 13.62 
 
 90 
 
 1.52786583 
 
 9.8597314 
 
 61 38 
 
 57.20 0.08465948 
 
 3 
 
 12 
 
 16.55 
 
 120 
 
 1.53974201 
 
 9.8626537 
 
 91 29 
 
 35.875 0.08634022 
 
 2 
 
 33 
 
 43.11 
 
 150 
 
 1.54723847 
 
 9.8644870 
 
 121 8 
 
 43.24 0.08741009 
 
 2 
 
 9 
 
 29.44 
 
 180 
 
 1.54824407 
 
 9.8645724 
 
 150 43 
 
 29.14 0.08768573 
 
 1 
 
 52 
 
 5.43 
 
 210 
 
 1.54243719 
 
 9.8629683 
 
 180 22 
 
 35.176 0.08700647 
 
 1 
 
 54 
 
 16.46 
 
 240 
 
 1.53142595 
 
 9.8603568 
 
 210 12 
 
 5.09 0.08536569 
 
 2 
 
 37 
 
 52.10 
 
 270 ' 
 
 1.51826327 
 
 9.8576205 
 
 240 13 
 
 17.66 0.08309965 
 
 4 
 
 8 
 
 28.42 
 
 300 
 
 1.50652764 
 
 9.8554195 
 
 270 23 
 
 50.222 0.08090038 
 
 5 
 
 56 
 
 51.93 
 
 330 
 
 1.49931255 
 
 9.8540767 
 
 300 40 
 
 21.95 0.07954648 
 
 7 
 
 7 
 
 42.75 
 
 Z, 
 
 9.13923084* 
 
 9.1535280 
 
 1085 23 
 
 41.42 0.50242554 
 
 24 
 
 20 
 
 41.81 
 
 2 2 
 
 9.13923113 
 
 9.1535281 
 
 905 23 
 
 41.40 0.50242550 
 
 24 
 
 20 
 
 42.41 
 
 E 
 
 1000 X r' 
 
 lOOOXs 
 
 G 
 
 G' 1000000 X G" 
 
 
 e 
 
 
 O 
 
 
 
 
 
 O 
 
 , 
 
 ,, 
 
 
 
 2.7380635 
 
 0.002936418 
 
 0.97594422 
 
 0.522484505 5.79 
 
 47 
 
 1 
 
 40.84 
 
 30 
 
 2.3200483 
 
 0.000006753 
 
 0.98353171 
 
 0.520557558 0.04 
 
 46 
 
 40 
 
 41.61 
 
 60 
 
 1.7893068 
 
 0.003473631 
 
 0.99543723 
 
 0.519388564 6.67 
 
 46 
 
 14 
 
 52.33 
 
 90 
 
 1.3770128 
 
 0.009946420 
 
 1.00773136 
 
 0.520128988 18.98 
 
 45 
 
 55 
 
 31.27 
 
 120 
 
 1.1340363 
 
 0.013007536 
 
 1.01774388 
 
 0.521998108 24.54 
 
 45 
 
 44 
 
 23.02 
 
 150 
 
 0.9732049 
 
 0.009615238 
 
 1.02406979 
 
 0.523162060 17.95 
 
 45 
 
 37 
 
 23.13 
 
 180 
 
 0.8464697 
 
 0.003140184 
 
 1.02571566 
 
 0.522509836 5.92 
 
 45 
 
 32 
 
 21.65 
 
 210 
 
 0.8529468 
 
 0.000000563 
 
 1.02173849 
 
 0.520674148 0.00 
 
 45 
 
 32 
 
 59.47 
 
 240 
 
 1.1449685 
 
 0.003259143 
 
 1.01199593 
 
 0.519411649 6.09 
 
 45 
 
 45 
 
 34.89 
 
 270 
 
 1.7299249 
 
 0.009581104 
 
 0.99828868 
 
 0.519968638 18.53 
 
 46 
 
 11 
 
 45.77 
 
 300 
 
 2.3843853 
 
 0.012589278 
 
 0.98467983 
 
 0.521847851 24.49 
 
 46 
 
 43 
 
 7.27 
 
 330 
 
 2.7841227 
 
 0.009256115 
 
 0.97615654 
 
 0.523149586 18.07 
 
 47 
 
 3 
 
 38.83 
 
 Zi 
 
 10.0372301 
 
 0.038406190 
 
 6.01151675 
 
 3.127640513 73.50 
 
 277 
 
 2 
 
 0.00 
 
 2 2 
 
 10.0372604 
 
 0.038406192 
 
 6.01151657 
 
 3.127640978 73.57 
 
 277 
 
 2 
 
 0.08 
 
 
 * 6o 2 + 3a 2 e 2 + 6[o' 2 - 
 
 2kaa'ce' cos K] = 9 
 
 13923110. 
 
 
 
 
 
128 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF VENUS ON THE EARTH. 
 
 E 
 
 logtfo 
 
 log Lo' 
 
 log AT log N log P 
 
 logO 
 
 
 
 0.26147483 
 
 0.60779696 
 
 0.54775634 0.2626427 0.8915846 
 
 0.8209715 
 
 30 
 
 0.25683878 
 
 0.60207812 
 
 0.54149917 0.2549482 0.8714495 
 
 0.8036590 
 
 60 
 
 0.25122074 
 
 0.59513848 
 
 0.53390180 0.2468811 0.8459860 
 
 0.7827661 
 
 90 
 
 0.24707162 
 
 0.59000660 
 
 0.52828039 0.2420422 0.8253429 
 
 0.7669697 
 
 120 
 
 0.24470700 
 
 0.58707937 
 
 0.52507272 0.2404872 0.8122693 
 
 0.7579113 
 
 150 
 
 0.24322990 
 
 0.58524987 
 
 0.52306751 0.2402489 0.8048244 
 
 0.7529792 
 
 180 
 
 0.24217345 
 
 0.58394092 
 
 0.52163264 0.2400756 0.8019574 
 
 0.7506787 
 
 210 
 
 0.24230579 
 
 0.58410492 
 
 0.52181242 0.2408209 0.8062462 
 
 0.7532935 
 
 240 
 
 0.24496050 
 
 0.58739327 
 
 0.52541673 0.2444416 0.8214721 
 
 0.7646769 
 
 270 
 
 0.25055055 
 
 0.59430993 
 
 0.53299440 0.2516543 0.8474359 
 
 0.7853846 
 
 300 
 
 0.25737182 
 
 0.60273603 
 
 0.54221917 0.2600985 0.8762226 
 
 0.8090118 
 
 330 
 
 0.26191246 
 
 0.60833646 
 
 0.54834645 0.2649133 0.8941948 
 
 0.8237322 
 
 2i 
 
 1.50190834 
 
 3.56408503 
 
 3.19599940 1.4946267 5.0494920 
 
 4.6860161 
 
 S 2 
 
 1.50190910 
 
 3.56408590 
 
 3.19600034 1.4946278 5.0494937 
 
 4.6860181 
 
 E 
 
 log V 
 
 Ji' 
 
 1000 X Ji 1000 X J 3 
 
 1000 X Ft 
 
 
 
 0.8209685 
 
 0.522862677 
 
 - 3.0632731 + 1 2.629050 
 
 + 1.2372987 
 
 30 
 
 0.8036589 
 
 0.521939851 
 
 -0.7336504 +24.877829 
 
 -0.0593369 
 
 60 
 
 0.7827628 
 
 0.521368465 
 
 +2.4504401 +30.338229 
 
 -1.3457284 
 
 90 
 
 0.7669603 
 
 0.521723193 
 
 +5.0835797 +27.547130 
 
 -2.2771885 
 
 120 
 
 0.7578992 
 
 0.522632205 
 
 +5.8345100 +17.252379 
 
 -2.6041325 
 
 150 
 
 0.7529704 
 
 0.523191642 
 
 +4.4234178 + 2.212471 
 
 -2.2389562 
 
 180 
 
 0.7506758 
 
 0.522862807 
 
 + 1.7386808 -13.542647 
 
 -1.2795084 
 
 210 
 
 0.7532935 
 
 0.521968683 
 
 -0.9069972 -25.791387 
 
 +0.017127o 
 
 240 
 
 0.7646739 
 
 0.521377068 
 
 -2.6859269 -31.251723 
 
 + 1.3035189 
 
 270 
 
 0.7853753 
 
 0.521675619 
 
 -3.5987100 -28.460585 
 
 +2.2349789 
 
 300 
 
 0.8089994 
 
 0.522582150 
 
 -4.0331250 -18.165873 
 
 + 2.5619225 
 
 330 
 
 0.8237230 
 
 0.523186452 
 
 -4.0263352 - 3.126030 
 
 + 2.1967465 
 
 2, 
 
 4.6859794 
 
 3.133685372* 
 
 +0.2413060 - 2.740585 
 
 -0.1266292 
 
 2 2 
 
 4.6859813 
 
 3.133685439 
 
 +0.2413047 - 2.740573 
 
 -0.1266288 
 
 
 * S,(J,' - G") = 3. 
 
 133611872. 
 
 
 
 
 2,(J,' - G") = 3. 
 
 133611869. 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 129 
 
 
 
 ACTION OF VENUS ON THE EARTH. 
 
 
 E 
 
 1000 XF 3 
 
 RO 100 x So w,, --RO 
 
 <> 
 
 
 
 a 
 
 
 
 
 +0.06535471 
 
 - 1.8283274 -1.0644424 +0.08413477 1.7976643 
 
 0.0000000 
 
 30 
 
 -0.00196302 
 
 - 1.7898617 -0.5109526 +0.15828193 1.7638656 
 
 -0.9081206 
 
 60 
 
 -0.00570449 
 
 - 1.7535729 +0.5420211 +0.18393250 1.7388680 
 
 -1.5314810 
 
 90 
 
 +0.05916590 
 
 - 1.7367356 +1.4494077 +0.16147390 1.7367356 
 
 -1.7367356 
 
 120 
 
 +0.12885901 
 
 - 1.7362036 +1.6510300 +0.09963433 1.7507629 
 
 -1.4910928 
 
 150 
 
 +0.13426042 
 
 - 1.7386892 +1.0760758 +0.01338361 1.7639423 
 
 -0.8568990 
 
 180 
 
 +0.06988985 
 
 - 1.7361351 +0.1682926 -0.07583153 1.7652520 
 
 0.0000000 
 
 210 
 
 -0.00059748 
 
 - 1.7337538 -0.5029615 -0.14614352 1.7589352 
 
 +0.8544665 
 
 240 
 
 -0.00787443 
 
 - 1.7442502 -0.6981624 -0.18183297 1.7588768 
 
 + 1.4980034 
 
 270 
 
 +0.05404190 
 
 - 1.7747204 -0.6225129 -0.17324740 1.7747204 
 
 + 1.7747204 
 
 300 
 
 +0.12215392 
 
 - 1.8154792 -0.6714245 -0.11610019 1.8002552 
 
 + 1.5855471 
 
 330 
 
 +0.12777090 
 
 - 1.8402326 -0.9613070 -0.01982985 1.8135046 
 
 +0.9336770 
 
 ?i 
 
 +0.37267857 
 
 -10.6139684 -0.0726856 -0.00606309 10.6116792 
 
 +0.0609767 
 
 2 2 
 
 +0.37267862 
 
 -10.6139933 -0.0722505 -0.00608133 10.6117037 
 
 +0.0611087 
 
 
 
 ,-, . Ra'cOS V 
 
 
 M 
 
 S 
 
 + (cost)+cosE)So + ( -sec ! ^+lJsint>/So 
 
 WL, sin ( + IT) 
 
 
 
 -0.010825990 
 
 -0.0212889 +1.8283274 -0.01513166 
 
 +0.08276286 
 
 30 
 
 -0.005184831 
 
 -0.9168211 +1.5373039 -0.10351881 
 
 +0.11973727 
 
 60 
 
 +0.005466048 
 
 -1.5259137 +0.8639706 -0.17411620 
 
 +0.05928500 
 
 90 
 
 +0.014494077 
 
 -1.7367344 -0.0001389 -0.15833153 
 
 -0.03170097 
 
 120 
 
 +0.016373004 
 
 -1.5075995 -0.8612810 -0.07497804 
 
 -0.06561473 
 
 150 
 
 +0.010606704 
 
 -0.8754611 -1.5022509 -0.00439303 
 
 -0.01264209 
 
 180 
 
 +0.001655167 
 
 -0.0033659 -1.7361351 -0.01363832 
 
 +0.07459502 
 
 210 
 
 -0.004957609 
 
 +0.8630786 -1.5036464 -0.09371241 
 
 +0.11214236 
 
 240 
 
 -0.006923567 
 
 + 1.5048615 -0.8818403 -0.17035373 
 
 +0.06358327 
 
 270 
 
 -0.006225129 
 
 + 1.7745751 -0.0173138 -0.17092102 
 
 -0.02829602 
 
 300 
 
 -0.006771025 
 
 + 1.5786949 +0.8963895 -0.08955312 
 
 -0.07388837 
 
 330 
 
 -0.009754749 
 
 +0.9169364 +1.5955426 -0.00682218 
 
 -0.01861936 
 
 2i 
 
 -0.001026363 
 
 +0.0253885 +0.1094311 -0.53777107 
 
 +0.14072305 
 
 2i 
 
 -0.001021537 
 
 +0.0255735 +0.1094965 -0.53769898 
 
 +0.14062119 
 
 
 sin <p \A\ W + cos (f 
 
 Bo (c> = - 0.0000000083. 
 
 
130 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 u log coeff. 
 
 [de/dt] w = + 5503.0089 m' p 3.7406002 
 
 [dx/dt\ = [dirfdt] = +1409586.4 TO' p 6.1490917 
 
 [dp/dt}^ = + 30388.832 TO' p 4.4827140 
 
 [dq/dt] m = - 116164.73 TO' n 5.0650743 
 
 = +4584354.6 TO' p 6.6612782 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF 
 
 [de/(ft] M = + 0.013483339 
 
 [dx/dt] = [dTr/dt] w = + 3.4537341 
 
 [dp/dt] w = + 0.074457966 
 
 [dq/dt]oo = - 0.28462399 
 
 [dL/dt] w = +11.232473 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 TO' 
 
 [de/dt] m 
 
 + 0.01344 
 
 +o'.01348 
 
 + 0.0134833 
 
 e[dw/dt] 
 
 + 0.05796 
 
 +0.05792 
 
 + 0.0579231 
 
 [dpldt] M 
 
 + 0.07450 
 
 +0.07446 
 
 + 0.0744580 
 
 [dq/dt]oo 
 
 - 0.28454 
 
 -0.28462 
 
 - 0.2846240 
 
 [dL/dt] w 
 
 + 11.2298 
 
 
 + 11.232473 
 
 NOTES. 
 
 This computation is of special interest because, notwithstanding the low eccen- 
 tricities of both the Earth and Venus, the perturbing function is but slowly con- 
 vergent for this case. In 1893, the computation was effected by MR. R. T. A. INNES 
 who employed HILL'S second modification of GAUSS'S method, using in the work 
 manuscript tables prepared by himself. (See M. N., Vol. LIII, No. 6. The tables 
 were afterward published in M. N., Vol. LIV, No. 5.) The values of [dp/dt] 00 and 
 [dq/dt]oo were also obtained by HILL in the "New Theory," pages 511 and 512. 
 
 As the results of INNES differed considerably in some cases from those hitherto 
 obtained, particularly in the case of [de/dt] o, which agreed to the first two figures 
 only with the values of LEVERRIER and NEWCOMB, and in the case of [dq/dt] QO , which 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 131 
 
 differed in the fourth figure from the value given by HILL, and in order to make the 
 comparison more exact, the roots in the present paper were computed by the formulas 
 of the second method, their values being afterward verified by those of the first. It 
 was found that the functions tabulated by MR. INNES are substantially correct, 
 though the last two significant figures of all functions from R to the end usually 
 differ, doubtless owing to the inaccuracy of the tables employed by MR. INNES. 
 Using the values as given by him, all of this part of his computation was duplicated, 
 with the result that an error was found in his value of [de/dt] 00 , while for [dq/dt] a 
 and the other coefficients his values were found to be substantially correct. The 
 various values here referred to are as follows: 
 
 Innes. Hill. 
 
 [de/dt]oo 
 e[dw/dt]m 
 [dp/dflw, 
 [dq/dtloo 
 
 + 0.013476* 
 + 0.057915 
 + 0.074459 
 - 0.284623 
 + 11.232490 
 
 a 
 +0.0744329 
 -0.2845280 
 
 It will be noticed that the results of INNES are in almost exact accordance with 
 those here given. The disagreement of the value of [dqldt] 00 as derived by GAUSS'S 
 method with that found by HILL is, however, a more serious matter, and is almost 
 the sole cause of the considerable disagreement of the values of this variation in the 
 complete perturbations of the Earth's orbit, the values of [dqldt] OQ from the action 
 of all of the other planets except Venus agreeing with those obtained by HILL very 
 exactly. Using the values tabulated on page 510 of the "New Theory " and the 
 formulas of page 511, I have duplicated the computation by HILL'S methods and 
 find the same results as he obtained. It is to be noticed that the theory of the motion 
 of the ecliptic here given by HILL was to serve a temporary purpose only, the numerical 
 values of the coefficients stated by LEVERRIER in the Annales, Vol. II, pages 94 to 96, 
 being employed without a re-computation of them. 
 
 * The uncorrected value was + 0".013156. 
 
132 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MARS ON THE EARTH. 
 
 E 
 
 A 
 
 B COS e 
 
 /; -in 
 
 g 
 
 h 
 
 
 
 3.12005845 
 
 -0.6857946 
 
 + 1.1901000 
 
 0.028604007 
 
 2.3106194 
 
 22.5 
 
 3.04885416 
 
 -1.0809801 
 
 +0.7480691 
 
 0.011301685 
 
 2.3059358 
 
 45 
 
 3.01959381 
 
 -1.2762880 
 
 +0.1890610 
 
 0.000721878 
 
 2.3030821 
 
 67.5 
 
 3.03677529 
 
 -1.2419833 
 
 -0.4018201 
 
 0.003260792 
 
 2.3034537 
 
 90 
 
 3.09782583 
 
 -0.9832901 
 
 -0.9346186 
 
 0.017641228 
 
 2.3070230 
 
 112.5 
 
 3.19346899 
 
 -0.5395909 
 
 -1.3282200 
 
 0.035628705 
 
 2.3122967 
 
 135 
 
 3.30912609 
 
 +0.0215645 
 
 -1.5227032 
 
 0.046826348 
 
 2.3167796 
 
 157.5 
 
 3.42714623' 
 
 +0.6147456 
 
 -1.4884581 
 
 0.044743825 
 
 2.3179851 
 
 180 
 
 3.52951891 
 
 + 1.1496460 
 
 -1.2306997 
 
 0.030588915 
 
 2.3148501 
 
 202.5 
 
 3.60064084 
 
 + 1.5448315 
 
 -0.7886687 
 
 0.012561720 
 
 2.3089820 
 
 225 
 
 3.62970226 
 
 + 1.7401394 
 
 -0.2296607 
 
 0.001065206 
 
 2.3043625 
 
 247.5 
 
 3.61232195 
 
 + 1.7058351 
 
 +0.3612206 
 
 0.002635147 
 
 2.3044350 
 
 270 
 
 3.55118902 
 
 + 1.4471417 
 
 +0.8940191 
 
 0.016141855 
 
 2.3088270 
 
 292.5 
 
 3.45562818 
 
 + 1.0034423 
 
 + 1.2876203 
 
 0.033483868 
 
 2.3139342 
 
 315 
 
 3.34017005 
 
 +0.4422870 
 
 + 1.4821032 
 
 0.044362582 
 
 2.3163153 
 
 337.5 
 
 3.22234876 
 
 -0.1508942 
 
 + 1.4478582 
 
 0.042336214 
 
 2.3148421 
 
 S, 
 
 26.59718442* 
 
 + 1.8554059f 
 
 -0.16239891 
 
 0.185952019 
 
 18.4818589 
 
 2 2 
 
 26.59718440 
 
 + 1.8554060 
 
 -0.1623987 
 
 0.185951956 
 
 18.4818645 
 
 * 8a s + 4a 8 e' + 8[a' ! - 2kaa'ee' cos A'] = 26.59718442. 
 t 8[a'V - kaa'e cos K] = + 1.8554056. 
 t - Sk'aa' cos v ' e sin A" = - 0.1623983. 
 
E 
 
 OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 ACTION OF MAES ON THK EARTH. 
 
 G G' G" 
 
 133 
 
 
 22.5 
 45 
 67.5 
 90 
 112.5 
 135 
 157.5 
 180 
 202.5 
 225 
 247.5 
 270 
 292.5 
 315 
 337.5 
 
 2, 
 
 ?2 
 
 0.7892434 
 0.7227227 
 0.6963160 
 0.7131259 
 0.7706071 
 0.8609765 
 0.9721508 
 . 1.0889654 
 1.1944730 
 1.2714631 
 1.3051440 
 1.2876912 
 1.2221663 
 1.1214982 
 1.0036591 
 0.8873110 
 
 7.9537596 
 7.9537539 
 
 2.3024091 0.8127396 0.0152860 36 42 22.96 
 2.3028298 0.7325284 0.0066997 34 27 16.75 
 2.3028870 0.6969608 0.0004498 33 23 4.72 
 2.3025627 0.7159948 0.0019779 33 55 44.11 
 2.3020189 0.7853689 0.0097577 35 54 24.65 
 2.3015508 0.8891330 0.0174106 38 41 59.61 
 2.3014739 1.0076482 0.0201918 41 42 38.47 
 2.3019609 1.1223087 0.0173190 44 30 20.12 
 2.3028661 1.2173683 0.0109112 46 46 8.30 
 2.3036994 1.2810025 0.0042567 48 15 58.20 
 2.3038996 1.3059610 0.0003540 48 50 43.66 
 2.3033085 1.2897047 0.0008871 48 27 7.65 
 2.3023363 1.2343370 0.0056800 47 8 13.64 
 2.3016065 1.1465149 0.0126889 45 3 3.25 
 2.3014626 1.0370980 0.0185863 42 25 11.06 
 2.3018396 0.9202987 0.0199852 39 31 20.00 
 
 18.4193535 8.0974817 0.0812168 332 52 47.46 
 18.4193582 8.0974855 0.0812251 332 52 49.69 
 
 ACTION OF MAKS ON THE EARTH. 
 
 E 
 
 log A' 
 
 log L ' 
 
 log N a 
 
 log AT 
 
 logP 
 
 logQ 
 
 
 
 0.14793515 
 
 0.46562072 
 
 0.39121913 
 
 9.5856600 
 
 9.3211682 
 
 9.6118228 
 
 22.5 
 
 0.12867513 
 
 0.44104101 
 
 0.36395436 
 
 9.5698260 
 
 9.2838199 
 
 9.5702568 
 
 45 
 
 0.12011431 
 
 0.43007010 
 
 0.35176641 
 
 9.5662159 
 
 9.2715711 
 
 9.5556249 
 
 67.5 
 
 0.12442259 
 
 0.43559482 
 
 0.35790544 
 
 9.5749532 
 
 9.2853792 
 
 9.5702743 
 
 90 
 
 0.14089818 
 
 0.45665643 
 
 0.38128232 
 
 9.5949794 
 
 9.3237440 
 
 9.6123158 
 
 112.5 
 
 0.16647531 
 
 0.48915019 
 
 0.41726416 
 
 9.6240919 
 
 9.3826551 
 
 9.6760625 
 
 135 
 
 0.19735020 
 
 0.52805532 
 
 0.46020914 
 
 9.6588907 
 
 9.4553466 
 
 9.7533001 
 
 157.5 
 
 0.22941061 
 
 0.56809820 
 
 0.50425204 
 
 9.6947360 
 
 9.5321279 
 
 9.8336349 
 
 180 
 
 0.25803549 
 
 0.60355500 
 
 0.54311538 
 
 9.7259997 
 
 9.6009117 
 
 9.9047936 
 
 202.5 
 
 0.27840311 
 
 0.62861983 
 
 0.57051120 
 
 9.7469169 
 
 9.6490816 
 
 9.9542006 
 
 225 
 
 0.28660968 
 
 0.63868139 
 
 0.58149048 
 
 9.7530543 
 
 9.6666751 
 
 9.9720145 
 
 247.5 
 
 0.28101715 
 
 0.63182707 
 
 0.57401211 
 
 9.7427949 
 
 9.6495832 
 
 9.9542876 
 
 270 
 
 0.26293394 
 
 0.60959543 
 
 0.54972342 
 
 9.7180756 
 
 9.6011932 
 
 9.9045601 
 
 292.5 
 
 0.23607945 
 
 0.57638319 
 
 0.51334444 
 
 9.683858o 
 
 9.5314041 
 
 9.8327841 
 
 315 
 
 0.20515940 
 
 0.53784176 
 
 0.47098807 
 
 9.6465515 
 
 9.4533990 
 
 9.7520425 
 
 337.5 
 
 0.17455362 
 
 0.49936282 
 
 0.42855178 
 
 9.6122456 
 
 9.3799495 
 
 9.6749680 
 
 V 
 1 
 
 1.61903635 
 
 4.27007615 
 
 3.72979435 
 
 7.2494269 
 
 5.6940086 
 
 8.0664741 
 
 2 2 
 
 1.61903696 
 
 4.27007713 
 
 3.72979553 
 
 7.2494229 
 
 5.6940004 
 
 8.0664686 
 
134 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MARS ON THE EARTH. 
 
 E 
 
 logF 
 
 Ji' 
 
 J 2 
 
 J, 
 
 Ft 
 
 
 
 9.6084099 
 
 2.3152378 
 
 +0.11029083 
 
 +0.061812671 
 
 -0.25644059 
 
 22.5 
 
 9.5687497 
 
 2.3059045 
 
 +0.06946998 
 
 +0.074883922 
 
 -0.16119255 
 
 45 
 
 9.5555233 
 
 2.2995328 
 
 +0.01805514 
 
 +0.077080482 
 
 -0.04073852 
 
 67.5 
 
 9.5698283 
 
 2.3016313 
 
 -0.03654213 
 
 +0.068068172 
 
 +0.08658346 
 
 90 
 
 9.6101298 
 
 2.3103260 
 
 -0.08632858 
 
 +0.049219187 
 
 +0.20138993 
 
 112.5 
 
 9.6721978 
 
 2.3186977 
 
 -0.12376087 
 
 +0.023403207 
 
 +0.28620250 
 
 135 
 
 9.7488617 
 
 2.3215901 
 
 -0.14288822 
 
 -0.005449606 
 
 +0.32810925 
 
 157.5 
 
 9.8298619 
 
 2.3181659 
 
 -0.14041555 
 
 -0.032946848 
 
 +0.32073022 
 
 180 
 
 9.9024333 
 
 2.3108630 
 
 -0.11642269 
 
 -0.054902532 
 
 +0.26518896 
 
 202.5 
 
 9.9532839 
 
 2.3034774 
 
 -0.07451078 
 
 -0.067974156 
 
 +0.16994090 
 
 225 
 
 9.9719383 
 
 2.2994233 
 
 -0.02127963 
 
 -0.070171605 
 
 +0.04948689 
 
 247.5 
 
 9.9540965 
 
 2.3004768 
 
 +0.03479475 
 
 -0.061160169 
 
 -0.07783516 
 
 270 
 
 9.9033316 
 
 2.3061702 
 
 +0.08485465 
 
 -0.042311544 
 
 -0.19264158 
 
 292.5 
 
 9.8300222 
 
 2.3139376 
 
 +0.12119708 
 
 -0.016495194 
 
 -0.27745411 
 
 315 
 
 9.7479652 
 
 2.3199983 
 
 +0.13850882 
 
 +0.012358484 
 
 -0.31936096 
 
 337.5 
 
 9.6705453 
 
 2.3208585 
 
 +0.13455789 
 
 +0.039856619 
 
 -0.31198183 
 
 s, 
 
 8.0485930 
 
 18.4831415* 
 
 -0.01520968 
 
 +0.027635537 
 
 +0.03499338 
 
 2 2 
 
 8.0485855 
 
 18.4831497 
 
 -0.01520963 
 
 +0.027635553 
 
 +0.03499343 
 
 *S,(y,' -G") = 18.4019247. 
 Zt(Ji' - G") = 18.4019246. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 135 
 
 ACTION OF MARS ON THE EARTH. 
 
 E 
 
 1000 X Ft 
 
 flo 
 
 -So 
 
 w. 
 
 RW 
 
 gw 
 
 
 
 - 5.019874 
 
 0.22207195 
 
 -0.00895625 
 
 +0.02403761 
 
 0.00000000 
 
 -0.00910901 
 
 22.5 
 
 - 1.341522 
 
 0.21055435 
 
 -0.00524961 
 
 +0.02748415 
 
 +0.08184381 
 
 -0.00533223 
 
 45 
 
 + 0.173061 
 
 0.20752103 
 
 -0.00112516 
 
 +0.02773157 
 
 +0.14850060 
 
 -0.00113867 
 
 67.5 
 
 - 1.404619 
 
 0.21281662 
 
 +0.00313244 
 
 +0.02500875 
 
 +0.19788699 
 
 +0.00315267 
 
 90 
 
 - 5.191525 
 
 0.22627677 
 
 +0.00726155 
 
 +0.01896286 
 
 +0.22627677 
 
 +0.00726155 
 
 112.5 
 
 - 9.004074 
 
 0.24749832 
 
 +0.01089521 
 
 +0.00882887 
 
 +0.22720045 
 
 +0.01082573 
 
 135 
 
 -10.631982 
 
 0.27522212 
 
 +0.01347761 
 
 -0.00609013 
 
 + 0.19233060 
 
 +0.01331965 
 
 157.5 
 
 - 9.129495 
 
 0.30647387 
 
 +0.01430898 
 
 -0.02537639 
 
 +0.11549298 
 
 +0.01409065 
 
 180 
 
 - 5.368216 
 
 0.33606721 
 
 +0.01279803 
 
 -0.04599728 
 
 0.00000000 
 
 +0.01258694 
 
 202.5 
 
 - 1.527816 
 
 0.35740451 
 
 +0.00883760 
 
 -0.06172287 
 
 -0.13468590 
 
 +0.00870276 
 
 225 
 
 + 0.177177 
 
 0.36477534 
 
 +0.00302208 
 
 -0.06569865 
 
 -0.25491209 
 
 +0.00298666 
 
 247.5 
 
 1.210722 
 
 0.35578971 
 
 -0.00342959 
 
 -0.05556595 
 
 -0.32661056 
 
 -0.00340772 
 
 270 
 
 - 4.837361 
 
 0.33265813 
 
 -0.00898150 
 
 -0.03579916 
 
 -0.33265813 
 
 -0.00898150 
 
 292.5 
 
 - 8.543560 
 
 0.30146830 
 
 -0.01237466 
 
 -0.01405700 
 
 -0.28031952 
 
 -0.01245460 
 
 315 
 
 -10.135232 
 
 0.26942963 
 
 -0.01319021 
 
 +0.00403826 
 
 -0.19280199 
 
 -0.01334851 
 
 337.5 
 
 - 8.672134 
 
 0.24201044 
 
 -0.01181400 
 
 +0.01658570 
 
 -0.09407098 
 
 -0.01199994 
 
 2, 
 
 -40.833952 
 
 2.23402218 
 
 +0.00430615 
 
 -0.07881492 
 
 -0.21326424 
 
 +0.00357711 
 
 2s 
 
 -40.833942 
 
 2.23401612 
 
 +0.00430637 
 
 -0.07881474 
 
 -0.21326273 
 
 +0.00357732 
 
 sin 
 
 <p l-Ai (a) + cos 
 
 V Bo> = + 
 
 0.0000000050. 
 
 
 
 
136 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MARS ON THE EARTH. 
 
 E 
 
 Ro sin v + 
 (cos v + cos E) So 
 
 ( - sec' if + ij sin vS a 
 
 Wo COS (!)+) 
 
 Wo sin (v + T ) 
 
 -2 a /J 
 
 
 
 -0.01791249 
 
 -0.22207195 
 
 -0.00432317 
 
 +0.02364565 
 
 -0.43669510 
 
 22.5 
 
 +0.07214538 
 
 -0.19805102 
 
 -0.01506202 
 
 +0.02298943 
 
 -0.41458381 
 
 45 
 
 +0.14689806 
 
 -0.14657925 
 
 -0.02300258 
 
 +0.01548939 
 
 -0.41012014 
 
 67.5 
 
 +0.20021147 
 
 -0.07256855 
 
 -0.02452836 
 
 +0.00487823 
 
 -0.42290136 
 
 90 
 
 +0.22612318 
 
 +0.01831801 
 
 -0.01859384 
 
 -0.00372284 
 
 -0.45255354 
 
 112.5 
 
 +0.21867465 
 
 +0.11830141 
 
 -0.00734127 
 
 -0.00490454 
 
 -0.49817351 
 
 135 
 
 +0.17313163 
 
 +0.21584084 
 
 +0.00340234 
 
 +0.00505112 
 
 -0.55697199 
 
 157.5 
 
 +0.08900258 
 
 +0.29475425 
 
 +0.00517812 
 
 +0.02484247 
 
 -0.62244500 
 
 180 
 
 -0.02559607 
 
 +0.33606721 
 
 -0.00827262 
 
 +0.04524724 
 
 -0.68340683 
 
 202.5 
 
 -0.15101812 
 
 +0.32435071 
 
 -0.03316015 
 
 +0.05205879 
 
 -0.72588458 
 
 225 
 
 -0.25917515 
 
 +0.25670929 
 
 -0.05360955 
 
 +0.03797800 
 
 -0.73820246 
 
 247.5 
 
 -0.32389097 
 
 +0.14753236 
 
 -0.05413656 
 
 +0.01252224 
 
 -0.71614631 
 
 270 
 
 -0.33246073 
 
 +0.02354205 
 
 -0.03531845 
 
 -0.00584698 
 
 -0.66531623 
 
 292.5 
 
 -0.28957300 
 
 -0.08808428 
 
 -0.01192486 
 
 -0.00744291 
 
 -0.59906678 
 
 315 
 
 -0.21131668 
 
 -0.16946344 
 
 +0.00233483 
 
 +0.00329486 
 
 -0.53246901 
 
 337.5 
 
 -0.11585773 
 
 -0.21387153 
 
 +0.00359251 
 
 +0.01619196 
 
 -0.47652121 
 
 s, 
 
 -0.30030825 
 
 +0.31236276 
 
 -0.13738304 
 
 +0.12113644 
 
 -4.47573530 
 
 2 2 
 
 -0.30030574 
 
 +0.31236335 
 
 -0.13738259 
 
 +0.12113567 
 
 -4.47572256 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 [dxldt] m = 
 
 = - 48641.893m' 
 = +3016769.1 m' 
 
 log coeff. 
 n 4.6870105 
 p 6.4795421 
 
 = + 
 [dq/dt] w = 
 
 = - 724628.93 
 
 19626.398 m' p 4.2928406 
 22258.695 m' n 4.3474997 
 
 m' n 5.8601 157 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'. 
 
 = -0.015723904 
 [d x /dt] m = [d7r/d<]oo = +0.97519611 
 
 [dpfdt] M = +0.0063443986 
 [dq/dt] m = -0.0071953108 
 [dL/dt] m = -0.23424243 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 137 
 
 Leverrier. 
 
 [<fe/cft]oo -0.01573 
 
 [dw/dt] m +0.9754 
 
 [dp/dt] w +0.00635 
 
 [dq/dtlw -0.00721 
 
 [dL/dt] w -0.2337 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Innes. Hall. Newcomb. Method of Gauss. 
 
 -0.015722 -0.0157232 -0.01572 -0.0157239 
 
 +0.975224 +0.9751387 +0.9755 +0.9751961 
 
 +0.0063401 +0.0063444 +0.00634 +0.0063444 
 
 -0.0071898 -0.0071952 -0.00719 -0.0071953 
 
 -0.23469 -0.2342416 -0.2342424 
 
 NOTES. 
 
 In the "New Theory," Page 511, HILL points out that the convergence of the 
 expansion of the perturbing function is slow in this case, the terms of the fifth order 
 in the inclinations and eccentricities amounting to one per cent, of those of the first 
 order. He stated that a computation by GAUSS'S method would be very desirable 
 and consequently this was effected by DR. ASAPH HALL, JR., in July, 1891 (A. J. 
 No. 244), and by INNES in November, 1891 (M. N., Vol. LII, Nos. 2 and 7). HALL'S 
 computation is the first application of GAUSS'S method made after the publication 
 of HILL'S memoir. 
 
 Both HALL and INNES employed the values of the elements stated by LEVERRIER; 
 Hall divided the orbit of the Earth into twelve parts and INNES into sixteen. The 
 values of [dpldt] 00 and [dq/dt] Q given by the latter were however in error owing to a 
 misprint that occurred in HILL'S original paper in the value of J 3 ] in M. N., Vol. LII, 
 No. 7, INNES pointed out this error but did not re-compute the variations. 
 
 The final results of the present paper were printed in A. J., No. 518, but the 
 values there given are all slightly incorrect owing to errors in some of the preliminary 
 constants, which remained undetected even in the duplication. Upon devising new 
 test equations these were always applied to all computations previously made and in 
 this way the errors affecting practically every figure of the present computation were 
 discovered. The work was then both repeated and duplicated so that it is hardly 
 possible that any errors can yet remain in it. 
 
 The latter part of INNES' computation was also duplicated, the values of J 3 , W Q , 
 [dp/dt] and [dq/dt] being freed from the errors referred to by him. It is these corrected 
 values which are given above. 
 
 It will be noticed that the agreement of the results here given with those of 
 HALL is very exact notwithstanding the difference of the original elements used in 
 the computation. The divergences from those of INNES are more considerable, 
 probably because the latter computer did not employ the accurate tables of HILL. 
 
138 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 The values obtained in the "New Theory " for the motion of the plane of the 
 ecliptic are, 
 
 [dp/dt] m = +0.0063362 
 
 [dq/dt]^ = -0.0072112 
 
 ACTION OF JUPITER ON THE EARTH. 
 
 E 
 
 A 
 
 B cos t 
 
 B sin t 
 
 g 
 
 
 
 28.04923872 
 
 + 1.4444306 
 
 +5.1077212 
 
 1.6444314 
 
 30 
 
 27.80097551 
 
 -1.1738070 
 
 +4.4818820 
 
 1.2661417 
 
 60 
 
 27.62449781 
 
 -3.1281533 
 
 +2.6317812 
 
 0.4365772 
 
 90 
 
 27.56719541 
 
 -3.8949449 
 
 +0.0531514 
 
 0.0001781 
 
 120 
 
 27.64447424 
 
 -3.2687173 
 
 -2.5630644 
 
 0.4140764 
 
 150 
 
 27.83557589 
 
 -1.4172691 
 
 -4.5158555 
 
 1.2854086 
 
 180 
 
 28.08919180 
 
 + 1.1633052 
 
 -5.2819683 
 
 1.7585429 
 
 210 
 
 28.33731437 
 
 +3.7815432 
 
 -4.6561285 
 
 1.3665053 
 
 240 
 
 28.51351085 
 
 +5.7358901 
 
 -2.8060283 
 
 0.4963014 
 
 270 
 
 28.57067253 
 
 +6.5026807 
 
 -0.2273984 
 
 0.0032594 
 
 300 
 
 28.49353432 
 
 +5.8764527 
 
 +2.3888176 
 
 0.3596892 
 
 330 
 
 28.30271399 
 
 +4.0250047 
 
 +4.3416085 
 
 1.1881266 
 
 2: 
 
 168.41444773* 
 
 +7.82320801 
 
 -0.52274091 
 
 5.1096185 
 
 Si 
 
 168.41444770 
 
 +7.8232076 
 
 -0.5227406 
 
 5.1096197 
 
 E 
 
 I 
 
 G 
 
 G' 
 
 G" 
 
 O 
 
 
 
 
 
 
 
 0.977739 
 
 27.0061276 
 
 1.0387013 
 
 0.0586223 
 
 30 
 
 0.729882 
 
 27.0062768 
 
 0.7909419 
 
 0.0592752 
 
 60 
 
 0.554312 
 
 27.0065429 
 
 0.5826671 
 
 0.0277442 
 
 90 
 
 0.497498 
 
 27.0066658 
 
 0.4975109 
 
 0.0000133 
 
 120 
 
 0.574342 
 
 27.0065199 
 
 0.6004570 
 
 0.0255347 
 
 150 
 
 0.764478 
 
 27.0062522 
 
 0.8240511 
 
 0.0577594 
 
 180 
 
 1.017523 
 
 27.0061314 
 
 1.0803044 
 
 0.0602760 
 
 210 
 
 1.266039 
 
 27.0062772 
 
 1.3067275 
 
 0.0387224 
 
 240 
 
 1.443204 
 
 27.0065561 
 
 1.4565395 
 
 0.0126169 
 
 270 
 
 1.500952 
 
 27.0066843 
 
 1.5010367 
 
 0.0000804 
 
 300 
 
 1.423446 
 
 27.0065354 
 
 1.4332595 
 
 0.0092925 
 
 330 
 
 1.231720 
 
 27.0062551 
 
 1.2681196 
 
 0.0346927 
 
 Si 
 
 5.990566 
 
 162.0384133 
 
 6.1919288 
 
 0.1940866 
 
 2 2 
 
 5.990569 
 
 162.0384114 
 
 6.1883877 
 
 0.1905434 
 
 
 * 6o 2 + 3aV + 6[a' 2 - 
 
 Zkaa'ee' cos K] = 168.41444773, 
 
 
 f 6[a' e' kaa'e r.os K] 
 
 = + 7.8232074. 
 
 
 
 - 
 
 | Gk'aa' cos <f>' e sin 
 
 K' - - 0.5227409. 
 
 
 
 
 h 
 
 
 27.008467 
 
 
 27.008061 
 
 
 27.007154 
 
 
 27.006666 
 
 
 27.007100 
 
 
 27.008066 
 
 
 27.008637 
 
 
 27.008243 
 
 
 27.007275 
 
 
 27.006689 
 
 
 27.007056 
 
 
 27.007962 
 
 
 162.045689 
 
 
 162.045687 
 
 O 
 
 e 
 
 / // 
 
 11 
 
 36 58.64 
 
 10 
 
 12 32.08 
 
 8 
 
 38 31.93 
 
 7 
 
 48 2.72 
 
 8 
 
 45 10.87 
 
 10 
 
 23 57.24 
 
 11 
 
 50 45.30 
 
 12 
 
 53 16.35 
 
 13 
 
 29 4.47 
 
 13 
 
 38 11.24 
 
 13 
 
 21 37.74 
 
 12 
 
 40 46.32 
 
 67 
 
 42 8.95 
 
 67 
 
 36 45.94 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 139 
 
 
 
 ACTION OF JUPITER ON THE EARTH. 
 
 
 E 
 
 logtfo 
 
 log Lo' log N, log N log P 
 
 logQ 
 
 
 
 0.01351621 
 
 0.29098220 0.19630730 7.8502195 5.2763937 
 
 6.6141228 
 
 30 
 
 0.01041684 
 
 0.28686620 0.19168189 7.8490834 5.2711158 
 
 6.6083484 
 
 60 
 
 0.00744942 
 
 0.28292144 0.18724767 7.8522627 5.2713543 
 
 6.6075955 
 
 90 
 
 0.00606351 
 
 0.28107775 0.18517479 7.8588567 5.276992T 
 
 6.6125603 
 
 120 
 
 0.00764272 
 
 0.28317853 0.18753669 7.8670773 5.2864976 
 
 6.6227349 
 
 150 
 
 0.01081185 
 
 0.28739101 0.19227173 7.8747484 5.2973550 
 
 6.6346279 
 
 180 
 
 0.01406123 
 
 0.29170555 0.19712004 7.8798617 5.3067059 
 
 6.6445510 
 
 210 
 
 0.01667410 
 
 0.29517157 0.20101379 7.8810685 5.3120661 
 
 6.6499953 
 
 240 
 
 0.01827428 
 
 0.29729273 0.20339625 7.8780194 5.3119679 
 
 6.6497435 
 
 270 
 
 0.01869373 
 
 0.29784858 0.20402051 7.8714850 5.3063886 
 
 6.6440330 
 
 300 
 
 0.01793523 
 
 0.29684340 0.20289160 7.8631935 5.2968004 
 
 6.6344668 
 
 330 
 
 0.01613325 
 
 0.29445437 0.20020816 7.8553923 5.2858029 
 
 6.6235786 
 
 Si 
 
 0.07887909 
 
 1.74292385 1.17449955 7.1906339 1.7497196 
 
 9.7732144 
 
 2 2 
 
 0.07879328 
 
 1.74280948 1.17437087 7.1906342 1.7497203 
 
 9.7731434 
 
 E 
 
 logF 
 
 Ji' Ji J, 
 
 F 2 
 
 
 
 6.6129514 
 
 27.064741582 +0.24634548 +0.02063093 
 
 -6.6623169 
 
 30 
 
 6.6071626 
 
 27.061450021 +0.21012324 +0.32748544 
 
 -5.8459973 
 
 60 
 
 6.6070396 
 
 27.022786983 +0.12128740 +0.54805936 
 
 -3.4327952 
 
 90 
 
 6.6125600 
 
 26.992014082 +0.00316191 +0.62325071 
 
 -0.0693287 
 
 120 
 
 6.6222233 
 
 27.021540830 -0.11740027 +0.53291210 
 
 +3.3431631 
 
 150 
 
 6.6334726 
 
 27.060747670 -0.21222668 +0.30124976 
 
 + 5.8903093 
 
 180 
 
 6.6433468 
 
 27.066395282 -0.25546365 -0.00966336 
 
 +6.8895984 
 
 210 
 
 6.6492222 
 
 27.041108427 -0.23112984 -0.31651859 
 
 +6.0732768 
 
 240 
 
 6.6494917 
 
 27.008005002 -0.14158677 -0.53709407 
 
 +3.6600761 
 
 270 
 
 6.6440314 
 
 26.992057684 -0.01087596 -0.61228634 
 
 +0.2966099 
 
 300 
 
 6.6342813 
 
 27.004932957 +0.12157000 -0.52194700 
 
 -3.1158827 
 
 330 
 
 6.6228858 
 
 27.037481542 +0.21569955 -0.29028297 
 
 -5.6630276 
 
 Zi 
 
 9.7693340 
 
 162.188402634* -0.02524781 +0.03289796 
 
 +0.6818428 
 
 S 2 
 
 9.7693345 
 
 162.184859426 -0.02524778 +0.03289801 
 
 +0.6818424 
 
 
 * s,(J,' - G") = 161 
 
 .994316084. 
 
 
 
 zl/i' - G") = 161 
 
 ,994316076. 
 
 
140 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 
 ACTION OF JUPITKK ON THE EARTH. 
 
 
 E 
 
 F, 
 
 1000 X fio 1000 X So 1000 X W 1000 X B (n) 
 
 1000 X S (n) 
 
 
 
 -0.14994500 
 
 3.5906058 -0.02485742 +0.00562843 0.0000000 
 
 -0.02528143 
 
 30 
 
 -0.11198359 
 
 3.5669877 -0.02409495 +0.13045170 +1.8097792 
 
 -0.02445007 
 
 60 
 
 -0.03628181 
 
 3.5812719 -0.01504685 +0.22107406 +3.1277000 
 
 -0.01517409 
 
 90 
 
 + 0.00006601 
 
 3.6318358 -0.00001620 +0.25540271 +3.6318358 
 
 -0.00001620 
 
 120 
 
 -0.04119166 
 
 3.7069905 +0.01547114 +0.22249821 +3.1836522 
 
 +0.01534248 
 
 150 
 
 -0.12070192 
 
 3.7863426 +0.02555514 +0.12714491 +1.8660682 
 
 +0.02518930 
 
 180 
 
 -0.16035007 
 
 3.8464128 +0.02722801 -0.00750003 0.0000000 
 
 +0.02677889 
 
 210 
 
 -0.12100025 
 
 3.8690539 +0.02153453 -0.14361296 -1.9068325 
 
 +0.02122624 
 
 240 
 
 -0.04149417 
 
 3.8484336 +0.01189802 -0.24048156 -3.3051260 
 
 +0.01179908 
 
 270 
 
 +0.00005467 
 
 3.7925083 +0.00121404 -0.26976404 -3.7925083 
 
 +0.00121404 
 
 300 
 
 -0.03599894 
 
 3.7183323 -0.00934064 -0.22557067 -3.2473996 
 
 -0.00941964 
 
 330 
 
 -0.11169659 
 
 3.6453213 -0.01884026 -0.12397381 -1.8495240 
 
 -0.01911793 
 
 s, 
 
 -0.46526165 
 
 22.2920469 +0.00535226 -0.02435156 -0.2411734 
 
 +0.00404529 
 
 S 2 
 
 -0.46526167 
 
 22.2920496 +0.00535230 -0.02435149 -0.2411816 
 
 +0.00404538 
 
 E 
 
 SSSL. 
 
 1000 X|-#i> cos v 
 .. 1000 XW a cos (+) 1000XW sin(y+ir) 
 
 OJ / f \ ' r 1 
 
 + ( sec- if + 1 1 sin r<So 
 \o / J 
 
 1000X-2-Ro 
 a 
 
 
 
 -0.0497148 
 
 -3.5906058 -0.00101227 +0.00553665 
 
 - 7.0607738 
 
 30 
 
 + 1.7678935 
 
 -3.0981989 -0.08531744 +0.09868428 
 
 - 7.0303590 
 
 60 
 
 +3.1124039 
 
 -1.7713812 -0.20927556 +0.07125641 
 
 - 7.1024820 
 
 90 
 
 +3.6313250 
 
 +0.0608776 -0.25043240 -0.05014132 
 
 - 7.2636717 
 
 120 
 
 +3.1675401 
 
 + 1.9264211 -0.16743708 -0.14652740 
 
 - 7.4761534 
 
 150 
 
 + 1.8214372 
 
 +3.3200889 -0.04173395 -0.12010043 
 
 - 7.6826719 
 
 180 
 
 -0.0544560 
 
 +3.8464128 -0.00134888 +0.00737774 
 
 - 7.8218420 
 
 210 
 
 -1.9439518 
 
 +3.3453092 -0.09208972 +0.11020057 
 
 - 7.8505000 
 
 240 
 
 -3.3167069 
 
 + 1.9516989 -0.22529979 +0.08409148 
 
 - 7.7614089 
 
 270 
 
 -3.7919952 
 
 +0.0611766 -0.26614166 -0.04405982 
 
 - 7.5850167 
 
 300 
 
 -3.2561646 
 
 -1.7957525 -0.17399247 -0.14355750 
 
 - 7.3742992 
 
 330 
 
 -1.8818157 
 
 -3.1224540 -0.04265147 -0.11640599 
 
 - 7.1847533 
 
 Zi 
 
 -0.3970983 
 
 +0.5667933 -0.77836605 -0.12182259 
 
 -44.5969593 
 
 2 2 
 
 -0.3971070 
 
 +0.5667994 -0.77836664 -0.12182271 
 
 -44.5969726 
 
 
 sin <f %A i ( *> + cos if 
 
 Bo w = - 0.0000000000093. 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 141 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 n log coeff. 
 
 [de/dilw = - 85.760340 m' n 1.9332865 
 
 [dx/dt]ao = [dv/dilw = +7298.7450 TO' p 3.8632482 
 
 [dp/dt] m = - 26.316855 TO' n 1.4202340 
 
 [dq/dt] m = - 168.14734 TO' n 2.2256900 
 
 [dL/dt] w = -9631.7202 TO' n 3.9837038 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 [de/dt] m = -0^081841849 
 [d x /dt] w = [dTr!dt] w = +6.9652565 
 
 [dp/dt]w = -0.025114405 
 
 [dq/dt] m = -0.16046446 
 
 [dLfdt] w = -9.1916336 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcoml). Method of Gauss. 
 
 [de/dt]oo 
 
 -0.08182 
 
 -0.08182 
 
 -0.0818418 
 
 e[dir/dl] 00 
 
 +0.11679 
 
 +0.11677 
 
 +0.1168153 
 
 [dp/dtlw 
 
 -0.02501 
 
 -0.02511 
 
 -0.0251144 
 
 [dq/dt] m 
 
 -0.16041 
 
 -0.16047 
 
 -0.1604644 
 
 [dL/dt] m 
 
 -9.1916 
 
 
 -9.1916336 
 
 NOTES. 
 
 The very close agreement of the sums toward the end of this computation is 
 owing to the circularity of the two orbits and to their small mutual inclination. It is 
 evident that a division into eight parts would have been sufficient, while the errors 
 arising from a division into only six parts would have been almost inappreciable. 
 
 In this, as in several other cases, the divergence from the last figure of NEW- 
 COMB'S results is rather larger than was to have been expected. The values stated 
 by NEWCOMB were computed to one more significant figure than was published to 
 insure the accuracy of the final figure given. The uncertainty of this figure is evi- 
 dently due to neglected terms in the series employed by LEVERRIER and NEWCOMB. 
 In the present case we obtain for [deldt] 00 , 
 
 Computed from the six even points of division 0".0818428 
 
 Computed from the six odd points of division .0818409, 
 
142 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 and the difference between any two corresponding values for any other coefficient is 
 even less than this. 
 
 The values of the coefficients which define the motion of the plane of the ecliptic 
 are stated by HILL as follows: 
 
 [dp/dt]^ = -0.0251149 
 
 [dq/dt] w = -0.1604628 
 
 E A 
 
 92.9909218 
 
 30 92.7594069 
 
 60 92.3168471 
 
 90 91.7819295 
 
 120 91.2980364 
 
 150 90.9947748 
 
 180 90.9533006 
 
 210 91.1846750 
 
 240 91.6269534 
 
 270 92.1617302 
 
 300 92.6457640 
 
 330 92.9493070 
 
 Zj 551.8318232* 
 
 2 2 551.8318234 
 
 E I 
 
 +2.001263 
 
 30 +1.768419 
 
 60 +1.323475 
 
 90 +0.787549 
 
 120 +0.304998 
 
 150 +0.004086 
 
 180 -0.036365 
 
 210 +0.193697 
 
 240 +0.633602 
 
 270 +1.167320 
 
 300 +1.652663 
 
 330 +1.958598 
 
 21 +5.879636 
 
 2 2 +5.879669 
 
 t 6[a'V - fcao'e cos K] = + 29.6608842. 
 t - 6k'aa' cos <p' e sin K' = - 0.1707000. 
 
 ACTION OF SATURN ON THE EARTH. 
 
 B cos . 
 
 B sin t 
 
 d 
 
 +14.3293908 
 
 + 1.6679163 
 
 0.795517 
 
 + 12.2250618 
 
 +6.1218324 
 
 10.716773 
 
 + 8.1696352 
 
 +8.9277816 
 
 22.792314 
 
 + 3.2497638 
 
 +9.3339213 
 
 24.913194 
 
 1.2162762 
 
 +7.2314183 
 
 14.953684 
 
 - 4.0318145 
 
 +3.1836393 
 
 2.898337 
 
 - 4.4424293 
 
 -1.7248166 
 
 0.850721 
 
 - 2.3380978 
 
 -6.1787321 
 
 10.916915 
 
 + 1.7173265 
 
 -8.9846857 
 
 23.083782 
 
 + 6.6371970 
 
 -9.3908217 
 
 25.217872 
 
 + 11.1032373 
 
 -7.2883175 
 
 15.189927 
 
 + 13.9187769 
 
 -3.2405388 
 
 3.002863 
 
 +29.6608843f 
 
 -0.1707036t 
 
 77.665945 
 
 +29.6608873 
 
 -0.1706996 
 
 77.665954 
 
 G 
 
 G' 
 
 G" 
 
 90.703603 
 
 2.0057342 
 
 0.0043727 
 
 90.703703 
 
 1.8341642 
 
 0.0644170 
 
 90.704603 
 
 1.4944314 
 
 0.1681447 
 
 90.705370 
 
 1.0517493 
 
 0.2611465 
 
 90.705257 
 
 0.5874560 
 
 0.2806341 
 
 90.704380 
 
 0.1809884 
 
 0.1765509 
 
 90.703605 
 
 0.0803976 
 
 0.1166593 
 
 90.703690 
 
 0.4578844 
 
 0.2628566 
 
 90.704569 
 
 0.9146652 
 
 0.2782376 
 
 90.705348 
 
 1.3729264 
 
 0.2025015 
 
 90.705263 
 
 1.7502260 
 
 0.0956818 
 
 90.704378 
 
 1.9757281 
 
 0.0167564 
 
 544.226899 
 
 6.8329103 
 
 0.9437302 
 
 544.226868 
 
 6.8734407 
 
 0.9842289 
 
 'ee' cos K] = 551.8318229. 
 
 
 h 
 
 
 90.703702 
 
 
 90.705031 
 
 
 90.707414 
 
 
 90.708424 
 
 
 90.707081 
 
 
 90.704732 
 
 
 90.703708 
 
 
 90.705020 
 
 
 90.707394 
 
 
 90.708453 
 
 
 90.707143 
 
 
 90.704751 
 
 
 544.236442 
 
 
 544.236411 
 
 
 e 
 
 O 
 
 i n 
 
 8 
 
 33 39.757 
 
 8 
 
 18 56.347 
 
 7 
 
 46 25.450 
 
 6 
 
 53 59.891 
 
 5 
 
 36 19.643 
 
 3 
 
 35 46.006 
 
 2 
 
 40 11.401 
 
 5 
 
 6 24.385 
 
 6 
 
 34 30.181 
 
 7 
 
 33 52.413 
 
 8 
 
 11 49.887 
 
 8 
 
 31 21.103 
 
 39 
 
 22 56.319 
 
 40 
 
 20.145 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 143 
 
 ACTION OF 
 
 SATUKN ON 
 
 THE 
 
 EARTH. 
 
 
 
 
 E 
 
 
 logKo 
 
 
 log La 
 
 
 logA^o 
 
 
 log AT 
 
 
 logP 
 
 logQ 
 
 
 
 0.00730944 
 
 
 
 .28273527 
 
 
 
 .18703836 
 
 7. 
 
 0561506 
 
 
 3.4235951 
 
 5.2855436 
 
 30 
 
 0.00689441 
 
 
 
 .28218320 
 
 
 
 .18641769 
 
 7.0572862 
 
 3.4236026 
 
 5.2857705 
 
 60 
 
 0.00602140 
 
 
 
 .28102171 
 
 
 
 .18511178 
 
 7.0610565 
 
 3.4252108 
 
 5.2877346 
 
 90 
 
 0.00473960 
 
 
 
 .27931572 
 
 
 
 .18319348 
 
 7.0664172 
 
 3.4279697 
 
 5.2907291 
 
 120 
 
 0.00312447 
 
 
 
 .27716505 
 
 
 
 .18077485 
 
 7.0719166 
 
 3.4311334 
 
 5.2937173 
 
 150 
 
 0.00128428 
 
 
 
 .27471328 
 
 
 
 .17801715 
 
 7. 
 
 0761000 
 
 
 3.4338677 
 
 5.2956443 
 
 180 
 
 0.00070759 
 
 
 
 .27394462 
 
 
 
 .17715249 
 
 7.0778799 
 
 3.4354590 
 
 5.2968496 
 
 210 
 
 0.00259230 
 
 0.27645617 
 
 
 
 .17997757 
 
 7. 
 
 0767946 
 
 
 3.4354874 
 
 5.2978905 
 
 240 
 
 0.00430241 
 
 
 
 .27873368 
 
 0.18253896 
 
 7.0731165 
 
 3.4339314 
 
 5.2966961 
 
 270 
 
 0.00570043 
 
 
 
 .28059458 
 
 
 
 .18463152 
 
 7. 
 
 0677983 
 
 
 3.4311901 
 
 5.2938284 
 
 300 
 
 0. 
 
 00669846 
 
 0.28192252 
 
 0.18612461 
 
 7.0622487 
 
 3.4279906 
 
 5.2902830 
 
 330 
 
 0.00724349 
 
 0.28264754 
 
 
 
 .18693973 
 
 7.0579727 
 
 3.4252033 
 
 5.2872040 
 
 Zi 
 
 0.02816377 
 
 1 
 
 .67552285 
 
 1.09874105 
 
 2.4023686 
 
 6.1773201 
 
 1.7508241 
 
 2, 
 
 0.02845451 
 
 1 
 
 .67591049 
 
 1 
 
 .09917714 
 
 2.4023690 
 
 6.1773208 
 
 1.7510668 
 
 E 
 
 
 log V 
 
 
 J,' 
 
 
 /, 
 
 
 
 
 /i 
 
 Ft 
 
 
 
 
 5.2855175 
 
 
 90.7004946 
 
 +0.12827478 
 
 -0.7968027 
 
 - 8.486434 
 
 30 
 
 
 5.2853861 
 
 
 90.7506534 
 
 +0.29200243 
 
 + 1.2416991 
 
 -31.148158 
 
 60 
 
 
 5.2867312 
 
 
 90.7742572 
 
 +0.41637229 
 
 +2.9535293 
 
 -45.424968 
 
 90 
 
 
 5.2891707 
 
 
 90.7991434 
 
 +0.49225830 
 
 +3 
 
 .8800058 
 
 -47.491413 
 
 120 
 
 
 5.2920417 
 
 
 90.8302700 
 
 +0.45906255 
 
 +3.7728791 
 
 -36.793788 
 
 150 
 
 
 5.2945889 
 
 
 90.8046167 
 
 +0.26449000 
 
 +2 
 
 .6608537 
 
 -16.198502 
 
 180 
 
 
 5.2961519 
 
 
 90.8127812 
 
 -0.06165231 
 
 +0.8418949 
 
 + 8.775944 
 
 210 
 
 
 5.2963206 
 
 
 90.9507883 
 
 -0.39625124 
 
 -1 
 
 .1966086 
 
 +31.437674 
 
 240 
 
 
 5.2950355 
 
 
 90.8893132 
 
 -0.59035199 
 
 -2 
 
 .9084413 
 
 +45.714495 
 
 270 
 
 
 5.2926201 
 
 
 90.7428396 
 
 -0.56521208 
 
 -3 
 
 .8349184 
 
 +47.780934 
 
 300 
 
 
 5.2897120 
 
 
 90.6423828 
 
 -0.36124653 
 
 -3 
 
 .7277897 
 
 +37.083286 
 
 330 
 
 
 5.2871040 
 
 
 90.6419564 
 
 -0.09682848 
 
 -2 
 
 .6157627 
 
 + 16.488008 
 
 Si 
 
 
 1.7451898 
 
 
 544.6494990* 
 
 -0.00954121 
 
 +0.1352696 
 
 + 0.868535 
 
 2 2 
 
 
 1.7451904 
 
 
 544.6899978 
 
 -0.00954107 
 
 +0.1352689 
 
 + 0.868543 
 
 
 *2,(J 
 
 i' - G") = 
 
 543.7057688. 
 
 
 z s (J 
 
 i' - G") = 
 
 543.7057689. 
 
144 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 E 
 
 
 30 
 60 
 90 
 120 
 150 
 180 
 210 
 240 
 270 
 300 
 330 
 
 Si 
 E 
 
 ACTION OF SATURN ON THE EARTH. 
 F, 1000 X Ro 100000 X So 100000 X Wo 10000 X R M 100000 X S<> 
 
 -0.3555837 0.57362783 +0.02247623 -1.5471185 0.0000000 +0.0228596 
 -1.2681621 0.57437868 -0.26276188 +2.3619011 +2.9142195 -0.2666346 
 -1.2912829 0.57674453 -0.40344654 +5.6791610 +5.0370180 -0.4068583 
 -0.3956591 0.58127800 -0.31429014 +7.5403759 +5.8127800 -0.3142901 
 +0.5220384 0.58774791 -0.09358683 +7.4052797 +5.0477180 -0.0928086 
 +0.5361282 0.59426527 +0.08130415 +5.2579217 +2.9287882 +0.0801402 
 -0.3802586 0.59796705 +0.11726800 +1.6546186 0.0000000 +0.1153337 
 -1.3248861 0.59688219 +0.07295990 -2.4035193 -2.9416851 +0.0719154 
 -1.3648579 0.59139747 +0.07709136 -5.7742147 -5.0790610 +0.0764503 
 -0.4663692 0.58408733 +0.18082475 -7.5353293 -5.8408733 +0.1808248 
 +0.4731390 0.57795107 +0.28959440 -7.2511167 -5.0475302 +0.2920434 
 +0.5221422 0.57454813 +0.25136029 -5.0525337 -2.9150792 +0.2550648 
 
 -2.3968057 3.50543586 +0.00939762 +0.1666094 -0.0418552 +0.0070201 
 -2.3968061 3.50543960 +0.00939707 +0.1688164 -0.0418499 +0.0070205 
 
 
 
 lOOOXtft sinv 1000 xT-flo cos w 
 
 
 + (coav+cosE)X } _ 
 
 if \ 
 
 h ( ~ sec 2 ^+ 1 I sin t'o'c 
 
 J 
 
 
 a 
 
 
 
 +0.00044952 
 
 -0.57362783 
 
 +0.002782496 
 
 -0.015218912 
 
 -0.0011280149 
 
 30 
 
 +0.28684099 
 
 -0.49762977 
 
 -0.015447198 
 
 +0.017867337 
 
 -0.0011320724 
 
 60 
 
 +0.49964759 
 
 -0.28807540 
 
 -0.053760700 
 
 +0.018305034 
 
 -0.0011438222 
 
 90 
 
 +0.58124893 
 
 +0.00316289 
 
 -0.073936353 
 
 -0.014803463 
 
 -0.0011625560 
 
 120 
 
 +0.50564826 
 
 +0.29959114 
 
 -0.055727122 
 
 -0.048767876 
 
 -0.0011853529 
 
 150 
 
 +0.29142597 
 
 +0.51791202 
 
 -0.017258560 
 
 -0.049666046 
 
 -0.0012057929 
 
 180 
 
 -0.00234536 
 
 +0.59796705 
 
 +0.002975836 
 
 -0.016276382 
 
 -0.0012159913 
 
 210 
 
 -0.29539381 
 
 +0.51865750 
 
 -0.015412218 
 
 +0.018443264 
 
 -0.0012111028 
 
 240 
 
 -0.50861524 
 
 +0.30174597 
 
 -0.054096837 
 
 + 0.020191247 
 
 -0.0011927133 
 
 270 
 
 -0.58403554 
 
 +0.00617932 
 
 -0.074341448 
 
 -0.012307244 
 
 -0.0011681747 
 
 300 
 
 -0.50182267 
 
 -0.28668159 
 
 -0.055931013 
 
 -0.046147489 
 
 -0.0011462092 
 
 330 
 
 -0.28712395 
 
 -0.49766091 
 
 -0.017382540 
 
 -0.047441087 
 
 -0.0011324063 
 
 S, 
 
 -0.00703790 
 
 +0.05091934 
 
 -0.213757340 
 
 -0.087914378 
 
 -0.0070121038 
 
 S, 
 
 -0.00703741 
 
 +0.05092105 
 
 -0.213778317 
 
 -0.087907239 
 
 -0.0070121051 
 
 
 sin <p yAi M + cos <f 
 
 Bo (e > = + 0.00000000000028. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 145 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 [de/dtlw = - 1.5163927 TO' n 0.1808117 
 
 [dx/dt] m = [dT/dfloo = + 655.70924 TO' p 2.8167113 
 
 [dp/dt] m = - 18.991017 TO' n 1.2785482 
 
 [dq/dt] m = - 46.179399 TO' n 1.6644483 
 
 = -1514.4911 TO' n 3.1802667 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF 
 
 [de/dt] M = -0.00043305713 
 [dx/df], = [dir/d4 = +0.18725991 
 
 [dp/dfloo = -0.0054235259 
 [dq/dt] M = -0.013188086 
 = -0.43251400 
 
 m 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Lcverrier. Newcomb. Method of Gauss. 
 
 [de/dt] w -0.00044 -0.00043 -0.00043306 
 
 e[dirldt} w +0.00315 +0.00314 +0.00314056 
 
 [dp/dt] m -0.00542 -0.00542 -0.00542353 
 
 [dq/dt] M -0.01317 -0.01318 -0.01318809 
 
 [dL/dt] M -0.4325 -0.43251400 
 
 NOTES. 
 
 Here, as in the previous case, the approximate tests completely fail with the 
 angle e, the roots G, G', G" , and with the functions which immediately depend upon 
 these quantities. The close agreement of the final sums shows, however, that the 
 expansion of the perturbing function is quite rapidly convergent for this case. 
 
 The values obtained by HILL in the "New Theory " are: 
 
 
 [dp/dt] w = -oo054237 [dq/dt]^ = -0.0131883 
 The agreement of the final results here obtained with all other values is satisfactory. 
 
146 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF URANUS ON THE EARTH. 
 
 E 
 
 A 
 
 B cos t B sin e g 
 
 h 
 
 
 
 369.9391833 
 
 +24.383407 -17,162615 247.29194 
 
 367.49698 
 
 45 
 
 370.9299673 
 
 +34.837638 - 7.159449 41.56718 
 
 367.49556 
 
 90 
 
 370.9628887 
 
 +34.937206 + 7.512108 45.76306 
 
 367.49557 
 
 135 
 
 370.0188613 
 
 +24.623780 +17.957661 261.51163 
 
 367.49708 
 
 180 
 
 368.6506847 
 
 + 9.938828 +18.058344 264.45220 
 
 367.49706 
 
 225 
 
 367.6596194 
 
 - 0.515405 + 7.755178 48.77249 
 
 367.49561 
 
 270 
 
 367.6264169 
 
 - 0.614971 - 6.916382 38.79263 
 
 367.49553 
 
 315 
 
 368.5707253 
 
 + 9.698453 -17.361932 244.44864 
 
 367.49688 
 
 Si 
 
 1477.1791736* 
 
 +68.644470f + 1.191455t 596.29983 
 
 1469.98514 
 
 Z 2 
 
 1477.1791733 
 
 +68.644466 + 1.191458 596.29994 
 
 1469.98513 
 
 E 
 
 / 
 
 G G' G" 
 
 e 
 
 O 
 
 
 O 
 
 i a 
 
 
 
 + 1.63126 
 
 367.495141 1.9739873 0.3408899 4 
 
 33 0.174 
 
 45 
 
 +2.62346 
 
 367.495250 2.6661957 0.0424235 4 
 
 55 28.953 
 
 90 
 
 +2.65637 
 
 367.495229 2.7027882 0.0460735 4 
 
 57 40.407 
 
 135 
 
 + 1.71085 
 
 367.495130 2.0584810 0.3456945 4 
 
 38 13.695 
 
 180 
 
 +0.34269 
 
 367.495095 1.0379451 0.6933002 3 
 
 55 55.015 
 
 225 
 
 -0.64694 
 
 367.495249 0.163776T 0.8103506 2 
 
 56 52.565 
 
 270 
 
 -0.68006 
 
 367.495243 0.1303078 0.8100787 2 
 
 53 46.998 
 
 315 
 
 +0.26290 
 
 367.495069 0.9586081 0.6938970 3 
 
 50 28.866 
 
 S, 
 
 +3.95026 
 
 1469.980708 5.8450284 1.8903423 16 
 
 20 22.594 
 
 2 2 
 
 +3.95027 
 
 1469.980698 5.8470609 1.8923656 16 
 
 21 4.079 
 
 
 
 ACTION OF URANUS ON THE EARTH. 
 
 
 E 
 
 log A' 
 
 log LO' log No log N log P 
 
 logQ 
 
 
 
 0.00205713 
 
 0.27574316 0.17917560 6.1388849 1.2833195 
 
 3.7524062 
 
 45 
 
 0.00241015 
 
 0.27621390 0.17970508 6.1440956 1.2897058 
 
 3.7584989 
 
 90 
 
 0.00244638 
 
 0.27626178 0.17975893 6.1514873 1.3001369 
 
 3.7689401 
 
 135 
 
 0.00213675 
 
 0.27584925 0.17929492 6.1638868 1.3084161 
 
 3.7775217 
 
 180 
 
 0.00153562 
 
 0.27504825 0.17839395 6.1668770 1.3097851 
 
 3.7792009 
 
 225 
 
 0.00086277 
 
 0.27415148 0.17738519 6.1617903 1.3035252 
 
 3.7729672 
 
 270 
 
 0.00083285 
 
 0.27411159 0.17734031 6.1515209 1.2932165 
 
 3.7626532 
 
 315 
 
 0.00146560 
 
 0.27195493 0.17828898 6.1419973 1.2848106 
 
 3.7542155 
 
 S, 
 
 0.00687198 
 
 1.10116478 0.71466879 4.611770(1 5.1864579 
 
 3.0632003 
 
 2 2 
 
 0.00687557 
 
 1.10116956 0.71467417 4.6117700 5.1864577 
 
 5.0632033 
 
 
 * 4a 2 + 2aV + 4[o' 2 
 
 2kaa'ee' cos A:] = 1477.1791732. 
 
 
 t 4[a'V - kaa'e cos K] = + 68.644468. 
 t + 4fc'aa' cos v' e sin K' = + 1.191454. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 147 
 
 
 
 ACTION OF URANUS ON THE EARTH 
 
 . 
 
 
 E 
 
 logF 
 
 Ji' J* 
 
 Ji 
 
 Fl 
 
 
 
 3.7519032 
 
 367.8221780 -0.84734235 
 
 +2.2088876 
 
 +301.43340 
 
 45 
 
 3.7584363 
 
 367.4766488 -0.35515557 
 
 +4.6641774 
 
 + 123.58384 
 
 90 
 
 3.7688721 
 
 367.4892672 +0.38057614 
 
 +4.3579828 
 
 -129.67133 
 
 135 
 
 3.7770116 
 
 367.8350259 +0.86234922 
 
 + 1.4696662 
 
 -309.97871 
 
 180 
 
 3.7781780 
 
 368.1745883 +0.82112849 
 
 -2.3088335 
 
 -311.71664 
 
 225 
 
 3.7717715 
 
 368.2455011 +0.34414218 
 
 -4.7641198 
 
 -133.86711 
 
 270 
 
 3.7614579 
 
 368.2514548 -0.29842856 
 
 -4.4579225 
 
 + 119.38809 
 
 315 
 
 3.7531918 
 
 368.1823028 -0.79535630 
 
 -1.5696087 
 
 + 299.69549 
 
 Si 
 
 5.0604111 
 
 1471.7374883* +0.05593372 
 
 -0.1998856 
 
 - 20.56648 
 
 2 2 
 
 5.0604112 
 
 1471.7394786 +0.05597953 
 
 -0.1998849 
 
 - 20.56649 
 
 E 
 
 r, 
 
 1000000 X Ra 1000000 X So 1000000 X W, 
 
 1000000 X -R<"> 
 
 1000000 X S<"> 
 
 
 
 +3.5560766 
 
 68.949325 +0.10018820 +1.2544323 
 
 0.000000 
 
 +0.10189713 
 
 45 
 
 +0.4865778 
 
 69.825413 +0.03716997 +2.6752556 
 
 +49.966575 
 
 +0.03761606 
 
 90 
 
 +0.8230258 
 
 71.523975 -0.03529126 +2.5611571 
 
 +71.523975 
 
 -0.03529126 
 
 135 
 
 +4.0393764 
 
 73.045932 -0.11453773 +0.8877061 
 
 +51.045918 
 
 -0.11319535 
 
 180 
 
 +3.8028425 
 
 73.443068 -0.14342145 -1.3776247 
 
 0.000000 
 
 -0.14105578 
 
 225 
 
 +0.5809413 
 
 72.487200 -0.06580265 -2.8156195 
 
 -50.655465 
 
 -0.06503154 
 
 270 
 
 +0.709709S 
 
 70.794418 +0.06221357 -2.5724979 
 
 -70.794418 
 
 +0.06221357 
 
 315 
 
 +3.7847601 
 
 69.352325 +0.12685775 -0.8818747 
 
 -49.628040 
 
 +0.12838021 
 
 Si 
 
 +8.8916547 
 
 284.710786 -0.01631094 -0.1345332 
 
 + 0.729557 
 
 -0.01223634 
 
 2* 
 
 +8.8916556 
 
 284.710870 -0.01631266 -0.1345325 
 
 + 0.728988 
 
 -0.01223053 
 
 
 
 1 0ftOOOn V 1 7?n rct<* u 
 
 
 
 E 
 
 1 000000 X[R sin v 
 
 X L 1000000 1000000 
 
 1000x-2 r -K 
 
 
 + (cosv+cosE)S l 
 
 l] +^Bec'>+l)<rfaS.] X^ cos ("+") 
 
 KWosm(v-\-ir) 
 
 a 
 
 
 
 + 0.200376 
 
 -68.949325 -0.2256099 
 
 + 1.2339773 
 
 -0.13558593 
 
 45 
 
 +50.011805 
 
 -48.728590 -2.2190515 
 
 + 1.4942570 
 
 -0.13799470 
 
 90 
 
 +71.514500 
 
 + 1.128956 -2.5113156 
 
 -0.5028130 
 
 -0.14304794 
 
 135 
 
 + 51.201671 
 
 +52.095595 -0.4959295 
 
 -0.7362581 
 
 -0.14782435 
 
 180 
 
 + 0.286843 
 
 +73.443068 -0.2477661 
 
 + 1.3551611 
 
 -0.14934959 
 
 225 
 
 -50.554733 
 
 +51.949429 -2.2975222 
 
 + 1.6276075 
 
 -0.14669365 
 
 270 
 
 -70.785500 
 
 + 1.062876 -2.5379547 
 
 -0.4201590 
 
 -0.14158884 
 
 315 
 
 -49.442739 
 
 -48.631449 -0.5098809 
 
 -0.7195307 
 
 -0.13705976 
 
 ft 
 
 + 1.216219 
 
 + 6.685575 -5.5226463 
 
 + 1.6661664 
 
 -0.56957230 
 
 2 2 
 
 + 1.216004 
 
 + 6.684985 -5.5223841 
 
 + 1.6660757 
 
 -0.56957246 
 
 
 sin <p \Ai ( *> + cos . 
 
 , p . J3 <c> = - 0.00000000000025. 
 
 
 
 * 2,(J,' - G") = 1469.8471460. 
 
 2(Ji' - G") = 1469.8471130. 
 
148 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 n log coeff. 
 
 [<fe/tt]oo = + 0.39395664 TO' p 9.5954484 
 
 [dxfdtlw, = [drldt] M = +129.13143 TO' p 2.1110320 
 
 [dp/dt] w = + 0.53988815 TO' p 9.7323038 
 
 [dq/dt] w = - 1.7895101 TO' n 0.2527342 
 
 [dL/(ft]oo =-184.51950 TO' n 2.2660422 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF 
 
 TO 
 
 = +0.000017278801 
 [d x /dt] M = [Ar/dfloo = +0.0056636605 
 = +0.000023679306 
 = -0.000078487295 
 = -0.0080929604 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [(fe/tft]oo +0^00002 +0^00002 +o!oOOO 172788 
 
 e[d7r/d<]oo +0.00009 +0.00010 +0.0000949860 
 
 [dpldt] m +0.00002 +0.00002 +0.0000236793 
 
 [dq/dt] m -0.00008 -0.00008 -0.0000784873 
 
 -0.0081 -0.0080929604 
 
 NOTES. 
 
 It will be noticed that, owing to the very small mutual inclination, the approxi- 
 mate tests are here more exactly satisfied than even in the case of Saturn, where 
 twelve points of division were employed. It is therefore evident that eight points 
 are fully sufficient and that the greatest error arising from a division into only four 
 points (which occurs with the coefficient [dir/dt] 00 ), could not amount to more than 
 1 /20,000th of the whole. 
 
 The results obtained by HILL are : 
 
 [dpldt] M = +0.0000237 [dqldt] m = -0.0000785 
 exactly agreeing with those here given. 
 
OP THE ORBITS OF THE FOUR INNER PLANETS. 
 
 149 
 
 ACTION OF NEPTUNE ON THE EARTH. 
 
 E 
 
 A 
 
 
 
 905.47785591 
 
 45 
 
 905.10315254 
 
 90 
 
 904.80486595 
 
 135 
 
 904.75792710 
 
 180 
 
 904.98963355 
 
 225 
 
 905.36405558 
 
 270 
 
 905.66206098 
 
 315 
 
 905.70928102 
 
 Zi 
 
 3620.93441639* 
 
 2 2 
 
 3620.93441624 
 
 E 
 
 I 
 
 
 
 
 
 
 1.23952 
 
 45 
 
 0.86483 
 
 90 
 
 0.56594 
 
 135 
 
 0.51893 
 
 180 
 
 0.75119 
 
 225 
 
 1.12575 
 
 270 
 
 1.42319 
 
 315 
 
 1.47027 
 
 2, 
 
 3.97984 
 
 2 2 
 
 3.97977 
 
 E 
 
 log A' 
 
 
 
 0.00047205 
 
 45 
 
 0.00035874 
 
 90 
 
 0.00022606 
 
 135 
 
 0.00019146 
 
 180 
 
 0.00031307 
 
 225 
 
 0.00044509 
 
 270 
 
 0.00052334 
 
 315 
 
 0.00053136 
 
 2i 
 
 0.00153452 1 
 
 y 
 
 0.00152666 1 
 
 B cos e B sin g 
 
 h 
 
 +23.748411 +24.815277 40.194861 
 
 904.17306 
 
 + 1.127297 +28.979673 54.817494 
 
 904.17306 
 
 -17.814345 +15.920217 16.543607 
 
 904.17365 
 
 -21.980762 - 6.713038 2.941510 
 
 904.17372 
 
 - 8.931313 -25.661837 42.984089 
 
 904.17317 
 
 + 13.689802 -29.826236 58.066960 
 
 904.17304 
 
 +32.631450 -1.6.766778 18.349806 
 
 904.17361 
 
 +36.797860 + 5.866476 2.246398 
 
 904.17374 
 
 +29.634203f - 1.693121f 118.072363 
 
 3616.69349 
 
 +29.634196 - 1.693125 118.072362 
 
 3616.69355 
 
 G G' G" 
 
 e 
 
 
 O 1 it 
 
 904.17301 1.274452 0.034882 
 
 2 10 50.926 
 
 904.17299 0.930076 0.065186 
 
 1 54 4.344 
 
 904.17363 0.596627 0.030667 
 
 1 30 33.476 
 
 904.17372 0.525125 0.006195 
 
 1 23 20.558 
 
 904.17312 0.809936 0.058696 
 
 1 46 33.999 
 
 904.17297 1.180229 0.054414 
 
 2 7 3.526 
 
 904.17359 1.437325 0.014120 
 
 2 17 46.322 
 
 904.17374 1.471958 0.001688 
 
 2 18 49.394 
 
 3616.69335 4.118340 0.138365 
 
 7 45 44.723 
 
 3616.69342 4.107388 0.127483 
 
 7 43 17.822 
 
 ACTION OF NEPTUNE ON THE EARTH. 
 
 
 og/V log Wo logtf logP 
 
 logQ 
 
 '363061 0.17679925 5.5513789 9.9124729 
 
 2.7719098 
 
 347955 0.17662931 5.5555725 9.9164864 
 
 2.7759190 
 
 330267 0.17643033 5.5658262 9.9265957 
 
 2.7859899 
 
 '325655 0.17637844 5.5760491 9.9367960 
 
 2.7961727 
 
 '341867 0.17656083 5.5803398 9.9411989 
 
 2.8006208 
 
 '359467 0.17675882 5.5762686 9.9373080 
 
 2.7967497 
 
 '369899 0.17687617 5.5661355 9.9273173 
 
 2.786753T 
 
 370968 0.17688820 5.5557902 9.9169945 
 
 2.7764257 
 
 405094 0.70666658 2.2636803 9.7075847 
 
 1.1452735 
 
 404045 0.70665477 2.2636803 9.7075848 
 
 1.145267T 
 
 * 4a" + 2aV + 4[a' 2 - 2kaa'ee' cos K\ = 3620.93441628. 
 t4[a'V-fraa'ecos A] = +29.634198. 
 | - 4k'aa' cos *.'. e sin A"' = - 1.693118. 
 
150 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 
 ACTION OF NEPTUNE ON THE EARTH. 
 
 
 E 
 
 logF 
 
 Ji' Ji Ja 
 
 F 2 
 
 
 
 2.7718889 
 
 903.9916758 +0.58828709 -13.723633 
 
 -190.54625 
 
 45 
 
 2.7758798 
 
 904.1719863 +0.01502688 + 7.632175 
 
 -222.52292 
 
 90 
 
 2.7859714 
 
 903.5306465 -0.23467770 +24.659009 
 
 -122.24477 
 
 135 
 
 2.7961690 
 
 903.3686400 +0.17398234 +27.382788 
 
 + 51.54664 
 
 180 
 
 2.8005856 
 
 904.0154903 +0.15962945 +14.207956 
 
 + 197.04666 
 
 225 
 
 2.7967170 
 
 904.1717548 -0.46647484 - 7.147853 
 
 +229.02334 
 
 270 
 
 2.7867446 
 
 903.5394237 -0.52719220 -24.174694 
 
 + 128.74518 
 
 315 
 
 2.7764247 
 
 903.3535982 +0.26287497 -26.898466 
 
 - 45.04625 
 
 2! 
 
 1.145190o 
 
 3615.0772363* -0.01395336 + 0.968638 
 
 + 13.00082 
 
 2 2 
 
 1.1451905 
 
 3615.0659793 -0.01459065 + 0.968644 
 
 + 13.00081 
 
 E 
 
 F, 
 
 100000 X Ro 10000000 X So 100000 X We 100000 X #<"> 
 
 10000000 X -S ( "> 
 
 
 
 - 5.0649535 
 
 1.7793453 +0.19215285 -0.08120439 0.0000000 
 
 + 0.19543047 
 
 45 
 
 - 6.5852856 
 
 1.7972262 -0.17462602 +0.04549983 +1.2860826 
 
 -0.17672179 
 
 90 
 
 - 1.8342882 
 
 1.8362237 -0.24659977 +0.15062685 +1.8362237 
 
 -0.24659977 
 
 135 
 
 - 0.4453113 
 
 1.8790643 +0.15337661 +0.17125249 +1.3131267 
 
 +0.15157902 
 
 180 
 
 - 5.4164244 
 
 1.9016607 +0.27295016 +0.08971979 0.0000000 
 
 +0.26844799 
 
 225 
 
 - 6.9853355 
 
 1.8852158 -0.09387083 -0.04482073 -1.3174254 
 
 -0.09277065 
 
 270 
 
 - 2.0485736 
 
 1.8383392 -0.21373143 -0.14796412 -1.8383392 
 
 -0.21373143 
 
 315 
 
 - 0.3483062 
 
 1.7940628 +0.11988971 -0.16075330 -1.2838186 
 
 +0.12132856 
 
 Si 
 
 - 14.3642397 
 
 7.3555689 +0.00477181 +0.01117813 -0.0021155 
 
 + 0.00354726 
 
 2 2 
 
 - 14.3642386 
 
 7.3555691 +0.00476947 +0.01117829 -0.0020347 
 
 + 0.00341514 
 
 E 
 
 100000X[osin, I00000x[-Rocos, imm l(mQQ 
 + (eosv+cosE)St,\ . (r , ,\ r, 1 XWoCos (W+JT) XlFosin (W+JT) 
 
 I- I ^i<l<- ,n -I- 1 1 Q1T1 JlX., 1 
 
 100000x-2-flo 
 
 a 
 
 
 
 v / .1 
 
 
 
 
 +0.0038431 
 
 -1.7793453 +0.01460463 -0.07988026 
 
 - 3.4990076 
 
 45 
 
 + 1.2834469 
 
 -1.2580636 -0.03774087 +0.02541381 
 
 - 3.5518258 
 
 90 
 
 + 1.8360068 
 
 +0.0258636 -0.14769558 -0.02957146 
 
 - 3.6724475 
 
 135 
 
 + 1.3107603 
 
 + 1.3464279 -0.09567261 -0.14203579 
 
 - 3.8026901 
 
 180 
 
 -0.0054590 
 
 + 1.9016607 +0.01613612 -0.08825681 
 
 - 3.8671076 
 
 225 
 
 -1.3159048 
 
 + 1.3499919 -0.03657334 +0.02590924 
 
 - 3.8151452 
 
 270 
 
 -1.8380449 
 
 +0.0351057 -0.14597725 -0.02416657 
 
 - 3.6766784 
 
 315 
 
 -1.2819528 
 
 -1.2550750 -0.09294409 -0.13116027 
 
 - 3.5455738 
 
 2i 
 
 -0.0036540 
 
 +0.1832847 -0.26293208 -0.22187510 
 
 -14.7152411 
 
 2 2 
 
 -0.0036504 
 
 +0.1832812 -0.26293091 -0.22187301 
 
 -14.7152409 
 
 
 sin if \A i (> > + cos <f 
 
 . B = + 0.0000000000000014. 
 
 
 * 2,(J,' - G") = 3614.9388718. 
 
 2,(J,' - G") = 3614.9384968. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 151 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 [de/dt] m = -0.011831221m' n 8.0730296 
 
 [dx/dt] m = [dTr/dt]^ = +35.402545 TO' p 1.5490345 
 
 [dp/e&]oo = -- 0.71895833 TO' n 9.8567037 
 
 [dq/dt] m = -- 0.85200049 TO' n 9.9304399 
 
 [dL/dt] w = -47.671428 TO' n 1.6782582 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO' 
 
 [de/dt] w = -0.00000060056972 
 
 [d x !dt] w = [dw/dt] w = +0.0017970838 
 
 [dp/dtlao = -0.000036495344 
 
 [dg/dt] m = -0.000043248757 
 
 [dL/dt] m = -0.0024198698 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt} 00 0.00000 0.00000 -0.00000060057 
 
 e[drldt]oo +0.00003 +0.00003 +0.00003013915 
 
 [dpldt] m -0.00004 -0.00004 -0.00003649534 
 
 [dg/dt] m -0.00004 -0.00004 -0.00004324876 
 
 NOTES. 
 
 The mutual inclination is here nearly twice as great as in the case of Uranus, and 
 yet the convergence of the perturbing function is more rapid because the eccentricity 
 of Neptune is so much smaller than that of Uranus. Hence, although the sums of 
 e, G, G', G", etc., are in great disagreement, the final sums from which the differential 
 coefficients are obtained are almost identical. The greatest error arising from a divi- 
 sion into only four parts occurs with the coefficient [dp/dt] 00 and amounts to but 
 0". 000000000002 
 
 The results of HILL are : 
 
 
 = -00000366 [dqldt] n = -0.0000435 
 
152 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 E 
 
 
 
 30 
 
 60 
 
 90 
 
 120 
 
 150 
 
 180 
 
 210 
 
 240 
 
 270 
 
 300 
 
 330 
 
 MARS. 
 
 ACTION OF MERCURY ON MARS. 
 
 A 
 
 B sin ( 
 
 B COS e 
 
 1000 x g 
 
 2.01395126 
 
 -0.51051059 
 
 -0.07779H4 
 
 1.6508943 
 
 2.19162773 
 
 -0.49427471 
 
 +0.22547307 
 
 1.5475600 
 
 2.44453125 
 
 -0.33152771 
 
 +0.47957033 
 
 0.6962245 
 
 2.71228886 
 
 +0.06587754 
 
 +0.61641571 
 
 0.0274906 
 
 2.92685114 
 
 +0.23149508 
 
 +0.59934132 
 
 0.3394636 
 
 3.02703005 
 
 +0.48090911 
 
 +0.43292228 
 
 1.4649936 
 
 2.97859089 
 
 +0.61553465 
 
 +0.16175058 
 
 2.4000193 
 
 2.79081655 
 
 +0.59929870 
 
 -0.14151362 
 
 2.2750785 
 
 2.51771727 
 
 +0.43655177 
 
 -0.39561101 
 
 1.2072061 
 
 2.23986183 
 
 +0.17090155 
 
 -0.53245622 
 
 0.1850128 
 
 2.03539750 
 
 -0.12647099 
 
 -0.51538178 
 
 0.1013191 
 
 1.95541421 
 
 -0.37588517 
 
 -0.34896288 
 
 0.8949940 
 
 14.91703931* 
 
 +0.31507221f 
 
 +0.25187830J 
 
 6.9351269 
 
 14.91703923 
 
 +0.31507194 
 
 +0.25187834 
 
 6.9351295 
 
 1.87197670 
 2.04795479 
 2.29696077 
 2.56269605 
 2.77862654 
 2.88207295 
 2.83609064 
 2.64795925 
 2.37176681 
 2.09074126 
 1.88662892 
 1.81063180 
 
 14.04205038 
 14.04205609 
 
 G' 
 
 G" 
 
 
 
 0.13564010 
 
 1.87146851 
 
 0.14234545 
 
 0.00619717 
 
 16 
 
 20 
 
 8.69 
 
 30 
 
 0.13733849 
 
 2.04755912 
 
 0.14301881 
 
 0.00528466 
 
 15 
 
 35 
 
 30.16 
 
 60 
 
 0.14123602 
 
 2.29682015 
 
 0.14348918 
 
 0.00211253 
 
 14 
 
 34 
 
 33.48 
 
 90 
 
 0.14325835 
 
 2.56269162 
 
 0.14333762 
 
 0.00007484 
 
 13 
 
 41 
 
 0.67 
 
 120 
 
 0.14189014 
 
 2.77858020 
 
 0.14279207 
 
 0.00085559 
 
 13 
 
 8 
 
 25.31 
 
 150 
 
 0.13862265 
 
 2.88188764 
 
 0.14237834 
 
 0.00357038 
 
 12 
 
 59 
 
 49.56 
 
 180 
 
 0.13616579 
 
 2.83577714 
 
 0.14242175 
 
 0.00594246 
 
 13 
 
 12 
 
 30.22 
 
 210 
 
 0.13652285 
 
 2.64761705 
 
 0.14287917 
 
 0.00601413 
 
 13 
 
 42 
 
 7.60 
 
 240 
 
 0.13961601 
 
 2.37153873 
 
 0.14339402 
 
 0.00354993 
 
 14 
 
 24 
 
 9.49 
 
 270 
 
 0.14278611 
 
 2.09069583 
 
 0.14344844 
 
 0.00061690 
 
 15 
 
 12 
 
 58.81 
 
 300 
 
 0.14243412 
 
 1.88659813 
 
 0.14284089 
 
 0.0003759S 
 
 15 
 
 59 
 
 29.51 
 
 330 
 
 0.13844795 
 
 1.81033610 
 
 0.14221981 
 
 0.00347617 
 
 16 
 
 27 
 
 51.54 
 
 2! 
 
 0.83698218 
 
 14.04078286 
 
 0.85728335 
 
 0.01903366 
 
 87 
 
 39 
 
 16.70 
 
 2 a 
 
 0.83697639 
 
 14.04078736 
 
 0.85728218 
 
 0.01903708 
 
 87 
 
 39 
 
 18.34 
 
 * 6a 2 + 3a 2 e ! + 6[a' s - Zkaa'ee' cos K] = 14.91703924. 
 t - Cfc'oo' cos >' e sin K' = + 0.31507212. 
 I 6[a'V - A-oaV cos A'] = + 0.25187831. 
 
OF THE ORBITS OF THE FOUK INNER PLANETS. 
 
 153 
 
 
 
 ACTION 
 
 OF MERCURY ON 
 
 MARS. 
 
 
 E 
 
 log /Co 
 
 log V 
 
 logtfo 
 
 log N log /' 
 
 logQ 
 
 
 
 0.02698253 
 
 0.30881667 
 
 0.21633345 
 
 0.0802043 9.8417843 
 
 0.0229194 
 
 30 
 
 0.02453876 
 
 0.30558607 
 
 0.21270775 
 
 0.0315423 9.7124165 
 
 9.9318941 
 
 60 
 
 0.02139874 
 
 0.30143126 
 
 0.20804354 
 
 9.9863199 9.5646986 
 
 9.8328372 
 
 90 
 
 0.01882471 
 
 0.29802213 
 
 0.20421541 
 
 9.9544525 9.4350567 
 
 9.7499589 
 
 120 
 
 0.01734193 
 
 0.29605696 
 
 0.20200830 
 
 9.9396879 9.3478315 
 
 9.6977395 
 
 150 
 
 0.01696133 
 
 0.29555239 
 
 0.20144156 
 
 9.9427992 9.3179221 
 
 9.6840260 
 
 180 
 
 0.01752419 
 
 0.29629857 
 
 0.20227967 
 
 9.9632970 9.3524331 
 
 9.7119954 
 
 210 
 
 0.01887658 
 
 0.29809086 
 
 0.20429260 
 
 9.9992756 9.4496853 
 
 9.7797276 
 
 240 
 
 0.02088535 
 
 0.30075154 
 
 0.20728036 
 
 0.0456467 9.5950386 
 
 9.8772472 
 
 270 
 
 0.02335208 
 
 0.30401640 
 
 0.21094579 
 
 0.0914150 9.7545935 
 
 9.9819419 
 
 300 
 
 0.02583676 
 
 0.30730232 
 
 0.21463400 
 
 0.1193985 9.8751689 
 
 0.0582665 
 
 330 
 
 0.02741722 
 
 0.30939106 
 
 0.21697800 
 
 fl.1150661 9.9072726 
 
 0.0734518 
 
 Si 
 
 0.12996950 
 
 1.81065732 
 
 1.25057932 
 
 0.1345541 7.5769549 
 
 9.2010052 
 
 2 2 
 
 0.12997068 
 
 1.81065891 
 
 1.25058111 
 
 0.1345506 7.5769465 
 
 9.2010002 
 
 E 
 
 log V 
 
 /i' 
 
 J 2 
 
 J, 
 
 Ft 
 
 
 
 0.0211422 
 
 0.148654076 
 
 -0.10722959 
 
 -0.015479687 
 
 +0.015330048 
 
 30 
 
 9.9305073 
 
 0.148321182 
 
 -0.10335718 
 
 -0.012048361 
 
 +0.014842503 
 
 60 
 
 9.8323421 
 
 0.145601872 
 
 -0.06932482 
 
 -0.006505466 
 
 +0.009955397 
 
 90 
 
 9.7499432 
 
 0.143412552 
 
 -0.01386724 
 
 -0.000353153 
 
 +0.001978227 
 
 120 
 
 9.6975735 
 
 0.143648964 
 
 +0.04858051 
 
 +0.004751607 
 
 -0.006951532 
 
 150 
 
 9.6833581 
 
 0.145958760 
 
 +0.10133761 
 
 +0.007449454 
 
 -0.014441148 
 
 180 
 
 9.7108663 
 
 0.148399366 
 
 +0.13001805 
 
 +0.007034457 
 
 -0.018483798 
 
 210 
 
 9.7785043 
 
 0.148942861 
 
 +0.12663289 
 
 +0.003626261 
 
 -0.017996253 
 
 240 
 
 9.8764415 
 
 0.146960494 
 
 +0.09193479 
 
 -0.001870362 
 
 -rO.013109151 
 
 270 
 
 9.9817831 
 
 0.144077345 
 
 -0.03534552 
 
 -0.007999542 
 
 -0.005131979 
 
 300 
 
 0.0581594 
 
 0.143355345 
 
 -0.02759166 
 
 -0.013127432 
 
 +0.003797779 
 
 330 
 
 0.0724209 
 
 0.145910972 
 
 0.07970449 
 
 -0.015871553 
 
 +0.011287401 
 
 2, 
 
 9.1965248 
 
 0.876620117* 
 
 +0.06638728 
 
 -0.025196883 
 
 -0.009461257 
 
 2 2 
 
 9.1965168 
 
 0.876623672 
 
 +0.06638711 
 
 -0.025196894 
 
 -0.009461249 
 
 
 * 2,(J,' - G") = 
 
 0.857586462. 
 
 
 
 
 
 S,(J,' - G") = 
 
 0.857586592. 
 
 
 
 
154 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF MERCURY ON MARS. 
 
 E 
 
 1000 X F, Ro So W,, fl<> 
 
 8* 
 
 
 
 +0.3749543 -1.1968194 -0.10192940 -0.015991438 0.00000000 
 
 -0.07377742 
 
 30 
 
 +0.9451901 -1.0712026 -0.08041942 -0.009779348 -0.38240323 
 
 -0.05741705 
 
 60 
 
 +0.8478648 -0.9676664 -0.04346887 -0.004110828 -0.57689880 
 
 -0.02992414 
 
 90 
 
 +0.1645035 -0.9003960 -0.00725842 -0.000153771 -0.59093068 
 
 -0.00476371 
 
 120 
 
 -0.4045215 -0.8699380 +0.02266377 +0.002278066 -0.47241848 
 
 +0.01421151 
 
 150 
 
 -0.2449197 -0.8749740 +0.04587694 +0.003542283 -0.26566469 
 
 +0.02785885 
 
 180 
 
 +0.5450969 -0.9160775 +0.06265312 +0.003737622 0.00000000 
 
 +0.03761136 
 
 210 
 
 + 1.2365773 -0.9949389 +0.07097316 +0.002525788 +0.30208776 
 
 +0.04309857 
 
 210 
 
 + 1.1824210 -1.1083446 +0.06401101 -0.000941853 +0.60188486 
 
 +0.04013864 
 
 270 
 
 +0.4525829 -1.2337308 +0.03097697 -0.007413721 +0.80969859 
 
 +0.02033021 
 
 300 
 
 -0.2401087 -1.3158841 -0.02869644 -0.015188691 +0.78449755 
 
 -0.01975474 
 
 330 
 
 -0.2482279 -1.2994033 -0.08505077 -0.018952193 +0.46386745 
 
 -0.06072369 
 
 Si 
 
 +2.3057068 -6.3747320 -0.02476681 -0.030217122 +0.33706513 
 
 -0.03149479 
 
 Z 2 
 
 +2.3057062 6.3746456 -0.02490154 -0.030230962 +0.33665520 
 
 -0.03161682 
 
 
 D . Ro cos v 
 
 
 E 
 
 RO sin v 
 + (cos ,+cos E)S a + g BCC' ,+l) sin S. 
 
 -2 -Bo 
 a 
 
 
 
 -0.2038588 +1.1968194 -0.004111709 +0.015453801 
 
 2.1703888 
 
 30 
 
 -0.7173754 +0.8165787 -0.007231890 +0.006582964 
 
 1.9693580 
 
 60 
 
 -0.9154636 +0.3357086 -0.004043881 +0.000738872 
 
 1.8450804 
 
 90 
 
 -0.8957943 -0.0984951 -0.000144266 -0.000053225 
 
 1.8007921 
 
 120 
 
 -0.7408612 -0.4547261 +0.001481632 +0.001730419 
 
 1.8210138 
 
 150 
 
 -0.4834776 -0.7324563 +0.000768358 +0.003457946 
 
 1.8912954 
 
 180 
 
 -0.1253062 -0.9160795 -0.000961015 +0.003611962 
 
 2.0030408 
 
 210 
 
 +0.3338218 -0.9514367 -0.001700736 +0.001867379 
 
 2.1505951 
 
 240 
 
 +0.8448003 -0.7366597 +0.000887111 -0.000316419 
 
 2.3200626 
 
 270 
 
 + 1.2254640 -0.1770222 +0.007311027 +0.001229687 
 
 2.4674617 
 
 300 
 
 + 1.1635310 +0.6123066 +0.011609114 +0.009794123 
 
 2.5090379 
 
 330 
 
 +0.5585553 +1.1811278 +0.005822238 +0.018035720 
 
 2.3888945 
 
 Si 
 
 +0.0228415 +0.0373693 +0.004861252 +0.031012758 
 
 12.6686243 
 
 2 2 
 
 +0.0211938 +0.0382962 +0.004824731 +0.031120471 
 
 12.6683967 
 
 
 sin p JA,<'> + cos v Bo (e) = + 0".000000008. 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 155 
 
 DEFERENTIAL COEFFICIENTS. 
 
 u log coeff. 
 
 [de/dtlw = + 2517.5250 m' p 3.4009738 
 
 [d x /dt] w = + 46380.761 TO' p 4.6663379 
 
 [di/dt] w =+ 558.61256 m' p 2.7471107 
 
 [dn/di]oo =+110961.28 TO' p 5.0451714 
 
 [dir/dt] M = + 46438.628 TO' p 4.6668794 
 
 =+1455134.1 TO' p 6.1629030 
 
 TO 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF 
 
 [de/dt]^ = +0.00033567000 
 
 [dx/dt] 00 = +0.0061841007 
 
 [di/dt] w = +0.000074481672 
 
 [cKl/dt] w = +0.014794833 
 
 [dw/dt] m = +0.0061918174 
 
 [dL/dt] w = +0.19401785 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt} 00 +0.00036 +0.00033 +0.0003357 
 
 e[dTr/dt] 00 +0.00058 +0.00057 +0.0005775 
 
 [di/dt]w +0.00008 +0.00007 +0.0000745 
 
 sin i [dtt/dt] w +0.00047 +0.00048 +0.0004778 
 
 NOTES. 
 
 On account of the large eccentricities of both orbits and the high mutual incli- 
 nation, the coefficients of the expansion diminish but slowly. Thus the combined 
 effect of all terms from the 6th to the llth orders is l/30th of the whole with [de/dt]o , 
 l/90th with [dw/dt] o, and 1 /200th with [di/dt] <>. Yet all of the variations are very 
 small on account of the smallness of the mass of Mercury. A comparison with the 
 computation of Mars on Mercury renders it evident that a division into twelve parts 
 is sufficient and that terms of orders above the eleventh are wholly inappreciable. 
 
156 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF VENUS ON MARS. 
 
 E 
 
 A 
 
 B COS e 
 
 B sin e 
 
 1000 xg 
 
 h 
 
 
 
 2.41946745 
 
 -0.9101348 
 
 -0.4038413 
 
 0.003995803 
 
 1.8967820 
 
 30 
 
 2.47732532 
 
 -0.5530947 
 
 -0.8457642 
 
 0.017525935 
 
 1.9549495 
 
 60 
 
 2.63301427 
 
 -0.0217107 
 
 -1.0499345 
 
 0.027008907 
 
 2.1104659 
 
 90 
 
 2.85220984 
 
 +0.5416339 
 
 -0.9616453 
 
 0.022657521 
 
 2.3292431 
 
 120 
 
 3.07987480 
 
 +0.9859909 
 
 -0.6045536 
 
 0.008954722 
 
 2.5566759 
 
 150 
 
 3.25131035 
 
 + 1.1922952 
 
 -0.0743414 
 
 0.000135408 
 
 2.7282304 
 
 180 
 
 3.31318850 
 
 + 1.1052676 
 
 +0.4869209 
 
 0.005808973 
 
 2.7904410 
 
 210 
 
 3.24523275 
 
 +0.7482272 
 
 +0.9288436 
 
 0.021138190 
 
 2.7227529 
 
 240 
 
 3.06934808 
 
 +0.2168441 
 
 + 1.1330145 
 
 0.031452377 
 
 2.5468467 
 
 270 
 
 2.84005464 
 
 -0.3465004 
 
 + 1.0447250 
 
 0.026741552 
 
 2.3172324 
 
 300 
 
 2.62248756 
 
 -0.7908574 
 
 +0.6876330 
 
 0.011585000 
 
 2.0993294 
 
 330 
 
 2.47124772 
 
 -0.9971620 
 
 +0.1574211 
 
 0.000607167 
 
 1.9481304 
 
 2, 
 
 17.13738066* 
 
 +0.5853997f 
 
 +0.2492390t 
 
 0.088805782 
 
 14.0005408 
 
 2 2 
 
 17.13738062 
 
 +0.5854002 
 
 +0.2492388 
 
 0.088805773 
 
 14.0005387 
 
 G 
 
 G' 
 
 10000 X G" 
 
 
 
 0.5226610 
 
 1.8967805 
 
 0.5226665 
 
 0.0403053 
 
 31 
 
 39 
 
 49.77 
 
 30 
 
 0.5223513 
 
 1.9549433 
 
 0.5223748 
 
 0.1716188 
 
 31 
 
 7 
 
 34.86 
 
 60 
 
 0.5225239 
 
 2.1104579 
 
 0.5225564 
 
 0.2449047 
 
 29 
 
 50 
 
 30.35 
 
 90 
 
 0.5229423 
 
 2.3292377 
 
 0.5229663 
 
 0.1860051 
 
 28 
 
 17 
 
 2.93 
 
 120 
 
 0.523174o 
 
 2.5566741 
 
 0.5231829 
 
 0.0669458 
 
 26 
 
 53 
 
 44.43 
 
 150 
 
 0.5230555 
 
 2.7282304 
 
 0.5230555 
 
 0.0009489 
 
 25 
 
 58 
 
 2.26 
 
 180 
 
 0.5227230 
 
 2.7904401 
 
 0.5227279 
 
 0.0398246 
 
 25 
 
 38 
 
 46.94 
 
 210 
 
 0.5224554 
 
 2.7227494 
 
 0.5224737 
 
 0.1485921 
 
 25 
 
 58 
 
 48.56 
 
 240 
 
 0.5224769 
 
 2.5468406 
 
 0.5225067 
 
 0.2363523 
 
 26 
 
 55 
 
 59.83 
 
 270 
 
 0.5227977 
 
 2.3172260 
 
 0.5228262 
 
 0.2207297 
 
 28 
 
 21 
 
 35.72 
 
 300 
 
 0.5231337 
 
 2.0993259 
 
 0.5231478 
 
 0.1054853 
 
 29 
 
 56 
 
 49.75 
 
 330 
 
 0.5230928 
 
 1.9481302 
 
 0.5230936 
 
 0.0059582 
 
 31 
 
 12 
 
 36.97 
 
 2i 
 
 3.1366929 
 
 14.0005190 
 
 3.1367881 
 
 0.7338180 
 
 170 
 
 55 
 
 41.07 
 
 Si 
 
 3.1366950 
 
 14.0005170 
 
 3.1367900 
 
 0.7338528 
 
 170 
 
 55 
 
 41.30 
 
 * 6o 2 + 3aV + 6[o' 2 - Zkaa'ee' cos K\ = 17.13738065. 
 t 6[a'V - kaa'c cos A'] = + 0.5854002. 
 \ - 6fc'aa' cos <p' e sin K' = + 0.2492389. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 157 
 
 ACTION OF VENUS ON MAKS. 
 
 I 
 
 logtfo 
 
 log LO 
 
 log # log N log P log Q 
 
 
 
 0.10710693 
 
 0.41334627 
 
 0.33316554 0.1537292 0.0110396 0.2088768 
 
 30 
 
 0.10323050 
 
 0.40834940 
 
 0.32760279 0.1420610 9.9681345 0.1785258 
 
 60 
 
 0.09431348 
 
 0.39683240 
 
 0.31477271 0.1149515 9.8630204 0.1053425 
 
 90 
 
 0.08414014 
 
 0.38365383 
 
 0.30007678 0.0820056 9.7312249 0.0148651 
 
 120 
 
 0.07564158 
 
 0.37216263 
 
 0.28775224 0.0524079 9.609667o 9.9324836 
 
 150 
 
 0.07024983 
 
 0.36559243 
 
 0.27991042 0.0325881 9.5264184 9.8766175 
 
 180 
 
 0.06843912 
 
 0.36323215 
 
 0.27727293 0.0260737 9.4979593 9.8576733 
 
 210 
 
 0.07032296 
 
 0.36568773 
 
 0.28001691 0.0339679 9.5296356 9.8789748 
 
 240 
 
 0.07586487 
 
 0.37290310 
 
 0.28807661 0.0551373 9.6180289 9.9372082 
 
 270 
 
 0.08461922 
 
 0.38427537 
 
 0.30077024 0.0858518 9.7401822 0.0216495 
 
 300 
 
 0.09502686 
 
 0.39775494 
 
 0.31580088 0.1191144 9.8727052 0.1128332 
 
 330 
 
 0.10383003 
 
 0.40912260 
 
 0.32846370 0.1449403 9.9748268 0.1837860 
 
 Zi 
 
 0.51639284 
 
 2.31668149 
 
 1.81684091 0.5214139 8.4704208 0.154417o 
 
 S 2 
 
 0.51639268 
 
 2.31668136 
 
 1.81684084 0.5214145 8.4704222 0.1544186 
 
 E 
 
 log V 
 
 Ji' 
 
 J 2 J 3 f\ 
 
 
 
 0.2088756 
 
 0.52281610 
 
 -0.0030557479 -0.012760938 +0.0014450479 
 
 30 
 
 0.1785212 
 
 0.52261080 
 
 -0.0058421642 -0.016264386 +0.0030263615 
 
 60 
 
 0.1053364 
 
 0.52273704 
 
 -0.0069459413 -0.015093265 +0.0037569353 
 
 90 
 
 0.0148609 
 
 0.52303324 
 
 -0.0063117580 -0.009561780 +0.0034410135 
 
 120 
 
 9.9324822 
 
 0.52319006 
 
 -0.0041023613 -0.001152260 +0.0021632475 
 
 150 
 
 9.8766174 
 
 0.52307981 
 
 -0.0007324438 +0.007882145 +0.0002660128 
 
 180 
 
 9.8576725 
 
 0.52281605 
 
 +0.0030484592 +0.015121070 -0.0017423280 
 
 210 
 
 9.8789719 
 
 0.52262308 
 
 +0.0062562314 +0.018625039 -0.0033236408 
 
 240 
 
 9.9372033 
 
 0.52266847 
 
 +0.0079298727 +0.017454980 -0.0040542164 
 
 270 
 
 0.0216445 
 
 0.52292962 
 
 +0.0074476695 +0.011924014 -0.0037382940 
 
 300 
 
 0.1128306 
 
 0.52316808 
 
 +0.0048298806 +0.003513972 -0.0024605273 
 
 330 
 
 0.1837858 
 
 0.52311928 
 
 +0.0008867096 -0.005521489 -0.0005632929 
 
 Si 
 
 0.1544005 
 
 3.13739580* 
 
 +0.0017041620 +0.007083559 -0.0008918410 
 
 2i 
 
 0.1544017 
 
 3.13739582 
 
 +0.0017042445 +0.007083543 -0.0008918399 
 
 
 * 2,(J/ - G") = 3, 
 
 13732242. 
 
 
 
 2,(J,' - G") = 3. 
 
 13732243. 
 
 
158 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 
 ACTION OF VENUS ON MARS. 
 
 
 
 E 
 
 100000 X F, 
 
 Ro 1000 X So 1000 X W<> 
 
 <' 
 
 S<"> 
 
 
 
 - 2.705324 
 
 -1.4244790 -3.460778 -20.67006 
 
 0.0000000 
 
 -0.002504942 
 
 30 
 
 - 0.803003 
 
 -1.3866027 -6.000157 -24.54092 
 
 -0.4949963 
 
 -0.004283931 
 
 60 
 
 + 5.500746 
 
 -1.3028006 -6.111878 -19.19609 
 
 -0.7766973 
 
 -0.004207440 
 
 90 
 
 + 9.816975 
 
 -1.2077652 -4.678320 - 9.84176 
 
 -0.7926572 
 
 -0.003070385 
 
 120 
 
 + 7.633806 
 
 -1.1282518 -2.631099 - 0.95528 
 
 -0.6126954 
 
 -0.001649853 
 
 150 
 
 + 0.880725 
 
 -1.0779056 -0.461908 + 5.93579 
 
 -0.3272800 
 
 -0.000280495 
 
 180 
 
 - 3.932916 
 
 -1.0618130 +1.648220 +10.88332 
 
 0.0000000 
 
 +0.000989445 
 
 210 
 
 - 2.161955 
 
 -1.0812437 +3.609363 +14.08781 
 
 +0.3282934 
 
 +0.002191792 
 
 240 
 
 + 4.374565 
 
 -1.1352348 +5.187600 +15.12313 
 
 +0.6164876 
 
 +0.003252928 
 
 270 
 
 + 9.225323 
 
 -1.2184712 +5.773047 +12.58406 
 
 +0.7996835 
 
 +0.003788855 
 
 300 
 
 + 7.735212 
 
 -1.3155488 +4.427365 + 4.61417 
 
 +0.7842976 
 
 +0.003047815 
 
 330 
 
 + 1.648024 
 
 -1.3961376 +0.822269 - 8.41473 
 
 +0.4984001 
 
 +0.000587075 
 
 Si 
 
 + 18.606089 
 
 -7.3681280 -0.940570 -10.20081 
 
 +0.0113925 
 
 -0.001072047 
 
 2 2 
 
 + 18.606089 
 
 -7.3681260 -0.935706 -10.18975 
 
 +0.0114435 
 
 -0.001067089 
 
 E 
 
 . Rt, cos v 
 Rosmv 
 
 + (cos v + cos )S +T- sec 2 v +lj sin vS, 
 
 Wo sin u 
 
 -2^0 
 a 
 
 
 
 -0.0069216 
 
 + 1.4244790 -0.005314674 
 
 +0.019975127 
 
 2.5832417 
 
 30 
 
 -0.7611745 
 
 + 1.1593980 -0.018148167 
 
 +0.016519714 
 
 2.5492068 
 
 60 
 
 -1.1839523 
 
 +0.5449664 -0.018883467 
 
 +0.003450266 
 
 2.4840909 
 
 90 
 
 -1.2020643 
 
 -0.1220026 -0.009233404 
 
 -0.003406539 
 
 2.4155306 
 
 120 
 
 -0.9266826 
 
 -0.6439879 -0.000621304 
 
 -0.000725630 
 
 2.3617340 
 
 150 
 
 -0.4956901 
 
 -0.9571936 +0.001287534 
 
 +0.005794466 
 
 2.3299411 
 
 180 
 
 -0.0032964 
 
 -1.0618130 -0.002798312 
 
 +0.010517416 
 
 2.3216920 
 
 210 
 
 +0.4917082 
 
 -0.9631868 -0.009486007 
 
 +0.010415471 
 
 2.3371562 
 
 240 
 
 +0.9297081 
 
 -0.6522758 -0.014244152 
 
 +0.005080676 
 
 2.3763511 
 
 270 
 
 + 1.2126214 
 
 -0.1251906 -0.012409750 
 
 -0.002087273 
 
 2.4369424 
 
 300 
 
 + 1.1939212 
 
 +0.5533939 -0.003526734 
 
 -0.002975358 
 
 2.5083988 
 
 330 
 
 +0.7575012 
 
 + 1.1728178 +0.002585060 
 
 +0.008007817 
 
 2.5667361 
 
 2i 
 
 +0.0027764 
 
 +0.1647626 -0.045388643 
 
 +0.035322497 
 
 14.6355085 
 
 2 2 
 
 +0.0029019 
 
 +0.1646422 -0.045404734 
 
 +0.035243656 
 
 14.6355132 
 
 
 sin<f> j4i (<) + coaip 
 
 Bo (c) = O."0000000073. 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 159 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 [de/dt] w = + 324.6318 m' p 2.5113911 
 
 [dx/dfloo = + 201915.56 TO' p 5.3051698 
 
 [di/dt]oo 5236.2608 m' n 3.7190213 
 
 [dtt/dt] m + 126021.28 TO' p 5.1004439 
 
 [dTT/dtlw = + 201981.28 m' p 5.3053112 
 
 [dL/dt] m = +1681713.6 m' p 6.2257520 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'. 
 
 [dg/dfloo = +0.0007954049 
 
 [dx/dfloo = +0.49472856 
 
 [di/di]oo = -0.012829757 
 
 [dO/d<]oo = +0.30877426 
 
 [dr/dfloo = +0.49488961 
 
 [dL/dt] m = +4.1204933 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 loo +0.00080 +0.00079 +0.000795 
 
 e[drldt]ao +0.04618 +0.04614 +0.0461574 
 
 [difdt] m -0.01280 -0.01284 -0.012830 
 
 sin i [dfl/d<]oo +0.00993 +0.00998 +0.009972 
 
 [dL/dt] M +4.117 +4.120493 
 
 NOTES. 
 
 The very close agreement of the sums of the functions near the beginning of the 
 computation is caused by the great circularity of the orbit of Venus. The discrepan- 
 cies increase however as the work proceeds because of the high eccentricity of Mars and 
 the rather large mutual inclination. All terms from the 6th to the llth orders, in- 
 clusive, produce an effect equal to l/30th of the whole in the very small coefficient 
 [de/dt]oo, and 1 /1000th of the whole in [dttfdt} 00 . Yet it is evident that terms of the 
 twelfth and higher orders are wholly inappreciable. 
 
160 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 E 
 
 A 
 
 
 
 2.88085183 
 
 30 
 
 2.95824702 
 
 60 
 
 3.13279096 
 
 90 
 
 3.36510700 
 
 120 
 
 3.59664230 
 
 150 
 
 3.76166099 
 
 180 
 
 3.80855449 
 
 210 
 
 3.72106141 
 
 240 
 
 3.52632175 
 
 270 
 
 3.28390784 
 
 300 
 
 3.06247042 
 
 330 
 
 2.91764743 
 
 Si 
 
 20.00763175* 
 
 S 2 
 
 20.00763169 
 
 ACTION OF THE 
 
 EARTH ON MARS. 
 
 
 B cos f 
 
 B sin 
 
 1000 X g 
 
 -0.8153552 
 
 -1.1018750 
 
 0.34149937 
 
 -0.0872040 
 
 -1.3962265 
 
 0.54832354 
 
 +0.6917421 
 
 -1.2860904 
 
 0.46523043 
 
 + 1.3127653 
 
 -0.8009781 
 
 0.18045398 
 
 + 1.6094626 
 
 -0.0708746 
 
 0.00141289 
 
 + 1.5023341 
 
 +0.7085891 
 
 0.14122581 
 
 + 1.0200854 
 
 + 1.3285566 
 
 0.49646115 
 
 +0.2919341 
 
 + 1.6229077 
 
 0.74082034 
 
 -0.4870119 
 
 + 1.5127725 
 
 0.64368353 
 
 -1.1080347 
 
 + 1.0276597 
 
 0.29704612 
 
 -1.4047322 
 
 +0.2975561 
 
 0.02490366 
 
 -1.2976042 
 
 -0.4819076 
 
 0.06532098 
 
 +0.6141908J 
 
 +0.6800452f 
 
 1.97319103 
 
 +0.6141906 
 
 +0.6800443 
 
 1.97319077 
 
 1.8833414 
 1.9597848 
 2.1332651 
 2.3654511 
 2.5975998 
 2.7632686 
 2.8103218 
 2.7223893 
 2.5269386 
 2.2841910 
 2.0633533 
 1.9197321 
 
 14.0148199 
 14.0148169 
 
 E 
 
 I' 
 
 G 
 
 G' 
 
 G' 
 
 o 
 
 
 
 0.9972292 
 
 1.8831367 
 
 0.9976157 
 
 0.00018178 
 
 
 
 46 
 
 42 
 
 32.02 
 
 30 
 
 0.9981810 
 
 1.9594937 
 
 0.9987522 
 
 0.00028018 
 
 45 
 
 33 
 
 35.15 
 
 60 
 
 0.9992447 
 
 2.1330727 
 
 0.9996552 
 
 0.00021818 
 
 43 
 
 12 
 
 19.69 
 
 90 
 
 0.9993747 
 
 2.3653953 
 
 0.9995068 
 
 0.00007633 
 
 40 
 
 32 
 
 44.90 
 
 120 
 
 0.9987612 
 
 2.5975995 
 
 0.9987622 
 
 0.00000054 
 
 38 
 
 19 
 
 18.09 
 
 150 
 
 0.9981111 
 
 2.7632397 
 
 0.9981913 
 
 0.00005120 
 
 36 
 
 56 
 
 40.43 
 
 180 
 
 0.9979514 
 
 2.8102243 
 
 0.9982259 
 
 0.00017698 
 
 36 
 
 35 
 
 10.16 
 
 210 
 
 0.9983909 
 
 2.7222314 
 
 0.9988211 
 
 0.00027246 
 
 37 
 
 17 
 
 7.37 
 
 240 
 
 0.9991020 
 
 2.5267718 
 
 0.9995236 
 
 0.00025487 
 
 38 
 
 58 
 
 33.58 
 
 270 
 
 0.9994356 
 
 2.2840897 
 
 0.9996671 
 
 0.00013009 
 
 41 
 
 25 
 
 15.69 
 
 300 
 
 0.9988359 
 
 2.0633420 
 
 0.9988593 
 
 0.00001208 
 
 44 
 
 5 
 
 20.00 
 
 330 
 
 0.9976341 
 
 1.9196952 
 
 0.9977050 
 
 0.00003411 
 
 46 
 
 7 
 
 50.45 
 
 2, 
 
 5.9911243 
 
 14.0141470 
 
 5.9926419 
 
 0.00081443 
 
 247 
 
 53 
 
 13.54 
 
 2 2 
 
 5.9911274 
 
 14.0141449 
 
 5.9926435 
 
 0.00084437 
 
 247 
 
 53 
 
 13.98 
 
 * Go 2 + 3o 2 e 2 + 6[a' 2 - Zkaa'ee' cos K] = 20.00763172. 
 t 6[a'V - fcaa'e cos A'] = + 0.6141907. 
 t - 6fcW cos <p' e sin K' = + 0.6800448. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 161 
 
 
 
 ACTION 
 
 OF THE EARTH ON MABS. 
 
 E 
 
 log A' 
 
 log Lo' 
 
 lOgtfo 
 
 log N log P 
 
 logQ 
 
 
 
 0.25724275 
 
 0.60257672 
 
 0.54204483 
 
 0.3085062 0.3612353 
 
 0.5756272 
 
 30 
 
 0.24243067 
 
 0.58425966 
 
 0.52198206 
 
 0.2796590 0.2795065 
 
 0.5094350 
 
 60 
 
 0.21406633 
 
 0.54897788 
 
 0.48324184 
 
 0.2277018 0.1185796 
 
 0.3818936 
 
 90 
 
 0.18497596 
 
 0.51250382 
 
 0.44306081 
 
 0.1727907 9.9374590 
 
 0.2419337 
 
 120 
 
 0.16284593 
 
 0.48455413 
 
 0.41218099 
 
 0.1292685 9.7846780 
 
 0.1268772 
 
 150 
 
 0.15007482 
 
 0.46834273 
 
 0.39423491 
 
 0.1040949 9.6895845 
 
 0.0569033 
 
 180 
 
 0.14686278 
 
 0.16425584 
 
 0.38970669 
 
 0.0998549 9.6665742 
 
 0.0407933 
 
 210 
 
 0.15317152 
 
 0.47227918 
 
 0.39859494 
 
 0.1168787 9.7192209 
 
 0.0805051 
 
 240 
 
 0.16915889 
 
 0.49254542 
 
 0.42101792 
 
 0.1535252 9.8408510 
 
 0.1719333 
 
 270 
 
 0.19422097 
 
 0.52412777 
 
 0.45588060 
 
 0.2048055 0.0114576 
 
 0.3019483 
 
 300 
 
 0.22440852 
 
 0.56187400 
 
 0.49741669 
 
 0.2597585 0.1924850 
 
 0.4426015 
 
 330 
 
 0.24970710 
 
 0.59326698 
 
 0.53185208 
 
 0.3003845 0.3271715 
 
 0.5489966 
 
 2, 
 
 1.17458520 
 
 3.15478399 
 
 2.74560896 
 
 1.1786151 9.9644030 
 
 1.7397260 
 
 2 2 
 
 1.17458104 
 
 3.15478014 
 
 2.74560540 
 
 1.1786131 9.9643999 
 
 1.7397219 
 
 E 
 
 logF 
 
 JV 
 
 J 2 
 
 J, 
 
 F, 
 
 
 
 0.5755790 
 
 0.99892684 
 
 -0.018227992 
 
 +0.028809050 
 
 +0.018467466 
 
 30 
 
 0.5093633 
 
 0.99952648 
 
 -0.022898545 
 
 +0.020515873 
 
 +0.023400805 
 
 60 
 
 0.3818419 
 
 0.99990323 
 
 -0.021379458 
 
 +0.006120062 
 
 +0.021554920 
 
 90 
 
 0.2419173 
 
 0.99967015 
 
 -0.013761084 
 
 -0.010518960 
 
 +0.013424123 
 
 120 
 
 0.1268771 
 
 0.99911770 
 
 -0.001690224 
 
 -0.024941756 
 
 +0.001187862 
 
 150 
 
 0.0568937 
 
 0.99877639 
 
 +0.011675876 
 
 -0.033284760 
 
 -0.011875978 
 
 180 
 
 0.0407608 
 
 0.99892204 
 
 +0.022549333 
 
 -0.033314577 
 
 -0.022266656 
 
 210 
 
 0.0804536 
 
 0.99942130 
 
 +0.027739796 
 
 -0.025024224 
 
 -0.027199994 
 
 240 
 
 0.1718817 
 
 0.99985593 
 
 +0.025696938 
 
 -0.010634071 
 
 -0.025354120 
 
 270 
 
 0.3019193 
 
 0.99982014 
 
 +0.017055202 
 
 +0.006002126 
 
 -0.017223615 
 
 300 
 
 0.4425985 
 
 0.99929729 
 
 +0.004464507 
 
 +0.020427743 
 
 -0.004987052 
 
 330 
 
 0.5489877 
 
 0.99880864 
 
 -0.008398230 
 
 +0.028776417 
 
 +0.008076789 
 
 2, 
 
 1.7395389 
 
 5.99602303* 
 
 + 0.011413104 
 
 -0.013533549 
 
 -0.011397580 
 
 Zo 
 
 1.7395349 
 
 5.99602310 
 
 +0.011413015 
 
 -0.013533528 
 
 -0.011397570 
 
 *Z,(Ji' - G") = 5.99517860. 
 2 2 (J/ - G") = 5.99517873. 
 
162 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 
 
 ACTION OF THE EARTH ON MARS. 
 
 
 
 E 
 
 1000 XF, 
 
 
 n c* "nr 
 
 ft/Q Ofl If 
 
 R<> 
 
 fl 
 
 
 
 -0.13911285 
 
 -2 
 
 .0302096 -0.026171976 +0.10810004 
 
 0.0000000 
 
 -0.018943511 
 
 30 
 
 -0.51397488 
 
 -1 
 
 .9019956 -0.029451014 +0.06531256 
 
 -0.6789837 
 
 -0.021027135 
 
 60 
 
 -0.65316258 
 
 -1 
 
 .6889696 -0.023181583 +0.01388518 
 
 -1.0069217 
 
 -0.015958290 
 
 90 
 
 -0.40694037 
 
 -1 
 
 .4884239 -0.012395858 -0.01871310 
 
 -0.9768538 
 
 -0.008135411 
 
 120 
 
 -0.02612649 
 
 -1 
 
 .3462165 -0.001540204 -0.03342035 
 
 -0.7310608 
 
 -0.000965798 
 
 150 
 
 +0.08995598 
 
 -1 
 
 .2702099 +0.007499119 -0.03789964 
 
 -0.3856685 
 
 +0.004553852 
 
 180 
 
 -0.20223795 
 
 -1 
 
 .2578222 +0.014435140 -0.03668660 
 
 0.0000000 
 
 +0.008665574 
 
 210 
 
 -0.63957154 
 
 -1 
 
 .3082364 +0.019136251 -0.03045222 
 
 +0.3972143 
 
 +0.011620521 
 
 240 
 
 -0.80757787 
 
 -1 
 
 .4237326 +0.020598-119 -0.01635704 
 
 +0.7731558 
 
 +0.012916412 
 
 270 
 
 -0.54879816 
 
 -1 
 
 .6023546 +0.016496265 +0.01146539 
 
 + 1.0516264 
 
 +0.010826511 
 
 300 
 
 -0.11741630 
 
 -1 
 
 .8174946 +0.004601712 +0.05641742 
 
 + 1.0835451 
 
 +0.003167836 
 
 330 
 
 +0.07369507 
 
 -1 
 
 .9931947 -0.012572900 +0.10202141 
 
 +0.7115406 
 
 -0.008976672 
 
 2, 
 
 -1.94563404 
 
 -9 
 
 .5644451 -0.011258492 +0.09193865 
 
 +0.1187184 
 
 -0.011117777 
 
 2 2 
 
 -1.94563390 
 
 -9.5644151 -0.011288137 +0.09173440 
 
 +0.1188753 
 
 -0.011138334 
 
 
 
 
 Ro COS V 
 
 
 
 E 
 
 Ro sin v 
 + (c,osv + cosE)S 
 
 . (r \ . Wa cos u 
 + ( - seo 2 <f + 1 j sin i'.S 
 
 W sin u 
 
 2 RQ 
 
 a 
 
 
 
 -0.0523440 
 
 
 +2.0302096 +0.027794619 
 
 -0.10446568 
 
 3.6817114 
 
 30 
 
 -1.0803151 
 
 
 + 1.5681913 +0.048299056 
 
 -0.04396512 
 
 3.4967331 
 
 60 
 
 -1.5490313 
 
 
 +0.6794308 +0.013659050 
 
 -0.00249569 
 
 3.2204119 
 
 90 
 
 -1.4807799 
 
 
 -0.1636143 -0.017556368 
 
 -0.00647718 
 
 2.9768478 
 
 120 
 
 -1.1074128 
 
 
 -0.7656902 -0.021736270 
 
 -0.02538610 
 
 2.8179922 
 
 150 
 
 -0.5982290 
 
 
 - 1 . 1 202 1 78 - 0.008220822 
 
 -0.03699731 
 
 2.7456152 
 
 180 
 
 -0.0288703 
 
 
 - 1 .2578222 +0.009432839 
 
 -0.03545319 
 
 2.7502731 
 
 210 
 
 +0.5690362 
 
 
 -1.1796147 +0.020504958 
 
 -0.02251409 
 
 2.8278111 
 
 240 
 
 + 1.1509410 
 
 
 -0.8419068 +0.015406346 
 
 -0.00549521 
 
 2.9802545 
 
 270 
 
 + 1.5938315 
 
 
 -0.1824413 -0.011306572 
 
 -0.00190172 
 
 3.2047092 
 
 300 
 
 + 1.6480563 
 
 
 +0.7072282 -0.043121307 
 
 -0.03637964 
 
 3.4654744 
 
 330 
 
 + 1.0579845 
 
 
 + 1.6887208 -0.031341647 
 
 -0.09708793 
 
 3.6643987 
 
 2, 
 
 +0.0613389 
 
 
 +0.6114494 +0.001435277 
 
 -0.20967551 
 
 18.9161075 
 
 2 2 
 
 +0.0615274 
 
 
 +0.6110240 +0.000378605 
 
 -0.20894335 
 
 18.9161151 
 
 
 sin ip |Ai (>) + cos 
 
 f 
 
 B <> = + 0.000000102. 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 163 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 [de/dt] w = + 7024.3393 TO' p 3.8466055 
 [d x /dt] w = + 749340.69 TO' p 5.8746793 
 [di/dt]<n = + 104.61082 TO' p 2.0195766 
 [dtt/dt] w = - - 747594.66 TO' n 5.8736662 
 [dvldt] m = + 748950.76 TO' p 5.8744532 
 [dL/dt] m = +2175235.9 TO' p 6.3375064 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 [<fe/cft]oo = +0.021481158 
 
 [d x /dt] w = +2.2915614 
 
 [dildt] m = +0.00031991074 
 
 [dQ/<ft]oo = -2.2862242 
 
 [dr/dt] m = +2.2903688 
 
 *dLldt] m = +6.6520970 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Neweomb. Method of Gauss. 
 
 [de/dt]oo 
 
 +0.02151 
 
 +o!o2148 
 
 +o!o2148116 
 
 e[dirldt] M 
 
 +0.21276 
 
 +0.21374 
 
 +0.21361818 
 
 [difdt] m 
 
 +0.00030 
 
 +0.00032 
 
 +0.00031991 
 
 sin i [dtt/dt] m 
 
 -0.07391 
 
 -0.07379 
 
 -0.07383093 
 
 [dL/diloo 
 
 +6.638 
 
 
 +6.6520970 
 
 NOTES. 
 
 As in all cases in which Mars is the disturbed body, the gradual increase in the 
 discrepancies in the sums of the functions as the computation proceeds is caused prin- 
 cipally by the large value of e. The greatest effect which is here produced by the 
 inclusion of all terms from the fifth to the eleventh orders occurs with the coefficient 
 [dirldt] QQ and amounts to 0".0007. It is evident that a division into twelve parts is 
 fully sufficient. 
 
164 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF JUPITER ON MARS. 
 
 E 
 
 A 
 
 B cos f 
 
 B sin f 
 
 g 
 
 h 
 
 
 
 29.52014024 
 
 +6.924444 
 
 -4.477639 
 
 1.2637451 
 
 27.008366 
 
 30 
 
 29.73057269 
 
 + 8.555992 
 
 -0.736229 
 
 0.0341655 
 
 27.006182 
 
 60 
 
 29.83402815 
 
 +8.090126 
 
 +3.325863 
 
 0.6972206 
 
 27.007669 
 
 90 
 
 29.81017790 
 
 +5.651672 
 
 +6.620206 
 
 2.7625134 
 
 27.011679 
 
 120 
 
 29.66910846 
 
 + 1.894010 
 
 +8.264081 
 
 4.3047762 
 
 27.014415 
 
 150 
 
 29.44492347 
 
 -2.175995 
 
 +7.817013 
 
 3.8516159 
 
 27.013253 
 
 180 
 
 29.19030067 
 
 -5.467791 
 
 +5.398794 
 
 1.8371936 
 
 27.009343 
 
 210 
 
 28.96977017 
 
 -7.099343 
 
 + 1.657384 
 
 0.1731141 
 
 27.006389 
 
 2-10 
 
 28.84611897 
 
 -6.633477 
 
 -2.404708 
 
 0.3644905 
 
 27.006977 
 
 270 
 
 28.85987156 
 
 -4.195019 
 
 -5.699050 
 
 2.0472288 
 
 27.010168 
 
 300 
 
 29.01103882 
 
 -0.437358 
 
 -7.342925 
 
 3.3985953 
 
 27.012567 
 
 330 
 
 29.25541959 
 
 +3.632647 
 
 -6.895857 
 
 2.9973538 
 
 27.011662 
 
 2, 
 
 176.07073531* 
 
 +4.369955t 
 
 +2.7634661 
 
 11.8660213 
 
 162.059336 
 
 2 2 
 
 176.07073538 
 
 +4.369955 
 
 +2.763467 
 
 11.8660215 
 
 162.059333 
 
 E 
 
 o 
 
 
 
 30 
 
 60 
 
 90 
 
 120 
 
 150 
 
 180 
 
 210 
 
 240 
 
 270 
 
 300 
 
 330 
 
 2, 
 
 I 
 
 G 
 
 G' 
 
 G" 
 
 27.0064605 
 27.0061300 
 27.0066036 
 27.0074648 
 27.0078870 
 27.0074649 
 27.0066097 
 27.0061336 
 27.0064416 
 27.0071624 
 27.0075479 
 27.0071916 
 
 162.0415503 
 162.0415473 
 
 * 6a 2 + SaV + 6[o' ! - 2fcaa'ee' cos A'] = 176.07073528. 
 t 6[a'V - kaa'e cos K] = + 4.369954. 
 t - Qk'aa' cos *>' e sin K' = + 2.763466. 
 
 2.448742 
 2.661358 
 2.763328 
 2.735467 
 2.591662 
 2.368639 
 2.117926 
 1.900349 
 1.77611T 
 1.786671 
 1.935440 
 2.180725 
 
 13.633208 
 13.633209 
 
 2.4695960 
 
 0.0189481 
 
 
 
 17 
 
 39' 
 
 53.68 
 
 2.6618860 
 
 0.0004753 
 
 18 
 
 17 
 
 56.75 
 
 2.7737001 
 
 0.0093077 
 
 18 
 
 43 
 
 15.01 
 
 2.7765210 
 
 0.0368400 
 
 18 
 
 48 
 
 58.35 
 
 2.6581522 
 
 0.0599626 
 
 18 
 
 28 
 
 29.57 
 
 2.4330416 
 
 0.0586151 
 
 17 
 
 39 
 
 45.34 
 
 2.1522663 
 
 0.0316074 
 
 16 
 
 30 
 
 39.82 
 
 1.9039718 
 
 0.0033673 
 
 15 
 
 24 
 
 39.18 
 
 1.7842098 
 
 0.0075644 
 
 14 
 
 55 
 
 27.23 
 
 1.8310752 
 
 0.0413982 
 
 15 
 
 15 
 
 16.64 
 
 2.0032755 
 
 0.0628165 
 
 16 
 
 2 
 
 14.95 
 
 2.2348561 
 
 0.0496603 
 
 16 
 
 53 
 
 32.61 
 
 13.841199!) 
 
 0.1902067 
 
 102 
 
 20 
 
 0.26 
 
 13.8413516 
 
 0.1903562 
 
 102 
 
 20 
 
 8.87 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 165 
 
 ACTION OF JUPITEU ON MARS. 
 
 E 
 
 logtfo 
 
 log L ' 
 
 log No log N log P 
 
 logQ 
 
 
 
 0.03165483 
 
 0.31498612 
 
 0.22325507 8.3476456 5.7990872 
 
 7.1391284 
 
 30 
 
 0.03402451 
 
 0.31811150 
 
 0.22676030 8.3623570 5.8175286 
 
 7.1576473 
 
 60 
 
 0.03565117 
 
 0.32025551 
 
 0.22916442 8.3954324 5.8524487 
 
 7.1929772 
 
 90 
 
 0.03602468 
 
 0.32074766 
 
 0.22971622 8.4366023 5.8931984 
 
 7.2342427 
 
 120 
 
 0.0346975G 
 
 0.31899875 
 
 0.22775524 8.4742980 5.9283892 
 
 7.2695995 
 
 150 
 
 0.03164635 
 
 0.31497492 
 
 0.22324250 8.4991683 5.9492925 
 
 7.2899854 
 
 180 
 
 0.02757619 
 
 0.30960108 
 
 0.21721367 8.5057537 5.9513987 
 
 7.2909894 
 
 210 
 
 0.02396325 
 
 0.30482489 
 
 0.21185335 8.4928484 5.9346402 
 
 7.2731852 
 
 240 
 
 0.02244999 
 
 0.30282273 
 
 0.20960576 8.4633475 5.9029922 
 
 7.2413643 
 
 270 
 
 0.02347170 
 
 0.30417466 
 
 0.21112344 8.4239469 5.8638333 
 
 7.2029262 
 
 300 
 
 0.02598821 
 
 0.30750252 
 
 0.21485868 8.3844577 5.8269721 
 
 7.1668224 
 
 330 
 
 0.02889111 
 
 0.31133796 
 
 0.21916251 8.3560126 5.8027960 
 
 7.1428978 
 
 ?! 
 
 0.17801795 
 
 1.87416671 
 
 1.32185284 0.5709348 5.2612881 
 
 3.3008810 
 
 2-2 
 
 0.17802160 
 
 1.87417159 
 
 1.32185832 0.5709353 5.2612888 
 
 3.3008846 
 
 E 
 
 logF 
 
 li' 
 
 Ji J, 
 
 F 2 
 
 
 
 7.1387522 
 
 27.0207890 
 
 -0.20899481 +0.30968500 
 
 + 5.840160 
 
 30 
 
 7.1576379 
 
 27.0065721 
 
 -0.03630747 -0.02674093 
 
 + 0.960260 
 
 60 
 
 7.1927926 
 
 27.0099958 
 
 +0.15284461 -0.36413185 
 
 - 4.337906 
 
 90 
 
 7.2335125 
 
 27.0290264 
 
 +0.31348417 -0.61207993 
 
 - 8.634698 
 
 120 
 
 7.2684108 
 
 27.0492502 
 
 + 0.40064241 -0.70414548 
 
 -10.778797 
 
 150 
 
 7.2888221 
 
 27.0526590 
 
 +0.38540712 -0.61566134 
 
 -10.195688 
 
 180 
 
 7.2903609 
 
 27.0334483 
 
 +0.26823852 -0.37034175 
 
 - 7.041617 
 
 210 
 
 7.2731181 
 
 27.0094582 
 
 +0.08088102 -0.03392190 
 
 - 2.161717 
 
 240 
 
 7.2112135 
 
 27.0106120 
 
 -0.12278062 +0.30345694 
 
 + 3.136448 
 
 270 
 
 7.2021019 
 
 27.0363209 
 
 -0.28345023 +0.55139880 
 
 + 7.433241 
 
 300 
 
 7.1655732 
 
 27.0521350 
 
 -0.35632057 +0.64347029 
 
 + 9.577337 
 
 330 
 
 7.1419111 
 
 27.0423420 
 
 -0.32638726 +0.55499853 
 
 + 8.991231 
 
 2i 
 
 3.2971031 
 
 162.1762303* 
 
 +0.13362954 -0.18200685 
 
 - 3.604375 
 
 V 
 
 3.2971036 
 
 162.1763786 
 
 +0.13362735 -0.18200677 
 
 - 3.604371 
 
 
 * S,(J,' - G") = 161 
 
 .9860236. 
 
 
 
 
 2(Ji' - G") = 161 
 
 .9860224. 
 
 
 
166 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 E 
 
 30 
 60 
 90 
 120 
 150 
 ISO 
 210 
 240 
 270 
 300 
 330 
 
 Si 
 E 
 
 F, 
 -0.11464084 
 -0.02209150 
 +0.08535418 
 +0.09095508 
 -0.02126155 
 -0.14774803 
 -0.16666131 
 -0.05846157 
 +0.07437981 
 +0.10831696 
 +0.01978452 
 -0.09401607 
 
 -0.12304519 
 -0.12304513 
 
 fto sin v + 
 (cos v + oos E)f><, 
 
 ACTION OF JUPITER ON MARS. 
 # 1000 X -So 1000 X W 1000 X R M 1000 X <S ( "> 
 0.011523729 +0.08005004 +0.4190397 0.000000 +0.05794093 
 0.011964482 +0.01088811 -0.0398940 +4.271140 +0.00777378 
 0.012921829 -0.07058125 -0.5615348 +7.703673 -0.04858841 
 0.014185469 -0.13852471 -1.0407895 +9.309934 -0.09091387 
 0.015433331 -0.17072189 -1.3081943 +8.381048 -0.10705259 
 0.016299442 -0.15776374 -1.2103390 +4.948932 -0.09580229 
 0.016504164 -0.10615120 -0.7376107 0.000000 -0.06372374 
 0.015978601 -0.03427562 -0.0686500 -4.851517 -0.02081393 
 0.014896574 +0.03689409 +0.5347724 -8.089562 +0.02313475 
 0.013593080 +0.09184683 +0.8860645 -8.921148 +0.06027915 
 0.012429971 +0.12131684 +0.9434385 -7.410440 +0.08351498 
 0.011688531 +0.11863615 +0.7635196 -4.172629 +0.08470265 
 
 0.083709598 -0.10919337 -0.7100892 +0.584719 -0.05477408 
 0.083709605 -0.10919298 -0.7100884 +0.584712 -0.05477451 
 
 i 
 
 flo COS V + 
 
 Ir \ . 1000 X Wo cosu 1000 X JTo sin u -2-R,, 
 
 
 
 +0.000160100 
 
 -0.011523729 
 
 +0.10774325 
 
 -0.4049515 
 
 -0.020897865 
 
 30 
 
 +0.006498115 
 
 -0.010046690 
 
 -0.02950189 
 
 +0.0268547 
 
 -0.021996162 
 
 60 
 
 +0.011621455 
 
 -0.005638033 
 
 -0.55239000 
 
 +0.1009292 
 
 -0.024638457 
 
 90 
 
 +0.014136555 
 
 +0.001045999 
 
 -0.97645422 
 
 -0.3602495 
 
 -0.028370938 
 
 120 
 
 +0.012896599 
 
 +0.008458998 
 
 -0.85083667 
 
 -0.9937043 
 
 -0.032306104 
 
 150 
 
 +0.007784436 
 
 +0.014315488 
 
 -0.26253500 
 
 -1.1815227 
 
 -0.035231972 
 
 180 
 
 +0.000212302 
 
 +0.016504164 
 
 +0.18965406 
 
 -0.7128120 
 
 -0.036086942 
 
 210 
 
 -0.007299886 
 
 +0.014215603 
 
 +0.04622538 
 
 -0.0507547 
 
 -0.034538456 
 
 240 
 
 -0.012311629 
 
 +0.008381401 
 
 -0.50369057 
 
 +0.1796589 
 
 -0.031182525 
 
 270 
 
 -0.013542396 
 
 +0.001084104 
 
 -0.87379071 
 
 -0.1469683 
 
 -0.027186159 
 
 300 
 
 -0.011129593 
 
 -0.005518211 
 
 -0.72109467 
 
 -0.6083574 
 
 -0.023700620 
 
 330 
 
 -0.006127613 
 
 -0.009949901 
 
 -0.23455823 
 
 - 0.7265980 
 
 -0.021488839 
 
 Si 
 
 +0.001449234 
 
 +0.010664590 
 
 -2.33061460 
 
 -2.4392371 
 
 -0.168812513 
 
 2,. 
 
 +0.001449211 
 
 +0.010664603 
 
 -2.33061467 
 
 -2.4392385 
 
 -0.168812526 
 
 
 sin v> Mi w + cos <p 
 
 B (c) = - 0.00000000019. 
 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 167 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 [<fe/cB]oo = + 165.70584 m' p 2.2193378 
 [dx/dtln =+13074.175 m' p 4.1164143 
 [di/dt] m = - 268.82366m' n 2.4294675 
 [dQ/dt] w = -- 8712.2760 m' n 3.9401316 
 [dT/dfla, = +13069.631 TO' p 4.1162634 
 [dL/dt] 00 = -19334.282 TO' n 4.2863281 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 [de/dt] m = + 0.15813453 
 [dxldt] m +12.476799 
 
 [di/dt] m = -- 0.25654077 
 
 [dQ/dflflo = - 8.3142000 
 
 [dirfdt] w = +12.472464 
 
 [dL/dt] w = -18.450874 
 
 COMPARISON WITH OTHER RESULTS. 
 
 LeveiTier. Newcomb. Method of Gauss. 
 
 [de/dt}oo 
 
 + 0.15810 
 
 +0.15818 
 
 + 0.1581345 
 
 e[dirfdt] m 
 
 + 1.16323 
 
 + 1.16372 
 
 + 1.1632822 
 
 [di/dfloo 
 
 - 0.25648 
 
 -0.25655 
 
 - 0.2565408 
 
 sin i [dttjdt] w 
 
 - 0.26864 
 
 -0.26850 
 
 - 0.2684974 
 
 [dL/dt]oo 
 
 -18.450 
 
 
 -18.450874 
 
 NOTES. 
 
 The very exact agreement of the final sums shows that for this case the expansion 
 of the perturbing function is highly convergent. The greatest effect of all terms 
 from the sixth to the eleventh orders inclusive occurs with [de/dt] 00 , and amounts to 
 but l/100000th of the whole variation. 
 
 This computation has been twice effected by DR. ARTHUR B. TURNER from the 
 same elements as are here employed. His first computation was made by HILL'S 
 first method, exactly as here, and was presented as a Thesis to the Faculty of the 
 
168 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 Graduate School of the University of Pennsylvania, 1902. The values of the functions 
 in this computation agree practically throughout with those here given, but the last 
 two figures usually differ because eight place logarithms are here employed in certain 
 parts of the work. 
 
 DR. TURNER'S second computation, (A. N., 3065), was made according to the 
 method developed by DR. Louis ARNDT (SO) . The two papers taken together are of 
 high value since they afford a means of comparing the labor and accuracy appertaining 
 to the two very different methods. It is DR. TURNER'S opinion that while the form- 
 ulas of ARNDT'S method are presented in a more symmetric form yet they are less 
 accurate in application than those of DR. HILL. This is confirmed by the circum- 
 stance that the residual from the equation [da/dt}^ = is 300 times larger with the 
 former method than with the latter. 
 
 DR. TURNER'S results from ARNDT'S method, which agree almost exactly with 
 his earlier values and with those here obtained, are as follows: 
 
 [de/diloo = + 0.1581330 
 [dx/<ttjoo = +12.47677 
 [dildt] M = - 0.2565480 
 
 [dti/dt]* = - 8.314194 
 ldr/dt]m = +12.47244 
 [dL/dtloo = -18.45083 
 
 ACTION OF SATURN ON MARS. 
 
 E 
 
 A 
 
 B sin ( 
 
 B cos < 
 
 9 
 
 h 
 
 
 
 92.2326164 
 
 -11.747198 
 
 - 0.835808 
 
 39.461155 
 
 90.708401 T 
 
 30 
 
 93.1075498 
 
 - 13.263728 
 
 + 6.495252 
 
 50.307460 
 
 90.7105920 
 
 60 
 
 94.0543836 
 
 -10.902482 
 
 + 13.616320 
 
 33.990063 
 
 90.7106532 
 
 90 
 
 94.8268069 
 
 - 5.296150 
 
 + 18.619311 
 
 8.020876 
 
 90.7086748 
 
 120 
 
 95.2215463 
 
 + 2.053054 
 
 +20.163679 
 
 1.205319 
 
 90.7070348 
 
 150 
 
 95.1291338 
 
 + 9.175915 
 
 + 17.835613 
 
 24.076867 
 
 90.7078833 
 
 180 
 
 94.5669406 
 
 + 14.163873 
 
 + 12.258914 
 
 57.367408 
 
 90.7108082 
 
 210 
 
 93.6819094 
 
 + 15.680401 
 
 + 4.927852 
 
 70.309758 
 
 90.7131088 
 
 240 
 
 92.7148797 
 
 + 13.319160 
 
 - 2.193216 
 
 50.728826 
 
 90.7125327 
 
 270 
 
 91.9323586 
 
 + 7.712824 
 
 - 7.196207 
 
 17.010933 
 
 90.7095237 
 
 300 
 
 91.5477178 
 
 + 0.363621 
 
 - 8.740573 
 
 0.037809 
 
 90.7066608 
 
 330 
 
 91.6603255 
 
 - 6.759241 
 
 - 6.412507 
 
 13.064630 
 
 90.7063166 
 
 ^ 
 
 560.3380844* 
 
 + 7.250028f 
 
 + 34.2693 16} 
 
 182.790580 
 
 544.256090(1 
 
 2 2 
 
 560.3380840 
 
 + 7.250021 
 
 +34.269314 
 
 182.790524 
 
 544.2560991 
 
 * 6d 2 + SaV + 6[a' 2 - 
 t Gfc'aa' cos if' e sin 
 \ 6[a'V - kaa'e cos A'] 
 
 Ikaa'ee' cos A'] = + 500.3380843. 
 A' = + 7.250024. 
 = + 34.269314. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 169 
 
 E 
 
 ACTION OF SATUKN ON MARS. 
 G G' 
 
 G" 
 
 
 30 
 60 
 90 
 120 
 150 
 180 
 210 
 240 
 270 
 300 
 330 
 
 Zi 
 
 z, 
 
 E 
 
 1.2382581 
 2.1110005 
 3.0577732 
 3.8321748 
 4.2285543 
 4.1352933 
 3.5701752 
 2.6828133 
 1.7163897 
 0.9368777 
 0.5550998 
 0.6680516 
 
 14.3662501 
 14.3662411 
 
 log Ko 
 
 90.70353S2 1.5278682 0.2847474 8 
 90.7043316 2.3529760 0.2357148 9 
 90.7063778 3.1798913 0.1178425 10 
 90.7076570 3.8561238 0.0229312 11 
 90.7068811 4.2318481 0.0031400 12 
 90.7048170 4.2015371 0.0631774 12 
 90.7035495 3.7462608 0.1688274 11 
 90.7043024 2.9540530 0.2624034 10 
 90.7062481 2.0020241 0.2793497 9 
 90.7074346 1.1081936 0.1692270 6 
 90.7066561 0.5558543 0.0007499 4 
 90.7047169 0.8409314 0.1712800 6 
 
 544.2332508 15.2437468 0.8546569 57 
 544.2332595 15.3138149 0.9247338 57 
 
 log LO' log No log N log I' 
 
 6 50.425 
 42 47.930 
 59 5.400 
 55 59.135 
 28 42.340 
 31 7.643 
 58 46.967 
 50 18.088 
 6 39.626 
 48 32.560 
 29 34.176 
 3 29.499 
 
 9 38.934 
 52 14.855 
 
 logQ 
 
 
 
 0.00656257 
 
 0.28174175 
 
 0.18592136 
 
 7.5317335 
 
 3.8955043 
 
 5.7586694 
 
 30 
 
 0.00942345 
 
 0.28554606 
 
 0.19019807 
 
 7.5468281 
 
 3.9148639 
 
 5.7782711 
 
 60 
 
 0.01207449 
 
 0.28906810 
 
 0.19415617 
 
 7.5819826 
 
 3.9546474 
 
 5.8179375 
 
 90 
 
 0.01427105 
 
 0.29198398 
 
 0.19743287 
 
 7.6263318 
 
 4.0028084 
 
 5.8660110 
 
 120 
 
 0.01561994 
 
 0.29377356 
 
 0.19944338 
 
 7.6674180 
 
 4.0458810 
 
 5.9092061 
 
 150 
 
 0.01572227 
 
 0.29390930 
 
 0.19959586 
 
 7.6949830 
 
 4.0730267 
 
 5.9366461 
 
 180 
 
 0.01438391 
 
 0.29213375 
 
 0.19760114 
 
 7.7028806 
 
 4.0781506 
 
 5.9420499 
 
 210 
 
 0.01175197 
 
 0.28863978 
 
 0.19367515 
 
 7.6895882 
 
 4.0604630 
 
 5.9243808 
 
 240 
 
 0.00828439 
 
 0.28403181 
 
 0.18849593 
 
 7.6581064 
 
 4.0241928 
 
 5.8876296 
 
 270 
 
 0.00461510 
 
 0.27914998 
 
 0.18300711 
 
 7.6156278 
 
 3.9778730 
 
 5.8401825 
 
 300 
 
 0.00200564 
 
 0.27567456 
 
 0.17909843 
 
 7.5727522 
 
 3.9331412 
 
 5.7942078 
 
 330 
 
 0.00365091 
 
 0.27786617 
 
 0.18156337 
 
 7.5415145 
 
 3.9024823 
 
 5.7646287 
 
 V 
 
 ^-1 
 
 0.05893094 
 
 1.71642353 
 
 1.14471671 
 
 5.7148732 
 
 3.9315172 
 
 5.1097001 
 
 2 2 
 
 0.05943475 
 
 1.71709527 
 
 1.14547243 
 
 5.7148733 
 
 3.9315172 
 
 5.1101201 
 
170 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 ACTION OF SATUKN 
 
 ON MARS. 
 
 
 
 E 
 
 logF 
 
 /' 
 
 J'Z 
 
 J 3 
 
 F 2 
 
 F, 
 
 
 
 5.7569714 
 
 90.9876364 
 
 -0.64905242 
 
 +0.2442109 
 
 + 59.776028 
 
 -2.2333582 
 
 30 
 
 5.7768667 
 
 90.9027545 
 
 -0.80993896 
 
 -1.6448282 
 
 + 67.492953 
 
 -2.2386206 
 
 60 
 
 5.8172358 
 
 90.7031717 
 
 -0.67748734 
 
 -3.0908929 
 
 +55.477667 
 
 -1.0572468 
 
 90 
 
 5.8658745 
 
 90.5722076 
 
 -0.29321392 
 
 -3.7064154 
 
 +26.949646 
 
 +0.0320703 
 
 120 
 
 5.9091874 
 
 90.5928360 
 
 +0.18262899 
 
 -3.3264146 
 
 -10.447038 
 
 -0.2313819 
 
 150 
 
 5.9362703 
 
 90.7254544 
 
 +0.58278840 
 
 -2.0527628 
 
 -46.691968 
 
 -1.7836963 
 
 180 
 
 5.9410456 
 
 90.8717164 
 
 +0.80450796 
 
 -0.2268316 
 
 -72.073367 
 
 -3.2467872 
 
 210 
 
 5.9228190 
 
 90.9409399 
 
 +0.82304924 
 
 + 1.6620728 
 
 -79.790278 
 
 -3.2597888 
 
 240 
 
 5.8859649 
 
 90.8876171 
 
 +0.67261531 
 
 +3.1078694 
 
 -67.775016 
 
 -1.8125358 
 
 270 
 
 5.8391722 
 
 90.7221965 
 
 +0.40924575 
 
 +3.7232565 
 
 -39.246977 
 
 -0.2549584 
 
 300 
 
 5.7942033 
 
 90.5707339 
 
 +0.07363159 
 
 +3.3433908 
 
 - 1.850296 
 
 +0.0267586 
 
 330 
 
 5.7636057 
 
 90.8202137 
 
 -0.30508990 
 
 +2.0700081 
 
 +34.394639 
 
 -1.0495556 
 
 Si 
 
 5.1046084 
 
 544.6137115* 
 
 +0.40684409 
 
 +0.0513320 
 
 -36.892022 
 
 -8.5545513 
 
 2 2 
 
 5.1046084 
 
 544.6837666 
 
 +0.40684061 
 
 +0.0513310 
 
 -36.891985 
 
 -8.5545494 
 
 E 
 
 R* 
 
 1000000 X So 
 
 1000 X W a 
 
 1000 X B ( "> 
 
 100000 X S ( "> 
 
 1000 X[flo sin v+ 
 (cos v-\- cos E) So] 
 
 
 
 0.0017097618 
 
 + 9.903284 
 
 +0.01219946 
 
 0.0000000 
 
 +0.7168085 
 
 +0.0198066 
 
 30 
 
 0.0017745243 
 
 + 7.025377 
 
 -.0.10023833 
 
 +0.6334785 
 
 +0.5015908 
 
 +0.9730086 
 
 60 
 
 0.0019262900 
 
 + 5.499323 
 
 -0.20387007 
 
 + 1.1484062 
 
 +0.3785755 
 
 + 1.7472855 
 
 90 
 
 0.0021375714 
 
 + 5.593738 
 
 -0.27213044 
 
 + 1.4028896 
 
 +0.3671175 
 
 +2.1277319 
 
 120 
 
 0.0023569201 
 
 + 3.205770 
 
 -0.27013289 
 
 + 1.2799221 
 
 +0.2010205 
 
 + 1.9382857 
 
 150 
 
 0.0025168546 
 
 - 4.917236 
 
 -0.17936963 
 
 +0.7641820 
 
 -0.2986000 
 
 + 1.1679255 
 
 180 
 
 0.0025606191 
 
 - 16.044422 
 
 -0.02369074 
 
 0.0000000 
 
 -0.9631644 
 
 +0.0320888 
 
 210 
 
 0.0024720520 
 
 -22.805150 
 
 +0.13539869 
 
 -0.7505788 
 
 -1.3848465 
 
 -1.0986739 
 
 240 
 
 0.0022843230 
 
 -19.928906 
 
 +0.23709997 
 
 -1.2404980 
 
 -1.2496588 
 
 -1.8606368 
 
 270 
 
 0.0020608533 
 
 - 9.038484 
 
 +0.25685358 
 
 -1.3525394 
 
 -0.5931965 
 
 -2.0510270 
 
 300 
 
 0.0018652798 
 
 + 2.997952 
 
 +0.20817962 
 
 -1.1120336 
 
 +0.2063802 
 
 -1.6842325 
 
 330 
 
 0.0017413378 
 
 + 9.774776 
 
 +0.11927113 
 
 -0.6216313 
 
 +0.6978895 
 
 -0.9263634 
 
 2, 
 
 0.0127031938 
 
 - 14.366999 
 
 -0.04021465 
 
 +0.0757967 
 
 -0.7100385 
 
 +0.1925973 
 
 2 
 
 0.0127031934 
 
 -14.366879 
 
 -0.04021500 
 
 +0.0758006 
 
 -0.7100452 
 
 +0.1926017 
 
 
 * 2,(J,' - G") = 
 
 543.7590546. 
 
 
 
 
 
 
 2 Z (J,' - G") = 
 
 543.7590328. 
 
 
 
 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 171 
 
 
 1000 X |~- flocosv 
 L 
 
 
 
 f 
 
 E 
 
 ( \ -i 
 
 1000 X W cos u 
 
 1000 X W sin u 
 
 1000X-2-flo 
 
 
 - sec 2 ip + 1 1 sin vSo 
 
 
 
 a 
 
 
 
 -1.7097618 
 
 +0.00313672 
 
 -0.01178931 
 
 - 3.1005914 
 
 30 
 
 -1.4844381 
 
 -0.07412689 
 
 +0.06747540 
 
 - 3.2623831 
 
 60 
 
 -0.8120508 
 
 -0.20054993 
 
 +0.03664321 
 
 - 3.6729182 
 
 90 
 
 +0.2105546 
 
 -0.25530900 
 
 -0.09419278 
 
 - 4.2751426 
 
 120 
 
 + 1.3414125 
 
 -0.17569179 
 
 -0.20519295 
 
 - 4.9336670 
 
 150 
 
 + 2.2292256 
 
 -0.03890712 
 
 -0.17509913 
 
 - 5.4402943 
 
 180 
 
 +2.5606191 
 
 +0.00609135 
 
 -0.02289425 
 
 - 5.5988851 
 
 210 
 
 + 2.2161495 
 
 -0.09117052 
 
 +0.10010366 
 
 - 5.3434506 
 
 240 
 
 + 1.3285850 
 
 -0.22331936 
 
 +0.07965468 
 
 - 4.7817011 
 
 270 
 
 +0.2102889 
 
 -0.25329570 
 
 -0.04260337 
 
 - 4.1217062 
 
 300 
 
 -0.8010985 
 
 -0.15911711 
 
 -0.13424046 
 
 - 3.5565885 
 
 330 
 
 -1.4740745 
 
 -0.03664087 
 
 -0.11350351 
 
 - 3.2013711 
 
 38, 
 
 + 1.9077055 
 
 -0.74945012 
 
 -0.25781908 
 
 -25.6443513 
 
 2 2 
 
 + 1.9077060 
 
 -0.74945010 
 
 -0.25781973 
 
 -25.6443479 
 
 sin 
 
 + cos <p (c) = + 0.000000000031. 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 [deldt] m 
 [d*/dt] w 
 [di/dt] m 
 
 = + 22.022051 m' 
 = +2338.7360 TO' 
 86.444970 TO' 
 - 920.85894 TO' 
 [dw/dt] w = +2338.2557 m' 
 -2935.3283 TO' 
 
 log coeff. 
 p 1.3428578 
 p 3.3689812 
 n 1.9367398 
 n 2.9641931 
 p 3.3688920 
 n 3.4676567 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 = +0.0062891406 
 = +0.66790508 
 [di/dt] M = -0.024687281 
 [dtt/dt] M = -0.26298236 
 [d7r/d<]oo = +0.66776785 
 = -0.83828212 
 
172 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt] w +0.00627 +0.00629 +0.0062891 
 
 e[dw/dt] w +0.06226 +0.06226 +0.0622814 
 
 [di/dt] m -0.02467 -0.02468 -0.0246873 
 
 sin i [dtt/dt] m -0.00852 -0.00849 -0.0084927 
 
 [dL/dt} w -0.838 -0.8382821 
 
 NOTES. 
 
 As in other similar cases, the great disagreement of the sums of the functions 
 near the beginning of the computation arises principally from the term a' 2 e', but the 
 remarkably close agreement of the final sums shows that the expansion of the per- 
 turbing function for this case is very convergent. The greatest effect produced by all 
 terms from the sixth to the eleventh orders inclusive here occurs with [de/dt] 00 and 
 amounts to but l/100000th of the value of this coefficient. 
 
 DR. SAMUEL G. BARTON has effected this computation from the same elements 
 as are here employed, making use of the formulas developed by DR. ARNDT (SO) . (A 
 Thesis presented to the Faculty of the Graduate School of the University of Pennsylvania, 
 1906}. The results* obtained by him are as follows: 
 
 [de/dt] m = +0.0062897 
 
 e[dw/dt] w = +0.0622817 
 
 [di/dl] w = -0.0246873 
 
 sin i [dtt/dt] M = -0.0084927 
 
 [dL/dt] w = -0.8382857 
 
 The agreement is thus practically exact. 
 
 It is the conclusion of DR. BARTON that in spite of the greater symmetry of the 
 equations employed in the method of ARNDT, computations effected by them are 
 somewhat less accurate than when the methods of HILL are employed. His residual 
 arising from the equation derived from the constancy of the major axis is eight times 
 greater than that here obtained. (See the notes to the computation of the action of 
 Jupiter on Mars, where it is shown that DR. TURNER came to the same conclusion.) 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 173 
 
 ACTION OF URANUS ON MARS. 
 
 E 
 
 A 
 
 B cos t B sin e g 
 
 h 
 
 
 
 367.8110283 
 
 - 8.331256 + 6.828502 37.81310 
 
 367.49598 
 
 45 
 
 368.2057075 
 
 - 5.369694 -15.243795 188.44191 
 
 367.49649 
 
 90 
 
 370.1698856 
 
 + 12.404411 -28.797937 672.53375 
 
 367.50044 
 
 135 
 
 372.5672534 
 
 +34.579222 -25.894092 543.74175 
 
 367.50030 
 
 180 
 
 373.9791848 
 
 + 48.165033 - 8.233286 54.97149 
 
 367.49639 
 
 225 
 
 373.5643097 
 
 +45.203469 +13.839014 155.31082 
 
 367.49635 
 
 270 
 
 371.5799361 
 
 +27.429370 +27.393154 608.52085 
 
 367.49979 
 
 315 
 
 369.2027636 
 
 + 5.254555 +24.489302 486.34483 
 
 367.49949 
 
 Zi 
 
 1483.5400347* 
 
 +79.667558f - 2.809567J: 1373.83919 
 
 1469.99259 
 
 2 2 
 
 1483.5400342 
 
 +79.667552 - 2.809571 1373.83931 
 
 1469.99263 
 
 
 * 4a 2 + 2a 2 c + 4[a' 2 - 
 
 2kaa'ee' cos K] = 1483.5400348. 
 
 
 
 t 4[a' e' kaa'e cos A'] 
 
 = + 79.667564. 
 
 
 | - 4fcW cos ?' e sin A" = - 2.809567. 
 
 E 
 
 I 
 
 G G' G" 
 
 8 
 
 
 
 
 O 
 
 1 II 
 
 
 
 - 0.49589 
 
 367.49570 0.157536 0.653148 2 
 
 41 22.746 
 
 45 
 
 - 0.10173 
 
 367.49510 0.667671 0.768004 3 
 
 34 47.145 
 
 90 
 
 + 1.85851 
 
 367.49543 2.57438(1 0.710869 5 
 
 25 12.441 
 
 135 
 
 + 4.25601 
 
 367.49622 4.582935 0.322847 6 
 
 37 54.392 
 
 180 
 
 + 5.67185 
 
 367.49598 5.698515 0.026250 7 
 
 10 10.482 
 
 225 
 
 + 5.25702 
 
 367.49519 5.337362 0.079182 6 
 
 58 20.716 
 
 270 
 
 + 3.26920 
 
 367.49525 3.718991 0.445245 6 
 
 6 25.010 
 
 315 
 
 + 0.89233 
 
 367.49588 1.682505 0.786567 4 
 
 41 47.833 
 
 Si 
 
 + 10.30367 
 
 1469.98230 12.149421 1.835512 21 
 
 23 10.679 
 
 2 2 
 
 + 10.30363 
 
 1469.98238 12.270473 1.956599 21 
 
 * 
 
 52 50.086 
 
 
 
 ACTION OF URANUS ON MARS. 
 
 
 E 
 
 log A, 
 
 logLo' logATo log AT logP 
 
 logQ 
 
 
 
 0.00071814 
 
 0.27395868 0.17716832 6.6153321 1.7572439 
 
 4.2264770 
 
 45 
 
 0.00127262 
 
 0.27469773 0.17799967 6.6411665 1.7838479 
 
 4.2533081 
 
 90 
 
 0.00292086 
 
 0.2768938-1 0.18046982 6.7024753 1.8471866 
 
 4.3168538 
 
 135 
 
 0.00437720 
 
 0.27883326 0.18265093 6.7600912 1.9076561 
 
 4.3771079 
 
 180 
 
 0.00511859 
 
 0.27982020 0.18376077 6.7833377 1.9325907 
 
 4.4018149 
 
 225 
 
 0.00484000 
 
 0.27944936 0.18334377 6.7609874 1.9097463 
 
 4.3789860 
 
 270 
 
 0.00371007 
 
 0.27794494 0.18165196 6.7037350 1.8501248 
 
 4.3196094 
 
 315 
 
 0.00219200 
 
 0.27592287 0.17937772 6.6423516 1.7859125 
 
 4.2555483 
 
 2, 
 
 0.01246766 
 
 1.10861766 0.72305087 6.804880T 7.3871460 
 
 7.2647550 
 
 2 2 
 
 0.01268182 
 
 1.10890322 0.72337209 6.8048967 7.3871627 
 
 7.2649502 
 
174 
 
 THE SECULAR VARIATIONS OP THE ELEMENTS 
 
 
 
 ACTION OF UBANUS ON MARS. 
 
 
 
 E 
 
 logF 
 
 Jl' J, 
 
 J, 
 
 F, 
 
 
 
 4.2255130 
 
 368.030126 +0.3910957 
 
 +5.9910041 
 
 -117.85624 
 
 45 
 
 4.2521750 
 
 368.257943 -0.6874254 
 
 + 1.3122864 
 
 +263.09961 
 
 90 
 
 4.3158056 
 
 368.150829 -1.4287110 
 
 -4.4833753 
 
 +497.03672 
 
 135 
 
 4.3766319 
 
 367.663776 -1.2456453 
 
 -8.0008361 
 
 +446.91789 
 
 180 
 
 4.4017762 
 
 367.403228 -0.3170169 
 
 -7.1797383 
 
 + 142.10202 
 
 225 
 
 4.3788692 
 
 367.559090 +0.6967658 
 
 -2.5011942 
 
 -238.85387 
 
 270 
 
 4,3189529 
 
 367.911230 +1.2249811 
 
 +3.2942931 
 
 -472.79092 
 
 315 
 
 4.2543883 
 
 368.137688 +1.1030033 
 
 +6.8119278 
 
 -422.67209 
 
 Zi 
 
 7.2620477 
 
 1471.495412* -0.1296511 
 
 -2.3778169 
 
 + 48,19158 
 
 2, 
 
 7.2620644 
 
 1471.618496 -0.1333016 
 
 -2.3778161 
 
 + 48.49154 
 
 E 
 
 F 3 
 
 1000 X fto 1000000 X So 1000000 X W 
 
 t 1000 X -K (n) 
 
 1000000 X S (B > 
 
 
 
 + 1.1362797 
 
 0.2059026 -0.0165525 +10.07610 
 
 0.0000000 
 
 -0.0119808 
 
 45 
 
 - 5.0397791 
 
 0.2189646 +0.3708684 + 2.31469 
 
 +0.1087907 
 
 +0.2605870 
 
 90 
 
 - 8.3838519 
 
 0.2523915 +0.5397194 - 9.33604 
 
 +0.1656447 
 
 +0.3542182 
 
 135 
 
 - 1.8404722 
 
 0.2886690 +0.6481209 -19.05928 
 
 +0.1256756 
 
 +0.3990450 
 
 180 
 
 + 1.6518880 
 
 0.3050752 +0,4171470 -18.09445 
 
 0.0000000 
 
 +0.2504179 
 
 225 
 
 - 5.0490913 
 
 0.2898966 -0.2732646 - 6.02536 
 
 -0.1262101 
 
 -0.1682478 
 
 270 
 
 - 8.9126276 
 
 0.2535451 -0.7948804 + 6.80306 
 
 -0.1664018 
 
 -0.5216806 
 
 315 
 
 - 2.5789640 
 
 0.2193839 -0.6003932 +12.22078 
 
 -0.1089990 
 
 -0.4218605 
 
 2, 
 
 -14.5083118 
 
 1.0169144 +0.1454335 -10.55133 
 
 -0.0007571 
 
 +0.0709747 
 
 S, 
 
 -14.5083066 
 
 1.0169141 +0.1453315 -10.54917 
 
 -0.0007428 
 
 +0.0695237 
 
 E 
 
 1000 X [flo sin v + 
 (cos v + cos E)So] 
 
 1000 xl-flocosv + 
 / r x -, 1000000 X W cos u 1000000 X T^o sin w 
 ( sec 8 <p + 1 J sin vSo 1 
 
 1000 X -2 ^flo 
 
 
 
 -0.00003310 
 
 -0.20590256 +2.590763 
 
 - 9.737343 
 
 -0.37339681 
 
 45 
 
 +0.16554695 
 
 -0.14335624 +2.077126 
 
 1.021440 
 
 -0.40904755 
 
 90 
 
 +0.25124093 
 
 +0.02461951 -8.758941 
 
 - 3.231492 
 
 -0.50478291 
 
 135 
 
 +0.18971131 
 
 +0.21763709 -8.485224 
 
 -17.006255 
 
 -0.61541380 
 
 180 
 
 -0.00083429 
 
 +0.30507521 +4.652436 
 
 -17.486113 
 
 -0.66705785 
 
 225 
 
 -0.19106859 
 
 + 0.21804505 +5.009009 
 
 - 3.348852 
 
 -0.61803079 
 
 270 
 
 -0.25236576 
 
 +0.02523742 -6.708828 
 
 1.128399 
 
 -0.50709023 
 
 315 
 
 -0.16617611 
 
 -0.14329578 -6.836536 
 
 -10.129621 
 
 -0.40983085 
 
 Si 
 
 -0.00199222 
 
 +0.14902958 -8.224570 
 
 -31.583347 
 
 -2.05232780 
 
 2 2 
 
 -0.00198644 
 
 +0.14903012 -8.235625 
 
 -31.566168 
 
 -2.05232299 
 
 
 sin if \A i '*' + cos ip 
 
 BO M = - 0.0000000000013. 
 
 
 
 * 2,(Ji' - G") = 1469.659900. 
 
 2i(Ji - G") = 1469.661897. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 175 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 [de/dt] w = - o'.341 19354m' n 9.5330008 
 [dx/dt]w = +274.05283 TO' p 2.4378343 
 [di/dt] w = - 1.4239452 m' n 0.1534933 
 [dn/dfe]oo = -169.16430 TO' n 2.2283087 
 [dTT/dt] M = +273.96460 TO' p 2.4376945 
 [dL/dt] m = -352.43262 TO' n 2.5470761 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'. 
 
 [de/dt] w = -o!o00014964631 
 
 [dx/dt]w = +0.012019862 
 
 [di/dtlw = -0.000062453743 
 
 = -0.0074194879 
 
 = +0.012015994 
 
 [dL/dt] w = -0.015457573 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt]m -0.00001 -0.00001 -0.000014964631 
 
 e[dirfdt] m +0.00112 +0.00112 +0.0011207080 
 
 [di/dt]m -0.00007 -0.00006 -0.000062453743 
 
 sin f [da/dtfo -0.00023 -0.00024 -0.00023960370 
 
 [dL/dt] w -0.015 -0.015457573 
 
 NOTES. 
 
 The greatest error produced in this case by a division into only four parts occurs 
 with the coefficient [dx/dt] 00 and amounts to but 0". 0000001. It is evident that, 
 notwithstanding the disagreement of the sums of the functions in the first part of 
 the computation, a division into eight parts is fully sufficient. 
 
176 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 
 ACTION OF NEPTUNE ON MARS. 
 
 
 E 
 
 A B cos e B sin e g 
 
 ft 
 
 
 
 906.38891911 +21.92636 -39.01745 99.36880 
 
 904.17365 
 
 45 
 
 906.94215394 +47.61371 -15.38170 15.44333 
 
 904.17339 
 
 90 
 
 907.26271705 +49.05172 +19.61543 25.11471 
 
 904.17633 
 
 135 
 
 907.17710690 +25.39796 +45.47311 134.97137 
 
 904.17696 
 
 180 
 
 906.72119273 - 9.49146 +47.04426 144.45927 
 
 904.17395 
 
 225 
 
 906.14776209 -35.17881 +23.40851 35.76678 
 
 904.17317 
 
 270 
 
 905.80700342 -36.61681 -11.58863 8.76590 
 
 904. 17576 
 
 315 
 
 905.91280912 -12.96307 -37.44631 91.52721 
 
 904.17627 
 
 2i 
 
 3626.17983231* +24.86981t +16.05361J 277.70868 
 
 3616.69969 
 
 S 2 
 
 3626.17983205 +24.86979 +16.05361 277.70869 
 
 3616.69979 
 
 E 
 
 1 G G' G" 
 
 e 
 
 
 
 
 i tt 
 
 
 
 2.15000 904.17353 2.20007 0.049953 2 
 
 51 33.458 
 
 45 
 
 2.70349 904.17337 2.70981 0.006303 3 
 
 8 30.703 
 
 90 
 
 3.02112 904.17630 3.03031 0.009166 3 
 
 19 25.738 
 
 135 
 
 2.93488 904.17680 2.98505 0.050008 3 
 
 19 16.746 
 
 180 
 
 2.48197 904.17377 2.54493 0.062779 3 
 
 4 42.112 
 
 225 
 
 1.90932 904.17313 1.92986 0.020498 2 
 
 39 43.146 
 
 270 
 
 1.56597 904.17575 1.57215 0.006167 2 
 
 23 40.258 
 
 315 
 
 1.67127 904.17616 1.72990 0.058516 2 
 
 32 56.175 
 
 2, 
 
 9.21906 3616.69935 9.34746 0.128065 11 
 
 39 21.566 
 
 S 2 
 
 9.21896 3616.69945 9.35462 0.135325 11 
 
 40 26.770 
 
 
 ACTION OF NEPTUNE ON MARS. 
 
 
 E 
 
 log Kn log LO' log Na log N log P 
 
 logQ 
 
 
 
 0.00081164 0.27408332 0.17730851 6.0300471 0.3915788 
 
 3.2510798 
 
 45 
 
 0.00098011 0.27430787 0.17756111 6.0560291 0.4178275 
 
 3.2773355 
 
 90 
 
 0.00109700 0.27446368 0.17773639 6.1154021 0.4773508 
 
 3.3368810 
 
 135 
 
 0.00109536 0.27446148 0.17773391 6.1708446 0.5327514 
 
 3.3923012 
 
 180 
 
 0.00094087 0.27425556 0.17750228 6.1926623 0.5543538 
 
 3.4138826 
 
 225 
 
 0.00070344 0.27393908 0.17714626 6.1704766 0.5318928 
 
 3.3913614 
 
 270 
 
 0.00056913 0.27376004 0.17694484 6.1148768 0.4761252 
 
 3.3355658 
 
 315 
 
 0.00064494 0.27386109 0.17705853 6.0556543 0.4169531 
 
 3.2764317 
 
 s, 
 
 0.00341864 1.09656260 0.70949202 4.4529882 1.8994085 
 
 3.3374091 
 
 2 2 
 
 0.00342385 1.09656952 0.70949981 4.4530046 1.8994248 
 
 3.3374297 
 
 * 4a 2 + 2aV + 4[a' 2 - Ikaa'ee' cos K] = 3626.17983218. 
 t 4[a'V - kaa'e cos K] = + 24.869793. 
 t - 4fc'aa' cos <p' e sin K' = + 16.05361. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 177 
 
 
 ACTION OF NEPTUNE ON MAES. 
 
 
 
 E 
 
 log V JY J, 
 
 J, 
 
 Ft 
 
 
 
 3.2510498 903.898094 +0.3009875 
 
 + 15.528734 
 
 +299.48600 
 
 45 
 
 3.2773317 903.965295 -0.6661748 
 
 - 12.963466 
 
 + 118.06523 
 
 90 
 
 3.3368755 902.840060 -0.3701497 
 
 -34.790528 
 
 -150.56206 
 
 135 
 
 3.3922712 902.874950 +0.9173304 
 
 -37.166389 
 
 -349.03766 
 
 180 
 
 3.4138449 903.910920 +1.0320306 
 
 -18.699362 
 
 -361.09725 
 
 225 
 
 3.3913491 904.099942 -0.1708692 
 
 + 9.792741 
 
 -179.67656 
 
 270 
 
 3.3355621 903.071970 -0.8044972 
 
 +31.619710 
 
 + 88.95077 
 
 315 
 
 3.2763966 902.764465 +0.0451253 
 
 +33.995664 
 
 +287.42636 
 
 2, 
 
 3.3373322 3613.721044* +0.1583712 
 
 - 6.341446 
 
 -123.22254 
 
 2 2 
 
 3.3373486 3613.704652 +0.1254117 
 
 - 6.341450 
 
 - 123.22263 
 
 E 
 
 F, 1000 X Ro 1000000 X S 1000000 X TFo 
 
 1000 X fl ( "> 
 
 1000000 X S<"> 
 
 
 
 -10.033551 0.05357158 +0.12743633 +2.7656535 
 
 0.00000000 
 
 +0.09223957 
 
 45 
 
 - 4.253985 0.05690845 -0.09526013 -2.4561209 
 
 +0.02827449 
 
 -0.06693360 
 
 90 
 
 + 2.323772 0.06500834 -0.12559190 -7.5560767 
 
 +0.04266503 
 
 -0.08242607 
 
 135 
 
 - 5.662284 0.07385617 +0.10733754 -9.1730340 
 
 +0.03215420 
 
 +0.06608723 
 
 180 
 
 -14.586465 0.07791030 +0.13821863 -4.8544449 
 
 0.00000000 
 
 +0.08297417 
 
 225 
 
 - 7.712269 0.07407227 -0.10322243 +2.4086872 
 
 -0.03224828 
 
 -0.06355357 
 
 270 
 
 + 1.985936 0.06494182 -0.14759132 +6.8479047 
 
 -0.04262137 
 
 -0.09686429 
 
 315 
 
 - 2.681773 0.05659771 +0.08359968 +6.4235154 
 
 -0.02812010 
 
 +0.05874051 
 
 2, 
 
 -20.310308 0.26143204 -0.00752826 -2.7969634 
 
 +0.00004366 
 
 -0.00407662 
 
 2 2 
 
 -20.310311 0.26143460 -0.00754534 -2.7969523 
 
 +0.00006031 
 
 -0.00565943 
 
 E 
 
 1 000 V 1 /? rrm v 
 lOOOXtflosinv ^L 1000000 
 
 1000000 
 
 1000 X 2 T -R 
 
 
 + (oos i)+cos E)So] fr 2 \ . , cl XWaCosu 
 
 X Wo sin u 
 
 a 
 
 
 
 +0.000254873 -0.053571580 + 0.7111033 
 
 - 2.6726716 
 
 -0.09715013 
 
 45 
 
 +0.042763861 -0.037538560 - 2.2040406 
 
 + 1.0838516 
 
 -0.10631064 
 
 90 
 
 +0.064736687 +0.005812013 - 7.0890057 
 
 - 2.6153925 
 
 -0.13001668 
 
 135 
 
 +0.048623034 +0.055602449 - 4.0838509 
 
 - 8.2138132 
 
 -0.15745404 
 
 180 
 
 -0.000276437 +0.077910304 + 1.2481721 
 
 - 4.6912376 
 
 -0.17035369 
 
 225 
 
 -0.048771753 +0.055759058 - 2.0023922 
 
 + 1.3387309 
 
 -0.15791475 
 
 270 
 
 -0.064644985 +0.006352181 - 6.7530485 
 
 - 1.1358370 
 
 -0.12988365 
 
 315 
 
 -0.042545549 -0.037317274 - 3.5934362 
 
 - 5.3243555 
 
 -0.10573015 
 
 2l 
 
 +0.000070138 +0.036502918 -11.8827788 
 
 -11.1151387 
 
 -0.52740415 
 
 s. 
 
 +0.000069593 +0.036505673 -11.8837299 
 
 -11.1155862 
 
 -0.52740958 
 
 
 sin v iAi (s > + cos <p Bo M = + 0.0000000000012. 
 
 
 
 
 * 2,(J/ - G") = 3613.592979. 
 
 
 
 
 2t(Ji - G") = 3613.569327. 
 
 
 
178 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 DIFFERENTIAL COEFFICIENTS. 
 
 log coeff. 
 
 [de/dt] w = + 0.011982 m' p 8.0785350 
 [dx/dt] M = +67.128215 TO' p 1.8269051 
 [di/dt] m = -- 2.0560028m' n 0.3130237 
 [dl2/cfc]oo = -59.551438 m' n 1.7748923 
 [dT/dt] w = +67.097154 TO' p 1.8267041 
 = -90.590942 TO' n 1.9570848 
 
 FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'. 
 
 [de/dtlw = +o!()0000060823 
 
 [dx/dt] w = +0.0034075236 
 
 [dt/<ft]oo = -0.00010436562 
 
 [dtt/dt] w = -0.0030229161 
 
 [dir/dtlw, = +0.0034059472 
 
 = -0.0045985255 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt] w +0^00000 +0^00000 +o!o0000060823 
 
 +0.00032 +0.00032 +0.00031766599 
 
 -0.00011 -0.00011 -0.00010436562 
 
 sin i [dtt/dt] m -0.00009 -0.00010 -0.000097621545 
 
 [dL/dt] w -0.004 -0.0045985255 
 
 NOTES. 
 
 The agreement of the sums of the functions is much more exact throughout than 
 in the preceding computation because e' is here so much smaller. The greatest effect 
 produced by all terms from the fourth to the seventh orders inclusive is but 0". 000001, 
 and it is evident that the terms of the eighth and higher orders are wholly inappreci- 
 able. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 179 
 
 11. THE FINAL VALUES OF THE PERTURBATIONS. 
 
 Combining the results of the preceding pages, we now obtain the values of the 
 perturbations stated in the following tables. For comparison with these, the results 
 obtained by LEVERRIER (V) and NEWCOMB (IS) are added, all of the results being reduced 
 to the values of the masses here adopted and stated in Article 6. 
 
 SECULAR PERTURBATIONS OF MERCURY. 
 (Epoch 1850.0, G. M. T.) 
 
 Action of 
 
 r*i 
 
 LdtJoo 
 
 r&i 
 
 ldt], 
 
 r*-. 
 
 Udoo 
 
 Venus 
 
 +0^027739414 
 
 +2.7763615 
 
 -0.14811133 
 
 Earth 
 
 +0.011476557 
 
 +0.91448833 
 
 -0.014040890 
 
 Mars 
 
 -0.000607428 
 
 +0.02486334 
 
 -0.000301945 
 
 Jupiter 
 
 +0.00319413 
 
 + 1.5400720 
 
 -0.049056191 
 
 Saturn 
 
 +0.000531095 
 
 +0.07312263 
 
 -0.004212776 
 
 Uranus 
 
 +0.000009638 
 
 +0.00142135 
 
 -0.000024450 
 
 Neptune 
 
 +0.000003320 
 
 +0.00041901 
 
 -0.000020027 
 
 
 
 
 
 
 -0.04234673 
 
 +5.3307482 
 
 -0.21576761 
 
 Action of 
 
 rda-i 
 
 LdUoo 
 
 f-1 
 
 LdUoo 
 
 1-dL-l 
 
 LJ 
 
 Venus 
 
 -1.9420214 
 
 +2.7618772 
 
 -3.2505323 
 
 Earth 
 
 -1.0037245 
 
 +0.90700208 
 
 -1.1935233 
 
 Mars 
 
 -0.01926435 
 
 +0.02471966 
 
 -0.03293324 
 
 Jupiter 
 
 -1.4795642 
 
 + 1.5290366 
 
 -2.2066350 
 
 Saturn 
 
 -0.06979662 
 
 +0.07260205 
 
 -0.10657405 
 
 Uranus . . 
 
 -0.00134987 
 
 +0.00141128 
 
 -0.00201139 
 
 Neptune 
 
 -0.00044314 
 
 +0.00041570 
 
 -0.00060031 
 
 
 
 
 
 
 -4.5161641 
 
 +5.2970646 
 
 -6.7928096 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt]oo 
 
 e[dir/dt]oo 
 
 [di/dt]oo 
 
 sin i [dtt/dt] m 
 
 [dL,'dt] w 
 
 * Exclusive of the action of Uranus and Neptune. 
 
 f This unexpectedly large difference is a gradual accumulation from all of the computations Thus, the residuals, 
 Newcomb-Gauss, are, in the several cases: 0".00300, 0".00151, 0".0000.3, 0".00227, and 0".00010, the sum of which is 
 the difference as found above. 
 
 +0.04246 
 
 +0.04234 
 
 +010423467 
 
 + 1.08946 
 
 + 1.09601 
 
 + 1.0891018t 
 
 -0.21586 
 
 -0.21570 
 
 -0.2157676 
 
 -0.55017 
 
 -0.55041 
 
 -0.5505495 
 
 -6.8190* 
 
 
 -6.79281 
 
180 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 SECULAR PERTURBATIONS OF VENUS. 
 (Epoch 1850.0, G. M. T.) 
 
 Action of 
 
 (-del 
 LdUoo 
 
 r*n 
 
 L<ftJoo 
 
 r-A-i 
 
 UJoo 
 
 Mercury 
 
 -01)13012279 
 
 l'l893992 
 
 +o'6o94965089 
 
 Earth 
 
 0.04898290 
 
 56289701 
 
 +0 000044940 
 
 Mars 
 
 0.001963988 
 
 +0 74594759 
 
 +0 001304280 
 
 Jupiter 
 
 0031162921 
 
 +6 5654682 
 
 038659982 
 
 Saturn 
 
 0000675363 
 
 +0 07935156 
 
 0052327048 
 
 Uranus 
 
 +0 000005263 
 
 +0 00278176 
 
 -1-0 0000018240 
 
 Neptune 
 
 -0.000000278 
 
 +0.00110440 
 
 -0.0000283988 
 
 
 -0.09579247 
 
 +0.5762842 
 
 -0.033057385 
 
 Action of 
 
 ran 
 
 1 dUoo 
 
 r d *~i 
 
 Udoo 
 
 nun 
 
 LdUoo 
 
 Mercury 
 
 + o'()897732 
 
 1 1892420 
 
 + / 7454252 t i 
 
 Earth 
 
 - 7.293993 
 
 56417558 
 
 5 4005288 
 
 Mars 
 
 - 0.0473504 
 
 +0 74586465 
 
 09940123 
 
 Jupiter 
 
 - 2.7242270 
 
 +6 5606924 
 
 5 5347410 
 
 Saturn 
 
 - 0.0824657 
 
 +0 07920700 
 
 26491624 
 
 Uranus 
 
 - 0028813 
 
 +0 00277671 
 
 00496096 
 
 Neptune 
 
 0.0007780 
 
 +000110304 
 
 00148569 
 
 
 
 
 
 
 -10.061922 
 
 +0.5586460 
 
 -10.5606087 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 [de/dt] M - 0.09558 
 
 e[dTr/dt] M + 0.00366 
 
 [di/diloo - 0.03318 
 
 sin i [dO/di]oo - 0.59530 
 
 [dL/dt] M -10.549 
 
 -0.09576 
 +0.00392 
 -0.03306 
 -0.59551 
 
 - 0.0957925 
 + 0.0038229 
 
 - 0.0330574 
 
 - 0.5955192 
 -10.5606087 
 
 SECULAR PERTURBATIONS OF THE EARTH. 
 (Epoch 1850.0, G. M. T.) 
 
 Action of 
 
 I"*] 
 
 L<Joo 
 
 f-1 =F^1 
 LdUoo LrfUoo 
 
 [dp-] 
 
 I dt Joo 
 
 Mercury 
 
 -o'6oi 1613570 
 
 - OJ0999815 
 
 +o'o025085775 
 
 Venus 
 
 +0.013483339 
 
 + 3.4537341 
 
 +0 074457966 
 
 Mars 
 
 -0.015723904 
 
 + 097519611 
 
 +0 0063443986 
 
 Jupiter 
 
 -0.081841849 
 
 + 6 9652565 
 
 -0 025114405 
 
 Saturn 
 
 -0.0004330571 
 
 + 0.18725991 
 
 -0.0054235259 
 
 Uranus 
 
 +0.0000172788 
 
 + 0.00566366 
 
 +0.0000236793 
 
 Neptune 
 
 -0.0000006006 
 
 + 0.00179708 
 
 -0.0000364953 
 
 
 
 
 
 
 -0.085660150 
 
 + 11.4789092 
 
 +0.052760195 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 
 
 181 
 
 Action of 
 
 l-dj-j 
 Ldt Joo 
 
 rdL-l 
 1 dt Joo 
 
 Mercury 
 
 -(X002098681 
 
 + 0^3930935 
 
 Venus 
 
 -0.28462399 
 
 + 11.232473 
 
 Mars 
 
 -0.007195311 
 
 - 0.2342424 
 
 Jupiter 
 
 -0.16046446 
 
 - 9.1916336 
 
 Saturn 
 
 -0.013188086 
 
 - 0.4325140 
 
 Uranus 
 
 -0.0000784873 
 
 - 0.0080930 
 
 Neptune 
 
 -0.0000432488 
 
 - 0.0024199 
 
 
 
 
 
 -0.46769226 
 
 + 1.756664 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverrier. Newcomb. Method of Gauss. 
 
 -0.08569 
 e[dir/dt] w +0.19254 
 [dpldt] M +0.05290 
 [dq/dt] m -0.46754 
 [dL/dt]w +1.7570* 
 
 -0.08563 
 +0.19248 
 +0.05276 
 -0.46768 
 
 -0.085660 
 +0.192514 
 +0.052760 
 -0.467692 
 + 1.756664 
 
 The values of [dp/dt}oo and [dq/dt} QO obtained by HILL in the "New Theory" 
 are given below. These were regarded as provisional results only, and were derived 
 from the numerical values of the coefficients in the expansion of the perturbing 
 function stated by LEVERRIER in the Annales, Vol. II. 
 
 It may also be of interest to add the results obtained by the first application 
 ever made of the method of GAUSS. This was a computation by NICOLAI of the 
 secular perturbations of the Earth, the final values only being published, in BODE'S 
 Berliner Jahrbuch, 1820, pages 224-226 (Aug. 30, 1817). These results are here 
 reduced to the values of the masses stated in Article 6. 
 
 [de/dt] w 
 
 e[dirjdt]w 
 
 [dp/dt]oo 
 
 [dq/dt] w 
 
 Hill. 
 
 +0.0527225 
 -0.4676079 
 
 Nicolai. 
 
 -o!()8606 
 +0.19283 
 +0.05182 
 -0.46738 
 
 * Exclusive of the action of Neptune. If the value of this found above is included, we have [dL/<i(]oo = 1".7546 ; 
 -a less exact agreement. 
 
182 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 SECULAR PERTURBATIONS OF MARS. 
 (Epoch 1850.0, G. M. T.) 
 
 Action of 
 
 f-1 
 
 Lddoo 
 
 [41 
 
 UtJoo 
 
 r*-] 
 
 UUoo 
 
 Mercury 
 
 +o!6o0335670 
 
 + 0061841 
 
 -l-o'bo0074482 
 
 Venus 
 
 +0.000795405 
 
 + 4947286 
 
 0128 < >Q7 1 V7 
 
 Earth 
 
 +0.021481158 
 
 + 2 2915614 
 
 -un 000*31 QQ11 
 
 Jupiter 
 
 +0.15813453 
 
 + 12 476799 
 
 2 s )fi l i4077 
 
 Saturn 
 
 +0.006289141 
 
 + 6679051 
 
 024687281 
 
 Uranus 
 
 -0.000014965 
 
 + 00120199 
 
 000062454 
 
 Neptune 
 
 +0.000000608 
 
 + 00034075 
 
 000104366 
 
 
 
 
 
 
 +0.18702155 
 
 + 15.952606 
 
 -0.29383023 
 
 Action of 
 
 rdo-l 
 
 LdiJoo 
 
 Fl 
 
 UUoo 
 
 f-1 
 ldt] w 
 
 Mercury 
 
 + o'6l479483 
 
 + o'6o61918 
 
 + o"l940178 
 
 Venus 
 
 + 0.30877426 
 
 + 0.4948896 
 
 + 4 1 9 04933 
 
 Earth 
 
 - 2.2862242 
 
 + 2.2903688 
 
 + 6 6520970 
 
 Jupiter 
 
 - 8.3142000 
 
 + 12.472464 
 
 18 450874 
 
 Saturn . ... 
 
 - 0.26298236 
 
 + 6677678 
 
 8382821 
 
 Uranus 
 
 - 0.00741949 
 
 + 0120160 
 
 0154576 
 
 Neptune 
 
 - 0.00302292 
 
 + 0.0034059 
 
 00045985 
 
 
 
 
 
 
 -10.5502799 
 
 + 15.947104 
 
 - 8.342604 
 
 COMPARISON WITH OTHER RESULTS. 
 
 Leverripr. Neweomb. Method of Gauss. 
 
 +0.18703 
 + 1.48645 
 [K/ctt]oo -0.29375 
 sin i [dtt/dfloo -0.34099 
 -8.358* 
 
 +0.18706 
 + 1.48787 
 -0.29385 
 -0.34066 
 
 +0.187022 
 + 1.487355 
 -0.293830 
 -0.340709 
 -8.34260 
 
 12. COMPARISON WITH THE RESULTS OF OBSERVATION. 
 
 From a discussion of all the available observations of the planets and of the 
 Sun, NEWCOMB has derived the most probable values of the preceding coefficients 
 based upon observations alone. These will be found summarized in a convenient 
 form on pages 107 and 108 of The Elements of the Four Inner Planets and the Funda- 
 mental Constants of Astronomy (Supplement to the American Ephemeris and Nautical 
 Almanac, 1897). 
 
 * The value of [dLldt]oo arising from the action of Mercury was not stated by Leverrier. The value as found above 
 has been added to his series of values in order to obtain this sum. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 183 
 
 In order to compare the values here obtained with those given by NEWCOMB 
 it is necessary to notice that the values of i and fl stated by him are measured from 
 the movable equator and equinox and that it is therefore necessary to free the values 
 of [di/dt] 00 and [dQ/dt] 00 here given from the changes caused by the motion of the 
 ecliptic itself. For this purpose we first compute p and L from the equations, 
 
 [dp-] \dq1 
 
 p sin L = and p cos L = Mr , 
 
 L dt Joo L dt Joo 
 
 the secular variations being those which belong to the Earth's orbit, and then add 
 the quantities p cos (L fl) to the several determinations of [di/dt] Q o and p X 
 cos i sin (L Q) to those above given for sin i [dQ/dt]o . In this manner the values 
 stated in the following tabulation are obtained. 
 
 In a similar way it might appear necessary to add the correction, 
 
 e tan \i ( sin i ^7 + P sin (L ft) J 
 
 to the values obtained for e [dwjdt] o, the first part arising from the change due to the 
 plane of the orbit and the second from that produced by the motion of the ecliptic. 
 And in the case of the Earth's perihelion, there is a secular motion due to the lack of 
 sphericity of the Earth-moon system which is expressed by the equation, 
 
 dir^ mm' (a '\ 2 . 
 
 TT = I*** ' 7 7\} 'I ) > ' 
 
 dt Joo 2 (m + m') 2 \a/' 
 
 the accented letters applying to the moon (Annales de I'Observatoire de Paris, Vol. IV, 
 pages 42-46). Employing the values of a' and m' given in the Astronomical Papers 
 of the American Ephemeris, Vol. IV, page 11, this correction is found to be 
 +0".0157884. But these last two corrections need not here be applied because the 
 values of the variations obtained by NEWCOMB from observation have already been 
 freed from their effects. 
 
 MERCURY. 
 
 Newcomb. Method of Gauss. Observ. 5i Si t 
 
 [de/dt] M 
 e [dv/dt]m 
 [dildt}oo 
 
 +0.0423 
 + 1.0960 
 +0.0676 
 
 +0.0423 
 + 1.0891 
 +0.0674 
 
 +o!t)336 
 + 1.1824 
 +0.0714 
 
 -o!()087 
 +0.0864 
 +0.0038 
 
 -0.0087 
 +0.0933 
 +0.0040 
 
 0.0050 
 0040 
 
 0.0080 
 
 sin i [dn/dfloo -0.9250 -0.9234 -0.9189 +0.0061 +0.0045 0.0045 
 
184 
 
 THE SECULAR VARIATIONS OP THE ELEMENTS 
 
 VENUS. 
 Newcomb. Method of Gauss. Observ. Si 
 
 [de/dt} w 
 
 -0.0958 
 
 -0.0958 
 
 -0.0946 
 
 +0.0012 
 
 +0.0012 
 
 0.0020 
 
 e [dv/dt]oo 
 
 +0.0039 
 
 +0.0038 
 
 +0.0029 
 
 -0.0010 
 
 -0.0009 
 
 0.0020 
 
 [dildt]oo 
 
 +0.0034 
 
 +0.0034 
 
 +0.0029 
 
 -0.0005 
 
 -0.0005 
 
 0.0030 
 
 sin i [dttldt]m 
 
 -1.0600 
 
 -1.0603 
 
 -1.0540 
 
 +0.0060 
 
 +0.0063 
 
 0.0012 
 
 EARTH. 
 
 Newcomb. Method of Gauss. Observ. 
 
 i 
 
 [de/dt]^ 
 
 -0.0856 
 
 -0.0857 
 
 -0.0855 
 
 +0.0001 
 
 +0.0002 
 
 0.0009 
 
 e. [dir/dt}oo 
 
 +0.1925 
 
 +0.1925 
 
 +0.1948 
 
 +0.0023 
 
 + 0.0023 
 
 0.0012 
 
 [d*!dt} M 
 
 -0.4677 
 
 -0.4677 
 
 -0.4711 
 
 -0.0034 
 
 -0.0034 
 
 0.0023 
 
 Newcomb. 
 
 MARS. 
 
 Method of Gauss. Observ. 
 
 [de/dt] M 
 
 +0.1871 
 
 +0.1870 
 
 +0.1900 
 
 +0.0029 
 
 +0.0030 
 
 0.0027 
 
 e [dw/dt] M 
 
 + 1.4879 
 
 + 1.4874 
 
 + 1.4955 
 
 +0.0076 
 
 +0.0081 
 
 0.0035 
 
 [di/dt] w 
 
 -0.0225 
 
 -0.0229 
 
 -0.0226 
 
 -0.0001 
 
 +0.0003 
 
 0.0020 
 
 sin i [dn/dt]oo 
 
 -0.7263 
 
 -0.7251 
 
 -0.7260 
 
 +0.0003 
 
 -0.0009 
 
 0.0020 
 
 In the above tabulation the column headed 61 expresses the residuals from the 
 computation of NEWCOMB and that headed 5 2 states the residuals from the results 
 here obtained. It will be noticed that the differences are very minute throughout, 
 the only appreciable improvement arising from the more accurate computation 
 occurring in the case of the node of Mercury, where the residual is reduced by its 
 fourth part. 
 
 The last column contains the mean errors of the observational results. If we 
 multiply these by 0.6745 to reduce them to probable errors, we observe that in seven 
 cases the residuals are less than the probable errors; in five cases they vary from one 
 to three times as great but that in each of these cases where the divergence is greatest 
 a slight change in the value of the masses will correct the disagreement, and that 
 in the remaining three cases the difference is very much greater than can be ascribed 
 to errors either in the adopted masses, the computation, or to errors in the obser- 
 vations themselves. These three cases are: 
 
 1. The motion of the perihelion of Mercury. 
 
 2. The motion in the node of Venus. 
 
 3. The motion of the perihelion of Mars. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 185 
 
 The first of these is the well-known discordance. The second is well established, 
 the discordance between observation and theory being nearly eight times the probable 
 error, nor can the uncertainty remaining in the values of the masses account for more 
 than a small part of the discrepancy. NEWCOMB estimates the mean error of the 
 computed value arising from this uncertainty as not more than 0".0012, so that with 
 this included the residual is nearly six times the probable error. The third dis- 
 cordance is the least of the three, but as the masses of Jupiter and Saturn, the principal 
 disturbing planets for this case, are accurately known, the uncertainty of the com- 
 puted results is almost negligible. NEWCOMB estimates the mean error of the result 
 of computation arising from the uncertainties in the masses of all the planets as here 
 but 0".0004, so that the residual remains between three and four times as large as 
 the probable error. 
 
 13. COMPARISON WITH SEELIGER'S HYPOTHESIS ON THE CON- 
 STITUTION OF THE ZODIACAL LIGHT. 
 
 Many hypotheses have been made for the purpose of explaining the discrepancies 
 shown in the preceding article. In general, either the assumption is made that 
 NEWTON'S Law of Gravitation is not strictly accurate* or else that certain additional 
 matter in the solar system must be considered whose attraction has not hitherto 
 been allowed for.| The most recent and the most plausible investigation of the 
 second kind is that effected by SEELIGER (IO)I (11)i (12> who seeks to account for all of 
 the appreciable discrepancies by the perturbing effect of the cloud of particles known 
 as the zodiacal light. 
 
 What the true form of this cloud is, and still more, what the law of the dis- 
 tribution of its density is, is very uncertain. J SEELIGER assumes that it can be 
 roughly conceived as made up of two homogeneous ellipsoids of revolution whose 
 semi axes have the values 0.24 and 1.20, respectively. Both the eccentricities of 
 these ellipsoids and the position of the equator of the outer one can vary within wide 
 limits without greatly altering the values of the perturbations which they produce; 
 the distance from the focus to the center in each of them is arbitrarily chosen as 
 equal in length to ten times the semi minor axis, and the equator of the outer one is 
 assumed to be coincident with the plane of the equator of the sun. The respective 
 densities and also the two constants which define the equatorial plane of the first 
 ellipsoid remain as unknown quantities whose values are to be determined. 
 
 * See Tisserand's Mecanique Celeste, Vol. IV, Pages 494-542. 
 
 fSee Newcomb's " Astronomical Constants. . . ." (1 ", Pages 110-120. 
 
 t See the article, "The Zodiacal Light" by Newcomb, in the Encyclopaedia Britannica, Vol. XXVIII. 
 
186 
 
 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 From the known formulas which express the attraction exerted by an ellipsoid 
 upon a point either wholly within or without its surface, the expression for the per- 
 turbing force in any case can readily be written, and from this the equations for the 
 variations of the various elements are derived, each equation containing five unknown 
 quantities whose values are to be so determined as to best account for the excess of 
 the variations observed over those heretofore obtained from the .theory. As the 
 ellipsoids are assumed to be symmetrical with respect to their axes of rotation, 
 however, they will cause no appreciable perturbation of any eccentricity. The 
 variation of the obliquity of the Earth's orbit was also not considered by SEELIGER. 
 
 There remain therefore but ten discrepancies to be represented; namely, those of 
 the four perihelia, those of the three nodes and those of the three inclinations. These 
 ten discrepancies form the absolute terms of ten corresponding equations which con- 
 tain five unknown quantities. It is to be noticed that in the "Astronomical Constants 
 ..." two tables of the theoretical variations are stated by NEWCOMB; the first, 
 on page 109, are those computed from the values of the various masses assumed in 
 Chapter V; the second, on page 185, are those computed from the definitively adopted 
 masses. The latter values of the masses are in closer accordance with those assumed 
 in the present paper than the former; the first values are, however, the ones adopted 
 by SEELIGER in the computation. 
 
 The final results are as in the following table: 
 
 
 Newcomb. 
 
 Method of 
 Gauss. 
 
 Per. caused by 
 Zod.L't. 
 
 Final Residuals. 
 
 Prob. Errors. 
 
 Newcomb. 
 
 Meth. of Gauss. 
 
 MERCURY. 
 
 // 
 
 // 
 
 // 
 
 // 
 
 // 
 
 // 
 
 edit 
 
 +8.64 
 
 +9.33 
 
 +8.49 
 
 +0.15 
 
 +0.84 
 
 0.29 
 
 sin i<Kl 
 
 +0.61 
 
 +0.45 
 
 +0.62 
 
 +0.01 
 
 -0.17 
 
 0.54 
 
 di 
 
 +0.38 
 
 +0.40 
 
 +0.49 
 
 -0.11 
 
 -0.09 
 
 0.35 
 
 VENUS. 
 
 
 
 
 
 
 
 edw 
 
 -0.10 
 
 -0.09 
 
 +0.05 
 
 -0.15 
 
 -0.14 
 
 0.17 
 
 sin idU 
 
 +0.60 
 
 +0.63 
 
 +0.60 
 
 0.00 
 
 +0.03 
 
 0.22 
 
 di 
 
 -0.05 
 
 -0.05 
 
 +0.20 
 
 -0.25 
 
 -0.25 
 
 0.11 
 
 EARTH. 
 
 
 
 
 
 
 
 edit 
 
 +0.23 
 
 +0.23 
 
 +0.09 
 
 +0.14 
 
 +0.14 
 
 0.09 
 
 MAKS. 
 
 
 
 
 
 
 
 edw 
 
 +0.76 
 
 +0.81 
 
 + 0.56 
 
 +0.20 
 
 +0.25 
 
 0.24 
 
 sin idSl 
 
 +0.03 
 
 -0.09 
 
 +0.21 
 
 -0.18 
 
 -0.30 
 
 0.14 
 
 di 
 
 -0.01 
 
 +0.03 
 
 -0.01 
 
 0.00 
 
 +0.04 
 
 0.15 
 
 The first two columns of the table contain the residuals from the masses employed 
 in the present paper; the third column states the perturbations caused by the zodiacal 
 light when its elements are derived from the residuals of NEWCOMB 's first tabulation. 
 As the five elements were so determined as to represent NEWCOMB'S first residuals as 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 187 
 
 accurately as possible, their agreement with these is naturally more exact than with 
 the values here stated. Thus the first agreement for the motion of Mercury's peri- 
 helion is exact while here the discrepancy is considerable. On the other hand, the 
 greatest discrepancy when the results are compared with the first tabulation, and 
 which occurs in the motion of the node of Mars, is slightly lessened when the new 
 masses are employed. 
 
 As the five elements were determined to represent the ten residuals of NEWCOMB'S 
 computation as accurately as possible, the numbers of the fourth column are, as 
 might have been expected, generally smaller than those of the fifth. It may justly 
 be inferred, however, that SEELIGER'S hypothesis is capable of greatly reducing those 
 discrepancies whose values are sufficiently large to establish their reality, without 
 at the same time unduly increasing any of the smaller ones. 
 
 The last column contains NEWCOMB'S estimate of the total probable errors 
 arising both from the errors of observation and from the uncertainties in the values 
 of the adopted masses. 
 
 The elements of the zodiacal light derived by SEELIGER are as follows: 
 
 Density of inner ellipsoid = 2.52 X 10~ n times the Sun's density. 
 
 Density of outer ellipsoid = 0.0026 X 10~ n times the Sun's density. 
 
 Total mass = 35000 X 10" 11 times the Sun's mass. 
 
 Inclination of equator of I = 6. 95 
 
 Longitude of node of I = 40.03. 
 
 The unit of time throughout this article is the Julian Century. 
 
 BIBLIOGRAPHY. 
 
 LIST OF WORKS ON GAUSS'S METHOD AND RELATED SUBJECTS WHICH ARE REFERRED 
 
 TO IN THE PRECEDING PAGES. 
 
 1. GAUSS. Determinatio Attractionis quam in punctum quodvis positionis datae exerceret planeta si 
 
 ejus massa per tot am orbitam ratione temporis quo singulae partes describuntur uniformiter 
 esset dispertita. Werke, Vol. Ill, pages 333-357. 
 
 2. NICOLAI. Neue Berechnung der Secular Anderungen der Erdbahn. Bode's Astronomische Jahrbuch, 
 
 1820, pages 224-226. 
 
 3. CLAUSEN. Alia solutio problematis a celeberrimo Gauss in opera " Determinatio attractionis. . . " 
 
 tractati. Crelle's Journal, Vol. VI, 1830, page 290. 
 
 4. - Bestimmung der Bahn und der Umlaufszeit des Tuttle'schen Cometen. Beobachtungen der 
 
 Kaiserlichen Univcrsitats Sternwarte Dorpat, Vol. XVI. 
 
 5. BOUK, EDMOND. Thesis presentees a la faculte des sciences de Paris, 1855. 
 
188 THE SECULAR VARIATIONS OF THE ELEMENTS 
 
 6. ADAMS. On the November meteors. Monthly Notices, Vol. XXVII; Collected Works, Vol. II, pages 
 
 194-200. 
 
 7. LEVERHIER. The secular perturbations of the elements of the orbits of the planets. Annales de 1'Ob- 
 
 servatoire de Paris. Mercury, Vol. V, pages 6 and 7; Venus, Vol. VI, page 6; The Earth, Vol. IV, 
 pages 11 and 12; Mars, Vol. VI, page 189. 
 
 8. HILL. On Gauss's method of computing secular perturbations. Astronomical Papers of the American 
 
 Ephemeris, Vol. I, pages 317-361. 
 
 9. SEELIGER. Ueber das von Gauss herruhrende Theorem die Sacularstorungen betreffend. Astrono- 
 
 mische Nachrichten, Vol. XCIV, 1879. 
 
 10. - Ueber die sogenannte absolute Bewegung. Sitzungsberichte der konigliche Akademie der Wis- 
 
 senschaften zu Munchen. Vol. XXXVI, pages 85-137. 
 
 11. - Ueber die empirischen Gleider in der Theorie der Bewegung der Planeten Merkur, Venus, Erde 
 
 und Mars. Vierteljahrschrift der Astronomischen Gesellschaft, Vol. XLI, pages 234-240. 
 
 12. - Das Zodiakallicht und die empirischeii Gleider in der Bewegung der inneren Planeten. Sitzungs- 
 
 berichte der koniglichliche Akademie der Wissenschaften zu Munchen. Vol. XXXVI, pages 
 595-622. 
 
 13. CALLANDREAU. Calcul des variations seculaires des elements des orbites. Annales de 1'Observatoire 
 
 de Paris, 1885, Vol. XVIII. 
 
 14. TISSERAND. Traite de Mecanique Celeste. Vol. I, pages 431-442; Vol. IV, pages 494-542. 
 
 15. NEWCOMB. Secular variations of the orbits of the four inner planets. Astronomical Papers of the 
 
 American Ephemeris, Vol. V, pages 301-378. 
 
 16. HILL. A new theory of Jupiter and Saturn. Astronomical papers of the American Ephemeris, Vol. IV 
 
 17. NEWCOMB. The elements of the four inner planets and the fundamental constants of astronomy. 
 
 1895. Supplement to the American Ephemeris and Nautical Almanac, 1897. 
 
 18. HALL, ASAPH, JR. Secular perturbations of the Earth from the action of Mars. Astronomical Journal, 
 
 No 244. 
 
 19. SEE. Secular perturbations of Uranus from the action of Neptune. Astronomical Journal, No. 316. 
 
 20. INNES. Secular perturbations of the Earth from the action of Mars. Monthly Notices, Vol. LII, 
 
 Nos. 2 and 7. 
 
 21. - Secular perturbations of the Earth from the action of Venus. Monthly Notices, Vol. LIII, No. 6. 
 
 22. - Tables to facilitate the application of Gauss's method. Monthly Notices, Vol. LIV, Nos. 5 and 6. 
 
 23. TURNER, ARTHUR B. Secular perturbations of Mars from the action of Jupiter. Thesis presented to 
 
 the Faculty of Philosophy of the Graduate School of the University of Pennsylvania, 1902. 
 
 24. - Secular perturbations of Mars from the action of Jupiter, computed by the method of Arndt. 
 
 Astronomische Nachrichten, No. 3065. 
 
 25. BARTON, SAMUEL B. Secular perturbations of Mars from the action of Saturn. Thesis presented to 
 
 the Faculty of Philosophy of the Graduate School of the University of Pennsylvania. 1906. 
 
 26. MERFIELD, C. J. The secular perturbations of Ceres from the action of Jupiter. Astronomische Nach- 
 
 richten, No. 4215. 
 
 27. DZIEWULSKI, W. Sakulare Marstorungen des Eros. Cracovi, 1906. 
 
 28. HALPHEN. Traite des fonctions elliptiques. Part II, pages 310-328. 
 
 29. BRUNS. Ueber die Perioden der elliptischen Integrale erster und zweiter Gattung. Dorpat, 1875. 
 
 30. ARNDT, Louis. Recherches sur le calcul des forces perturbatrices dans la theorie des perturbations 
 
 seculaires. Bulletin de la Societe des sciences naturelles de Neuchatel, Vol. XXIV, 1896. 
 
 31. INNES. The computation of secular perturbations. Monthly Notices, Vol. LXVII, pages 427-443. 
 
 32. ROBBINS, FRANK. Tables for the application of Mr. Innes' method. Monthly Notices, Vol. LXVII, 
 
 pages 444-447. 
 
OF THE ORBITS OF THE FOUR INNER PLANETS. 189 
 
 33. MERFIELD, C. J. Extension of Mr. Robbing' tables to the value i = 180. Monthly Notices, Vol. 
 
 XLVIII, pages 605-608. 
 
 34. - The secular perturbations of Eros. Astronomische Nachrichten, Nos. 4178-4179. 
 
 35. The secular perturbations of Iris. Astronomische Nachrichten, No. 4337. 
 
 36. - The secular perturbations of Ceres. Monthly Notices, Vol. XLVII, pages 551-560. 
 
 37. HILL, G. W. The secular perturbations of the planets. American Journal of Mathematics, Vol. XXIII, 
 
 page 317. 
 
 38. - On the use of the sphero-conic in astronomy. Astronomical Journal, No. 511. 
 
 39. INNES. Jacobi's nome, q, in astronomical formulas, with numerical tables. Monthly Notices, Vol. 
 
 LXII, pages 494-503. 
 

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