{
  "id": "3893f25ebbb34c5d2333c73aa4d786555b5d66a0",
  "text": "On the Theory of the Moon.\n\nThe method pursued by Clairaut in the solution of this important problem of Physical Astronomy, consists in the integration of the differential equations furnished by the principles of dynamics, upon the hypothesis that in the gravitation of the celestial bodies the force varies inversely as the square of the distance, and in which the true longitude of the moon is the independent variable; the time is thus obtained in terms of the true longitude, and by the reversion of series the longitude is afterwards obtained in terms of the time, which is necessary for the purpose of forming astronomical tables. But while on the one hand this method possesses the advantage, that the disturbing function can be developed with somewhat greater facility in terms of the true longitude of the moon than in terms of the mean longitude, yet on the other hand, the differential equations in which the true longitude is the independent variable are far more complicated than those in which the time is the independent variable. The latter equations are used in the planetary theory; so that the method of Clairaut has the additional inconvenience, that while the lunar theory is a particular case of the problem of the three bodies, one system of equations is used in this case, and another in the case of the planets.\n\nThe method of Clairaut has been adopted, however, by Mayer, by Laplace, and by M. Damoiseau. The last-mentioned author has arranged his results with remarkable clearness, so that any part of his processes may be easily verified by any one who does not shrink from this gigantic undertaking; and the immense labour which this method requires, when all sensible quantities\nare retained, may be seen in his invaluable memoir. Mr. Brice Bronwin has recently communicated to the Society a lunar theory, in which the same method is adopted.\n\nHaving reflected much upon the difficulties of this problem, I am led to believe that the integration of the differential equations in which the time is the independent variable, is at least as easy as the method hitherto, I think, solely employed, and I now have the honour to submit to the Society a lunar theory founded upon this integration, which is in fact merely an extension of the equations given in my Researches in Physical Astronomy, already printed, by embracing those terms which, in consequence of the magnitude of the eccentricity of the moon's orbit, are sensible; and the suppression of those, on the other hand, which are insensible on account of the great distance of the sun, the disturbing body. By means of the Table which I have given (Table II.), the developments may all be effected at once with the greatest facility.\n\nThe first column contains the indices, which I have employed to distinguish the inequalities. The numbers in the second column are the indices affixed by M. Damoiseau, in the Mém. sur la Théor. de la Lune, p. 547. to the inequalities of longitude.\n\n\\[ t^* = n_t - n_t t, \\quad x = c_n t - \\varphi, \\quad z = n_t t - \\varphi, \\quad y = g_n t - \\nu. \\]\n\n|   |   |   |   |   |\n|---|---|---|---|---|\n| 0 | .. | 0 | 21 | 45 | 2t - 3x |\n| 1 | 30 | 2t† | 22 | 46 | 2t + 3x |\n| 2 | 1 | x | 23 | 21 | 2x + z |\n| 3 | 31 | 2t - x† | 24 | 53 | 2t - 2x - z |\n| 4 | 32 | 2t + x | 25 | 54 | 2t + 2x + z |\n| 5 | 16 | z§ | 26 | 20 | 2x - z |\n| 6 | 33 | 2t - z | 27 | 51 | 2t - 2x + z |\n| 7 | 34 | 2t + z | 28 | 52 | 2t + 2x - z |\n| 8 | 2 | 2x | 29 | 23 | x + 2z |\n| 9 | 35 | 2t - 2x | 30 | 59 | 2t - x - 2z |\n| 10 | 36 | 2t + 2x | 31 | .. | 2t + x + 2z |\n| 11 | 19 | x + z | 32 | 22 | x - 2z |\n| 12 | 41 | 2t - x - z | 33 | 61 | 2t - x + 2z |\n| 13 | 42 | 2t + x + z | 34 | 60 | 2t + x - 2z |\n| 14 | 18 | x - z | 35 | .. | 3z |\n| 15 | 39 | 2t - x + z | 36 | .. | 2t - 3z |\n| 16 | 40 | 2t + x - z | 37 | .. | 2t + 3z |\n| 17 | 17 | 2z | 38 | 9 | 4x |\n| 18 | 43 | 2t - 2z | 39 | 67 | 2t - 4x |\n| 19 | 44 | 2t + 2z | 40 | .. | 2t + 4x |\n| 20 | 4 | 3x | 41 | 27 | 3x + z |\n\n* Inconvenience arises from using the letter \\( t \\) in this acceptation. I have done so in order to conform to the notation of M. Damoiseau. † Variation. ‡ Evection. § Annual Equation.\n|   |   |   |   |   |   |\n|---|---|---|---|---|---|\n| 63 | 37 | $2t - 2y$ | 105 | 84 | $t + z$ |\n| 64 | 38 | $2t + 2y$ | 106 | 85 | $t - 2x$ |\n| 65 | 5  | $x - 2y$ | 107 | 86 | $t + 2x$ |\n| 66 | 6  | $x + 2y$ | 108 | 91 | $t - x - z$ |\n| 67 | 49 | $2t - x - 2y$ | 109 | 92 | $t + x + z$ |\n| 68 | 47 | $2t - x + 2y$ | 110 | 89 | $t - x + z$ |\n| 69 | 48 | $2t + x - 2y$ | 111 | ... | $t + x - z$ |\n| 70 | 50 | $2t + x + 2y$ | 112 | ... | $t - 2z$ |\n| 71 | 24 | $z - 2y$ | 113 | ... | $t + 2z$ |\n| 72 | 25 | $z + 2y$ | 114 | ... | $t - 2y$ |\n| 73 | 57 | $2t - z - 2y$ | 115 | ... | $t + 2y$ |\n| 74 | 56 | $2t - z + 2y$ | 116 | 100 | $3t$ |\n| 75 | 55 | $2t + z - 2y$ | 117 | 101 | $3t - x$ |\n| 76 | 58 | $2t + z + 2y$ | 118 | 102 | $3t + x$ |\n| 77 | 7  | $2x - 2y$ | 119 | 103 | $3t - z$ |\n| 78 | 8  | $2x + 2y$ | 120 | 104 | $3t + z$ |\n| 79 | 65 | $2t - 2x - 2y$ | 121 | ... | $3t - 2x$ |\n| 80 | 63 | $2t - 2x + 2y$ | 122 | ... | $3t + 2x$ |\n| 81 | 64 | $2t + 2x - 2y$ | 123 | ... | $3t - x - z$ |\n| 82 | ... | $2t + 2x + 2y$ | 124 | ... | $3t + x + z$ |\n| 83 | ... | $x + z - 2y$ | 125 | ... | $3t - x + z$ |\n| 84 | ... | $x + z + 2y$ | 126 | ... | $3t + x - z$ |\n| 85 | ... | $2t - x - z - 2y$ | 127 | ... | $3t - 2z$ |\n| 86 | ... | $2t - x - z + 2y$ | 128 | ... | $3t + 2z$ |\n| 87 | ... | $2t + x + z - 2y$ | 129 | ... | $3t - 2y$ |\n| 88 | ... | $2t + x + z + 2y$ | 130 | ... | $3t + 2y$ |\n| 89 | ... | $x - z - 2y$ | 131 | 120 | $4t$ |\n| 90 | ... | $x - z + 2y$ | 132 | 121 | $4t - x$ |\n| 91 | ... | $2t - x + z - 2y$ | 133 | 122 | $4t + x$ |\n| 92 | ... | $2t - x + z + 2y$ | 134 | 123 | $4t - z$ |\n| 93 | ... | $2t + x - z - 2y$ | 135 | 124 | $4t + z$ |\n| 94 | ... | $2t + x - z + 2y$ | 136 | 125 | $4t - 2x$ |\n| 95 | ... | $2z - 2y$ | 137 | 126 | $4t + 2x$ |\n| 96 | ... | $2z + 2y$ | 138 | 131 | $4t - x - z$ |\n| 97 | ... | $2t - 2x - 2y$ | 139 | ... | $4t + x + z$ |\n| 98 | ... | $2t - 2x + 2y$ | 140 | 129 | $4t - x + z$ |\n| 99 | ... | $2t + 2z - 2y$ | 141 | ... | $4t + x - z$ |\n| 100| ... | $2t + 2z + 2y$ | 142 | ... | $4t - 2z$ |\n| 101| 80 | $t^*$ | 143 | ... | $4t + 2z$ |\n| 102| 81 | $t - x$ | 144 | 127 | $4t - 2y$ |\n| 103| 82 | $t + x$ | 145 | ... | $4t + 2y$ |\n| 104| 83 | $t - z$ |     |     |     |\n\n$$\\cos 2t \\cos 2t = \\frac{1}{2} \\cos 4t + \\frac{1}{2}$$\n\n$$[131] \\quad [0]$$\n\n$$\\cos 2t \\cos x = \\frac{1}{2} \\cos (2t + x) + \\frac{1}{2} \\cos (-2t + x)$$\n\n$$[4] \\quad [-3]$$\n\nHence the multiplication of $\\cos 2t$ by $\\cos 2t$ produces the arguments 131 and 0, similarly the multiplication of $\\cos x$ by $\\cos 2t$ produces the arguments 4 and -3; proceeding in this way the following Table was formed, by writing down the indices instead of the arguments themselves.\n\n* Parallactic inequality.\nTable I.\n\nShowing the arguments which result from the combination of the arguments 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 20, 35, 62, 101, 146 and 147, with the arguments 1, 2, 3, &c. by addition and subtraction.\n\n|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 |\n|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----|\n| 1 | 131 | 4 | 132 | 133 | 7 | 134 | 135 | 10 | 136 | 137 | 13 | 138 | 16 | 19 | 22 | 37 | 64 | 116 | 148 | 146 |\n| 2 | -3 | 8 | 1 | 10 | 11 | 16 | 13 | 20 | 3 | 22 | 23 | 6 | 26 | 29 | 38 | 53 | 66 | 103 | 150 | 153 |\n| 3 | -2 | 9 | 1 | 136 | 131 | 15 | 138 | 140 | 4 | 133 | 7 | 6 | 33 | 10 | 57 | 68 | 117 | 152 | 155 |\n| 4 | 2 | 1 | 8 | 131 | 137 | 13 | 141 | 139 | 22 | 132 | 25 | 23 | 7 | 34 | 9 | 58 | 69 | 118 | 154 | 150 |\n| 5 | -6 | 11 | 15 | 13 | 17 | 1 | 19 | 23 | 27 | 25 | 29 | 3 | 2 | 35 | 41 | 59 | 72 | 105 | 156 | 159 |\n| 6 | -5 | 12 | 14 | 11 | 18 | 0 | 17 | 24 | 26 | 23 | 30 | 31 | 132 | 4 | 37 | 61 | 76 | 120 | 160 | 156 |\n| 7 | 5 | 15 | 11 | 14 | 1 | 17 | 0 | 27 | 23 | 26 | 3 | 29 | 33 | 6 | 45 | 18 | 75 | 105 | 159 | 156 |\n| 8 | -9 | 20 | 4 | 22 | 23 | 28 | 25 | 38 | 1 | 41 | 16 | 44 | 47 | 78 | 107 | 162 | 165 | 163 | 163 |\n| 9 | -8 | 21 | -2 | 20 | 24 | -26 | -23 | 39 | 0 | 38 | 42 | -14 | 45 | 48 | 79 | 106 | 163 | 161 | 161 |\n| 10 | 8 | 4 | 20 | 2 | 28 | 23 | 26 | 1 | 38 | 0 | 16 | 41 | 13 | 52 | 82 | 122 | 166 | 165 | 165 |\n| 11 | -12 | 5 | -24 | 6 | 2 | -30 | -3 | 14 | -42 | -16 | 0 | -48 | 17 | 14 | 84 | 109 | 168 | 171 | 169 |\n| 12 | -11 | 24 | -5 | 23 | 30 | -2 | -29 | 42 | 14 | -41 | 48 | 0 | 9 | 54 | 85 | 108 | 169 | 167 | 167 |\n| 13 | 11 | 7 | 23 | 5 | 4 | 29 | 2 | 15 | 41 | -14 | 1 | 47 | 19 | 16 | 88 | 124 | 172 | 168 | 168 |\n| 14 | -15 | 6 | 26 | 6 | 28 | 2 | 34 | 4 | 44 | 12 | 46 | 8 | 18 | 50 | 11 | 90 | 111 | 174 | 177 |\n| 15 | -14 | 27 | 5 | -26 | 3 | -32 | -2 | 45 | 11 | -44 | 9 | 17 | 51 | 12 | 92 | 125 | 176 | 175 | 173 |\n| 16 | 14 | 6 | 26 | -5 | 34 | 2 | 32 | 12 | 44 | -11 | 18 | 8 | 1 | 58 | 93 | 111 | 177 | 174 | 16 |\n| 17 | -18 | 29 | 33 | 31 | 35 | 7 | 37 | 47 | 51 | 49 | 53 | 15 | 11 | 59 | 96 | 113 | 180 | 183 | 17 |\n| 18 | -17 | 30 | 32 | 29 | 36 | -5 | 35 | 48 | 50 | -47 | 54 | 14 | 12 | 60 | 97 | 112 | 181 | 179 | 18 |\n| 19 | 17 | 33 | 29 | 32 | 7 | 35 | 5 | 51 | 47 | -50 | 15 | 53 | 57 | 1 | 99 | 113 | 183 | 180 | 19 |\n| 20 | -21 | 8 | -39 | 9 | 44 | | | | | | | | | | | | | | | | |\n| 21 | -20 | 9 | -8 | 38 | 42 | | | | | | | | | | | | | | | | |\n\n---\n\nMR. LUBBOCK'S RESEARCHES\n|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 |\n|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----|\n| 22 | 20 | 40 | 38 | 8 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |\n| 23 | -24 | 41 | 11 | -42 | -12 | 8 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 24 | -23 | 12 | 42 | -11 | -41 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 |\n| 25 | -23 | 43 | 139 | 41 | 11 | 10 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 26 | -27 | 28 | 44 | 16 | 46 | 8 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 27 | -26 | 15 | 45 | -14 | -44 | 9 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 28 | -26 | 46 | 141 | 14 | 14 | 52 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 29 | -30 | 31 | 47 | 19 | 49 | 53 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 30 | -29 | 18 | 48 | -17 | -47 | 54 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 31 | -29 | 49 | 143 | 17 | 13 | 55 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 32 | -33 | 34 | 50 | 18 | 52 | 14 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 33 | -32 | 19 | 51 | 17 | -50 | 15 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 34 | -32 | 52 | 142 | 16 | 58 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 |\n| 35 | -36 | 37 | 53 | 57 | 58 | 59 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 36 | -35 | 58 | 54 | 56 | -53 | 60 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 37 | -35 | 55 | 57 | 53 | -56 | 19 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 |\n| 62 | -63 | 64 | 66 | 68 | 70 | 72 | 74 | 76 | 78 | 80 | 82 | 84 | 86 | 89 | 95 | 115 | 114 | 146 | 148 | 62 |\n| 63 | -62 | 144 | 69 | 75 | 81 | 87 | 93 | 99 | 114 | 129 | 147 | 148 | 63 |\n| 64 | -62 | 145 | 70 | 76 | 82 | 88 | 94 | 100 | 115 | 130 | 148 | 64 |\n| 65 | -68 | 69 | 77 | 83 | 89 | 95 | 101 | 107 | 122 | 138 | 148 | 65 |\n| 66 | -67 | 70 | 78 | 84 | 90 | 96 | 102 | 108 | 124 | 140 | 66 |\n| 67 | -66 | 63 | 144 | 91 | 97 | 103 | 109 | 125 | 141 | 67 |\n| 68 | -65 | 64 | 145 | 92 | 98 | 104 | 110 | 126 | 142 | 68 |\n| 69 | -65 | 81 | 144 | 87 | 93 | 109 | 115 | 131 | 147 | 69 |\n|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 |\n|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----|\n| 70 | 66 | 64 | 78 | 62 | 94 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 71 | 75 | 83 | 91 | 87 | 95 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 72 | 76 | 84 | 92 | 88 | 96 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 73 | 72 | 85 | 89 | 84 | 97 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 74 | 71 | 86 | 90 | 83 | 98 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 75 | 71 | 87 | 91 | 83 | 90 | 63 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 76 | 72 | 88 | 92 | 84 | 89 | 64 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 101 | 116 | 103 | 117 | 118 | 105 | 119 | 120 | 107 | 121 | 122 | 109 | 123 | 111 | 113 |    |    |    |    |    |    |\n| 102 | 117 | 101 | 121 | 116 | 110 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 103 | 118 | 107 | 116 | 122 | 109 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 104 | 119 | 111 | 123 | 126 | 101 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 105 | 120 | 109 | 125 | 124 | 113 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 116 | 101 | 118 | 117 | 103 | 102 | 119 | 105 | 104 | 121 | 107 | 106 | 123 | 109 | 125 | 127 |    |    |    |    |\n| 117 | 102 | 116 | 121 | 101 | 106 | 123 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 118 | 103 | 122 | 116 | 107 | 101 | 126 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 119 | 104 | 126 | 123 | 111 | 108 | 127 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 120 | 105 | 124 | 125 | 109 | 110 | 116 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 131 | 133 | 132 | 4 | 3 | 135 | 137 | 136 | 10 | 9 | 139 | 141 | 143 |    |    |    |    |    |    |    |\n| 132 | 131 | 136 | 1 | 9 | 138 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 133 | 137 | 131 | 10 | 1 | 141 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 134 | 141 | 138 | 16 | 12 | 142 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 135 | 139 | 140 | 13 | 15 | 131 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n| 146 | 148 | 150 | 152 | 154 | 156 | 158 | 160 | 162 | 164 | 166 | 168 | 170 | 174 | 180 |    |    |    |    |    |\n| 147 | 153 | 151 | 149 | 150 | 157 | 155 | 156 | 163 | 161 | 162 | 169 | 175 | 181 |    |    |    |    |    |    |    |\n### Table I. (Continued.)\n\n|   | 1  | 2  | 3  | 4  | 5  | 6  | 7  | 8  | 9  | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 |\n|---|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|-----|-----|-----|\n| 148 | 146 | 152 | 150 | -149 | 158 | 156 | -155 | 164 | 162 | -161 | 170 | 168 | 176 | 182 | ... | ... | ... | 147 | 186 | 64 | 131 | 62 | 148 |\n| 149 | 153 | 161 | 147 | 165 | 167 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 149 |\n| 150 | -152 | -146 | -164 | -148 | 173 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 150 |\n| 151 | 154 | 162 | 148 | 166 | 168 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 151 |\n| 152 | -151 | 146 | -163 | -147 | 174 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 152 |\n| 153 | 147 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 153 |\n| 154 | -150 | 163 | -146 | -162 | 169 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 154 |\n| 155 | 148 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 155 |\n| 156 | -149 | 164 | 146 | -161 | 170 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 156 |\n| 157 | 165 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 157 |\n| 158 | 149 | 147 | 161 | -146 | 177 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 158 |\n| 159 | 166 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 159 |\n| 160 | -158 | -174 | -176 | -178 | -146 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 160 |\n\n|   | 38 | 59 |\n|---|----|----|\n| 1 | 40 | 61 |\n|   | 39 | 60 |\n\n1\nTable II.\n\nShowing the arguments which, by their combination with the arguments 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 20, 35, 62, 101, 146, 147, produce the arguments, 12, 3, &c. in the left hand column. This Table is formed from the preceding, by making the numbers in the left hand column in that Table change places with the rest. A full stop is placed after the figure where it does not occupy the same cell as in the preceding Table.\n\n|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 |\n|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----|\n| 1 | 0 | 3 | 2 | 133 | 6 | 5 | ..... | 9 | 8 | ..... | 12 | 11 | 15 | 18 | ..... | 63 | 101 | 147 | 146 | 1 |\n| 2 | - | 4 | 0 | 9 | 10 | 14 | 16 | - | 15 | ..... | 3 | - | 5 | ..... | ..... | ..... | ..... | ..... | 2 |\n| 3 | - | 2 | 9 | 0 | 131 | 12 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 3 |\n| 4 | - | 2 | 1 | 8 | 0 | 16 | 11 | 14 | 3 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 4 |\n| 5 | - | 7 | - | 14 | 15 | - | 16 | 0 | 18 | 19 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 5 |\n| 6 | - | 134 | 12 | 14 | ..... | 18 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 6 |\n| 7 | - | 5 | 15 | 11 | ..... | 1 | 17 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 7 |\n| 8 | - | 10 | 2 | - | 21 | 22 | 26 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 8 |\n| 9 | - | 8 | 21 | ..... | 132 | 24 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 9 |\n| 10 | - | 8 | 4 | 20 | 2 | 28 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 10 |\n| 11 | - | 13 | 5 | - | 24 | 25 | 2 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 11 |\n| 12 | - | 11 | 24 | ..... | 134 | 30 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 12 |\n| 13 | - | 11 | 7 | 23 | 5 | 4 | ..... | 2 | 15 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 13 |\n| 14 | - | 16 | 26 | - | 27 | 28 | 32 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 14 |\n| 15 | - | 14 | 27 | 5 | 135 | 3 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 15 |\n| 16 | - | 14 | 6 | 26 | ..... | 34 | 2 | ..... | 12 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 16 |\n| 17 | - | 19 | 32 | 33 | - | 34 | 5 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 17 |\n| 18 | - | 17 | 30 | 32 | ..... | 36 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 18 |\n| 19 | - | 17 | 33 | 29 | ..... | 7 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 19 |\n| 20 | - | 22 | 8 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 20 |\n|   | 1  | 2  | 3  | 4  | 5  | 6  | 7  | 8  | 9  | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 |\n|---|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|-----|-----|-----|\n| 21 | -20 | 9  | -8 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 22 | 20  | 10 | 8  |    |    |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 23 | -24 | 11 | 13 | -12 | 8  | 10 | -9 | 5  | 7  | -6 | 2  | 4  |    |    |    |    |    |     |     |     |\n| 24 | -23 | 12 | -11 | 9  | -8 | 6  | -5 | 3  | -2 |    |    |    |    |    |    |    |    |     |     |     |\n| 25 | 23  | 13 | 11 | 10 | 8  | 7  | 5  | 4  |    |    |    |    |    |    |    |    |    |     |     |     |\n| 26 | -27 | 14 | 16 | -15 | 8  | 9  | 10 | -5 | 6  | -7 | 2  |    |    |    |    |    |    |     |     |     |\n| 27 | -26 | 15 | -14 | 9  | 8  | 7  | 5  |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 28 | 26  | 16 | 14 | 10 | 8  | 6  |    | -5 |    |    |    |    |    |    |    |    |    |     |     |     |\n| 29 | -30 | 31 | 17 | 19 | -18 | 11 | 13 | -12 |    |    |    |    |    |    |    |    |    |     |     |     |\n| 30 | -29 | 18 | -17 | 12 | -11 |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 31 | 29  | 19 | 17 | 13 | 11 |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 32 | -33 | 17 | 18 | -19 | 14 | -15 | 16 |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 33 | -32 | 19 | 17 | 15 |    | -14 |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 34 | 32  | 18 | 17 | 16 | 14 |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 35 | -36 | 37 | 17 |    | 19 | -18 |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 36 | -35 |    |    |    | 18 | -17 |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 37 | 35  |    |    |    | 19 | 17 |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 38 | 20  | 22 | -21 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 39 | 21  | -20 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 40 | 22  | 20 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 41 | 23  | 25 | -24 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 42 | 24  | -23 | 21 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 43 | 25  | 23 | 22 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n| 44 | 26  | 28 | -27 | 20 |    |    |    |    |    |    |    |    |    |    |    |    |    |     |     |     |\n\nMDCCCXXXI.\n|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 |\n|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----|\n| 45 | ... | 27 | -26 | ... | 21 | ... | ... | 15 | -14 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 45 |\n| 46 | ... | 28 | ... | 26 | ... | 22 | ... | 16 | ... | 14 | ... | ... | ... | ... | ... | ... | ... | ... | 46 |\n| 47 | ... | 29 | ... | 31 | -30 | ... | 23 | ... | 17 | ... | 19 | -18 | ... | ... | ... | ... | ... | ... | 47 |\n| 48 | ... | 30 | -29 | ... | 24 | ... | ... | 18 | -17 | ... | 12 | -11 | ... | ... | ... | ... | ... | ... | 48 |\n| 49 | ... | 31 | ... | 29 | ... | 25 | ... | 19 | ... | 17 | 13 | ... | ... | ... | ... | ... | ... | ... | 49 |\n| 50 | ... | 32 | ... | 34 | -33 | ... | 26 | ... | -17 | ... | 18 | -19 | ... | ... | ... | ... | ... | ... | 50 |\n| 51 | ... | 33 | -32 | ... | 27 | ... | ... | 19 | ... | 17 | ... | ... | ... | ... | ... | ... | ... | ... | 51 |\n| 52 | ... | 34 | ... | 32 | ... | 28 | ... | 18 | ... | -17 | ... | ... | ... | ... | ... | ... | ... | ... | 52 |\n| 53 | ... | 35 | ... | 37 | -36 | ... | 29 | ... | ... | ... | 17 | ... | 19 | ... | ... | ... | ... | ... | 53 |\n| 54 | ... | 36 | -35 | ... | 30 | ... | ... | ... | ... | ... | 18 | -17 | ... | ... | ... | ... | ... | ... | 54 |\n| 55 | ... | 37 | ... | 31 | ... | ... | ... | ... | ... | ... | 19 | ... | ... | ... | ... | ... | ... | ... | 55 |\n| 56 | ... | -35 | ... | 36 | -37 | ... | 32 | ... | ... | ... | ... | -17 | ... | ... | ... | ... | ... | ... | 56 |\n| 57 | ... | 37 | ... | 35 | ... | ... | ... | ... | ... | ... | ... | ... | 15 | ... | ... | ... | ... | ... | 57 |\n| 58 | ... | 36 | ... | 35 | ... | ... | ... | ... | ... | ... | ... | ... | ... | 18 | ... | ... | ... | ... | 58 |\n| 59 | ... | ... | ... | 35 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 17 | ... | ... | ... | 59 |\n| 60 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 18 | ... | ... | ... | 60 |\n| 61 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 19 | ... | ... | 61 |\n| 62 | ... | 64 | ... | 66 | ... | 68 | ... | 70 | ... | 72 | ... | ... | ... | ... | ... | ... | ... | ... | 62 |\n| 63 | ... | -63 | ... | -65 | ... | -67 | ... | -69 | ... | -71 | ... | ... | ... | ... | ... | ... | ... | ... | 63 |\n| 64 | ... | 67 | ... | 65 | ... | 73 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 64 |\n| 65 | ... | -62 | ... | 69 | ... | -66 | ... | 75 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 65 |\n| 66 | ... | 62 | ... | 68 | ... | 66 | ... | 74 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 66 |\n| 67 | ... | -67 | ... | 70 | ... | -65 | ... | 76 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 67 |\n| 68 | ... | -66 | ... | 63 | ... | -62 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 68 |\n|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 |\n|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----|\n| 69 |   |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 70 |   |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 71 | -75 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 72 | -73 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 73 | -72 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 74 | -71 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 75 | 71 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 76 | 72 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 77 | 65 | 69 |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 78 | 66 | 70 | -67 |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 79 | 67 | -66 |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 80 | 68 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 81 | 69 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 82 | 70 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 83 | 71 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 84 | 72 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 85 | 73 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 86 | 74 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 87 | 75 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 88 | 76 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 89 | -72 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 90 | -71 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 91 | 75 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n| 92 | 76 |   |   |   |   |   |   |   |   |    |    |    |    |    |    |    |    |    |    |    |\n|   | 1   | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10  | 11  | 12  | 14  | 17  | 20  | 35  | 62  | 101 | 146 | 147 |\n|---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|\n| 93 |     | 73  |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n| 94 |     | 74  |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n| 95 |     |     | 71  |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n| 96 |     |     |     | 72  |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n| 97 |     |     |     |     | 73  |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n| 98 |     |     |     |     |     | 74  |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n| 99 |     |     |     |     |     |     | 75  |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|100 |     |     |     |     |     |     |     | 76  |     |     |     |     |     |     |     |     |     |     |     |     |\n|101 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|102 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|103 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|104 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|105 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|106 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|107 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|108 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|109 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|110 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|111 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|112 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|113 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|114 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|115 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|116 |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |\n|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 |\n|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----|\n|117| 102| 101| 116|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|118| 103| 116| 101|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|119| 104|    | 116| 101|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|120| 105|    | 116| 101|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|121|    | 102|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|122|    | 117|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|123|    | 103|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|124|    | 116|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|125|    | 101|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|126|    | 116|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|127|    | 116|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|128|    | 116|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|129|    | 116|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|130|    | 116|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|131|    | 132|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|132|    | 133|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|133|    | 134|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|134|    | 135|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|135|    | 136|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|136|    | 137|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|137|    | 138|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|138|    | 139|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|139|    | 140|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |\n|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 62 | 101 | 146 | 147 |\n|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|\n|141| 16| 134| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|142| 18| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|143| 19| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|144| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|145| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|146| -147| 150| 152| 154| 156| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|147| -146| 153| 149| ...| 157| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|148| ...| 152| 150| ...| 158| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|149| -152| -146| 147| -148| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|150| -151| 146| 148| ...| -147| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|151| -150| 147| -146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|152| -149| 148| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|153| ...| 149| 147| ...| -146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|154| ...| 150| 148| ...| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|155| -159| -158| ...| ...| -146| 147| -148| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|156| -160| -157| ...| ...| 146| 148| -147| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|157| ...| -156| ...| ...| 147| -146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|158| ...| -155| ...| ...| 148| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|159| ...| 155| ...| ...| 147| ...| -146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|160| ...| 156| ...| ...| 148| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|161| ...| 149| 153| -152| ...| -146| 147| -148| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|162| ...| 150| 154| -151| ...| 146| ...| 148| -147| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|163| ...| 151| -150| ...| ...| 147| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|164| ...| 152| -149| ...| ...| 147| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...|\n|   | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 62 | 101 | 146 | 147 |\n|---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|\n|165|...|153|...|149|...|...|...|...|-146|...|...|...|...|...|...|...|...|...|\n|166|...|154|...|150|...|...|...|148|146|...|...|...|...|...|...|...|...|...|\n|167|...|155|...|159|-158|...|...|...|-146|147|...|...|...|...|...|...|...|...|\n|168|...|156|...|160|-157|...|...|...|...|146|148|...|...|...|...|...|...|...|\n|169|...|157|-156|...|151|...|...|...|...|147|-146|...|...|...|...|...|...|...|\n|170|...|158|-155|...|152|...|...|...|...|148|146|...|...|...|...|...|...|...|\n|171|...|159|...|...|153|...|...|...|...|147|...|...|...|...|...|...|...|...|\n|172|...|160|...|156|154|...|...|...|...|148|...|...|...|...|...|...|...|...|\n|173|...|-156|157|-160|149|...|...|...|...|-146|...|...|...|...|...|...|...|...|\n|174|...|-155|158|-159|150|...|...|...|...|146|...|...|...|...|...|...|...|...|\n|175|...|155|...|151|...|...|...|...|147|...|...|...|...|...|...|...|...|...|\n|176|...|156|...|152|...|...|...|...|148|...|...|...|...|...|...|...|...|...|\n|177|...|157|...|-156|153|...|...|...|...|147|...|...|...|...|...|...|...|...|\n|178|...|158|...|-155|154|...|...|...|...|148|...|...|...|...|...|...|...|...|\n|179|...|...|...|155|...|...|...|...|-146|...|...|...|...|...|...|...|...|...|\n|180|...|...|...|156|...|...|...|...|146|...|...|...|...|...|...|...|...|...|\n|181|...|...|...|157|...|...|...|...|147|...|...|...|...|...|...|...|...|...|\n|182|...|...|...|158|...|...|...|...|148|...|...|...|...|...|...|...|...|...|\n|183|...|...|...|159|...|...|...|...|147|...|...|...|...|...|...|...|...|...|\n|184|...|...|...|160|...|...|...|...|148|...|...|...|...|...|...|...|...|...|\n|185|...|...|...|...|...|...|...|...|...|147|-146|...|...|...|...|...|...|...|\n|186|...|...|...|...|...|...|...|...|...|146|148|...|...|...|...|...|...|...|\n\n|38|59|38|59|\n|---|---|---|---|\n|39|1|...|39|60|1|...|60|\n|40|1|...|40|61|1|...|61|\nTable II. may be used in forming the developments required in the method employed by MM. Laplace and Damoiseau; for this purpose it is only necessary to make \\( t = \\lambda' - \\lambda \\) instead of \\( n t - n_1 t \\)\n\n\\[\nx = c \\lambda' - \\varpi \\quad \\ldots \\quad c n t - \\varpi \\\\\nz = c_1 \\lambda' - \\varpi_1 \\quad \\ldots \\quad c_1 n t - \\varpi_1 \\\\\n\\text{and } y = g \\lambda' - \\nu \\quad \\ldots \\quad g n t - \\nu\n\\]\n\nThe notation throughout is the same as that used Phil. Trans. 1830, p. 328, with the exception of the indices of the arguments.\n\nIn the elliptic movement;\n\n\\[\na^5 r^{-5} = 1 + 5 e^2 \\left( 1 + \\frac{21}{8} e^2 \\right) + 5 e \\left( 1 + \\frac{27}{8} e^2 \\right) \\cos x + 10 e^2 \\left( 1 + \\frac{31}{12} e^2 \\right) \\cos 2 x \\\\\n+ \\frac{145}{8} e^3 \\cos 3 x + \\frac{745}{48} e^4 \\cos 4 x\n\\]\n\n\\[\na^4 r^{-4} = 1 + 3 e^2 + 4 e \\cos x + 7 e^2 \\cos 2 x\n\\]\n\n\\[\na^3 r^{-3} = 1 + \\frac{3}{2} e^2 \\left( 1 + \\frac{5}{4} e^2 \\right) + 3 e \\left( 1 + \\frac{9}{8} e^2 \\right) \\cos x + \\frac{9}{2} e^2 \\left( 1 + \\frac{7}{9} e^2 \\right) \\cos 2 x \\\\\n+ \\frac{53}{8} e^3 \\cos 3 x + \\frac{77}{8} e^4 \\cos 4 x\n\\]\n\n\\[\na^2 r^{-2} = 1 + \\frac{e^2}{2} \\left( 1 + \\frac{3}{4} e^2 \\right) + 2 e \\left( 1 + \\frac{3}{8} e^2 \\right) \\cos x + \\frac{5}{2} e^2 \\left( 1 + \\frac{2}{15} e^2 \\right) \\cos 2 x \\\\\n+ \\frac{13}{4} e^3 \\cos 3 x + \\frac{103}{24} e^4 \\cos 4 x\n\\]\n\n\\[\na r^{-1} = 1 + e \\left( 1 - \\frac{e^2}{8} \\right) \\cos x + e^2 \\left( 1 - \\frac{e^2}{3} \\right) \\cos 2 x + \\frac{9}{8} e^3 \\cos 3 x + \\frac{4}{3} e^4 \\cos 4 x\n\\]\n\n\\[\n\\frac{r}{a} = 1 + \\frac{e^2}{2} - e \\left( 1 - \\frac{3 e^2}{8} \\right) \\cos x - \\frac{e^2}{2} \\left( 1 - \\frac{2 e^2}{3} \\right) \\cos 2 x - \\frac{3 e^3}{8} \\cos 3 x - \\frac{e^4}{3} \\cos 4 x\n\\]\n\n\\[\n\\frac{r^2}{a^2} = 1 + \\frac{3 e^2}{2} - 2 e \\left( 1 - \\frac{e^2}{8} \\right) \\cos x - \\frac{e^2}{2} \\left( 1 - \\frac{e^2}{3} \\right) \\cos 2 x - \\frac{e^3}{4} \\cos 3 x - \\frac{e^4}{6} \\cos 4 x\n\\]\n\n\\[\n\\frac{r^3}{a^3} = 1 + 3 e^2 \\left( 1 + \\frac{e^2}{8} \\right) - 3 e \\left( 1 + \\frac{3}{8} e^2 \\right) \\cos x - \\frac{5}{8} e^4 \\cos 2 x + \\frac{e^3}{8} \\cos 3 x + \\frac{e^4}{8} \\cos 4 x\n\\]\n\n\\[\n\\frac{r^4}{a^4} = 1 + 5 e^2 - 4 e \\cos x + e^2 \\cos 2 x\n\\]\n\n\\[\n\\frac{a}{r} = r_0 \\\\\n+ r_1 \\cos 2 t \\\\\n+ e r_2 \\cos x \\\\\n+ e r_3 \\cos (2 t - x) \\\\\n+ e r_4 \\cos (2 t + x) \\\\\n+ e r_5 \\cos z \\\\\n+ e r_6 \\cos (2 t - z) + &c. &c.\n\\]\nλ = n t\n+ λ₁ cos 2 t\n+ e λ₂ cos x\n+ e λ₃ cos (2 t - x)\n+ e λ₄ cos (2 t + x)\n+ e λ₅ cos z &c. &c.\n\nThe quantities λ correspond to the quantities b in M. Damoiseau's notation.\n\ns = γ s₁₄₆ sin y\n+ γ s₁₄₇ sin (2 t - y)\n+ γ s₁₄₈ sin (2 t + y)\n+ eγ s₁₄₉ sin (x - y) &c. &c.\n\nγ = tan t\n\n\\[ R = m_i \\left\\{ \\frac{r^3 r_i \\cos (\\lambda - \\lambda_i)}{r_i^3} - \\frac{1}{\\{r^2 - 2 r^3 r_i \\cos (\\lambda' - \\lambda_i) + r_i^2\\}^{3/2}} \\right\\} \\]\n\n\\[ = m_i \\left\\{ -\\frac{1}{r_i} + \\frac{r^2}{2 r_i^3} - \\frac{3}{8} \\left\\{ 2 r^3 r_i \\cos (\\lambda' - \\lambda_i) - r_i^2 \\right\\}^2 - \\frac{15}{48} \\left\\{ 2 r^3 r_i \\cos (\\lambda' - \\lambda_i) - r_i^2 \\right\\}^3 \\right\\} \\]\n\n\\[ = m_i \\left\\{ -\\frac{1}{r_i} + \\frac{r^2}{2 r_i^3} - \\frac{3}{2} \\frac{r^3 r_i^2}{r_i^3} \\cos (\\lambda' - \\lambda_i)^2 + \\frac{3}{2} \\frac{r^3 r_i^2}{r_i^5} \\cos (\\lambda' - \\lambda_i) - \\frac{5}{2} \\frac{r^3 r_i^2}{r_i^7} \\cos (\\lambda' - \\lambda_i)^3 \\right\\} \\]\n\n\\[ = m_i \\left\\{ -\\frac{1}{r_i} - \\frac{r^2}{4 r_i^3} \\left\\{ 1 + 3 \\cos (2 \\lambda' - 2 \\lambda_i) - 2 s^2 \\right\\} \\right\\} \\]\n\n\\[ - \\frac{r^3}{8 r_i^4} \\left\\{ 3 (1 - 4 s^2) \\cos (\\lambda' - \\lambda_i) + 5 \\cos (3 \\lambda' - 3 \\lambda_i) \\right\\} \\]\n\n\\[ r^3 r_i \\cos (\\lambda' - \\lambda_i) = rr_i \\left\\{ \\cos^2 \\frac{\\lambda'}{2} \\cos (\\lambda - \\lambda_i) + \\sin^2 \\frac{\\lambda'}{2} \\cos (\\lambda + \\lambda_i - 2 \\nu) \\right\\} \\]\n\n\\[ = * a a_i \\cos^2 \\frac{\\lambda'}{2} \\left( 1 - \\frac{e^2}{2} - \\frac{e^4}{64} \\right) \\left( 1 - \\frac{e^2}{2} - \\frac{e^4}{64} \\right) \\cos t - \\frac{3}{2} e \\left( 1 - \\frac{e^2}{2} \\right) \\cos (t - x) \\]\n\n\\[ + \\frac{e}{2} \\left( 1 - \\frac{3}{4} e^2 \\right) \\left( 1 - \\frac{e^2}{2} \\right) \\cos (t + x) + \\frac{3}{8} e^2 \\left( 1 - e^2 \\right) \\left( 1 - \\frac{e^2}{2} \\right) \\cos (t + 2 x) \\]\n\n\\[ + \\frac{e^3}{3} \\cos (t + 3 x) + \\frac{125}{384} e^4 \\cos (t + 4 x) + \\frac{e^2}{8} \\left( 1 + \\frac{e^2}{3} \\right) \\left( 1 - \\frac{e^2}{2} \\right) \\cos (t - 2 x) \\]\n\n\\[ + \\frac{e^3}{24} \\cos (t - 3 x) + \\frac{3}{128} e^4 \\cos (t - 4 x) - \\frac{3}{2} e_i \\left( 1 - \\frac{e^2}{2} \\right) \\cos (t + z) \\]\n\n\\[ + \\frac{9}{4} e e_i \\cos (t - x + z) - \\frac{3}{4} e e_i \\left( 1 - \\frac{3}{4} e^2 \\right) \\cos (t + x + z) \\]\n\n\\[ - \\frac{9}{16} e^2 e_i \\cos (t + 2 x + z) - \\frac{e^3 e_i}{2} \\cos (t + 3 x + z) - \\frac{3}{16} e^2 e_i \\cos (t - 2 x + z) \\]\n\n* See Phil. Trans. 1830, p. 343.\n\\[\n-\\frac{e^3}{16} e_i \\cos(t - 3x + z) + \\frac{e_i}{2} \\left(1 - \\frac{3}{4} e_i^2\\right) \\left(1 - \\frac{e^2}{2}\\right) \\cos(t - z)\n\\]\n\\[\n- \\frac{3}{4} e e_i \\left(1 - \\frac{3}{4} e_i^2\\right) \\cos(t - x - z)\n\\]\n\\[\n+ \\frac{e e_i}{4} \\left(1 - \\frac{3}{4} e^2\\right) \\left(1 - \\frac{3}{4} e_i^2\\right) \\cos(t + x - z) + \\frac{3}{16} e^2 e_i \\cos(t + 2x - z)\n\\]\n\\[\n+ \\frac{e^3}{6} e_i \\cos(t + 3x - z) + \\frac{e^2}{16} e_i \\cos(t - 2x - z) + \\frac{e^3}{48} e_i \\cos(t - 3x - z)\n\\]\n\\[\n+ \\frac{3}{8} e_i^2 \\left(1 - e_i^2\\right) \\left(1 - \\frac{e^2}{2}\\right) \\cos(t - 2z) - \\frac{9}{16} e e_i^2 \\cos(t - x - 2z)\n\\]\n\\[\n+ \\frac{3}{16} e e_i^2 \\cos(t + x - 2z) + \\frac{9}{64} e^2 e_i^2 \\cos(t + 2x - 2z)\n\\]\n\\[\n+ \\frac{3}{64} e^2 e_i^2 \\cos(t - 2x - 2z) + \\frac{e^3}{3} \\cos(t - 3z) - \\frac{e e_i^3}{2} \\cos(t - x - 3z)\n\\]\n\\[\n+ \\frac{e e_i^3}{6} \\cos(t + x - 3z) + \\frac{125}{384} e_i^4 \\cos(t - 4z)\n\\]\n\\[\n+ \\frac{e_i^3}{8} \\left(1 + \\frac{e^2}{3}\\right) \\left(1 - \\frac{e^2}{2}\\right) \\cos(t + 2z) - \\frac{3}{16} e e_i^2 \\cos(t - x + 2z)\n\\]\n\\[\n+ \\frac{e e_i^2}{16} \\cos(t + x + 2z) + \\frac{3}{64} e^2 e_i^2 \\cos(t + 2x + 2z) + \\frac{e^2 e_i^2}{64} \\cos(t - 2x + 2z)\n\\]\n\\[\n+ \\frac{e_i^3}{24} \\cos(t + 3z) - \\frac{e e_i^3}{16} \\cos(t - x + 3z) + \\frac{e e_i^3}{48} \\cos(t + x + 3z)\n\\]\n\\[\n+ \\frac{3}{128} e_i^4 \\cos(t + 4z)\n\\]\n\n\\[+ a a_i \\sin^2 \\frac{t}{2} \\left\\{ \\left(1 - \\frac{e^2 + e_i^2}{2}\\right) \\cos(t - 2y) - \\frac{3}{2} e \\cos(t + x - 2y) + \\frac{e}{2} \\cos(t - x - 2y)\\right.\\]\n\\[\n+ \\frac{3}{8} e^2 \\cos(t - 2x - 2y) + \\frac{e^2}{8} \\cos(t + 2x - 2y) - \\frac{3}{2} e_i \\cos(t + z - 2y)\n\\]\n\\[\n+ \\frac{9}{4} e e_i \\cos(t + x + z - 2y) - \\frac{3}{4} e e_i \\cos(t - x + z - 2y)\n\\]\n\\[\n+ \\frac{e_i}{2} \\cos(t - z - 2y) - \\frac{3}{4} e e_i \\cos(t + x - z - 2y) + \\frac{e e_i}{4} \\cos(t - x - z - 2y)\n\\]\n\n\\[r^2 r_i^2 \\cos(\\lambda' - \\lambda)^2 = a^2 a_i^2 \\cos^4 \\frac{t}{2} \\left\\{ \\frac{1}{2} + \\left(-\\frac{1}{2} + \\frac{9}{8} + \\frac{1}{8}\\right)(e^2 + e_i^2) + \\left(\\frac{1}{2} - \\frac{9}{8} - \\frac{1}{8} - \\frac{9}{8} + \\frac{81}{32} + \\frac{9}{32}\\right)\\right.\\]\n\\[\n- \\frac{1}{8} + \\frac{9}{32} + \\frac{1}{32}\\right\\} e^2 e_i^2 + \\left\\{\\frac{7}{64} - \\frac{3}{16} + \\frac{9}{128} + \\frac{1}{128}\\right\\}(e^4 + e_i^4)\\]\n\n[0]\n\\[ + \\left\\{ \\frac{1}{2} + \\left( -\\frac{1}{2} - \\frac{3}{4} \\right) (e^2 + e_i^2) + \\left( \\frac{1}{2} + \\frac{3}{4} + \\frac{3}{4} + \\frac{9}{16} + \\frac{9}{16} \\right) e^2 e_i^2 \\right\\} \\cos 2t \\]\n\n\\[ + \\left\\{ \\frac{7}{64} + \\frac{9}{16} + \\frac{3}{64} \\right\\} (e^4 + e_i^4) \\]\n\n\\[ + \\left\\{ -\\frac{3}{2} + \\frac{1}{2} + \\left( \\frac{3}{4} - \\frac{5}{8} - \\frac{3}{16} + \\frac{3}{16} \\right) e^2 \\right\\} e \\cos x \\]\n\n\\[ + \\left\\{ \\frac{3}{2} - \\frac{1}{2} - \\frac{27}{8} + \\frac{9}{8} - \\frac{3}{8} + \\frac{1}{8} \\right\\} e_i^2 \\]\n\n\\[ + \\left\\{ -\\frac{3}{2} + \\left( \\frac{3}{4} + \\frac{1}{16} \\right) e^2 + \\left( \\frac{3}{2} + \\frac{9}{8} + \\frac{9}{8} \\right) e_i^2 \\right\\} e \\cos (2t - x) \\]\n\n\\[ + \\left\\{ -\\frac{1}{2} + \\left( \\frac{5}{8} - \\frac{9}{16} \\right) e^2 + \\left( -\\frac{1}{2} - \\frac{3}{8} - \\frac{3}{8} \\right) e_i^2 \\right\\} e \\cos (2t + x) \\]\n\n\\[ + \\left\\{ \\frac{3}{8} + \\frac{1}{8} - \\frac{3}{4} + \\left( -\\frac{9}{16} - \\frac{1}{48} + \\frac{9}{16} - \\frac{1}{16} + \\frac{1}{6} \\right) e^2 \\right\\} e^2 \\cos 2x \\]\n\n\\[ + \\left\\{ -\\frac{3}{8} - \\frac{1}{8} + \\frac{3}{4} + \\frac{27}{32} + \\frac{9}{32} - \\frac{27}{16} + \\frac{3}{32} + \\frac{1}{32} - \\frac{3}{16} \\right\\} e_i^2 \\]\n\n\\[ + \\left\\{ \\frac{9}{8} + \\frac{1}{8} + \\left( -\\frac{1}{48} + \\frac{1}{48} \\right) e^2 \\right\\} e^2 \\cos (2t - 2x) \\]\n\n\\[ + \\left\\{ \\frac{1}{8} + \\frac{3}{8} + \\left( -\\frac{3}{16} - \\frac{9}{16} - \\frac{1}{2} \\right) e^2 \\right\\} e^2 \\cos (2t + 2x) \\]\n\n\\[ + \\left\\{ -\\frac{3}{4} - \\frac{3}{4} + \\frac{9}{4} + \\frac{1}{4} + \\left( \\frac{15}{16} + \\frac{3}{8} - \\frac{9}{8} - \\frac{3}{32} - \\frac{9}{32} - \\frac{5}{16} \\right) \\right\\} e e_i \\cos (x + z) \\]\n\n* The coefficient of argument 5 being the same, \\( e \\) and \\( e_i \\) changing places, that coefficient is not written down, in order to avoid useless repetition.\n\\[\n\\begin{align*}\n&+ \\left\\{ -\\frac{3}{4} - \\frac{3}{4} + \\left\\{ \\frac{3}{8} + \\frac{3}{8} + \\frac{1}{32} + \\frac{1}{32} \\right\\} e^2 \\right. \\\\\n&\\quad + \\left\\{ \\frac{15}{16} + \\frac{15}{16} + \\frac{27}{32} + \\frac{27}{32} \\right\\} e^2 e_i \\cos (2t - x - z) \\\\\n&\\left. \\quad + \\left\\{ \\frac{9}{4} + \\frac{1}{4} - \\frac{3}{4} + \\left\\{ -\\frac{9}{8} - \\frac{5}{16} + \\frac{9}{32} + \\frac{3}{8} + \\frac{15}{16} \\right. \\right. \\right. \\\\\n&\\quad + \\left. \\left. \\left. \\frac{3}{32} - \\frac{9}{32} - \\frac{3}{32} \\right\\} (e^2 + e_i^2) \\right\\} e e_i \\cos (x - z) \\right. \\\\\n&\\left. \\quad + \\left\\{ \\frac{9}{4} + \\frac{9}{4} + \\left\\{ -\\frac{9}{8} - \\frac{9}{8} - \\frac{3}{32} - \\frac{3}{32} \\right\\} (e^2 + e_i^2) \\right\\} ee_i \\cos (2t - x + z) \\right. \\\\\n&\\left. \\quad + \\left\\{ \\frac{1}{4} + \\frac{1}{4} + \\left\\{ -\\frac{5}{16} - \\frac{9}{32} - \\frac{5}{16} - \\frac{9}{32} \\right\\} (e_i + e_i^2) \\right\\} \\cos (2t + x - z) \\right. \\\\\n&\\left. \\quad + \\left\\{ \\frac{1}{3} + \\frac{1}{24} - \\frac{9}{16} + \\frac{1}{16} \\right\\} e \\cos 3x \\right. \\\\\n&\\left. \\quad + \\left\\{ \\frac{1}{24} - \\frac{3}{16} \\right\\} e^3 \\cos (2t - 3x) \\right. \\\\\n&\\left. \\quad + \\left\\{ \\frac{1}{3} + \\frac{3}{16} \\right\\} e^3 \\cos (2t + 3x) \\right. \\\\\n&\\left. \\quad + \\left\\{ -\\frac{9}{16} + \\frac{1}{16} + \\frac{9}{8} - \\frac{3}{8} + \\frac{3}{16} - \\frac{3}{16} \\right\\} e^2 e_i \\cos (2x + z) \\right. \\\\\n&\\left. \\quad + \\left\\{ \\frac{1}{16} + \\frac{9}{8} + \\frac{1}{16} \\right\\} e^2 e_i \\cos (2t - 2x - z) \\right. \\\\\n&\\left. \\quad + \\left\\{ -\\frac{9}{16} - \\frac{3}{8} - \\frac{9}{16} \\right\\} e^2 e_i \\cos (2t + 2x + z) \\right. \\\\\n&\\left. \\quad + \\left\\{ -\\frac{3}{16} + \\frac{3}{16} - \\frac{3}{8} + \\frac{9}{8} - \\frac{9}{16} + \\frac{1}{16} \\right\\} e^2 e_i \\cos (2x - z) \\right. \\\\\n&\\left. \\quad + \\left\\{ -\\frac{3}{16} - \\frac{27}{8} - \\frac{3}{16} \\right\\} e^2 e_i \\cos (2t - 2x + z) \\right. \\\\\n&\\left. \\quad + \\left\\{ \\frac{3}{16} + \\frac{1}{8} + \\frac{3}{16} \\right\\} e^2 e_i \\cos (2t + 2x - z) \\right. \\\\\n&\\left. \\quad + \\left\\{ \\frac{125}{384} + \\frac{3}{128} - \\frac{1}{2} + \\frac{1}{48} + \\frac{3}{64} \\right\\} e^4 \\cos 4x \\right.\n\\end{align*}\n\\]\n\\[ + \\left\\{ \\frac{1}{128} + \\frac{3}{128} - \\frac{1}{16} \\right\\} e^4 \\cos (2t - 4x) \\]\n\n\\[ + \\left\\{ \\frac{9}{128} + \\frac{125}{384} + \\frac{1}{6} \\right\\} e^4 \\cos (2t + 4x) \\]\n\n\\[ + \\left\\{ -\\frac{1}{2} + \\frac{1}{48} + \\frac{27}{32} + \\frac{1}{32} - \\frac{9}{32} + \\frac{1}{6} - \\frac{3}{32} - \\frac{1}{16} \\right\\} e^3 \\cos (3x + z) \\]\n\n\\[ + \\left\\{ \\frac{1}{48} - \\frac{3}{32} - \\frac{3}{32} + \\frac{1}{48} \\right\\} e^3 e_i \\cos (2t - 3x - z) \\]\n\n\\[ + \\left\\{ -\\frac{1}{2} - \\frac{9}{32} - \\frac{9}{32} - \\frac{1}{2} \\right\\} e^3 e_i \\cos (2t + 3x + z) \\]\n\n\\[ + \\left\\{ -\\frac{1}{16} + \\frac{1}{6} - \\frac{9}{32} - \\frac{3}{32} + \\frac{27}{32} - \\frac{1}{2} + \\frac{1}{32} + \\frac{1}{48} \\right\\} e^3 e_i \\cos (3x - z) \\]\n\n\\[ + \\left\\{ -\\frac{1}{16} + \\frac{9}{32} + \\frac{9}{32} - \\frac{1}{16} \\right\\} e^3 e_i \\cos (2t - 3x + z) \\]\n\n\\[ + \\left\\{ \\frac{1}{6} + \\frac{3}{32} + \\frac{3}{32} + \\frac{1}{6} \\right\\} e^3 e_i \\cos (2t + 3x - z) \\]\n\n\\[ + \\left\\{ \\frac{3}{64} + \\frac{3}{64} - \\frac{3}{32} - \\frac{9}{32} + \\frac{9}{64} + \\frac{1}{64} - \\frac{3}{32} + \\frac{9}{16} - \\frac{9}{32} \\right\\} e^3 e_i^3 \\cos (2x + 2z) \\]\n\n\\[ + \\left\\{ \\frac{9}{32} + \\frac{3}{64} + \\frac{27}{32} + \\frac{3}{64} + \\frac{1}{32} \\right\\} e^3 e_i^3 \\cos (2t - 2x + 2z) \\]\n\n\\[ + \\left\\{ \\frac{9}{32} + \\frac{3}{64} + \\frac{1}{32} + \\frac{3}{64} + \\frac{27}{32} \\right\\} e^3 e_i^3 \\cos (2t + 2x + 2z) \\]\n\n\\[ + \\left\\{ \\frac{9}{64} + \\frac{1}{64} - \\frac{9}{32} - \\frac{3}{32} + \\frac{3}{64} + \\frac{3}{64} - \\frac{9}{32} + \\frac{9}{16} - \\frac{3}{32} \\right\\} e^3 e_i^3 \\cos (2x - 2z) \\]\n\n\\[ + \\left\\{ \\frac{81}{32} + \\frac{1}{64} + \\frac{9}{32} + \\frac{1}{64} + \\frac{9}{32} \\right\\} e^3 e_i^3 \\cos (2t - 2x + 2z) \\]\n\n\\[ + \\left\\{ \\frac{1}{32} + \\frac{9}{64} + \\frac{3}{32} + \\frac{9}{64} + \\frac{3}{32} \\right\\} e^3 e_i^3 \\cos (2t + 2x - 2z) \\]\n\n\\[ + a^2 a_i^3 \\sin^3 \\frac{t}{2} \\cos^2 \\frac{t}{2} \\left\\{ 1 + \\left\\{ -1 - \\frac{3}{4} - \\frac{3}{4} \\right\\} e^3 + \\left\\{ -1 + \\frac{9}{4} + \\frac{1}{4} \\right\\} e_i^3 \\right\\} \\cos 2y \\]\n\n\\[ + \\left\\{ 1 + \\left\\{ -1 + \\frac{9}{4} + \\frac{1}{4} \\right\\} e^3 + \\left\\{ -1 - \\frac{3}{4} - \\frac{3}{4} \\right\\} e_i^3 \\right\\} \\cos (2t - 2y) \\]\n\\[ + \\left\\{ -\\frac{3}{2} - \\frac{3}{2} \\right\\} e \\cos (x - 2y) + \\left\\{ \\frac{1}{2} + \\frac{1}{2} \\right\\} e \\cos (x + 2y) \\]\n\n\\[ + \\left\\{ \\frac{1}{2} - \\frac{3}{2} \\right\\} e \\cos (2t - x - 2y) \\]\n\n\\[ + \\left\\{ -\\frac{3}{2} + \\frac{1}{2} \\right\\} e \\cos (2t + x - 2y) + \\left\\{ -\\frac{3}{2} + \\frac{1}{2} \\right\\} e_t \\cos (z - 2y) \\]\n\n\\[ + \\left\\{ \\frac{1}{2} - \\frac{3}{2} \\right\\} e_t \\cos (z + 2y) + \\left\\{ \\frac{1}{2} + \\frac{1}{2} \\right\\} e_t \\cos (2t - z - 2y) \\]\n\n\\[ + \\left\\{ -\\frac{3}{2} - \\frac{3}{2} \\right\\} e_t \\cos (2t + z - 2y) + \\left\\{ \\frac{1}{8} + \\frac{9}{4} + \\frac{1}{8} \\right\\} e^2 \\cos (2x - 2y) \\]\n\n\\[ + \\left\\{ \\frac{3}{8} + \\frac{1}{4} + \\frac{3}{8} \\right\\} e^2 \\cos (2x + 2y) \\]\n\n\\[ + \\left\\{ \\frac{3}{8} - \\frac{3}{4} + \\frac{1}{8} \\right\\} e^2 \\cos (2t - 2x - 2y) + \\left\\{ \\frac{1}{8} - \\frac{3}{4} + \\frac{3}{8} \\right\\} e^2 \\cos (2t + 2x - 2y) \\]\n\n\\[ + \\left\\{ \\frac{9}{4} + \\frac{9}{4} - \\frac{3}{4} - \\frac{3}{4} \\right\\} ee_t \\cos (x + z - 2y) \\]\n\n\\[ + \\left\\{ \\frac{1}{4} + \\frac{1}{4} - \\frac{3}{4} - \\frac{3}{4} \\right\\} ee_t \\cos (x + z + 2y) \\]\n\n\\[ + \\left\\{ \\frac{1}{4} - \\frac{3}{4} + \\frac{1}{4} - \\frac{3}{4} \\right\\} ee_t \\cos (2t - x - z - 2y) \\]\n\n\\[ + \\left\\{ \\frac{9}{4} - \\frac{3}{4} + \\frac{9}{4} - \\frac{3}{4} \\right\\} ee_t \\cos (2t + x + z + 2y) \\]\n\n\\[ + \\left\\{ -\\frac{3}{4} - \\frac{3}{4} + \\frac{9}{4} + \\frac{9}{4} \\right\\} ee_t \\cos (x - z - 2y) \\]\n\n\\[ + \\left\\{ -\\frac{3}{4} - \\frac{3}{4} + \\frac{1}{4} + \\frac{1}{4} \\right\\} ee_t \\cos (x - z + 2y) \\]\n\n\\[ + \\left\\{ -\\frac{3}{4} + \\frac{9}{4} - \\frac{3}{4} + \\frac{9}{4} \\right\\} ee_t \\cos (2t - x + z - 2y) \\]\n\n\\[ + \\left\\{ -\\frac{3}{4} + \\frac{1}{4} - \\frac{3}{4} + \\frac{1}{4} \\right\\} ee_t \\cos (2t + x - z - 2y) \\]\n\\[ r^2 r^2 \\cos (\\lambda' - \\lambda)^2 = a^2 e_i^2 \\cos^4 \\frac{t}{2} \\left\\{ \\frac{1}{2} + \\frac{3}{4} (e^2 + e_i^2) + \\frac{9}{8} e^2 e_i^2 + \\left\\{ \\frac{1}{2} - \\frac{5}{4} (e^2 + e_i^2) \\right\\} \\right. \\]\n\\[ + \\frac{23}{32} (e^4 + e_i^4) + \\frac{25}{8} e^2 e_i^2 \\} \\cos 2t + \\left\\{ -1 + \\frac{e^2}{8} - \\frac{3}{2} e_i^2 \\right\\} e \\cos x \\]\n\\[ + \\left\\{ -\\frac{3}{2} + \\frac{13}{16} e^2 + \\frac{15}{4} e_i^2 \\right\\} e \\cos (2t - x) + \\left\\{ \\frac{1}{2} - \\frac{19}{16} e^2 - \\frac{5}{4} e_i^2 \\right\\} e \\cos (2t + x) \\]\n\\[ + \\left\\{ -\\frac{1}{4} + \\frac{1}{12} e^2 - \\frac{3}{8} e_i^2 \\right\\} e^2 \\cos 2x \\]\n\\[ + \\left\\{ \\frac{5}{4} - \\frac{25}{8} e_i^2 \\right\\} e^2 \\cos (2t - 2x) + \\left\\{ \\frac{1}{2} - \\frac{5}{4} e^2 - \\frac{5}{4} e_i^2 \\right\\} e \\cos (2t + 2x) \\]\n\\[ + \\left\\{ 1 - \\frac{1}{8} (e^2 + e_i^2) \\right\\} ee_i \\cos (x + z) \\]\n\\[ + \\left\\{ -\\frac{3}{2} + \\frac{13}{16} e^2 + \\frac{57}{16} e_i^2 \\right\\} ee_i \\cos (2t - x - z) \\]\n\\[ + \\left\\{ 1 - \\frac{(e^2 + e_i^2)}{8} \\right\\} ee_i \\cos (x - z) \\]\n\\[ + \\left\\{ \\frac{9}{2} - \\frac{39}{16} (e^2 + e_i^2) \\right\\} ee_i \\cos (2t - x + z) \\]\n\\[ + \\left\\{ \\frac{1}{2} - \\frac{19}{16} (e^2 + e_i^2) \\right\\} ee_i \\cos (2t + x - z) - \\frac{e^3}{8} \\cos 3x \\]\n\\[ - \\frac{7}{48} e^3 \\cos (2t - 3x) + \\frac{25}{48} e^3 \\cos (2t + 3x) + \\frac{e^2 e_i}{4} \\cos (2x + z) \\]\n\\[\n\\begin{align*}\n&+ \\frac{5}{4} e^2 e_i \\cos (2t - 2x - z) - \\frac{25}{16} e^2 e_i \\cos (2t + 2x + z) + \\frac{e^2 e_i}{4} \\cos (2x - z) \\\\\n&[24] [31] & [25] [30] & [26] [32] \\\\\n&- \\frac{15}{4} e^2 e_i \\cos (2t - 2x + z) + \\frac{e^2 e_i}{2} \\cos (2t + 2x - z) \\\\\n&[27] [33] & [28] [34] \\\\\n&- \\frac{e^4}{12} \\cos 4x - \\frac{e^4}{32} \\cos (2t - 4x) + \\frac{9}{16} e^4 \\cos (2t + 4x) + \\frac{e^3 e_i}{8} \\cos (3x + z) \\\\\n&[38] [59] & [39] [61] & [40] [60] & [41] [53] \\\\\n&- \\frac{7}{48} e^3 e_i \\cos (2t - 3x - z) - \\frac{25}{16} e^3 e_i \\cos (2t + 3x + z) \\\\\n&[42] [55] & [43] [54] \\\\\n&+ \\frac{e^3 e_i}{8} \\cos (3x - z) + \\frac{7}{16} e^3 e_i \\cos (2t - 3x + z) + \\frac{25}{48} e^3 e_i \\cos (2t + 3x - z) \\\\\n&[44] [56] & [45] [57] & [46] [58] \\\\\n&+ \\frac{e^2 e_i}{16} \\cos (2x + 2z) + \\frac{5}{4} e^2 e_i \\cos (2t - 2x - 2z) \\\\\n&[47] & [48] \\\\\n&+ \\frac{5}{4} e^2 e_i \\cos (2t + 2x + 2z) + \\frac{e^2 e_i}{16} \\cos (2x - 2z) \\\\\n&[49] & [50] \\\\\n&+ \\frac{25}{8} e^2 e_i \\cos (2t - 2x + 2z) + \\frac{e^2 e_i}{2} \\cos (2t + 2x - 2z) \\\\\n&[51] & [52] \\\\\n&+ a^3 a_i^3 \\cos^3 \\frac{\\epsilon}{2} \\sin^2 \\frac{\\epsilon}{2} \\left\\{ \\left[ 1 - \\frac{5}{2} e^2 + \\frac{3}{2} e_i^2 \\right] \\cos 2y + \\left[ 1 + \\frac{3}{2} e^2 - \\frac{5}{2} e_i^2 \\right] \\cos (2t - 2y) \\right\\} \\\\\n&[62] & [63] \\\\\n&- 3e \\cos (x - 2y) + e \\cos (x + 2y) - e \\cos (2t - x - 2y) \\\\\n&[65] & [66] & [67] \\\\\n&- e \\cos (2t + x - 2y) - e_i \\cos (z - 2y) - e_i \\cos (z + 2y) \\\\\n&[69] & [71] & [72] \\\\\n&+ e_i \\cos (2t - z - 2y) - 3e_i \\cos (2t + z - 2y) \\\\\n&[73] & [75] \\\\\n&+ \\frac{5}{2} e^2 \\cos (2x - 2y) + e^2 \\cos (2x + 2y) \\\\\n&[77] & [78] \\\\\n&- \\frac{e^2}{4} \\cos (2t - 2x - 2y) - \\frac{e^2}{4} \\cos (2t + 2x - 2y) \\\\\n&[79] & [81]\n\\end{align*}\n\\]\n\\[ + 3 e e_i \\cos (x + z - 2y) - ee_i \\cos (x + z + 2y) \\]\n\n\\[ - ee_i \\cos (2t - x - z - 2y) + 3 ee_i \\cos (2t + x + z - 2y) \\]\n\n\\[ + 3 ee_i \\cos (x - z - 2y) - ee_i \\cos (x - z + 2y) \\]\n\n\\[ + 3 ee_i \\cos (2t - x + z - 2y) - ee_i \\cos (2t + x - z - 2y) \\]\n\n\\[ - \\frac{3}{8} e_i^2 \\cos (2z - 2y) - \\frac{5}{8} e_i^2 \\cos (2z + 2y) \\]\n\n\\[ + \\frac{5}{8} e_i^2 \\cos (2t - 2z - 2y) + \\frac{19}{8} e_i^2 \\cos (2t + 2z - 2y) \\}\n\n\\[ + a^2 a_i^2 \\sin^4 \\frac{t}{2} \\left\\{ \\frac{1}{2} + \\frac{1}{2} \\cos (2t - 2y) \\right\\} \\]\n\n\\[ \\frac{r^2}{2r_i^3} = \\frac{a^2}{a_i^3} \\left[ \\frac{1}{2} + \\frac{3}{4} e^2 + \\frac{3}{4} e_i^2 + \\frac{9}{8} e^2 e_i^2 + \\frac{15}{16} e_i^4 - e \\left\\{ 1 - \\frac{e^2}{8} + \\frac{3}{2} e_i^2 \\right\\} \\cos x \\right] \\]\n\n\\[ + \\frac{3}{2} e_i \\left\\{ 1 + \\frac{3}{2} e^2 + \\frac{9}{8} e_i^2 \\right\\} \\cos z - \\frac{e^2}{4} \\left\\{ 1 - \\frac{e^2}{3} + \\frac{3}{2} e_i^2 \\right\\} \\cos 2x \\]\n\n\\[ - \\frac{3}{2} ee_i \\left\\{ 1 - \\frac{e^2}{8} + \\frac{9}{8} e_i^2 \\right\\} \\cos (x + z) \\]\n\n\\[ - \\frac{3}{2} ee_i \\left\\{ 1 - \\frac{e^2}{8} + \\frac{9}{8} e_i^2 \\right\\} \\cos (x - z) \\]\n\n\\[ + \\frac{9}{4} e_i^2 \\left\\{ 1 + \\frac{7}{9} e_i^2 + \\frac{3}{2} e^2 \\right\\} \\cos 2z - \\frac{e^3}{8} \\cos 3x - \\frac{3}{8} e^3 e_i \\cos (2x + z) \\]\n\n\\[ - \\frac{3}{8} e^3 e_i \\cos (2x - z) - \\frac{9}{4} ee_i^3 \\cos (x + 2z) - \\frac{9}{4} ee_i^3 \\cos (x - 2z) \\]\n\n\\[ + \\frac{53}{16} e_i^3 \\cos 3z - \\frac{e^4}{12} \\cos 4x - \\frac{3}{16} e^3 e_i \\cos (3x + z) \\]\n\\[\n- \\frac{3}{16} e^3 e_i \\cos (3x - z) - \\frac{9}{16} e^2 e_i^2 \\cos (2x + 2z) - \\frac{9}{16} e^2 e_i \\cos (2x - 2z)\n\\]\n\n\\[\n- \\frac{53}{16} e e_i^3 \\cos (x + 3z) - \\frac{53}{16} e e_i^3 \\cos (x - 3z) + \\frac{77}{16} e^4 \\cos 4z\n\\]\n\nTerms in \\( R \\) multiplied by \\(- \\frac{3}{2} \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3}\\)\n\n\\[\n= \\left\\{ \\frac{1}{2} + \\frac{3}{4} (e^2 + e_i^2) + \\frac{9}{8} e^2 e_i^2 \\right\\} \\left\\{ 1 + 5 e_i^2 + \\frac{105}{8} e_i^4 \\right\\}\n\\]\n\n\\[\n+ \\left\\{ -1 + \\frac{e_i^2}{8} - \\frac{3}{2} e \\right\\} \\left\\{ \\frac{5}{2} e_i^2 + \\frac{135}{16} e_i^4 \\right\\} - \\frac{5}{4} e_i^4\n\\]\n\n\\[\n+ \\left\\{ \\frac{1}{2} - \\frac{5}{4} (e^2 + e_i^2) + \\frac{23}{32} (e^4 + e_i^4) + \\frac{25}{8} e^2 e_i^2 \\right\\} \\left\\{ 1 + 5 e_i^2 + \\frac{105}{8} e_i^4 \\right\\}\n\\]\n\n\\[\n+ \\left\\{ \\frac{1}{2} - \\frac{19}{16} e_i^2 - \\frac{5}{4} e^2 - \\frac{3}{2} + \\frac{13}{16} e_i^2 + \\frac{15}{4} e^2 \\right\\} \\left\\{ \\frac{5}{2} e_i^2 + \\frac{135}{16} e_i^4 \\right\\}\n\\]\n\n\\[\n+ \\left\\{ \\frac{5}{2} + \\frac{25}{4} \\right\\} e_i^4 \\cos 2t\n\\]\n\n\\[\n+ \\left\\{ -1 + \\frac{e_i^2}{8} - \\frac{3}{2} e_i^2 - 5 e_i^2 + \\frac{5}{2} e_i^2 + \\frac{5}{2} e_i^2 \\right\\} e \\cos x\n\\]\n\n\\[\n+ \\left\\{ -\\frac{3}{2} + \\frac{13}{16} e_i^2 + \\frac{15}{4} e_i^2 - \\frac{15}{2} e_i^2 - \\frac{15}{4} e_i^2 + \\frac{45}{4} e_i^2 \\right\\} e \\cos (2t - x)\n\\]\n\n\\[\n+ \\left\\{ \\frac{1}{2} - \\frac{19}{16} e_i^2 - \\frac{5}{4} e_i^2 + \\frac{5}{2} e_i^2 + \\frac{5}{4} e_i^2 - \\frac{15}{4} e_i^2 \\right\\} e \\cos (2t + x)\n\\]\n\n\\[\n+ \\left\\{ -1 + \\frac{e_i^2}{8} - \\frac{3}{2} e_i^2 - 5 e_i^2 + \\frac{5}{2} + \\frac{15}{4} e_i^2 + \\frac{15}{4} e_i^2 + \\frac{135}{16} e_i^2 - \\frac{5}{8} e_i^2 - 5 e_i^2 \\right\\} e_i \\cos z\n\\]\n\n\\[\n+ \\left\\{ \\frac{1}{2} - \\frac{19}{16} e_i^2 - \\frac{5}{4} e_i^2 + \\frac{5}{2} e_i^2 + \\frac{5}{4} e_i^2 + \\frac{5}{4} - \\frac{25}{8} e_i^2 - \\frac{25}{8} e_i^2 + \\frac{135}{32} e_i^2 - \\frac{15}{2} e_i^2 \\right\\} e_i \\cos (2t - z)\n\\]\n\n* This multiplication of \\( r^2 r_i^2 \\cos (\\lambda' - \\lambda)^2 \\) by \\( r_i^5 \\) may be effected at once by means of Table II.\n\\[ + \\left\\{ -\\frac{3}{2} + \\frac{13}{16} e_i^2 + \\frac{15}{4} e_i^3 - \\frac{15}{2} e_i^4 + \\frac{5}{4} - \\frac{25}{8} e_i^2 + \\frac{135}{32} e_i^3 + \\frac{25}{8} e_i^4 + \\frac{5}{2} e_i^5 \\right\\} e_i \\cos(2t + z) \\]\n\n\\[ + \\left\\{ -\\frac{1}{4} + \\frac{e_i^2}{12} - \\frac{3}{8} e_i^3 - \\frac{5}{4} e_i^4 + \\frac{5}{8} e_i^5 + \\frac{5}{8} e_i^6 \\right\\} e_i^2 \\cos 2x \\]\n\n\\[ + \\left\\{ \\frac{5}{4} - \\frac{25}{8} e_i^2 + \\frac{25}{4} e_i^3 + \\frac{25}{8} e_i^4 - \\frac{75}{8} e_i^5 \\right\\} e_i^3 \\cos (2t - 2x) \\]\n\n\\[ + \\left\\{ \\frac{1}{2} - \\frac{5}{4} e_i^2 - \\frac{5}{4} e_i^3 + \\frac{5}{2} e_i^4 + \\frac{5}{4} e_i^5 - \\frac{15}{4} e_i^6 \\right\\} e_i^2 \\cos (2t + 2x) \\]\n\n\\[ + \\left\\{ 1 - \\frac{e_i^2}{8} - \\frac{e_i^3}{8} + 5 e_i^4 - \\frac{5}{2} + \\frac{5}{16} - \\frac{15}{4} e_i^2 - \\frac{135}{16} e_i^3 + \\frac{5}{8} e_i^4 + 5 e_i^5 \\right\\} e_i e_i \\cos (x + z) \\]\n\n\\[ + \\left\\{ -\\frac{3}{2} + \\frac{13}{16} e_i^2 + \\frac{57}{16} e_i^3 - \\frac{15}{2} e_i^4 - \\frac{15}{4} e_i^5 - \\frac{15}{4} + \\frac{65}{32} e_i^2 + \\frac{75}{8} e_i^3 \\right. \\]\n\n\\[ - \\frac{405}{32} e_i^4 + \\frac{45}{2} e_i^5 \\right\\} e_i e_i \\cos (2t - x - z) + \\left\\{ -\\frac{3}{2} + \\frac{13}{16} e_i^2 + \\frac{57}{16} e_i^3 - \\frac{15}{2} e_i^4 + \\frac{5}{4} \\right. \\]\n\n\\[ - \\frac{95}{32} e_i^2 - \\frac{25}{8} e_i^3 + \\frac{135}{32} e_i^4 + \\frac{25}{8} e_i^5 + \\frac{5}{2} e_i^6 \\right\\} e_i e_i \\cos (2t + x + z) \\]\n\n\\[ + \\left\\{ 1 - \\frac{e_i^2}{8} - \\frac{e_i^3}{8} + 5 e_i^4 - \\frac{5}{2} + \\frac{5}{16} e_i^2 - \\frac{15}{4} e_i^3 - \\frac{135}{16} e_i^4 + 5 e_i^5 \\right\\} e_i e_i \\cos (x - z) \\]\n\n\\[ + \\left\\{ \\frac{9}{2} - \\frac{39}{16} e_i^2 - \\frac{39}{16} e_i^3 + \\frac{45}{2} e_i^4 - \\frac{15}{4} + \\frac{65}{32} e_i^2 + \\frac{75}{8} e_i^3 - \\frac{405}{32} e_i^4 \\right. \\]\n\n\\[ - \\frac{75}{8} e_i^5 - \\frac{15}{2} e_i^6 \\right\\} e_i e_i \\cos (2t - x + z) \\]\n\n\\[ + \\left\\{ \\frac{1}{2} - \\frac{19}{16} e_i^2 - \\frac{19}{16} e_i^3 + \\frac{5}{2} e_i^4 + \\frac{5}{4} e_i^5 + \\frac{5}{4} - \\frac{95}{32} e_i^2 - \\frac{25}{8} e_i^3 + \\frac{135}{32} e_i^4 - \\frac{15}{2} e_i^5 \\right\\} e_i e_i \\cos (2t + x - z) \\]\n\n\\[ + \\left\\{ -\\frac{1}{4} + \\frac{e_i^2}{12} - \\frac{3}{8} e_i^3 - \\frac{5}{4} e_i^4 - \\frac{5}{2} + \\frac{5}{16} e_i^2 - \\frac{15}{4} e_i^3 - \\frac{135}{16} e_i^4 - \\frac{5}{16} e_i^5 + 5 \\right. \\]\n\n\\[ + \\frac{15}{2} e_i^2 + \\frac{15}{2} e_i^3 + \\frac{155}{12} e_i^4 - \\frac{145}{16} e_i^5 \\right\\} e_i^3 \\cos 2z \\]\n\n\\[ + \\left\\{ \\frac{1}{2} - \\frac{5}{4} e_i^2 - \\frac{5}{4} e_i^3 + \\frac{5}{2} e_i^4 + \\frac{125}{96} e_i^5 + \\frac{5}{4} - \\frac{95}{32} e_i^2 - \\frac{25}{8} e_i^3 + \\frac{135}{32} e_i^4 + \\frac{5}{2} - \\frac{25}{4} e_i^2 - \\frac{25}{4} e_i^3 \\right. \\]\n\n\\[ + \\frac{155}{24} e_i^4 - \\frac{435}{32} e_i^5 \\right\\} e_i^3 \\cos (2t - 2z) \\]\n\\[ \\frac{5}{4} - \\frac{25}{8} e^3 + \\frac{25}{4} e_i^2 - \\frac{15}{4} + \\frac{65}{32} e_i^3 + \\frac{75}{8} e^2 - \\frac{405}{32} e_i^2 - \\frac{35}{96} e_i^2 + \\frac{5}{2} - \\frac{25}{4} e^2 - \\frac{25}{4} e_i^2 \\]\n\n\\[ + \\frac{155}{24} e_i^2 + \\frac{145}{32} e_i^2 \\] \\( e_i^2 \\cos (2t + 2z) \\)\n\n\\[ [19] \\]\n\n\\[ - \\frac{e^3}{8} \\cos 3x - \\frac{7}{48} e^3 \\cos (2t - 3x) + \\frac{25}{48} e^3 \\cos (2t + 3x) + \\left\\{ \\frac{1}{4} - \\frac{5}{8} \\right\\} e^3 e_i \\cos (2x + z) \\]\n\n\\[ [20] \\quad [21] \\quad [22] \\quad [23] \\]\n\n\\[ + \\left\\{ \\frac{5}{4} + \\frac{25}{8} \\right\\} e^3 e_i \\cos (2t - 2x - z) - \\left\\{ -\\frac{3}{2} + \\frac{5}{4} \\right\\} e^3 e_i \\cos (2t + 2x + z) \\]\n\n\\[ [24] \\quad [25] \\]\n\n\\[ + \\left\\{ \\frac{1}{4} - \\frac{5}{8} \\right\\} e^3 e_i \\cos (2x - z) + \\left\\{ -\\frac{15}{4} + \\frac{25}{8} \\right\\} e^3 e_i \\cos (2t - 2x + z) \\]\n\n\\[ [26] \\quad [27] \\]\n\n\\[ + \\left\\{ \\frac{1}{2} + \\frac{5}{4} \\right\\} e^3 e_i \\cos (2t + 2x - z) + \\left\\{ \\frac{1}{4} + \\frac{5}{2} - 5 \\right\\} e^3 e_i \\cos (x + 2z) \\]\n\n\\[ [28] \\quad [29] \\]\n\n\\[ + \\left\\{ -\\frac{3}{2} - \\frac{15}{4} - \\frac{15}{2} \\right\\} e^3 e_i \\cos (2t - x - 2z) \\]\n\n\\[ [30] \\]\n\n\\[ + \\left\\{ \\frac{5}{4} - \\frac{15}{4} + \\frac{5}{2} \\right\\} e^3 e_i \\cos (2t + x + 2z) + \\left\\{ \\frac{1}{4} + \\frac{5}{2} - 5 \\right\\} e^3 e_i \\cos (x - 2z) \\]\n\n\\[ [31] \\quad [32] \\]\n\n\\[ + \\left\\{ -\\frac{15}{4} + \\frac{45}{4} - \\frac{15}{2} \\right\\} e^3 e_i \\cos (2t - x + 2z) + \\left\\{ \\frac{1}{2} + \\frac{5}{4} + \\frac{5}{2} \\right\\} e^3 e_i \\cos (2t + x - 2z) \\]\n\n\\[ [33] \\quad [34] \\]\n\n\\[ + \\left\\{ -\\frac{1}{8} - \\frac{5}{8} - 5 + \\frac{145}{16} \\right\\} e^3 \\cos 3z + \\left\\{ \\frac{25}{48} + \\frac{5}{4} + \\frac{5}{2} + \\frac{145}{32} \\right\\} e^3 \\cos (2t - 3z) \\]\n\n\\[ [35] \\quad [36] \\]\n\n\\[ + \\left\\{ -\\frac{7}{48} + \\frac{25}{8} - \\frac{15}{2} + \\frac{145}{32} \\right\\} e^3 \\cos (2t + 3z) \\]\n\n\\[ [37] \\]\n\n\\[ - \\frac{e^4}{12} \\cos 4x - \\frac{e^4}{32} \\cos (2t - 4x) + \\frac{9}{16} e^4 \\cos (2t + 4x) \\]\n\n\\[ [38] \\quad [39] \\quad [40] \\]\n\n\\[ + \\left\\{ \\frac{1}{8} - \\frac{5}{16} \\right\\} e^3 e_i \\cos (3x + z) + \\left\\{ -\\frac{7}{48} - \\frac{35}{96} \\right\\} e^3 e_i \\cos (2t - 3x - z) \\]\n\n\\[ [41] \\quad [42] \\]\n\n\\[ + \\left\\{ -\\frac{25}{16} + \\frac{125}{96} \\right\\} e^3 e_i \\cos (2t - 3x - z) + \\left\\{ \\frac{1}{8} - \\frac{5}{16} \\right\\} e^3 e_i \\cos (3x - z) \\]\n\n\\[ [43] \\quad [44] \\]\n\\[\n+ \\left\\{ \\frac{7}{16} - \\frac{35}{96} \\right\\} e^3 e_i \\cos (2t - 3x + z)\n\\]\n\n\\[\n+ \\left\\{ \\frac{25}{48} + \\frac{125}{96} \\right\\} e^3 e_i \\cos (2t + 3x - z) + \\left\\{ \\frac{1}{16} + \\frac{5}{8} - \\frac{5}{4} \\right\\} e^2 e_i^2 \\cos (2x + 2z)\n\\]\n\n\\[\n+ \\left\\{ \\frac{5}{4} + \\frac{25}{8} + \\frac{25}{4} \\right\\} e^2 e_i^2 \\cos (2t - 2x - 2z) + \\left\\{ \\frac{5}{4} - \\frac{15}{4} + \\frac{5}{2} \\right\\} e^2 e_i^2 \\cos (2t + 2x + 2z)\n\\]\n\n\\[\n+ \\left\\{ \\frac{1}{16} + \\frac{5}{8} - \\frac{5}{4} \\right\\} e^2 e_i^2 \\cos (2x - 2z) + \\left\\{ \\frac{25}{8} - \\frac{75}{8} + \\frac{25}{4} \\right\\} e^2 e_i^2 \\cos (2t - 2x + 2z)\n\\]\n\n\\[\n+ \\left\\{ \\frac{1}{2} + \\frac{5}{4} + \\frac{5}{2} \\right\\} e^2 e_i^2 \\cos (2t + 2x - 2z) + \\left\\{ \\frac{1}{8} + \\frac{5}{8} + 5 - \\frac{145}{16} \\right\\} ee_i^3 \\cos (x + 3z)\n\\]\n\n\\[\n+ \\left\\{ -\\frac{25}{16} - \\frac{15}{4} - \\frac{15}{2} - \\frac{435}{32} \\right\\} ee_i^3 \\cos (2t - x - 3z)\n\\]\n\n\\[\n+ \\left\\{ -\\frac{7}{48} + \\frac{25}{8} - \\frac{15}{2} + \\frac{145}{32} \\right\\} ee_i^3 \\cos (2t + x + 3z)\n\\]\n\n\\[\n+ \\left\\{ \\frac{1}{8} + \\frac{5}{8} + 5 - \\frac{145}{16} \\right\\} ee_i^3 \\cos (x - 3z) + \\left\\{ \\frac{7}{16} - \\frac{75}{8} + \\frac{45}{2} - \\frac{435}{32} \\right\\} ee_i^3 \\cos (2t - x + 3z)\n\\]\n\n\\[\n+ \\left\\{ \\frac{25}{48} + \\frac{5}{4} + \\frac{5}{2} - \\frac{145}{32} \\right\\} ee_i^3 \\cos (2t + x - 3z) - \\left\\{ -\\frac{1}{12} - \\frac{5}{16} - \\frac{5}{4} - \\frac{145}{16} + \\frac{745}{96} \\right\\} e_i^4 \\cos 4z\n\\]\n\n\\[\n+ \\left\\{ \\frac{9}{16} + \\frac{125}{96} + \\frac{5}{2} + \\frac{145}{32} + \\frac{745}{192} \\right\\} e_i^4 \\cos (2t - 4z)\n\\]\n\n\\[\n+ \\left\\{ -\\frac{1}{32} - \\frac{35}{96} + \\frac{25}{4} - \\frac{435}{32} + \\frac{745}{192} \\right\\} e_i^4 \\cos (2t + 4z)\n\\]\n\nTerms in \\( R \\) multiplied by \\(-\\frac{3}{2} \\sin^2 \\frac{t}{2} \\cos^2 \\frac{t}{2} \\frac{a^2}{a_i^3}\\)\n\n\\[\n= \\left\\{ 1 - \\frac{5}{2} e^2 + \\frac{3}{2} e_i^2 + 5 e_i^2 - \\frac{5}{2} e_i^2 - \\frac{5}{2} e_i^2 \\right\\} \\cos 2y\n\\]\n\n\\[\n+ \\left\\{ 1 + \\frac{3}{2} e^2 - \\frac{5}{2} e_i^2 + 5 e_i^2 + \\frac{5}{2} e_i^2 - \\frac{15}{2} e_i^2 \\right\\} \\cos (2t - 2y)\n\\]\n\n\\[\n- 3e \\cos (x - 2y) + e \\cos (x + 2y) - e \\cos (2t - x - 2y) - e \\cos (2t + x - 2y)\n\\]\n\\[ + \\left\\{ -1 + \\frac{5}{2} \\right\\} e_i \\cos (z - 2y) + \\left\\{ -1 + \\frac{5}{2} \\right\\} e_i \\cos (z + 2y) + \\left\\{ 1 + \\frac{5}{2} \\right\\} e_i \\cos (2t - z - 2y) \\]\n\n\\[ + \\left\\{ -3 + \\frac{5}{2} \\right\\} e_i \\cos (2t + z - 2y) + \\frac{5}{2} e^2 \\cos (2x - 2y) + e^2 \\cos (2x + 2y) \\]\n\n\\[ - \\frac{e^2}{4} \\cos (2t - 2x - 2y) - \\frac{e^2}{4} \\cos (2t + 2x - 2y) \\]\n\n\\[ + \\left\\{ 3 - \\frac{15}{2} \\right\\} ee_i \\cos (x + z - 2y) + \\left\\{ -1 + \\frac{5}{2} \\right\\} ee_i \\cos (x + z + 2y) \\]\n\n\\[ + \\left\\{ -1 - \\frac{5}{2} \\right\\} ee_i \\cos (2t - x - z - 2y) \\]\n\n\\[ + \\left\\{ 3 - \\frac{5}{2} \\right\\} ee_i \\cos (2t + x + z - 2y) + \\left\\{ 3 - \\frac{15}{2} \\right\\} ee_i \\cos (2t - z - 2y) \\]\n\n\\[ + \\left\\{ -1 + \\frac{5}{2} \\right\\} ee_i \\cos (x - z + 2y) + \\left\\{ 3 - \\frac{5}{2} \\right\\} ee_i \\cos (2t - x + z - 2y) \\]\n\n\\[ + \\left\\{ -1 - \\frac{5}{2} \\right\\} ee_i \\cos (2t + x - z - 2y) + \\left\\{ -\\frac{3}{8} - \\frac{5}{2} + 5 \\right\\} e_i^2 \\cos (2z - 2y) \\]\n\n\\[ + \\left\\{ -\\frac{5}{8} - \\frac{5}{2} + 5 \\right\\} e_i^2 \\cos (2z + 2y) + \\left\\{ \\frac{5}{8} + \\frac{5}{2} + 5 \\right\\} e_i^2 \\cos (2t - 2z - 2y) \\]\n\n\\[ + \\left\\{ \\frac{19}{8} - \\frac{15}{2} + 5 \\right\\} e_i^2 \\cos (2t + 2z - 2y) \\]\n\nTerms in \\( R \\) multiplied by \\(-\\frac{3}{2} \\cos^4 \\frac{t}{2} a_i^3\\)\n\n\\[ = \\frac{1}{2} + \\frac{3}{4} e^2 + \\frac{3}{4} e_i^2 + \\frac{15}{16} e^4 + \\frac{9}{8} e^2 e_i^2 + \\left\\{ \\frac{1}{2} - \\frac{5}{4} e^2 - \\frac{5}{4} e_i^2 + \\frac{23}{32} e^4 + \\frac{13}{32} e_i^4 + \\frac{25}{8} e^2 e_i^2 \\right\\} \\cos 2t \\]\n\n\\[ + \\left\\{ 1 + \\frac{e^2}{8} - \\frac{3}{2} e_i^2 \\right\\} e \\cos x + \\left\\{ -\\frac{3}{2} + \\frac{13}{16} e^2 + \\frac{15}{4} e_i^2 \\right\\} e \\cos (2t - x) \\]\n\n\\[ + \\left\\{ \\frac{1}{2} - \\frac{19}{16} e^2 - \\frac{5}{4} e_i^2 \\right\\} e \\cos (2t + x) + \\left\\{ \\frac{3}{2} + \\frac{9}{4} e^2 + \\frac{27}{16} e_i^2 \\right\\} e_i \\cos z \\]\n\\[ + \\left\\{ \\frac{7}{4} - \\frac{35}{8} e^2 - \\frac{123}{32} e_i^2 \\right\\} e_i \\cos (2t - z) \\]\n\n\\[ + \\left\\{ -\\frac{1}{4} + \\frac{5}{8} e^2 + e_i^2 \\right\\} e_i \\cos (2t + z) + \\left\\{ -\\frac{1}{4} + \\frac{e^2}{12} - \\frac{3}{8} e_i^2 \\right\\} e^2 \\cos 2x \\]\n\n\\[ + \\left\\{ \\frac{5}{4} - \\frac{25}{8} e_i^2 \\right\\} e^2 \\cos (2t - 2x) + \\left\\{ \\frac{1}{2} - \\frac{5}{4} e^2 - \\frac{5}{4} e_i^2 \\right\\} e^2 \\cos (2t + 2x) \\]\n\n\\[ + \\left\\{ -\\frac{3}{2} + \\frac{3}{16} e^2 - \\frac{27}{16} e_i^2 \\right\\} ee_i \\cos (x + z) + \\left\\{ -\\frac{21}{4} + \\frac{91}{32} e^2 + \\frac{369}{32} e_i^2 \\right\\} ee_i \\cos (2t - x - z) \\]\n\n\\[ + \\left\\{ -\\frac{1}{4} + \\frac{19}{32} e^2 + \\frac{e_i^2}{32} \\right\\} ee_i \\cos (2t + x + z) + \\left\\{ -\\frac{3}{2} + \\frac{3}{16} e^2 - \\frac{27}{16} e_i^2 \\right\\} ee_i \\cos (x - z) \\]\n\n\\[ + \\left\\{ \\frac{3}{4} - \\frac{13}{32} e^2 - \\frac{3}{32} e_i^2 \\right\\} ee_i \\cos (2t - x + z) + \\left\\{ \\frac{7}{4} - \\frac{133}{32} e^2 - \\frac{123}{32} e_i^2 \\right\\} ee_i \\cos (2t + x - z) \\]\n\n\\[ + \\left\\{ \\frac{9}{4} + \\frac{27}{8} e^2 + \\frac{21}{12} e_i^2 \\right\\} e_i^2 \\cos 2z + \\left\\{ \\frac{17}{4} - \\frac{85}{8} e^2 - \\frac{115}{12} e_i^2 \\right\\} e_i^2 \\cos (2t - 2z) - \\frac{e^3}{8} \\cos 3x \\]\n\n\\[ - \\frac{7}{48} e^3 \\cos (2t - 3x) + \\frac{25}{48} e^3 \\cos (2t + 3x) - \\frac{3}{8} e^2 e_i \\cos (2x + z) \\]\n\n\\[ + \\frac{35}{8} e^2 e_i \\cos (2t - 2x - z) - \\frac{e^2 e_i}{4} \\cos (2t + 2x + z) - \\frac{3}{8} e^2 e_i \\cos (2x - z) \\]\n\n\\[ - \\frac{5}{8} e^2 e_i \\cos (2t - 2x + z) + \\frac{7}{4} e^2 e_i \\cos (2t + 2x - z) \\]\n\n\\[ - \\frac{9}{4} ee_i^2 \\cos (x + 2z) - \\frac{51}{4} ee_i^2 \\cos (2t - x - 2z) \\]\n\n\\[ - \\frac{9}{4} ee_i^2 \\cos (x - 2z) + \\frac{17}{4} ee_i^2 \\cos (2t + x - 2z) \\]\n\n\\[ + \\frac{53}{16} ee_i^3 \\cos 3z + \\frac{845}{96} ee_i^3 \\cos (2t - 3z) + \\frac{e^3 e_i}{96} \\cos (2t + 3z) - \\frac{e^4}{12} \\cos 4x - \\frac{e^4}{32} \\cos (2t - 4x) \\]\n\n* It is remarkable that the coefficient of argument 19 equals zero.\n\\[ \\frac{9}{16} e^t \\cos (2t + 4x) - \\frac{3}{16} e^3 e_i \\cos (3x + z) \\]\n\n\\[ - \\frac{49}{96} e^3 e_i \\cos (2t + 3x - z) - \\frac{25}{96} e^3 e_i \\cos (2t - 3x - z) - \\frac{3}{16} e^3 e_i \\cos (3x - z) \\]\n\n\\[ + \\frac{7}{96} e^3 e_i \\cos (2t - 3x + z) + \\frac{175}{96} e^3 e_i \\cos (2t + 3x - z) - \\frac{9}{16} e^2 e_i^2 \\cos (2x + 2z) \\]\n\n\\[ + \\frac{85}{8} e^2 e_i^2 \\cos (2t - 2x - 2z) - \\frac{9}{16} e^2 e_i^2 \\cos (2x - 2z) + \\frac{17}{4} e^2 e_i^2 \\cos (2t + 2x - 2z) \\]\n\n\\[ - \\frac{53}{16} e e_i^3 \\cos (x + 3z) - \\frac{845}{32} e e_i^3 \\cos (2t - x - 3z) + \\frac{ee_i^3}{96} \\cos (2t + x - 3z) \\]\n\n\\[ - \\frac{53}{16} e e_i^3 \\cos (x - 3z) - \\frac{ee_i^3}{32} \\cos (2t - x + 3z) - \\frac{25}{96} e e_i^3 \\cos (2t + x - 3z) \\]\n\n\\[ - \\frac{283}{96} e_i^4 \\cos 4z + \\frac{2453}{192} e_i^4 \\cos (2t - 4z) - \\frac{741}{192} e_i^4 \\cos (2t + 4z) \\]\n\nTerms in \\( R \\) multiplied by \\( -\\frac{3}{2} \\sin^2 \\frac{t}{2} \\cos^2 \\frac{t}{2} \\frac{a^2}{a_i^3} \\) or \\( -\\frac{3}{8} \\sin^2 \\frac{t}{2} \\frac{a^3}{a_i^3} \\)\n\n\\[ = \\left\\{ 1 - \\frac{5}{2} e^2 + \\frac{3}{2} e_i^2 \\right\\} \\cos 2y + \\left\\{ 1 + \\frac{3}{2} e^2 - \\frac{5}{2} e_i^2 \\right\\} \\cos (2t - 2y) - 3e \\cos (x - 2y) \\]\n\n\\[ + e \\cos (x + 2y) - e \\cos (2t - x - 2y) - e \\cos (2t + x - 2y) + \\frac{3}{2} e_i \\cos (z - 2y) + \\frac{3}{2} e_i \\cos (z + 2y) \\]\n\n\\[ + \\frac{7}{2} e_i \\cos (2t - z - 2y) - \\frac{e_i}{2} \\cos (2t + z - 2y) + \\frac{5}{2} e^2 \\cos (2x - 2y) + e^2 \\cos (2x + 2y) \\]\n\n\\[ - \\frac{e^2}{4} \\cos (2t - 2x - 2y) - \\frac{e^2}{4} \\cos (2t + 2x - 2y) - \\frac{9}{2} ee_i \\cos (x + z - 2y) \\]\n\n\\[ + \\frac{3}{2} ee_i \\cos (x + z + 2y) - \\frac{7}{2} ee_i \\cos (2t - x - z - 2y) + \\frac{ee_i}{2} \\cos (2t + x + z - 2y) \\]\n\\[\nR = m_i \\left\\{ -\\frac{1}{r_i} - \\frac{1}{4} \\left[ 1 + \\frac{3}{2} e^2 + \\frac{3}{2} e_i^2 + \\frac{9}{4} e^2 e_i^2 + \\frac{15}{8} e_i^4 - \\frac{3}{2} \\gamma^2 - \\frac{9}{4} \\gamma^2 e^2 - \\frac{9}{4} \\gamma^2 e_i^2 + \\frac{39}{8} \\gamma^4 \\right] \\frac{a^2}{a_i^3} \\right\\}\n\\]\n\n\\[\n- \\frac{3}{4} \\left[ 1 - \\frac{5}{2} e^2 - \\frac{5}{2} e_i^2 + \\frac{23}{16} e^4 + \\frac{25}{4} e^2 e_i^2 + \\frac{13}{16} e_i^4 \\right] \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} \\cos 2t\n\\]\n\n\\[\n+ \\frac{1}{2} \\left[ 1 - \\frac{e^2}{8} - \\frac{3}{2} e_i^2 - \\frac{3}{2} \\gamma^2 \\right] \\frac{a^2}{a_i^3} e \\cos x\n\\]\n\n\\[\n+ \\frac{9}{4} \\left[ 1 - \\frac{13}{24} e^2 - \\frac{5}{2} e_i^2 \\right] \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e \\cos (2t - x)\n\\]\n\n\\[\n- \\frac{3}{4} \\left[ 1 - \\frac{19}{8} e^2 - \\frac{5}{2} e_i^2 \\right] \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e \\cos (2t + x)\n\\]\n\n\\[\n- \\frac{3}{4} \\left[ 1 + \\frac{3}{2} e^2 + \\frac{9}{8} e_i^2 - \\frac{3}{2} \\gamma^2 \\right] \\frac{a^2}{a_i^3} e_i \\cos z\n\\]\n\n\\[\n- \\frac{21}{8} \\left[ 1 - \\frac{5}{2} e^2 - \\frac{123}{56} e_i^2 \\right] \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e_i \\cos (2t - z)\n\\]\n\n\\[\n+ \\frac{3}{8} \\left[ 1 - \\frac{5}{2} e^2 - 4 e_i^2 \\right] \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e_i \\cos (2t + z)\n\\]\n\n\\[\n+ \\frac{1}{8} \\left[ 1 - \\frac{e^2}{3} + \\frac{3}{2} e_i^2 - \\frac{3}{2} \\gamma^2 \\right] \\frac{a^2}{a_i^3} e^2 \\cos 2x\n\\]\n\n\\[\n- \\frac{15}{8} \\left[ 1 - \\frac{5}{2} e_i^2 \\right] \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e^2 \\cos (2t - 2x)\n\\]\n\n\\[\n- \\frac{3}{4} \\left[ 1 - \\frac{5}{2} e^2 - \\frac{5}{2} e_i^2 \\right] \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e^2 \\cos (2t + 2x)\n\\]\nDevelopment of $R$.\n\n\\[\n\\begin{align*}\n&+ \\frac{3}{4} \\left\\{ 1 - \\frac{e^2}{8} + \\frac{9}{8} e_i^2 - \\frac{3}{2} \\gamma^2 \\right\\} \\frac{a^2}{a_i^3} e e_i \\cos (x + z) \\\\\n&+ \\frac{63}{8} \\left\\{ 1 - \\frac{91}{168} e^2 - \\frac{123}{56} e_i^2 \\right\\} \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e e_i \\cos (2t - x - z) \\\\\n&+ \\frac{3}{8} \\left\\{ 1 - \\frac{19}{8} e^2 - \\frac{e_i^2}{8} \\right\\} \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e e_i \\cos (2t + x + z) \\\\\n&+ \\frac{3}{4} \\left\\{ 1 - \\frac{e^2}{8} + \\frac{9}{8} e_i^2 - \\frac{3}{2} \\gamma^2 \\right\\} \\frac{a^2}{a_i^3} e e_i \\cos (x - z) \\\\\n&- \\frac{9}{8} \\left\\{ 1 - \\frac{13}{24} e^2 - \\frac{e_i^2}{8} \\right\\} \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e e_i \\cos (2t - x + z) \\\\\n&- \\frac{21}{8} \\left\\{ 1 - \\frac{19}{8} e^2 - \\frac{123}{56} e_i^2 \\right\\} \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e e_i \\cos (2t + x - z) \\\\\n&- \\frac{9}{8} \\left\\{ 1 + \\frac{3}{2} e^2 + \\frac{7}{9} e_i^2 - \\frac{3}{2} \\gamma^2 \\right\\} \\frac{a^2}{a_i^3} e_i^2 \\cos 2z \\\\\n&- \\frac{51}{8} \\left\\{ 1 - \\frac{5}{2} e^2 - \\frac{115}{51} e_i^2 \\right\\} \\cos^4 \\frac{t}{2} \\frac{a^2}{a_i^3} e_i^2 \\cos (2t - 2z) \\\\\n&+ \\frac{1}{16} \\frac{a^2}{a_i^3} e^3 \\cos 3x + \\frac{7}{32} \\frac{a^2}{a_i^3} e^3 \\cos (2t - 3x) - \\frac{25}{32} \\frac{a^2}{a_i^3} e^3 \\cos (2t + 3x) \\\\\n&+ \\frac{3}{16} \\frac{a^2}{a_i^3} e^2 e_i \\cos (2x + z) - \\frac{105}{16} \\frac{a^2}{a_i^3} e^2 e_i \\cos (2t - 2x - z) + \\frac{3}{8} \\frac{a^2}{a_i^3} e^2 e_i \\cos (2t + 2x + z) \\\\\n&+ \\frac{3}{16} \\frac{a^2}{a_i^3} e^2 e_i \\cos (2x - z) + \\frac{15}{16} \\frac{a^2}{a_i^3} e^2 e_i \\cos (2t - 2x + z) - \\frac{21}{8} \\frac{a^2}{a_i^3} e^2 e_i \\cos (2t + 2x - z) \\\\\n&+ \\frac{9}{8} \\frac{a^2}{a_i^3} e e_i^2 \\cos (x + 2z) + \\frac{153}{8} \\frac{a^2}{a_i^3} e e_i^2 \\cos (2t - x - 2z) \\\\\n&+ \\frac{9}{8} \\frac{a^2}{a_i^3} e e_i^2 \\cos (x - 2z) - \\frac{51}{8} \\frac{a^2}{a_i^3} e e_i^2 \\cos (2t + x - 2z) - \\frac{53}{32} \\frac{a^2}{a_i^3} e_i^3 \\cos 3z \\\\\n&- \\frac{845}{64} \\frac{a^2}{a_i^3} e_i^3 \\cos (2t - 3z) - \\frac{1}{64} \\frac{a^2}{a_i^3} e_i^3 \\cos (2t + 3z) + \\frac{1}{24} \\frac{a^2}{a_i^3} e^4 \\cos 4x \\\\\n&+ \\frac{3}{64} \\frac{a^2}{a_i^3} e^4 \\cos (2t - 4x) - \\frac{27}{32} \\frac{a^2}{a_i^3} e^4 \\cos (2t + 4x) + \\frac{3}{32} \\frac{a^2}{a_i^3} e^3 e_i \\cos (3x + z)\n\\end{align*}\n\\]\n\\[\n\\begin{align*}\n&+ \\frac{49}{64} \\frac{a^2}{a_i^3} e^3 e_i \\cos (2t - 3x - z) + \\frac{25}{64} \\frac{a^2}{a_i^3} e^3 e_i \\cos (2t - 3x - z) + \\frac{3}{32} \\frac{a^2}{a_i^3} e^3 e_i \\cos (3x - z) \\\\\n&- \\frac{7}{64} \\frac{a^2}{a_i^3} e^3 e_i \\cos (2t - 3x + z) - \\frac{175}{64} \\frac{a^2}{a_i^3} e^3 e_i \\cos (2t + 3x - z) \\\\\n&+ \\frac{9}{32} \\frac{a^2}{a_i^3} e^2 e_i^2 \\cos (2x + 2z) - \\frac{255}{16} \\frac{a^2}{a_i^3} e^2 e_i^2 \\cos (2t - 2x - 2z) \\\\\n&+ \\frac{9}{32} \\frac{a^2}{a_i^3} e^2 e_i^2 \\cos (2x - 2z) - \\frac{51}{8} \\frac{a^2}{a_i^3} e^2 e_i^2 \\cos (2t + 2x - 2z) + \\frac{53}{32} \\frac{a^2}{a_i^3} e e_i^3 \\cos (x + 3z) \\\\\n&+ \\frac{2535}{64} \\frac{a^2}{a_i^3} e e_i^3 \\cos (2t - x - 3z) - \\frac{1}{64} \\frac{a^2}{a_i^3} e e_i^3 \\cos (2t + x + 3z) \\\\\n&+ \\frac{53}{32} \\frac{a^2}{a_i^3} e e_i^3 \\cos (x - 3z) + \\frac{3}{64} \\frac{a^2}{a_i^3} e e_i^3 \\cos (2t - x + 3z) \\\\\n&+ \\frac{45}{64} \\frac{a^2}{a_i^3} e e_i^3 \\cos (2t + x - 3z) + \\frac{591}{64} \\frac{a^2}{a_i^3} e_i^4 \\cos 4z \\\\\n&- \\frac{2453}{128} \\frac{a^2}{a_i^3} e_i^4 \\cos (2t - 4z) + \\frac{741}{128} \\frac{a^2}{a_i^3} e_i^4 \\cos (2t + 4z) \\\\\n&- \\frac{3}{8} \\left\\{ 1 - \\frac{5}{2} e^2 + \\frac{3}{2} e_i^2 \\right\\} \\frac{a^2}{a_i^3} \\gamma^2 \\cos 2y - \\frac{3}{8} \\left\\{ 1 + \\frac{3}{2} e^2 - \\frac{5}{2} e_i^2 + \\frac{\\gamma^2}{8} \\right\\} \\frac{a^2}{a_i^3} \\gamma^2 \\cos (2t - 2y) \\\\\n&+ \\frac{9}{8} \\frac{a^2}{a_i^3} \\gamma^2 e \\cos (x - 2y) \\\\\n&- \\frac{3}{8} \\frac{a^2}{a_i^3} \\gamma^2 e \\cos (x + 2y) + \\frac{3}{8} \\frac{a^2}{a_i^3} \\gamma^2 e \\cos (2t - x - 2y) \\\\\n&+ \\frac{3}{8} \\frac{a^2}{a_i^3} \\gamma^2 e \\cos (2t + x - 2y) - \\frac{9}{16} \\frac{a^2}{a_i^3} \\gamma^2 e_i \\cos (z - 2y) \\\\\n&- \\frac{9}{16} \\frac{a^2}{a_i^3} \\gamma^2 e_i \\cos (z + 2y) - \\frac{21}{16} \\frac{a^2}{a_i^3} \\gamma^2 e_i \\cos (2t - z - 2y) \\\\\n&+ \\frac{3}{16} \\frac{a^2}{a_i^3} \\gamma^2 e_i \\cos (2t + z - 2y) - \\frac{15}{16} \\frac{a^2}{a_i^3} \\gamma^2 e^2 \\cos (2x - 2y)\n\\end{align*}\n\\]\nDevelopment of $R$.\n\n\\[- \\frac{3}{8} \\frac{a^2}{a_i^3} \\gamma^2 e^3 \\cos (2x + 2y) + \\frac{3}{32} \\frac{a^2}{a_i^3} \\gamma^2 e^3 \\cos (2t - 2x - 2y) \\]\n\n\\[78\\] \\[79\\]\n\n\\[+ \\frac{3}{32} \\frac{a^2}{a_i^3} \\gamma^2 e^3 \\cos (2t + 2x - 2y) + \\frac{27}{16} \\frac{a^2}{a_i^3} \\gamma^2 e e_i \\cos (x + z - 2y) \\]\n\n\\[81\\] \\[83\\]\n\n\\[- \\frac{9}{16} \\frac{a^2}{a_i^3} \\gamma^2 e e_i \\cos (x + z + 2y) + \\frac{21}{16} \\frac{a^2}{a_i^3} \\gamma^2 e e_i \\cos (2t - x - z - 2y) \\]\n\n\\[84\\] \\[85\\]\n\n\\[- \\frac{3}{16} \\frac{a^2}{a_i^3} \\gamma^2 e e_i \\cos (2t + x + z - 2y) + \\frac{27}{16} \\frac{a^2}{a_i^3} \\gamma^2 e e_i \\cos (x - z - 2y) \\]\n\n\\[87\\] \\[89\\]\n\n\\[- \\frac{9}{16} \\frac{a^2}{a_i^3} \\gamma^2 e e_i \\cos (x - z + 2y) - \\frac{3}{16} \\frac{a^2}{a_i^3} \\gamma^2 e e_i \\cos (2t - x + z - 2y) \\]\n\n\\[90\\] \\[91\\]\n\n\\[+ \\frac{21}{16} \\frac{a^2}{a_i^3} \\gamma^2 e e_i \\cos (2t + x - z - 2y) \\]\n\n\\[93\\]\n\n\\[- \\frac{51}{64} \\frac{a^2}{a_i^3} \\gamma^2 e_i^2 \\cos (2z - 2y) - \\frac{45}{64} \\frac{a^2}{a_i^3} \\gamma^2 e_i^2 \\cos (2z + 2y) \\]\n\n\\[95\\] \\[96\\]\n\n\\[- \\frac{195}{64} \\frac{a^2}{a_i^3} \\gamma^2 e_i^2 \\cos (2t - 2z - 2y) + \\frac{3}{64} \\frac{a^2}{a_i^3} \\gamma^2 e_i^2 \\cos (2t + 2z - 2y) \\]\n\n\\[97\\] \\[99\\]\n\n\\[- \\frac{3}{8} \\left\\{ 1 + 3e^2 + 3e_i^2 - \\frac{11}{4} \\gamma^2 \\right\\} \\frac{a^3}{a_i^4} * \\cos t + \\frac{15}{16} \\frac{a^3}{a_i^4} e \\cos (t - x) \\]\n\n\\[101\\] \\[102\\]\n\n\\[+ \\frac{3}{16} \\frac{a^3}{a_i^4} e \\cos (t + x) - \\frac{9}{8} \\frac{a^3}{a_i^4} e_i \\cos (t - z) - \\frac{3}{8} \\frac{a^3}{a_i^4} e_i \\cos (t + x) \\]\n\n\\[103\\] \\[104\\] \\[105\\]\n\n\\[- \\frac{33}{64} \\frac{a^3}{a_i^4} e^2 \\cos (t - 2x) + \\frac{9}{64} \\frac{a^3}{a_i^4} e^2 \\cos (t + 2x) + \\frac{45}{16} \\frac{a^3}{a_i^4} e e_i \\cos (t - x - z) \\]\n\n\\[106\\] \\[107\\] \\[108\\]\n\n\\[+ \\frac{3}{16} \\frac{a^3}{a_i^4} e e_i \\cos (t + x + z) + \\frac{15}{16} \\frac{a^3}{a_i^4} e e_i \\cos (t - x + z) + \\frac{9}{16} \\frac{a^3}{a_i^4} e e_i \\cos (t + x - z) \\]\n\n\\[109\\] \\[110\\] \\[111\\]\n\n\\[- \\frac{159}{64} \\frac{a^3}{a_i^4} e_i^2 \\cos (t - 2z) - \\frac{33}{64} \\frac{a^3}{a_i^4} e_i^2 \\cos (t + 2z) - \\frac{9}{16} \\frac{a^3}{a_i^4} \\gamma^2 \\cos (t - 2y) \\]\n\n\\[112\\] \\[113\\] \\[114\\]\n\n\\[- \\frac{15}{32} \\frac{a^3}{a_i^4} \\sin^2 \\frac{t}{2} \\cos (t + 2y) - \\frac{5}{8} \\left\\{ 1 - 6e^2 - 6e_i^2 - \\frac{3}{4} \\gamma^2 \\right\\} \\frac{a^3}{a_i^4} \\cos 3t \\]\n\n\\[115\\] \\[116\\]\n\n* For the coefficients of the terms multiplied by $\\frac{a^3}{a_i^4}$ see p. 39.\n\\[\n+ \\frac{45}{16} \\frac{a^3}{a_i^4} e \\cos (3t - x) - \\frac{15}{16} \\frac{a^3}{a_i^4} e \\cos (3t + x) - \\frac{25}{8} \\frac{a^3}{a_i^4} e_i \\cos (3t - z)\n\\]\n\n\\[ [117] \\quad [118] \\quad [119] \\]\n\n\\[\n+ \\frac{5}{8} \\frac{a^3}{a_i^4} e_i \\cos (3t + z) - \\frac{285}{64} \\frac{a^3}{a_i^4} e^2 \\cos (3t - 2x) - \\frac{75}{64} \\frac{a^3}{a_i^4} e^2 \\cos (3t + 2x)\n\\]\n\n\\[ [120] \\quad [121] \\quad [122] \\]\n\n\\[\n- \\frac{225}{16} \\frac{a^3}{a_i^4} ee_i \\cos (3t - x - z) + \\frac{15}{16} \\frac{a^3}{a_i^4} ee_i \\cos (3t + x + z)\n\\]\n\n\\[ [123] \\quad [124] \\]\n\n\\[\n- \\frac{45}{16} \\frac{a^3}{a_i^4} ee_i \\cos (3t - x + z) - \\frac{75}{16} \\frac{a^3}{a_i^4} ee_i \\cos (3t + x - z)\n\\]\n\n\\[ [125] \\quad [126] \\]\n\n\\[\n- \\frac{635}{64} \\frac{a^3}{a_i^4} e_i^2 \\cos (3t - 2z) - \\frac{5}{64} \\frac{a^3}{a_i^4} e_i^2 \\cos (3t + 2z) - \\frac{15}{32} \\frac{a^3}{a_i^4} \\gamma^2 \\cos (3t - 2y)\n\\]\n\n\\[ [127] \\quad [128] \\quad [129] \\]\n\nIn the elliptic movement;\n\n\\[ s = \\gamma \\sin (g \\lambda - \\nu) \\]\n\n\\[ \\lambda = n t + 2e \\sin x + \\frac{5}{4} e^2 \\sin 2x \\]\n\n\\[ s = \\gamma (1 - e^2) \\sin y + \\gamma e \\sin (x - y) + \\gamma e \\sin (x + y) + \\gamma e^2 \\sin (2x - y) + \\frac{9}{8} \\gamma e^2 \\sin (2x + y)\n\\]\n\n\\[ [146] \\quad [149] \\quad [150] \\quad [161] \\quad [162] \\]\n\n\\[ s^2 = \\frac{\\gamma^2}{2} - \\frac{\\gamma^2}{2} (1 - 4e^2) \\cos 2y + \\gamma^2 e \\cos (x - 2y) - \\gamma^2 e \\cos (x + 2y)\n\\]\n\n\\[ [62] \\quad [65] \\quad [66] \\]\n\n\\[ + \\frac{5}{8} \\gamma^2 e^2 \\cos (2x - 2y) - \\frac{5}{8} \\gamma^2 e^2 \\cos (2x + 2y)\n\\]\n\n\\[ [77] \\quad [78] \\]\n\n\\[ z^* = a \\gamma \\left(1 - \\frac{e^2}{2}\\right) \\sin y + \\frac{3a \\gamma e}{2} \\sin (x - y) + \\frac{a \\gamma e}{2} \\sin (x + y)\n\\]\n\n\\[ [146] \\quad [149] \\quad [150] \\]\n\n\\[ - \\frac{a \\gamma e^2}{8} \\sin (2x - y) + \\frac{3a \\gamma e^2}{8} \\sin (2x + y)\n\\]\n\n\\[ [161] \\quad [162] \\]\n\n\\[ \\frac{s}{r} = \\frac{\\gamma}{a} (1 - e^2) \\sin y + \\frac{\\gamma e}{2a} \\sin (x - y) + \\frac{3 \\gamma e}{2a} \\sin (x + y)\n\\]\n\n\\[ [146] \\quad [149] \\quad [150] \\]\n\n* This quantity \\( z \\), which is one of the rectangular coordinates of the moon, must not be confounded with \\( z = n_t - w_t \\); this latter quantity should rather be \\( x \\), but I think it better to conform as far as possible to the notation of M. Damoiseau.\n\\[ \\frac{s}{r} \\delta \\cdot \\frac{1}{r} = \\left\\{ (1 - e^2) r_0 + \\frac{e^2}{2} r_2 \\right\\} \\frac{\\gamma}{a^2} \\sin y - \\left\\{ (1 - e^2) \\frac{r_1}{2} + \\frac{e^2}{4} r_3 - \\frac{3 e^2}{4} r_4 \\right\\} \\frac{\\gamma}{a^2} \\sin (2t - y) \\]\n\n\\[ + \\left\\{ (1 - e^2) \\frac{r_1}{2} - \\frac{e^2}{4} r_4 + \\frac{3 e^2}{4} r_3 \\right\\} \\frac{e \\gamma}{a^2} \\sin (2t + y) + \\frac{e \\gamma r_0}{2 a^2} \\sin (x - y) \\]\n\n\\[ + \\frac{3 r_0}{2 a^2} e \\gamma \\sin (x + y) + \\left\\{ - \\frac{r_3}{2} - \\frac{3 r_1}{4} \\right\\} \\frac{e \\gamma}{a^2} \\sin (2t - x - y) \\]\n\n\\[ + \\left\\{ \\frac{r_3}{2} - \\frac{r_1}{4} \\right\\} \\frac{e \\gamma}{a^2} \\sin (2t - x + y) + \\left\\{ - \\frac{r_4}{2} - \\frac{r_1}{4} \\right\\} \\frac{e \\gamma}{a^2} \\sin (2t + x - y) \\]\n\n\\[ + \\left\\{ \\frac{r_4}{2} + \\frac{3 r_1}{4} \\right\\} \\frac{e \\gamma}{a^2} \\sin (2t + x + y) + \\frac{r_5 e \\gamma}{2 a^2} \\sin (z - y) + \\frac{r_5 e \\gamma}{2 a^2} \\sin (z + y) \\]\n\n\\[ - \\frac{r_6 e \\gamma}{2 a^2} \\sin (2t - z - y) + \\frac{r_6 e \\gamma}{2 a^2} \\sin (2t - z + y) - \\frac{r_7 e \\gamma}{2 a^2} \\sin (2t + z - y) \\]\n\n\\[ + \\left\\{ - \\frac{r_9}{2} - \\frac{3 r_3}{4} - \\frac{17 r_1}{16} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2t - 2x - y) + \\left\\{ \\frac{r_9}{2} - \\frac{r_3}{4} - \\frac{r_1}{16} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2t - 2x + y) \\]\n\n\\[ + \\left\\{ - \\frac{r_{10}}{2} + \\frac{r_4}{4} + \\frac{r_1}{16} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2t + 2x - y) + \\left\\{ \\frac{r_{10}}{2} + \\frac{3 r_4}{4} + \\frac{17 r_1}{16} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2t + 2x + y) \\]\n\n\\[ + \\left\\{ - \\frac{r_{11}}{2} + \\frac{r_5}{4} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (x + z - y) + \\left\\{ \\frac{r_{11}}{2} + \\frac{3 r_5}{4} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (x + z + y) \\]\n\n\\[ + \\left\\{ - \\frac{r_{12}}{2} + \\frac{r_6}{4} - \\frac{3 r_6}{4} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (2t - x - z - y) + \\left\\{ \\frac{r_{12}}{2} + \\frac{r_6}{4} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (2t - x - z + y) \\]\n\n\\[ + \\left\\{ - \\frac{r_{13}}{2} + \\frac{r_7}{4} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (2t + x + z - y) + \\left\\{ \\frac{r_{13}}{2} + \\frac{3 r_7}{4} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (2t + x + z + y) \\]\n\n\\[ + \\left\\{ - \\frac{r_{14}}{2} + \\frac{r_8}{2} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (x - z - y) + \\left\\{ \\frac{r_{14}}{2} + \\frac{3 r_8}{4} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (x - z + y) \\]\n\n\\[ + \\left\\{ - \\frac{r_{15}}{2} - \\frac{3 r_7}{4} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (2t - x + z - y) + \\left\\{ \\frac{r_{15}}{2} - \\frac{r_7}{2} \\right\\} \\frac{e e \\gamma}{a^2} \\sin (2t - x + z + y) \\]\n\\[\n-\\frac{r_{16}e_e\\gamma}{2a^2}\\sin(2t+x-z-y) + \\left\\{\\frac{r_{16}}{2} + \\frac{3}{4}r_6\\right\\}\\frac{ee\\gamma}{a^2}\\sin(2t+x-z+y)\n\\]\n\n\\[\n-\\frac{r_{17}e_e^2\\gamma}{2a^2}\\sin(2z-y) + \\frac{r_{17}e_e^2\\gamma}{2a^2}\\sin(2z+y) - \\frac{r_{18}e_e^2\\gamma}{2a^2}\\sin(2t-2z-y)\n\\]\n\n\\[\n+\\frac{r_{18}e_e^2\\gamma}{2a^2}\\sin(2t-2z+y) - \\frac{r_{19}e_e^2\\gamma}{2a^2}\\sin(2t+2z-y) + \\frac{r_{19}e_e^2\\gamma}{2a^2}\\sin(2t+2z+y)\n\\]\n\n\\[\n\\frac{m_i z}{r^3} = \\frac{m_i a\\gamma}{a_i^3}\\left(1 + \\frac{3}{2}e_i^2 - \\frac{e_i^2}{2}\\right)\\sin y + \\frac{3m_i a\\gamma e_i}{2a_i^3}\\sin(x-y) + \\frac{m_i a\\gamma}{2a_i^3}\\sin(x+y)\n\\]\n\n\\[\n-\\frac{3m_i a\\gamma e_i}{2a_i^3}\\sin(z-y) + \\frac{3m_i a\\gamma e_i}{2a_i^3}\\sin(z+y) - \\frac{m_i a\\gamma e_i^2}{8a_i^3}\\sin(2x-y)\n\\]\n\n\\[\n+\\frac{3m_i a\\gamma e_i^2}{8a_i^3}\\sin(2x+y) + \\frac{9m_i a\\gamma ee_i}{4a_i^3}\\sin(x+z-y) + \\frac{3m_i a\\gamma ee_i}{4a_i^3}\\sin(x+z+y)\n\\]\n\n\\[\n+\\frac{9m_i a\\gamma ee_i}{4a_i^3}\\sin(x-z-y) + \\frac{3m_i a\\gamma ee_i}{4a_i^3}\\sin(x-z+y) - \\frac{9m_i a\\gamma ee_i}{4a_i^3}\\sin(2z-y)\n\\]\n\n\\[\n+\\frac{9m_i a\\gamma ee_i}{4a_i^3}\\sin(2z+y)\n\\]\n\n\\[\n\\frac{a^3}{r^3} = 1 + \\frac{3}{2}e^2 + 3e\\cos x + \\frac{9}{2}e^2\\cos 2x\n\\]\n\n\\(r\\) being the elliptic value of \\(r\\).\n\nIf \\(z = a\\gamma z_{146}\\sin y + a\\gamma z_{147}\\sin(2t-y) + a\\gamma z_{148}\\sin(2t+y)\\) &c.\n\n\\[\n\\frac{z^*}{r^3} = \\left\\{(1 + \\frac{3}{2}e^2)z_{146} + \\frac{3}{2}e^2z_{150} - \\frac{3}{2}e^2z_{149}\\right\\}\\frac{\\gamma}{a^2}\\sin y\n\\]\n\n\\[\n+\\left\\{(1 + \\frac{3}{2}e^2)z_{147} + \\frac{3}{2}e^2z_{151} + \\frac{3}{2}e^2z_{153}\\right\\}\\frac{\\gamma}{a^2}\\sin(2t-y)\n\\]\n\n\\[\n+\\left\\{(1 + \\frac{3}{2}e^2)z_{148} + \\frac{3}{2}e^2z_{152} + \\frac{3}{2}e^2z_{154}\\right\\}\\frac{\\gamma}{a^2}\\sin(2t+y)\n\\]\n\n*This multiplication of \\(z\\) by \\(r^{-3}\\) may be effected at once by means of Table II.*\n\\[ + \\left\\{ z_{149} - \\frac{3}{2} z_{146} \\right\\} \\frac{e \\gamma}{a^2} \\sin (x - y) + \\left\\{ z_{150} + \\frac{3}{2} z_{146} \\right\\} \\frac{e \\gamma}{a^2} \\sin (x + y) \\]\n\n\\[ + \\left\\{ z_{151} + \\frac{3}{2} z_{147} \\right\\} \\frac{e \\gamma}{a^2} \\sin (2t - x - y) + \\left\\{ z_{152} + \\frac{3}{2} z_{148} \\right\\} \\frac{e \\gamma}{a^2} \\sin (2t - x + y) \\]\n\n\\[ + \\left\\{ z_{153} + \\frac{3}{2} z_{147} \\right\\} \\frac{e \\gamma}{a^2} \\sin (2t + x - y) \\]\n\n\\[ + \\left\\{ z_{154} + \\frac{3}{2} z_{148} \\right\\} \\frac{e \\gamma}{a^2} \\sin (2t + x + y) + z_{155} \\frac{e \\gamma}{a^2} \\sin (z - y) + z_{156} \\frac{e \\gamma}{a^2} \\sin (z + y) \\]\n\n\\[ + \\left\\{ z_{161} + \\frac{3}{2} z_{149} - \\frac{9}{4} z_{146} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2x - y) + \\left\\{ z_{162} + \\frac{3}{2} z_{150} + \\frac{9}{4} z_{146} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2x + y) \\]\n\n\\[ + \\left\\{ z_{163} + \\frac{3}{2} z_{151} + \\frac{9}{4} z_{147} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2t - 2x - y) \\]\n\n\\[ + \\left\\{ z_{164} + \\frac{3}{2} z_{152} + \\frac{9}{4} z_{148} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2t - 2x + y) \\]\n\n\\[ + \\left\\{ z_{165} + \\frac{3}{2} z_{153} + \\frac{9}{4} z_{147} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2t + 2x - y) \\]\n\n\\[ + \\left\\{ z_{166} + \\frac{3}{2} z_{154} + \\frac{9}{4} z_{148} \\right\\} \\frac{e^2 \\gamma}{a^2} \\sin (2t + 2x + y) + \\text{&c.} \\]\n\n\\[ + \\left\\{ z_{167} + \\frac{3}{2} z_{155} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (x + z - y) + \\left\\{ z_{168} + \\frac{3}{2} z_{156} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (x + z + y) \\]\n\n\\[ + \\left\\{ z_{169} + \\frac{3}{2} z_{157} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (2t - x - z - y) + \\left\\{ z_{170} + \\frac{3}{2} z_{158} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (2t - x - z + y) \\]\n\n\\[ + \\left\\{ z_{171} + \\frac{3}{2} z_{159} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (2t + x + z - y) + \\left\\{ z_{172} + \\frac{3}{2} z_{160} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (2t + x + z + y) \\]\n\n\\[ + \\left\\{ z_{173} - \\frac{3}{2} z_{156} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (x - z - y) + \\left\\{ z_{174} - \\frac{3}{2} z_{155} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (x - z + y) \\]\n\n\\[ + \\left\\{ z_{175} + \\frac{3}{2} z_{159} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (2t - x + z - y) + \\left\\{ z_{176} + \\frac{3}{2} z_{160} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (2t - x + z + y) \\]\n\n\\[ + \\left\\{ z_{177} + \\frac{3}{2} z_{157} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (2t + x - z - y) + \\left\\{ z_{178} + \\frac{3}{2} z_{158} \\right\\} \\frac{ee_1 \\gamma}{a^2} \\sin (2t + x - z + y) \\]\n\\[ s = \\frac{z}{r} \\text{ nearly,} \\]\n\n\\[ = \\left\\{ z_{146} + \\frac{e^2}{2} z_{150} + \\frac{e^2}{2} z_{149} \\right\\} \\gamma \\sin y \\]\n\n\\[ + \\left\\{ z_{147} + \\frac{e^2}{2} z_{151} + \\frac{e^2}{2} z_{153} \\right\\} \\gamma \\sin (2t - y) \\]\n\n\\[ + \\left\\{ z_{148} + \\frac{e^2}{2} z_{152} + \\frac{e^2}{2} z_{154} \\right\\} \\gamma \\sin (2t + y) + \\&c. \\]\n\n\\[ \\frac{d^2 r^2}{2 \\cdot d t^2} - \\frac{\\mu}{r} + \\frac{\\mu}{a} + 2 \\int d R + r \\left( \\frac{d R}{d r} \\right) = 0 \\]\n\n\\[ \\frac{d^2 z}{d t^2} + \\frac{\\mu z}{r^3} + \\frac{m_i z}{r_i^3} \\left\\{ r^2 - 2rr' \\cos (\\lambda - \\lambda') + r_i^2 \\right\\}^{\\frac{3}{2}} \\]\n\n\\[ r^4 \\cdot \\frac{d \\lambda'^2}{d t^2} = h^2 - 2 \\int r^2 \\left( \\frac{d R}{d \\lambda'} \\right) d \\lambda' \\]\n\nNeglecting the square of the disturbing force\n\n\\[ - \\frac{d^2 r^2}{d t^2} \\cdot \\frac{1}{r} - \\mu \\delta \\cdot \\frac{1}{r} + 2 \\int d R + r \\left( \\frac{d R}{d r} \\right) = 0 \\]\n\n\\[ \\frac{d^2 z}{d t^2} + \\frac{\\mu z}{r^3} + \\frac{m_i z}{r_i^3} + \\frac{3 m_i z r' r \\cos (\\lambda' - \\lambda)}{r_i^5} = 0 \\]\n\n\\[ \\frac{d^2 \\delta z}{d t^2} + \\frac{3 \\mu s \\delta}{r} + \\frac{\\mu \\delta z}{r^3} + \\frac{m_i z}{r_i^3} + \\frac{3 \\mu_i z r' r \\cos (\\lambda' - \\lambda)}{r_i^5} = 0 \\]\n\n\\[ \\frac{d \\lambda'}{d t} = \\frac{h(1 + s^2)}{r^2} - \\frac{(1 + s^2)}{r^2} \\int \\left( \\frac{d R}{d \\lambda'} \\right) d t \\]\n\n\\[ r \\left( \\frac{d R}{d r} \\right) = a \\left( \\frac{d R}{d a} \\right), \\quad \\frac{d R}{d \\lambda'} = \\frac{d R}{d t}, \\quad (t \\text{ being used for } n_i - n_t). \\]\n\nIntegrating the equation of p. 270, line 9, by the method of indeterminate coefficients, neglecting the cubes and higher powers of \\( e \\) in order to obtain a first approximation, \\( m \\) being equal to \\( \\frac{n_i}{n} \\) as in the notation of M. Da-Moiseau;\n\n\\[ - r_0 - \\frac{m_i a^3}{2 \\mu a_i^3} \\left\\{ 1 + \\frac{3}{2} e^2 + \\frac{3}{2} e_i^2 - \\frac{3}{2} \\gamma^2 \\right\\} = 0 \\]\n\n\\[ 4 (1 - m)^2 \\left\\{ (1 + 3 e^2) r_1 - \\frac{3 e^2}{2} \\{ r_3 + r_i \\} \\right\\} - r_1 \\]\n\\[- \\frac{3 m_i a^3}{2 \\mu a^3} \\left\\{ 1 - \\frac{5}{2} e^2 - \\frac{5}{2} e_i^2 - \\frac{\\gamma^2}{2} \\right\\} \\left\\{ \\frac{1}{1-m} + 1 \\right\\} = 0\\]\n\n\\[c^2 \\left\\{ 1 - 3 r_0 \\right\\} - 1 + \\frac{2 m_i a^3}{\\mu a^3} = 0\\]\n\n\\[(2 - 2m - c)^2 \\left\\{ r_3 - \\frac{3}{2} r_1 \\right\\} - r_3 + \\frac{9}{2} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{2 - c}{2 - 2m - c} + 1 \\right\\} = 0\\]\n\n\\[(2 - 2m + c)^2 \\left\\{ r_4 - \\frac{3}{2} r_1 \\right\\} - r_4 - \\frac{3}{2} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{2 + c}{2 - 2m + c} + 1 \\right\\} = 0\\]\n\n\\[m^3 r_5 - r_5 - \\frac{3}{2} \\frac{m_i a^3}{\\mu a^3} = 0\\]\n\n\\[(2 - 3m)^2 r_6 - r_6 - \\frac{21}{4} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{2}{2 - 3m} + 1 \\right\\} = 0\\]\n\n\\[(2 - m)^2 r_7 - r_7 + \\frac{3}{4} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{2}{2 - m} + 1 \\right\\} = 0\\]\n\n\\[4 c^2 \\left\\{ (1 - 3r_0) r_8 - \\frac{3}{4} + 3 r_0 \\right\\} - r_8 + \\frac{m_i a^3}{2 \\mu a^3} = 0\\]\n\n\\[(2 - 2m - 2c)^2 \\left\\{ r_9 - \\frac{3}{2} r_3 \\right\\} - r_9 - \\frac{15}{4} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{2 - 2c}{2 - 2m - 2c} + 1 \\right\\} = 0\\]\n\n\\[(2 - 2m + 2c)^2 \\left\\{ r_{10} - \\frac{3}{2} r_4 \\right\\} - r_{10} - \\frac{3}{2} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{2 + 2c}{2 - 2m + 2c} + 1 \\right\\} = 0\\]\n\n\\[(c + m)^2 \\left\\{ r_{11} - \\frac{3}{2} r_5 \\right\\} - r_{11} + \\frac{3}{2} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{c}{c + m} + 1 \\right\\} = 0\\]\n\n\\[(2 - 3m - c)^2 \\left\\{ r_{12} - \\frac{3}{2} r_6 \\right\\} - r_{12} + \\frac{63}{4} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{c}{2 - 3m - c} + 1 \\right\\} = 0\\]\n\n\\[(2 - m + c)^2 \\left\\{ r_{13} - \\frac{3}{2} r_7 \\right\\} - r_{13} + \\frac{3}{4} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{2 + c}{2 - m + c} + 1 \\right\\} = 0\\]\n\n\\[(c - m)^2 \\left\\{ r_{14} - \\frac{3}{2} r_5 \\right\\} - r_{14} + \\frac{3}{2} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{c}{c - m} + 1 \\right\\} = 0\\]\n\n\\[(2 - m - c)^2 \\left\\{ r_{15} - \\frac{3}{2} r_7 \\right\\} - r_{15} - \\frac{9}{4} \\frac{m_i a^3}{\\mu a^3} \\left\\{ \\frac{2 - c}{2 - m - c} + 1 \\right\\} = 0\\]\n\n* The letter \\(c\\) does not strictly denote the same quantity as in the notation of M. Damoiseau, or in that of the Mathematical Tracts, p. 33.\n\\[(2 - 3m + c)^2 \\left\\{ r_{16} - \\frac{3}{2} r_6 \\right\\} - r_{16} - \\frac{21}{4} \\frac{m_i}{\\mu} \\frac{a^3}{a_i^3} \\left\\{ \\frac{2 + c}{2 - 3m + c} + 1 \\right\\} = 0\\]\n\n\\[4m^2 r_{17} - r_{17} - \\frac{9}{4} \\frac{m_i}{\\mu} \\frac{a^3}{a_i^3} = 0\\]\n\n\\[(2 - 4m)^2 r_{18} - r_{18} - \\frac{51}{4} \\frac{m_i}{\\mu} \\frac{a^3}{a_i^3} \\left\\{ \\frac{2}{2 - 4m} + 1 \\right\\}\\]\n\n\\[4r_{19} - r_{19} = 0\\]\n\nThe equation for determining \\(z\\) may be integrated in the same way.\n\n\\[-g^2 z_{146} + 3r_0 + z_{146} + \\frac{m_i}{\\mu} \\frac{a^3}{a_i^3} = 0\\]\n\n\\[-\\left\\{ 2(1-m) - g \\right\\} z_{147} - \\frac{3}{2} r_1 + z_{147} = 0\\]\n\n\\[-\\left\\{ 2(1-m) + g \\right\\} z_{148} + \\frac{3}{2} r_1 + z_{148} = 0\\]\n\n\\[-\\left\\{ c - g \\right\\} z_{149} + \\frac{3}{2} r_6 + z_{149} - \\frac{3}{2} z_{146} + \\frac{3m_i}{2\\mu} \\frac{a^3}{a_i^3} = 0\\]\n\n\\[-\\left\\{ c + g \\right\\} z_{150} + \\frac{9}{2} r_6 + z_{150} + \\frac{3}{2} z_{146} + \\frac{m_i}{\\mu} \\frac{a^3}{a_i^3} = 0\\]\n\n\\[-\\left\\{ 2(1-m) - c - g \\right\\} z_{147} + 3\\left\\{ -\\frac{3}{4} r_1 - \\frac{r_3}{2} \\right\\} + z_{151} + \\frac{3}{2} z_{147} = 0\\]\n\n\\[-\\left\\{ 2(1-m) - c + g \\right\\} z_{148} + 3\\left\\{ -\\frac{r_1}{4} + \\frac{r_3}{2} \\right\\} + z_{152} + \\frac{3}{2} z_{148} = 0\\]\n\n\\[-\\left\\{ 2(1-m) + c - g \\right\\} z_{149} + 3\\left\\{ \\frac{r_1}{4} - \\frac{r_4}{2} \\right\\} + z_{153} + \\frac{3}{2} z_{147} = 0\\]\n\n\\[-\\left\\{ 2(1-m) + c + g \\right\\} z_{150} + 3\\left\\{ \\frac{3}{4} r_1 + \\frac{r_4}{2} \\right\\} + z_{154} + \\frac{3}{2} z_{146} = 0\\]\n\n\\[-\\left\\{ m - g \\right\\} z_{151} + \\frac{3}{2} r_5 + z_{155} - \\frac{3m_i}{2\\mu} \\frac{a^3}{a_i^3} = 0\\]\n\n\\[-\\left\\{ m + g \\right\\} z_{152} + \\frac{3}{2} r_5 + z_{156} + \\frac{3m_i}{2\\mu} \\frac{a^3}{a_i^3} = 0\\]\n\n\\[-\\left\\{ 2(1-m) - m - g \\right\\} z_{153} - \\frac{3}{2} r_6 + z_{157} = 0\\]\n\n\\[-\\left\\{ 2(1-m) - m + g \\right\\} z_{154} + \\frac{3}{2} r_6 + z_{158} = 0\\]\n\n\\[-\\left\\{ 2(1-m) + m - g \\right\\} z_{155} - \\frac{3}{2} r_7 + z_{159} = 0\\]\n\\[- \\left\\{ 2(1-m) + m + g \\right\\}^2 z_{156} + \\frac{3}{2} r_7 + z_{160} = 0\\]\n\n\\[\\frac{d\\lambda}{dt} = \\frac{h}{r^2} + \\frac{2h}{r} \\cdot \\frac{1}{r} + \\frac{h^2}{r^4} - \\frac{(1+s^2)}{r^2} \\int \\left( \\frac{dR}{d\\lambda} \\right) dt\\]\n\n\\[\\lambda' = \\frac{h}{a^2} \\left\\{ 1 + \\frac{e^2}{2} + \\frac{\\gamma^2}{2} + 2r_0 \\right\\} t + \\frac{2e(1+r_0)}{c} \\sin x + \\frac{5e^2(1+r_0)}{4c} \\sin 2x\\]\n\n\\[+ \\left\\{ 2r_1 + e^2(r_3 + r_4) - \\left\\{ -(1 - \\frac{5}{2} e^2 - \\frac{5}{2} e_i^2 - \\frac{\\gamma^2}{2}) \\right\\} \\frac{3}{4(1-m)} + \\frac{9e^2}{2(2-2m-c)} \\right\\}\\]\n\n\\[- \\frac{3e^2}{2(2-2m+c)} \\frac{m_i a^3}{\\mu a_i^3} \\frac{1}{2(1-m)} \\sin 2t\\]\n\n\\[+ \\left\\{ 2r_3 + e^2 r_1 - \\left\\{ \\frac{9}{2(2-m-c)} - \\frac{3}{2(2-m)} \\right\\} \\frac{m_i a^3}{\\mu a_i^3} \\right\\} \\frac{e}{(2-2m-c)} \\sin (2t-x)\\]\n\n\\[+ \\left\\{ 2r_4 + e^2 r_1 - \\left\\{ -\\frac{3}{2(2-m+c)} - \\frac{3}{2(2-m)} \\right\\} \\frac{m_i a^3}{\\mu a_i^3} \\right\\} \\frac{e}{(2-m+c)} \\sin (2t+x)\\]\n\n\\[+ \\frac{2r_5}{m} \\sin z\\]\n\n\\[+ \\left\\{ 2r_6 + \\frac{21}{4(2-3m)} \\frac{m_i a^3}{\\mu a_i^3} \\right\\} \\frac{e_i}{(2-3m)} \\sin (2t-z)\\]\n\n\\[+ \\left\\{ 2r_7 - \\frac{3}{4(2-m)} \\frac{m_i a^3}{\\mu a_i^3} \\right\\} \\frac{e_i}{(2-m)} \\sin (2t+z)\\]\n\n\\[+ \\left\\{ 2r_9 + r_3 - \\left\\{ -\\frac{15}{4(2-2m-2c)} + \\frac{9}{2(2-2m-c)} \\right\\} \\frac{m_i a^3}{\\mu a_i^3} \\right\\} \\frac{e^2}{2(1-m-c)} \\sin (2t-2x)\\]\n\n\\[+ \\left\\{ 2r_{10} + r_4 - \\left\\{ -\\frac{3}{2(2-2m+2c)} - \\frac{3}{2(2-m+c)} \\right\\} \\frac{m_i a^3}{\\mu a_i^3} \\right\\} \\frac{e^2}{2(1-m+c)} \\sin (2t+2x)\\]\n\n\\[+ \\left\\{ 2r_{11} + r_5 \\right\\} \\frac{ee_i}{(c+m)} \\sin (x+z)\\]\n\n\\[+ \\left\\{ 2r_{12} + r_6 - \\left\\{ \\frac{63}{4(2-3m-c)} - \\frac{21}{4(2-3m)} \\right\\} \\frac{m_i a^3}{\\mu a_i^3} \\right\\} \\frac{ee_i}{(2-3m-c)} \\sin (2t-x-z)\\]\n\n\\[+ \\left\\{ 2r_{13} + r_7 - \\left\\{ \\frac{3}{4(2-m+c)} + \\frac{3}{4(2-m)} \\right\\} \\frac{m_i a^3}{\\mu a_i^3} \\right\\} \\frac{ee_i}{(2-m+c)} \\sin (2t+x+z)\\]\n\nConsidering the terms which depend on the square of the disturbing force\n\n\\[\\frac{d^2}{dt^2} \\frac{r^2}{2} - \\frac{\\mu}{r} + \\frac{\\mu}{a} + 2 \\int dR + r \\left( \\frac{dR}{dr} \\right) = 0\\]\n\\[ \\frac{d^2 r}{dt^2} = \\frac{d^2 r^3 \\delta}{dt^2} + \\frac{3 d^2 r^4 (\\delta \\cdot \\frac{1}{r})^2}{2 dt^2} - \\frac{\\mu}{r} + \\frac{\\mu}{a} + 2 \\int dR + r \\left( \\frac{dR}{dr} \\right) = 0 \\]\n\n\\[ \\frac{dz}{dt^2} + \\frac{\\mu z}{r^3} + \\frac{m_i z}{r_i^3} - \\frac{3 m_i z r r \\cos (\\lambda' - \\lambda)}{r_i^5} = 0. \\]\n\n\\[ \\frac{d\\lambda'}{dt} = \\frac{h}{r^2} \\left[ 1 - \\frac{1}{h} \\int \\left( \\frac{dR}{d\\lambda'} \\right) dt \\left\\{ 1 - \\frac{1}{h} \\int \\left( \\frac{dR}{d\\lambda'} \\right) dt \\right\\} - \\frac{1}{2 h^2} \\int \\left( \\frac{dR}{d\\lambda'} \\right) dt \\right]^2 \\]\n\n\\[ = \\frac{h (1 + s^2)}{r^2} - \\frac{(1 + s^2)}{r^2} \\int \\left( \\frac{dR}{d\\lambda'} \\right) dt + \\frac{(1 + s^2)}{2 r^2 h} \\left\\{ \\int \\left( \\frac{dR}{d\\lambda'} \\right) dt \\right\\}^2 \\]\n\n\\( dR \\) = the differential of \\( R \\), supposing \\( nt \\) only variable + the differential of \\( R \\), with regard to \\( n_t \\) only in as much as it is contained in the terms in \\( r, \\lambda \\) and \\( s \\) due to the perturbations; hence\n\n\\( dR \\) = the differential of \\( R \\), supposing only \\( nt \\) variable + \\( \\frac{dR}{da} \\cdot d.\\delta r + \\frac{dR}{dt} \\cdot d.\\delta \\lambda' \\)\n\n+ \\( \\frac{dR}{dr} \\cdot d.\\delta s; \\ d.\\delta r', \\ d.\\delta \\lambda, \\) and \\( d.\\delta s, \\) being restrained to mean the differentials of those quantities with regard to \\( n_t \\) only.\n\n\\[ \\delta R = \\left( \\frac{dR}{dr} \\right) \\delta r + \\left( \\frac{dR}{d\\lambda'} \\right) \\delta \\lambda' + \\left( \\frac{dR}{ds} \\right) \\delta s = -a \\left( \\frac{dR}{da} \\right) r \\delta \\cdot \\frac{1}{r} + \\left( \\frac{dR}{dt} \\right) \\delta \\lambda' + \\left( \\frac{dR}{ds} \\right) \\delta s, \\]\n\n\\( t \\) being used in the sense \\( nt - n_t. \\)\n\n\\[ \\left( \\frac{dR}{ds} \\right) \\delta s = \\frac{r^2}{2 r_i^3} s \\delta s \\text{ nearly.} \\]\n\n\\[ \\left( \\frac{dR}{dr} \\right) d.\\delta r + \\left( \\frac{dR}{d\\lambda'} \\right) d.\\delta \\lambda' + \\left( \\frac{dR}{ds} \\right) d.\\delta s = -a \\left( \\frac{dR}{da} \\right) d.r \\delta \\cdot \\frac{1}{r} + \\left( \\frac{dR}{dt} \\right) d.\\delta \\lambda' + \\left( \\frac{dR}{ds} \\right) d.\\delta s \\]\n\n\\( d.r \\delta \\frac{1}{r}, \\ d.\\delta \\lambda' \\) and \\( d.\\delta s \\) being restrained to mean the differentials of those quantities with regard to \\( n_t \\) only.\n\n\\[ \\delta r \\left( \\frac{dR}{dr} \\right) = d.\\frac{r}{dr} \\left( \\frac{dR}{dr} \\right) \\delta r + d.\\frac{r}{d\\lambda'} \\left( \\frac{dR}{dr} \\right) \\delta \\lambda' + d.\\frac{r}{ds} \\left( \\frac{dR}{dr} \\right) \\delta s \\]\n\n\\[ = -a d.\\frac{r}{da} \\left( \\frac{dR}{dr} \\right) r \\delta \\cdot \\frac{1}{r} + d.\\frac{r}{dt} \\left( \\frac{dR}{dr} \\right) \\delta \\lambda' + d.\\frac{r}{ds} \\left( \\frac{dR}{dr} \\right) \\delta s \\]\n\n\\[ \\delta \\left( \\frac{dR}{d\\lambda'} \\right) = -a d.\\frac{dR}{da} \\left( \\frac{dR}{d\\lambda'} \\right) r \\delta \\cdot \\frac{1}{r} + d.\\frac{dR}{dt} \\left( \\frac{dR}{d\\lambda'} \\right) \\delta \\lambda' + \\left( \\frac{dR}{ds} \\right) \\delta s \\]\nA similar theorem exists with the quantity $\\delta \\cdot \\frac{dR}{dz}$, and it will readily be seen that all the developments $\\delta R$, $\\delta \\cdot r \\left( \\frac{dR}{dr} \\right)$, $\\delta \\cdot \\left( \\frac{dR}{d\\lambda} \\right)$ and $\\delta \\cdot \\left( \\frac{dR}{dz} \\right)$ may be effected very easily by means of Table II.\n\nSimilarly, if $\\delta'$ denote the variation due to the disturbance of the earth by the moon,\n\n$$\\delta' R = -a_i d \\cdot \\left( \\frac{dR}{da_i} \\right) r_i \\delta \\cdot \\frac{1}{r_i} - d \\cdot \\left( \\frac{dR}{dt} \\right) \\delta \\lambda_i$$\n\nIn $dR$ the terms which arise from\n\n$$-a \\left( \\frac{dR}{da} \\right) d \\cdot r \\delta \\cdot \\frac{1}{r} + \\left( \\frac{dR}{dt} \\right) d \\cdot \\delta \\lambda + \\left( \\frac{dR}{ds} \\right) d \\cdot \\delta s$$\n\nare multiplied by the small quantity $m$.\n\nConsidering in $r \\left( \\frac{dR}{dr} \\right)$ and $R$ the terms multiplied by $\\frac{a^2}{a_i^3}$,\n\n$$r \\left( \\frac{dR}{dr} \\right) = 2R, \\quad \\delta \\cdot r \\left( \\frac{dR}{dr} \\right) = 2\\delta R;$$\n\nconsidering the terms multiplied by $\\frac{a^3}{a_i^4}$,\n\n$$r \\left( \\frac{dR}{dr} \\right) = 3R, \\quad \\delta \\cdot r \\left( \\frac{dR}{dr} \\right) = 3\\delta R$$\n\nHence the value of $r \\left( \\frac{dR}{dr} \\right)$ and $\\delta \\cdot r \\left( \\frac{dR}{dr} \\right)$ may at once be inferred from $R$ and $\\delta R$.\n\nI reserve the formation of these developments and of the final equations for determining the coefficients of the different inequalities to another opportunity. These equations are voluminous when all sensible quantities are taken into account; but they are formed with so much facility by means of Table II., that error is not likely to arise in this part of the process. Error is more, I think, to be apprehended in the terms of $R$ multiplied by the cubes and fourth powers of the eccentricities; the rest have been verified by an independent method. See p. 39.\n### Addition to Table I.\n\n|   | 146 | 149 | 150 |   | 146 | 149 | 150 |   | 146 | 149 | 150 |\n|---|-----|-----|-----|---|-----|-----|-----|---|-----|-----|-----|\n| 1 | 148 | 153 | 154 | 1 | 147 | 63  | 69  | 4 | 147 | 5  | 83  |\n|   | 147 | 152 | 151 |   | 147 | 63  | 67  | 147 | 155 | 71  | 14  |\n| 2 | 150 | 161 | 162 | 2 | 148 | 64  | 4   | 70  | 148 | 72  | 11  |\n|   | 149 | 162 | -146 |   | 148 | 64  | 68  | 3   | 148 | 72  | 11  |\n| 3 | 152 | 147 | 148 | 3 | 149 | 2   | 77  | 8   | 149 | 6   | 93  |\n|   | 151 | 164 | 163 |   | 149 | 2   | 65  | 0   | 149 | 73  | 12  |\n| 4 | 154 | 165 | 166 | 4 | 150 | 66  | 8   | 78  | 150 | 74  | 16  |\n|   | 153 | 148 | 147 |   | 150 | 66  | 62  | 0   | 150 | 74  | 16  |\n| 5 | 156 | 167 | 168 | 5 | 151 | 3   | 63  | 1   | 151 | 7   | 87  |\n|   | 155 | -1/3 | -174 |   | 151 | 3   | 67  | 9   | 151 | 75  | 15  |\n| 6 | 158 | 169 | 178 | 6 | 152 | 68  | 1   | 64  | 152 | 76  | 13  |\n|   | 157 | 170 | 169 |   | 152 | 68  | 80  | 9   | 152 | 76  | 13  |\n| 7 | 160 | 171 | 172 | 7 | 153 | 4   | 81  | 10  | 153 | 7   | 88  |\n|   | 159 | 176 | 175 |   | 153 | 4   | 69  | 1   | 153 | 7   | 88  |\n| 146 | 62 | 2   | 66  | 146 | 154 | 70  | 10  | 82  | 154 |\n\n### Addition to Table II.\n\n|   | 146 | 149 | 150 |   | 146 | 149 | 150 |   | 146 | 149 | 150 |\n|---|-----|-----|-----|---|-----|-----|-----|---|-----|-----|-----|\n| 1 | 147 | 152 | 151 | 1 | 10  | 165 | 154 | 153 | 10 | 64  | 148 |\n|   | 148 | 153 | 154 |   | 10  | 165 | 154 | 153 | 10 | 64  | 148 |\n| 2 | 149 | 146 | ..... | 2 | 11  | 167 | 156 | 155 | 11 | 65  | 149 |\n|   | 150 | ..... | -146 |   | 11  | 167 | 156 | 155 | 11 | 65  | 149 |\n| 3 | 151 | ..... | ..... | 3 | 12  | 169 | ..... | ..... | 12 | 66  | 150 |\n|   | 152 | ..... | ..... |   | 12  | 169 | ..... | ..... | 12 | 66  | 150 |\n| 4 | 153 | 148 | 147 | 4 | 13  | 171 | 160 | 159 | 13 | 67  | 151 |\n|   | 154 | ..... | ..... |   | 13  | 171 | 160 | 159 | 13 | 67  | 151 |\n| 5 | 155 | ..... | ..... | 5 | 14  | 173 | ..... | ..... | 14 | 68  | 152 |\n|   | 156 | ..... | ..... |   | 14  | 173 | ..... | ..... | 14 | 68  | 152 |\n| 6 | 157 | ..... | ..... | 6 | 15  | 175 | ..... | ..... | 15 | 69  | 153 |\n|   | 158 | ..... | ..... |   | 15  | 175 | ..... | ..... | 15 | 69  | 153 |\n| 7 | 159 | ..... | ..... | 7 | 16  | 177 | 158 | 157 | 16 | 70  | 154 |\n|   | 160 | ..... | ..... |   | 16  | 177 | 158 | 157 | 16 | 70  | 154 |\n| 8 | 161 | 150 | 149 | 8 | 62  | 146 | ..... | ..... | 62 | 71  | 155 |\n|   | 162 | ..... | ..... |   | 62  | 146 | ..... | ..... | 62 | 71  | 155 |\n| 9 | 163 | ..... | ..... | 9 | 63  | 147 | ..... | ..... | 63 | 72  | 156 |\n|   | 146 | 149 | 150 |   | 146 | 149 | 150 |   | 146 | 149 | 150 |\n|---|-----|-----|-----|---|-----|-----|-----|---|-----|-----|-----|\n| 73 | 157 |     |     | 73 | 178 | 158 |     | 94 | 10  |     | 4   |\n| 74 | 158 |     |     | 74 | 146 | 62  |     | 146 | 11  |     | 5   |\n| 75 | 159 |     |     | 75 | 147 | 63  |     | 147 | 11  |     | 5   |\n| 76 | 160 |     |     | 76 | 148 | 64  |     | 148 | 12  |     | 6   |\n| 77 | 161 | 149 |     | 77 | 149 | 2   |     | 149 | 12  |     | 6   |\n| 78 | 162 | 150 |     | 78 | 150 | 2   |     | 150 | 13  |     | 7   |\n| 79 | 163 | 151 |     | 79 | 151 | 3   |     | 151 | 13  |     | 7   |\n| 80 | 164 | 152 |     | 80 | 152 | 3   |     | 152 | 14  |     | 5   |\n| 81 | 165 | 153 |     | 81 | 153 | 4   |     | 153 | 14  |     | 5   |\n| 82 | 166 | 154 |     | 82 | 154 | 4   |     | 154 | 15  |     | 7   |\n| 83 | 167 | 155 |     | 83 | 155 | 5   |     | 155 | 15  |     | 7   |\n| 84 | 168 | 156 |     | 84 | 156 | 5   |     | 156 | 16  |     |     |\n| 85 | 169 | 157 |     | 85 | 157 | 6   |     | 157 | 16  |     |     |\n| 86 | 170 | 158 |     | 86 | 158 | 6   |     | 158 | 17  |     |     |\n| 87 | 171 | 159 |     | 87 | 159 | 7   |     | 159 | 17  |     |     |\n| 88 | 172 | 160 |     | 88 | 160 | 7   |     | 160 | 18  |     |     |\n| 89 | 173 | -156 |     | 89 | 161 | 8   |     | 161 | 18  |     |     |\n| 90 | 174 | -155 |     | 90 | 162 | 8   |     | 162 | 19  |     |     |\n| 91 | 175 | 159 |     | 91 | 163 | 9   |     | 163 | 19  |     |     |\n| 92 | 176 | 160 |     | 92 | 164 | 9   |     | 164 |     |     |     |\n| 93 | 177 | 157 |     | 93 | 165 | 10  |     | 165 |     |     |     |\n\n|   | 161 | 162 |   | 161 | 162 |   | 161 | 162 |   | 161 | 162 |\n|---|-----|-----|---|-----|-----|---|-----|-----|---|-----|-----|\n| 8 | 146 | -146 | 8 | 78 | 146 | 78 | 81 | 147 | 81 | 164 | 1 |\n| 9 | 147 | 148 | 9 | 79 | 147 | 79 | 82 | 148 | 82 | 165 | 1 |\n| 10| 148 | 147 | 10| 80 | 148 | 80 | 163 | 1 | 163 | 166 |\nOn the Precession of the Equinoxes, supposing the Earth to revolve in a resisting medium.\n\nIn my last paper on Physical Astronomy, I gave expressions for the variations of the six constants which enter into the solution of this problem, upon the hypothesis that the body revolves in a medium devoid of resistance.\n\nIf we suppose a plane to revolve in a resisting medium, about an axis perpendicular to itself, the resistance of the medium can produce no effect, and the phenomena will only be modified in a slight degree by the friction of the plane surface against the medium. If, however, the inclination of the plane on the axis of rotation differs from $90^\\circ$, the effect of the resistance of the medium becomes sensible, tending to retard the motion of the plane; the effect being greatest when the axis of rotation is parallel to the plane.\n\nThis principle is used in some machines, as in self-playing organs, to regulate the motion by means of a vane, of which the inclination to its axis of rotation can be varied at pleasure.\n\nIn the case of a sphere, whatever be the direction of the axis of rotation, this effect of the resistance is insensible; and also in the case of a solid of revolution when the axis of rotation coincides with the axis of the figure, but not otherwise. If the difference of the latitude of the axis of rotation from $90^\\circ$ (supposing the equator from which the latitudes are reckoned to coincide with the equator of the figure) be at any time small, the mathematical investigation appears to show, that the effect of the resistance of the medium is to diminish continually this difference. In the case of the earth, this quantity is now insensible; but as the probability is small that this was the case in the first instance, may this circumstance arise from the resistance of a medium of small density acting for a great length of time? and can the change of climate on the surface of the earth, a change of which the probability is indicated by many geological phenomena, be explained in the same manner? It may be remarked, however, that the effect of a resisting medium in diminishing the eccentricities of the orbits of the planets is of the same order, and that these, although for the most part small, are far from having reached zero. The tendency of a resisting medium is also to diminish the major axes of the orbits of the planets; these effects, if they exist, will probably be most sensible.\nin the case of comets, not only on account of their great eccentricity, but also on account of their small density, in the same manner as a flock of any light substance is wafted by the gentlest wind and prevented from reaching the ground. The eccentricity of the orbit of the comet of Halley in 1759 is known with great accuracy, and as its perturbations have been calculated with great care by MM. Damoiseau and de Pontecoulant, the eccentricity which it should have in 1835, when it will again visit this part of space, unless it be affected by a resisting medium, is also known with great precision. It is scarcely probable, however, that any change will be perceptible in one revolution, even if the cause exists; but the succeeding revolutions of this body will no doubt throw light upon this question. The ratio of the change of the semi-major axis to the change of the eccentricity, due to the action of the resisting medium, is known, being a function of the eccentricity, and independent of the constant, which depends upon the density of the medium; this ratio therefore may also tend to elucidate the question, if it can be determined by observation with sufficient accuracy.\n\nLet $x'$, $y'$, $z'$ be the co-ordinates of any point P corresponding to the elementary portion of the surface $d s$, and referred to axes passing through the centre of gravity and revolving with the body in motion.\n\nLet P be the point of which the co-ordinates are $x'$, $y'$, $z'$, A P the direction of the normal at the point P, B P perpendicular to the axis of instantaneous rotation, and cutting it in B, and C P the direction of motion of the point P. I suppose the resistance of the medium to create a force proportional to $v^2 \\cos A P C d s$, acting in the direction of the normal A P upon the point P, $v$ being the velocity of the point P.\n\nSuppose the straight line M O P L to revolve about an axis passing through O, and perpendicular to it, and in the direction L N, the action of the resisting medium will be in the direction N L, on one side only of the line O L, upon all the points P between O and L, and upon all the points between M P it will be in the contrary direction R M, and on the other side of the line.\nNow, let LSM'T be the section of a cylinder revolving about an axis, passing through O perpendicular to the plane LSM'T, and let the cylinder revolve in the direction LN. The action of the resisting medium will be in the direction ZP, perpendicular to OP upon all the points P between LS; and in the contrary direction KP upon all the points, P between TM. These remarks show that in what follows, the integrations must not be made throughout the whole surface of the body revolving: this consideration however does not affect the nature of the results.\n\nThe equation to a plane perpendicular to the axis of rotation, and passing through the centre of gravity of the body, is \\( px + qy + rz = 0 \\).\n\nLet the body revolving be a spheroid of which the equation is\n\n\\[\nx^2 + y^2 + z^2 (1 + e^2) = a^2 (1 + e^2)\n\\]\n\nThe equation to the tangent plane to the spheroid at the point \\( x, y, z \\) is\n\n\\[\nxx' + yy' + zz' (1 + e^2) = a^2 (1 + e^2)\n\\]\n\nThe equations to the planes from whose intersection the line PB results, are\n\n\\[\n* z(qz' - ry') + y(rx' - pz') + z(py' - qx') = 0 \\\\\npx + qy + rz = D\n\\]\n\n\\( D \\) being a constant. The equations to the line PC are\n\n\\[\nx\\{r(qz' - ry') - p(py' - qx')\\} + y\\{r(rx' - pz') - q(py' - qx')\\} = 0 \\\\\nx\\{q(qz' - ry') - p(rx' - pz')\\} + z\\{q(py' - qx') - r(rx' - pz')\\} = 0\n\\]\n\nand neglecting \\( p^2, q^2, pq \\),\n\n\\[\nx(qz' - ry') = y(pz' - rx') \\\\\nx(qy' + px') = z(pz' - rx')\n\\]\n\nThe equations to the direction of motion of the point P are\n\n\\[\nx(pz' - rx') = y(ry' - qx') \\\\\nx(qx' - py') = z(ry' - qz')\n\\]\n\nCos. angle, which the direction of motion of P makes with the normal to the surface or cos A PC\n\n\\[\n= \\frac{x'(ry' - qz') + y'(pz' - rx') + z'(1 + e^2)(qx' - py')}{\\sqrt{(ry' - qz')^2 + (pz' - rx')^2 + (qy' - px')^2}} \\sqrt{x'^2 + y'^2 + z'^2 (1 + 2e^2)}\n\\]\n\n* The notation is the same as p. 20, except that the accents at foot of \\( x_p, y_p, z_p \\) are omitted.\n\\[ \\frac{e^2 z' (q x' - p y')}{r \\sqrt{x'^2 + y'^2} \\sqrt{x'^2 + y'^2 + z'^2}} \\text{ nearly.} \\]\n\nThe resistance acting in the direction of the normal, and since the velocity\n\n\\[ = \\sqrt{x'^2 + y'^2} \\sqrt{(p^2 + q^2 + r^2)} \\text{ nearly;} \\]\n\n\\[ Cdr = 0 \\]\n\n\\[ B dq + (A - C) rp dt = dt \\int \\left\\{ \\frac{z' x' - x' z' (1 + e^2)}{r \\{ x'^2 + y'^2 + z'^2 \\}} \\right\\} e^2 z' (q x' - p y') \\sqrt{x'^2 + y'^2} ds (p^2 + q^2 + r^2) \\]\n\n\\[ A dp + (C - B) qr dt = dt \\int \\left\\{ \\frac{y' z' (1 + e^2) - z' y'}{r \\{ x'^2 + y'^2 + z'^2 \\}} \\right\\} e^2 z' (q x' - p y') \\sqrt{x'^2 + y'^2} ds (p^2 + q^2 + r^2) \\]\n\n\\[ \\sin \\frac{C - A}{A} (nt + \\gamma) dc + c \\frac{(C - A)}{A} \\cos \\frac{C - A}{A} (nt + \\gamma) d\\gamma \\]\n\n\\[ = - \\frac{n dt e^t}{A} \\int \\frac{x' z'^2 (q x' - p y') \\sqrt{x'^2 + y'^2}}{\\{ x'^2 + y'^2 + z'^2 \\}} ds \\]\n\n\\[ \\cos \\frac{C - A}{A} (nt + \\gamma) dc - c \\frac{(C - A)}{A} \\sin \\frac{C - A}{A} (nt + \\gamma) d\\gamma \\]\n\n\\[ = \\frac{n dt e^t}{A} \\int \\frac{y' z'^2 (q x' - p y') \\sqrt{x'^2 + y'^2}}{\\{ x'^2 + y'^2 + z'^2 \\}} ds \\]\n\nsince \\( \\int x'^2 z'^2 ds = \\int y'^2 z'^2 ds \\)\n\n\\[ dc = - \\frac{n dt e^t c}{A} \\int \\frac{x'^2 z'^2 \\sqrt{x'^2 + y'^2}}{\\{ x'^2 + y'^2 + z'^2 \\}} ds + \\frac{n dt e^t}{2A} \\sin 2 \\left( \\frac{C - A}{A} (nt + \\gamma) \\right) \\int \\frac{x' y' z'^2 \\sqrt{x'^2 + y'^2}}{\\{ x'^2 + y'^2 + z'^2 \\}} ds \\]\n\nneglecting the term which is periodic,\n\n\\[ dc = - nc \\frac{e^t dt}{A} \\int \\frac{x'^2 z'^2 \\sqrt{x'^2 + y'^2}}{\\{ x'^2 + y'^2 + z'^2 \\}} ds \\]\n\nLet \\( \\int \\frac{x'^2 z'^2 \\sqrt{x'^2 + y'^2}}{\\{ x'^2 + y'^2 + z'^2 \\}} ds = D \\)\n\n\\( D \\) being a positive quantity.\n\n\\[ dc = - \\frac{n D e^t dt}{A} \\quad e^c = \\frac{n De^t}{A}, \\quad e \\text{ being the base of Naperian logarithms.} \\]\n\nWhen \\( t \\) is infinite \\( c = 0 \\); hence the latitude of the axis of instantaneous rotation increases until it reaches 90°, which is its limit.\n\nHaving determined the variations of \\( c, \\gamma \\) and \\( n \\) by means of the above equations, the variations of the other constants \\( \\omega, \\psi_0 \\) and \\( \\phi_0 \\) may be determined from the equations\n\n\\[ p dt = \\sin \\phi \\sin \\theta d\\psi - \\cos \\phi d\\theta \\]\n\n\\[ q dt = \\cos \\phi \\sin \\theta d\\psi + \\sin \\phi d\\theta \\]\n\n\\[ r dt = d\\phi - \\cos \\theta d\\psi \\]",
  "source": "olmocr",
  "added": "2026-01-12",
  "created": "2026-01-12",
  "metadata": {
    "Source-File": "/home/jic823/projects/def-jic823/royalsociety/pdfs/107934.pdf",
    "olmocr-version": "0.3.4",
    "pdf-total-pages": 53,
    "total-input-tokens": 90464,
    "total-output-tokens": 63474,
    "total-fallback-pages": 0
  },
  "attributes": {
    "pdf_page_numbers": [
      [
        0,
        0,
        1
      ],
      [
        0,
        1709,
        2
      ],
      [
        1709,
        4157,
        3
      ],
      [
        4157,
        6867,
        4
      ],
      [
        6867,
        9450,
        5
      ],
      [
        9450,
        11977,
        6
      ],
      [
        11977,
        14953,
        7
      ],
      [
        14953,
        17183,
        8
      ],
      [
        17183,
        20325,
        9
      ],
      [
        20325,
        23237,
        10
      ],
      [
        23237,
        26257,
        11
      ],
      [
        26257,
        28840,
        12
      ],
      [
        28840,
        32140,
        13
      ],
      [
        32140,
        34775,
        14
      ],
      [
        34775,
        37282,
        15
      ],
      [
        37282,
        39276,
        16
      ],
      [
        39276,
        41615,
        17
      ],
      [
        41615,
        43975,
        18
      ],
      [
        43975,
        46207,
        19
      ],
      [
        46207,
        48038,
        20
      ],
      [
        48038,
        50171,
        21
      ],
      [
        50171,
        52355,
        22
      ],
      [
        52355,
        54093,
        23
      ],
      [
        54093,
        55438,
        24
      ],
      [
        55438,
        57144,
        25
      ],
      [
        57144,
        58604,
        26
      ],
      [
        58604,
        60645,
        27
      ],
      [
        60645,
        63401,
        28
      ],
      [
        63401,
        65747,
        29
      ],
      [
        65747,
        67903,
        30
      ],
      [
        67903,
        69767,
        31
      ],
      [
        69767,
        71725,
        32
      ],
      [
        71725,
        73415,
        33
      ],
      [
        73415,
        75027,
        34
      ],
      [
        75027,
        77289,
        35
      ],
      [
        77289,
        79230,
        36
      ],
      [
        79230,
        81627,
        37
      ],
      [
        81627,
        83847,
        38
      ],
      [
        83847,
        86400,
        39
      ],
      [
        86400,
        88184,
        40
      ],
      [
        88184,
        90695,
        41
      ],
      [
        90695,
        92674,
        42
      ],
      [
        92674,
        94802,
        43
      ],
      [
        94802,
        96726,
        44
      ],
      [
        96726,
        98813,
        45
      ],
      [
        98813,
        101743,
        46
      ],
      [
        101743,
        103659,
        47
      ],
      [
        103659,
        106284,
        48
      ],
      [
        106284,
        108255,
        49
      ],
      [
        108255,
        110703,
        50
      ],
      [
        110703,
        112925,
        51
      ],
      [
        112925,
        114829,
        52
      ],
      [
        114829,
        117089,
        53
      ]
    ],
    "primary_language": [
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en",
      "en"
    ],
    "is_rotation_valid": [
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true
    ],
    "rotation_correction": [
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0
    ],
    "is_table": [
      false,
      false,
      false,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      true,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      true,
      true,
      false,
      false,
      false,
      false
    ],
    "is_diagram": [
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      false,
      true,
      false
    ]
  },
  "jstor_metadata": {
    "identifier": "jstor-107934",
    "title": "Researches in Physical Astronomy",
    "authors": "J. W. Lubbock",
    "year": 1831,
    "volume": "121",
    "journal": "Philosophical Transactions of the Royal Society of London",
    "page_count": 53,
    "jstor_url": "https://www.jstor.org/stable/107934"
  }
}