Researches in Physical Astronomy

On the Theory of the Moon. The 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. The 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 are 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. Having 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. The 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. \[ t^* = n_t - n_t t, \quad x = c_n t - \varphi, \quad z = n_t t - \varphi, \quad y = g_n t - \nu. \] | | | | | | |---|---|---|---|---| | 0 | .. | 0 | 21 | 45 | 2t - 3x | | 1 | 30 | 2t† | 22 | 46 | 2t + 3x | | 2 | 1 | x | 23 | 21 | 2x + z | | 3 | 31 | 2t - x† | 24 | 53 | 2t - 2x - z | | 4 | 32 | 2t + x | 25 | 54 | 2t + 2x + z | | 5 | 16 | z§ | 26 | 20 | 2x - z | | 6 | 33 | 2t - z | 27 | 51 | 2t - 2x + z | | 7 | 34 | 2t + z | 28 | 52 | 2t + 2x - z | | 8 | 2 | 2x | 29 | 23 | x + 2z | | 9 | 35 | 2t - 2x | 30 | 59 | 2t - x - 2z | | 10 | 36 | 2t + 2x | 31 | .. | 2t + x + 2z | | 11 | 19 | x + z | 32 | 22 | x - 2z | | 12 | 41 | 2t - x - z | 33 | 61 | 2t - x + 2z | | 13 | 42 | 2t + x + z | 34 | 60 | 2t + x - 2z | | 14 | 18 | x - z | 35 | .. | 3z | | 15 | 39 | 2t - x + z | 36 | .. | 2t - 3z | | 16 | 40 | 2t + x - z | 37 | .. | 2t + 3z | | 17 | 17 | 2z | 38 | 9 | 4x | | 18 | 43 | 2t - 2z | 39 | 67 | 2t - 4x | | 19 | 44 | 2t + 2z | 40 | .. | 2t + 4x | | 20 | 4 | 3x | 41 | 27 | 3x + z | * 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. | | | | | | | |---|---|---|---|---|---| | 63 | 37 | $2t - 2y$ | 105 | 84 | $t + z$ | | 64 | 38 | $2t + 2y$ | 106 | 85 | $t - 2x$ | | 65 | 5 | $x - 2y$ | 107 | 86 | $t + 2x$ | | 66 | 6 | $x + 2y$ | 108 | 91 | $t - x - z$ | | 67 | 49 | $2t - x - 2y$ | 109 | 92 | $t + x + z$ | | 68 | 47 | $2t - x + 2y$ | 110 | 89 | $t - x + z$ | | 69 | 48 | $2t + x - 2y$ | 111 | ... | $t + x - z$ | | 70 | 50 | $2t + x + 2y$ | 112 | ... | $t - 2z$ | | 71 | 24 | $z - 2y$ | 113 | ... | $t + 2z$ | | 72 | 25 | $z + 2y$ | 114 | ... | $t - 2y$ | | 73 | 57 | $2t - z - 2y$ | 115 | ... | $t + 2y$ | | 74 | 56 | $2t - z + 2y$ | 116 | 100 | $3t$ | | 75 | 55 | $2t + z - 2y$ | 117 | 101 | $3t - x$ | | 76 | 58 | $2t + z + 2y$ | 118 | 102 | $3t + x$ | | 77 | 7 | $2x - 2y$ | 119 | 103 | $3t - z$ | | 78 | 8 | $2x + 2y$ | 120 | 104 | $3t + z$ | | 79 | 65 | $2t - 2x - 2y$ | 121 | ... | $3t - 2x$ | | 80 | 63 | $2t - 2x + 2y$ | 122 | ... | $3t + 2x$ | | 81 | 64 | $2t + 2x - 2y$ | 123 | ... | $3t - x - z$ | | 82 | ... | $2t + 2x + 2y$ | 124 | ... | $3t + x + z$ | | 83 | ... | $x + z - 2y$ | 125 | ... | $3t - x + z$ | | 84 | ... | $x + z + 2y$ | 126 | ... | $3t + x - z$ | | 85 | ... | $2t - x - z - 2y$ | 127 | ... | $3t - 2z$ | | 86 | ... | $2t - x - z + 2y$ | 128 | ... | $3t + 2z$ | | 87 | ... | $2t + x + z - 2y$ | 129 | ... | $3t - 2y$ | | 88 | ... | $2t + x + z + 2y$ | 130 | ... | $3t + 2y$ | | 89 | ... | $x - z - 2y$ | 131 | 120 | $4t$ | | 90 | ... | $x - z + 2y$ | 132 | 121 | $4t - x$ | | 91 | ... | $2t - x + z - 2y$ | 133 | 122 | $4t + x$ | | 92 | ... | $2t - x + z + 2y$ | 134 | 123 | $4t - z$ | | 93 | ... | $2t + x - z - 2y$ | 135 | 124 | $4t + z$ | | 94 | ... | $2t + x - z + 2y$ | 136 | 125 | $4t - 2x$ | | 95 | ... | $2z - 2y$ | 137 | 126 | $4t + 2x$ | | 96 | ... | $2z + 2y$ | 138 | 131 | $4t - x - z$ | | 97 | ... | $2t - 2x - 2y$ | 139 | ... | $4t + x + z$ | | 98 | ... | $2t - 2x + 2y$ | 140 | 129 | $4t - x + z$ | | 99 | ... | $2t + 2z - 2y$ | 141 | ... | $4t + x - z$ | | 100| ... | $2t + 2z + 2y$ | 142 | ... | $4t - 2z$ | | 101| 80 | $t^*$ | 143 | ... | $4t + 2z$ | | 102| 81 | $t - x$ | 144 | 127 | $4t - 2y$ | | 103| 82 | $t + x$ | 145 | ... | $4t + 2y$ | | 104| 83 | $t - z$ | | | | $$\cos 2t \cos 2t = \frac{1}{2} \cos 4t + \frac{1}{2}$$ $$[131] \quad [0]$$ $$\cos 2t \cos x = \frac{1}{2} \cos (2t + x) + \frac{1}{2} \cos (-2t + x)$$ $$[4] \quad [-3]$$ Hence 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. * Parallactic inequality. Table I. Showing 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. | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----| | 1 | 131 | 4 | 132 | 133 | 7 | 134 | 135 | 10 | 136 | 137 | 13 | 138 | 16 | 19 | 22 | 37 | 64 | 116 | 148 | 146 | | 2 | -3 | 8 | 1 | 10 | 11 | 16 | 13 | 20 | 3 | 22 | 23 | 6 | 26 | 29 | 38 | 53 | 66 | 103 | 150 | 153 | | 3 | -2 | 9 | 1 | 136 | 131 | 15 | 138 | 140 | 4 | 133 | 7 | 6 | 33 | 10 | 57 | 68 | 117 | 152 | 155 | | 4 | 2 | 1 | 8 | 131 | 137 | 13 | 141 | 139 | 22 | 132 | 25 | 23 | 7 | 34 | 9 | 58 | 69 | 118 | 154 | 150 | | 5 | -6 | 11 | 15 | 13 | 17 | 1 | 19 | 23 | 27 | 25 | 29 | 3 | 2 | 35 | 41 | 59 | 72 | 105 | 156 | 159 | | 6 | -5 | 12 | 14 | 11 | 18 | 0 | 17 | 24 | 26 | 23 | 30 | 31 | 132 | 4 | 37 | 61 | 76 | 120 | 160 | 156 | | 7 | 5 | 15 | 11 | 14 | 1 | 17 | 0 | 27 | 23 | 26 | 3 | 29 | 33 | 6 | 45 | 18 | 75 | 105 | 159 | 156 | | 8 | -9 | 20 | 4 | 22 | 23 | 28 | 25 | 38 | 1 | 41 | 16 | 44 | 47 | 78 | 107 | 162 | 165 | 163 | 163 | | 9 | -8 | 21 | -2 | 20 | 24 | -26 | -23 | 39 | 0 | 38 | 42 | -14 | 45 | 48 | 79 | 106 | 163 | 161 | 161 | | 10 | 8 | 4 | 20 | 2 | 28 | 23 | 26 | 1 | 38 | 0 | 16 | 41 | 13 | 52 | 82 | 122 | 166 | 165 | 165 | | 11 | -12 | 5 | -24 | 6 | 2 | -30 | -3 | 14 | -42 | -16 | 0 | -48 | 17 | 14 | 84 | 109 | 168 | 171 | 169 | | 12 | -11 | 24 | -5 | 23 | 30 | -2 | -29 | 42 | 14 | -41 | 48 | 0 | 9 | 54 | 85 | 108 | 169 | 167 | 167 | | 13 | 11 | 7 | 23 | 5 | 4 | 29 | 2 | 15 | 41 | -14 | 1 | 47 | 19 | 16 | 88 | 124 | 172 | 168 | 168 | | 14 | -15 | 6 | 26 | 6 | 28 | 2 | 34 | 4 | 44 | 12 | 46 | 8 | 18 | 50 | 11 | 90 | 111 | 174 | 177 | | 15 | -14 | 27 | 5 | -26 | 3 | -32 | -2 | 45 | 11 | -44 | 9 | 17 | 51 | 12 | 92 | 125 | 176 | 175 | 173 | | 16 | 14 | 6 | 26 | -5 | 34 | 2 | 32 | 12 | 44 | -11 | 18 | 8 | 1 | 58 | 93 | 111 | 177 | 174 | 16 | | 17 | -18 | 29 | 33 | 31 | 35 | 7 | 37 | 47 | 51 | 49 | 53 | 15 | 11 | 59 | 96 | 113 | 180 | 183 | 17 | | 18 | -17 | 30 | 32 | 29 | 36 | -5 | 35 | 48 | 50 | -47 | 54 | 14 | 12 | 60 | 97 | 112 | 181 | 179 | 18 | | 19 | 17 | 33 | 29 | 32 | 7 | 35 | 5 | 51 | 47 | -50 | 15 | 53 | 57 | 1 | 99 | 113 | 183 | 180 | 19 | | 20 | -21 | 8 | -39 | 9 | 44 | | | | | | | | | | | | | | | | | | 21 | -20 | 9 | -8 | 38 | 42 | | | | | | | | | | | | | | | | | --- MR. LUBBOCK'S RESEARCHES | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----| | 22 | 20 | 40 | 38 | 8 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | | 23 | -24 | 41 | 11 | -42 | -12 | 8 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 24 | -23 | 12 | 42 | -11 | -41 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | | 25 | -23 | 43 | 139 | 41 | 11 | 10 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 26 | -27 | 28 | 44 | 16 | 46 | 8 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 27 | -26 | 15 | 45 | -14 | -44 | 9 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 28 | -26 | 46 | 141 | 14 | 14 | 52 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 29 | -30 | 31 | 47 | 19 | 49 | 53 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 30 | -29 | 18 | 48 | -17 | -47 | 54 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 31 | -29 | 49 | 143 | 17 | 13 | 55 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 32 | -33 | 34 | 50 | 18 | 52 | 14 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 33 | -32 | 19 | 51 | 17 | -50 | 15 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 34 | -32 | 52 | 142 | 16 | 58 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | | 35 | -36 | 37 | 53 | 57 | 58 | 59 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 36 | -35 | 58 | 54 | 56 | -53 | 60 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 37 | -35 | 55 | 57 | 53 | -56 | 19 | 43 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | | 62 | -63 | 64 | 66 | 68 | 70 | 72 | 74 | 76 | 78 | 80 | 82 | 84 | 86 | 89 | 95 | 115 | 114 | 146 | 148 | 62 | | 63 | -62 | 144 | 69 | 75 | 81 | 87 | 93 | 99 | 114 | 129 | 147 | 148 | 63 | | 64 | -62 | 145 | 70 | 76 | 82 | 88 | 94 | 100 | 115 | 130 | 148 | 64 | | 65 | -68 | 69 | 77 | 83 | 89 | 95 | 101 | 107 | 122 | 138 | 148 | 65 | | 66 | -67 | 70 | 78 | 84 | 90 | 96 | 102 | 108 | 124 | 140 | 66 | | 67 | -66 | 63 | 144 | 91 | 97 | 103 | 109 | 125 | 141 | 67 | | 68 | -65 | 64 | 145 | 92 | 98 | 104 | 110 | 126 | 142 | 68 | | 69 | -65 | 81 | 144 | 87 | 93 | 109 | 115 | 131 | 147 | 69 | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----| | 70 | 66 | 64 | 78 | 62 | 94 | | | | | | | | | | | | | | | | | | 71 | 75 | 83 | 91 | 87 | 95 | | | | | | | | | | | | | | | | | | 72 | 76 | 84 | 92 | 88 | 96 | | | | | | | | | | | | | | | | | | 73 | 72 | 85 | 89 | 84 | 97 | | | | | | | | | | | | | | | | | | 74 | 71 | 86 | 90 | 83 | 98 | | | | | | | | | | | | | | | | | | 75 | 71 | 87 | 91 | 83 | 90 | 63 | | | | | | | | | | | | | | | | | 76 | 72 | 88 | 92 | 84 | 89 | 64 | | | | | | | | | | | | | | | | | 101 | 116 | 103 | 117 | 118 | 105 | 119 | 120 | 107 | 121 | 122 | 109 | 123 | 111 | 113 | | | | | | | | 102 | 117 | 101 | 121 | 116 | 110 | | | | | | | | | | | | | | | | | | 103 | 118 | 107 | 116 | 122 | 109 | | | | | | | | | | | | | | | | | | 104 | 119 | 111 | 123 | 126 | 101 | | | | | | | | | | | | | | | | | | 105 | 120 | 109 | 125 | 124 | 113 | | | | | | | | | | | | | | | | | | 116 | 101 | 118 | 117 | 103 | 102 | 119 | 105 | 104 | 121 | 107 | 106 | 123 | 109 | 125 | 127 | | | | | | 117 | 102 | 116 | 121 | 101 | 106 | 123 | | | | | | | | | | | | | | | | | 118 | 103 | 122 | 116 | 107 | 101 | 126 | | | | | | | | | | | | | | | | | 119 | 104 | 126 | 123 | 111 | 108 | 127 | | | | | | | | | | | | | | | | | 120 | 105 | 124 | 125 | 109 | 110 | 116 | | | | | | | | | | | | | | | | | 131 | 133 | 132 | 4 | 3 | 135 | 137 | 136 | 10 | 9 | 139 | 141 | 143 | | | | | | | | | 132 | 131 | 136 | 1 | 9 | 138 | | | | | | | | | | | | | | | | | | 133 | 137 | 131 | 10 | 1 | 141 | | | | | | | | | | | | | | | | | | 134 | 141 | 138 | 16 | 12 | 142 | | | | | | | | | | | | | | | | | | 135 | 139 | 140 | 13 | 15 | 131 | | | | | | | | | | | | | | | | | | 146 | 148 | 150 | 152 | 154 | 156 | 158 | 160 | 162 | 164 | 166 | 168 | 170 | 174 | 180 | | | | | | | 147 | 153 | 151 | 149 | 150 | 157 | 155 | 156 | 163 | 161 | 162 | 169 | 175 | 181 | | | | | | | | ### Table I. (Continued.) | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|-----|-----|-----| | 148 | 146 | 152 | 150 | -149 | 158 | 156 | -155 | 164 | 162 | -161 | 170 | 168 | 176 | 182 | ... | ... | ... | 147 | 186 | 64 | 131 | 62 | 148 | | 149 | 153 | 161 | 147 | 165 | 167 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 149 | | 150 | -152 | -146 | -164 | -148 | 173 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 150 | | 151 | 154 | 162 | 148 | 166 | 168 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 151 | | 152 | -151 | 146 | -163 | -147 | 174 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 152 | | 153 | 147 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 153 | | 154 | -150 | 163 | -146 | -162 | 169 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 154 | | 155 | 148 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 155 | | 156 | -149 | 164 | 146 | -161 | 170 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 156 | | 157 | 165 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 157 | | 158 | 149 | 147 | 161 | -146 | 177 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 158 | | 159 | 166 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 159 | | 160 | -158 | -174 | -176 | -178 | -146 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 160 | | | 38 | 59 | |---|----|----| | 1 | 40 | 61 | | | 39 | 60 | 1 Table II. Showing 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. | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----| | 1 | 0 | 3 | 2 | 133 | 6 | 5 | ..... | 9 | 8 | ..... | 12 | 11 | 15 | 18 | ..... | 63 | 101 | 147 | 146 | 1 | | 2 | - | 4 | 0 | 9 | 10 | 14 | 16 | - | 15 | ..... | 3 | - | 5 | ..... | ..... | ..... | ..... | ..... | 2 | | 3 | - | 2 | 9 | 0 | 131 | 12 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 3 | | 4 | - | 2 | 1 | 8 | 0 | 16 | 11 | 14 | 3 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 4 | | 5 | - | 7 | - | 14 | 15 | - | 16 | 0 | 18 | 19 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 5 | | 6 | - | 134 | 12 | 14 | ..... | 18 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 6 | | 7 | - | 5 | 15 | 11 | ..... | 1 | 17 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 7 | | 8 | - | 10 | 2 | - | 21 | 22 | 26 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 8 | | 9 | - | 8 | 21 | ..... | 132 | 24 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 9 | | 10 | - | 8 | 4 | 20 | 2 | 28 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 10 | | 11 | - | 13 | 5 | - | 24 | 25 | 2 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 11 | | 12 | - | 11 | 24 | ..... | 134 | 30 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 12 | | 13 | - | 11 | 7 | 23 | 5 | 4 | ..... | 2 | 15 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 13 | | 14 | - | 16 | 26 | - | 27 | 28 | 32 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 14 | | 15 | - | 14 | 27 | 5 | 135 | 3 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 15 | | 16 | - | 14 | 6 | 26 | ..... | 34 | 2 | ..... | 12 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 16 | | 17 | - | 19 | 32 | 33 | - | 34 | 5 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 17 | | 18 | - | 17 | 30 | 32 | ..... | 36 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 18 | | 19 | - | 17 | 33 | 29 | ..... | 7 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 19 | | 20 | - | 22 | 8 | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | ..... | 20 | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|-----|-----|-----| | 21 | -20 | 9 | -8 | | | | | | | | | | | | | | | | | | | 22 | 20 | 10 | 8 | | | | | | | | | | | | | | | | | | | 23 | -24 | 11 | 13 | -12 | 8 | 10 | -9 | 5 | 7 | -6 | 2 | 4 | | | | | | | | | | 24 | -23 | 12 | -11 | 9 | -8 | 6 | -5 | 3 | -2 | | | | | | | | | | | | | 25 | 23 | 13 | 11 | 10 | 8 | 7 | 5 | 4 | | | | | | | | | | | | | | 26 | -27 | 14 | 16 | -15 | 8 | 9 | 10 | -5 | 6 | -7 | 2 | | | | | | | | | | | 27 | -26 | 15 | -14 | 9 | 8 | 7 | 5 | | | | | | | | | | | | | | | 28 | 26 | 16 | 14 | 10 | 8 | 6 | | -5 | | | | | | | | | | | | | | 29 | -30 | 31 | 17 | 19 | -18 | 11 | 13 | -12 | | | | | | | | | | | | | | 30 | -29 | 18 | -17 | 12 | -11 | | | | | | | | | | | | | | | | | 31 | 29 | 19 | 17 | 13 | 11 | | | | | | | | | | | | | | | | | 32 | -33 | 17 | 18 | -19 | 14 | -15 | 16 | | | | | | | | | | | | | | | 33 | -32 | 19 | 17 | 15 | | -14 | | | | | | | | | | | | | | | | 34 | 32 | 18 | 17 | 16 | 14 | | | | | | | | | | | | | | | | | 35 | -36 | 37 | 17 | | 19 | -18 | | | | | | | | | | | | | | | | 36 | -35 | | | | 18 | -17 | | | | | | | | | | | | | | | | 37 | 35 | | | | 19 | 17 | | | | | | | | | | | | | | | | 38 | 20 | 22 | -21 | | | | | | | | | | | | | | | | | | | 39 | 21 | -20 | | | | | | | | | | | | | | | | | | | | 40 | 22 | 20 | | | | | | | | | | | | | | | | | | | | 41 | 23 | 25 | -24 | | | | | | | | | | | | | | | | | | | 42 | 24 | -23 | 21 | | | | | | | | | | | | | | | | | | | 43 | 25 | 23 | 22 | | | | | | | | | | | | | | | | | | | 44 | 26 | 28 | -27 | 20 | | | | | | | | | | | | | | | | | MDCCCXXXI. | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----| | 45 | ... | 27 | -26 | ... | 21 | ... | ... | 15 | -14 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 45 | | 46 | ... | 28 | ... | 26 | ... | 22 | ... | 16 | ... | 14 | ... | ... | ... | ... | ... | ... | ... | ... | 46 | | 47 | ... | 29 | ... | 31 | -30 | ... | 23 | ... | 17 | ... | 19 | -18 | ... | ... | ... | ... | ... | ... | 47 | | 48 | ... | 30 | -29 | ... | 24 | ... | ... | 18 | -17 | ... | 12 | -11 | ... | ... | ... | ... | ... | ... | 48 | | 49 | ... | 31 | ... | 29 | ... | 25 | ... | 19 | ... | 17 | 13 | ... | ... | ... | ... | ... | ... | ... | 49 | | 50 | ... | 32 | ... | 34 | -33 | ... | 26 | ... | -17 | ... | 18 | -19 | ... | ... | ... | ... | ... | ... | 50 | | 51 | ... | 33 | -32 | ... | 27 | ... | ... | 19 | ... | 17 | ... | ... | ... | ... | ... | ... | ... | ... | 51 | | 52 | ... | 34 | ... | 32 | ... | 28 | ... | 18 | ... | -17 | ... | ... | ... | ... | ... | ... | ... | ... | 52 | | 53 | ... | 35 | ... | 37 | -36 | ... | 29 | ... | ... | ... | 17 | ... | 19 | ... | ... | ... | ... | ... | 53 | | 54 | ... | 36 | -35 | ... | 30 | ... | ... | ... | ... | ... | 18 | -17 | ... | ... | ... | ... | ... | ... | 54 | | 55 | ... | 37 | ... | 31 | ... | ... | ... | ... | ... | ... | 19 | ... | ... | ... | ... | ... | ... | ... | 55 | | 56 | ... | -35 | ... | 36 | -37 | ... | 32 | ... | ... | ... | ... | -17 | ... | ... | ... | ... | ... | ... | 56 | | 57 | ... | 37 | ... | 35 | ... | ... | ... | ... | ... | ... | ... | ... | 15 | ... | ... | ... | ... | ... | 57 | | 58 | ... | 36 | ... | 35 | ... | ... | ... | ... | ... | ... | ... | ... | ... | 18 | ... | ... | ... | ... | 58 | | 59 | ... | ... | ... | 35 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 17 | ... | ... | ... | 59 | | 60 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 18 | ... | ... | ... | 60 | | 61 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 19 | ... | ... | 61 | | 62 | ... | 64 | ... | 66 | ... | 68 | ... | 70 | ... | 72 | ... | ... | ... | ... | ... | ... | ... | ... | 62 | | 63 | ... | -63 | ... | -65 | ... | -67 | ... | -69 | ... | -71 | ... | ... | ... | ... | ... | ... | ... | ... | 63 | | 64 | ... | 67 | ... | 65 | ... | 73 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 64 | | 65 | ... | -62 | ... | 69 | ... | -66 | ... | 75 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 65 | | 66 | ... | 62 | ... | 68 | ... | 66 | ... | 74 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 66 | | 67 | ... | -67 | ... | 70 | ... | -65 | ... | 76 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 67 | | 68 | ... | -66 | ... | 63 | ... | -62 | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | 68 | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----| | 69 | | | | | | | | | | | | | | | | | | | | | | 70 | | | | | | | | | | | | | | | | | | | | | | 71 | -75 | | | | | | | | | | | | | | | | | | | | | 72 | -73 | | | | | | | | | | | | | | | | | | | | | 73 | -72 | | | | | | | | | | | | | | | | | | | | | 74 | -71 | | | | | | | | | | | | | | | | | | | | | 75 | 71 | | | | | | | | | | | | | | | | | | | | | 76 | 72 | | | | | | | | | | | | | | | | | | | | | 77 | 65 | 69 | | | | | | | | | | | | | | | | | | | | 78 | 66 | 70 | -67 | | | | | | | | | | | | | | | | | | | 79 | 67 | -66 | | | | | | | | | | | | | | | | | | | | 80 | 68 | | | | | | | | | | | | | | | | | | | | | 81 | 69 | | | | | | | | | | | | | | | | | | | | | 82 | 70 | | | | | | | | | | | | | | | | | | | | | 83 | 71 | | | | | | | | | | | | | | | | | | | | | 84 | 72 | | | | | | | | | | | | | | | | | | | | | 85 | 73 | | | | | | | | | | | | | | | | | | | | | 86 | 74 | | | | | | | | | | | | | | | | | | | | | 87 | 75 | | | | | | | | | | | | | | | | | | | | | 88 | 76 | | | | | | | | | | | | | | | | | | | | | 89 | -72 | | | | | | | | | | | | | | | | | | | | | 90 | -71 | | | | | | | | | | | | | | | | | | | | | 91 | 75 | | | | | | | | | | | | | | | | | | | | | 92 | 76 | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----| | 93 | | 73 | | | | | | | | | | | | | | | | | | | | 94 | | 74 | | | | | | | | | | | | | | | | | | | | 95 | | | 71 | | | | | | | | | | | | | | | | | | | 96 | | | | 72 | | | | | | | | | | | | | | | | | | 97 | | | | | 73 | | | | | | | | | | | | | | | | | 98 | | | | | | 74 | | | | | | | | | | | | | | | | 99 | | | | | | | 75 | | | | | | | | | | | | | | |100 | | | | | | | | 76 | | | | | | | | | | | | | |101 | | | | | | | | | | | | | | | | | | | | | |102 | | | | | | | | | | | | | | | | | | | | | |103 | | | | | | | | | | | | | | | | | | | | | |104 | | | | | | | | | | | | | | | | | | | | | |105 | | | | | | | | | | | | | | | | | | | | | |106 | | | | | | | | | | | | | | | | | | | | | |107 | | | | | | | | | | | | | | | | | | | | | |108 | | | | | | | | | | | | | | | | | | | | | |109 | | | | | | | | | | | | | | | | | | | | | |110 | | | | | | | | | | | | | | | | | | | | | |111 | | | | | | | | | | | | | | | | | | | | | |112 | | | | | | | | | | | | | | | | | | | | | |113 | | | | | | | | | | | | | | | | | | | | | |114 | | | | | | | | | | | | | | | | | | | | | |115 | | | | | | | | | | | | | | | | | | | | | |116 | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 20 | 35 | 62 | 101 | 146 | 147 | |---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----|----|----| |117| 102| 101| 116| | | | | | | | | | | | | | | | | | |118| 103| 116| 101| | | | | | | | | | | | | | | | | | |119| 104| | 116| 101| | | | | | | | | | | | | | | | | |120| 105| | 116| 101| | | | | | | | | | | | | | | | | |121| | 102| | | | | | | | | | | | | | | | | | | |122| | 117| | | | | | | | | | | | | | | | | | | |123| | 103| | | | | | | | | | | | | | | | | | | |124| | 116| | | | | | | | | | | | | | | | | | | |125| | 101| | | | | | | | | | | | | | | | | | | |126| | 116| | | | | | | | | | | | | | | | | | | |127| | 116| | | | | | | | | | | | | | | | | | | |128| | 116| | | | | | | | | | | | | | | | | | | |129| | 116| | | | | | | | | | | | | | | | | | | |130| | 116| | | | | | | | | | | | | | | | | | | |131| | 132| | | | | | | | | | | | | | | | | | | |132| | 133| | | | | | | | | | | | | | | | | | | |133| | 134| | | | | | | | | | | | | | | | | | | |134| | 135| | | | | | | | | | | | | | | | | | | |135| | 136| | | | | | | | | | | | | | | | | | | |136| | 137| | | | | | | | | | | | | | | | | | | |137| | 138| | | | | | | | | | | | | | | | | | | |138| | 139| | | | | | | | | | | | | | | | | | | |139| | 140| | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 62 | 101 | 146 | 147 | |---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----| |141| 16| 134| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |142| 18| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |143| 19| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |144| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |145| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |146| -147| 150| 152| 154| 156| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |147| -146| 153| 149| ...| 157| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |148| ...| 152| 150| ...| 158| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |149| -152| -146| 147| -148| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |150| -151| 146| 148| ...| -147| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |151| -150| 147| -146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |152| -149| 148| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |153| ...| 149| 147| ...| -146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |154| ...| 150| 148| ...| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |155| -159| -158| ...| ...| -146| 147| -148| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |156| -160| -157| ...| ...| 146| 148| -147| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |157| ...| -156| ...| ...| 147| -146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |158| ...| -155| ...| ...| 148| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |159| ...| 155| ...| ...| 147| ...| -146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |160| ...| 156| ...| ...| 148| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |161| ...| 149| 153| -152| ...| -146| 147| -148| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |162| ...| 150| 154| -151| ...| 146| ...| 148| -147| ...| ...| ...| ...| ...| ...| ...| ...| ...| |163| ...| 151| -150| ...| ...| 147| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| |164| ...| 152| -149| ...| ...| 147| 146| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| ...| | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 14 | 17 | 62 | 101 | 146 | 147 | |---|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----|----|----| |165|...|153|...|149|...|...|...|...|-146|...|...|...|...|...|...|...|...|...| |166|...|154|...|150|...|...|...|148|146|...|...|...|...|...|...|...|...|...| |167|...|155|...|159|-158|...|...|...|-146|147|...|...|...|...|...|...|...|...| |168|...|156|...|160|-157|...|...|...|...|146|148|...|...|...|...|...|...|...| |169|...|157|-156|...|151|...|...|...|...|147|-146|...|...|...|...|...|...|...| |170|...|158|-155|...|152|...|...|...|...|148|146|...|...|...|...|...|...|...| |171|...|159|...|...|153|...|...|...|...|147|...|...|...|...|...|...|...|...| |172|...|160|...|156|154|...|...|...|...|148|...|...|...|...|...|...|...|...| |173|...|-156|157|-160|149|...|...|...|...|-146|...|...|...|...|...|...|...|...| |174|...|-155|158|-159|150|...|...|...|...|146|...|...|...|...|...|...|...|...| |175|...|155|...|151|...|...|...|...|147|...|...|...|...|...|...|...|...|...| |176|...|156|...|152|...|...|...|...|148|...|...|...|...|...|...|...|...|...| |177|...|157|...|-156|153|...|...|...|...|147|...|...|...|...|...|...|...|...| |178|...|158|...|-155|154|...|...|...|...|148|...|...|...|...|...|...|...|...| |179|...|...|...|155|...|...|...|...|-146|...|...|...|...|...|...|...|...|...| |180|...|...|...|156|...|...|...|...|146|...|...|...|...|...|...|...|...|...| |181|...|...|...|157|...|...|...|...|147|...|...|...|...|...|...|...|...|...| |182|...|...|...|158|...|...|...|...|148|...|...|...|...|...|...|...|...|...| |183|...|...|...|159|...|...|...|...|147|...|...|...|...|...|...|...|...|...| |184|...|...|...|160|...|...|...|...|148|...|...|...|...|...|...|...|...|...| |185|...|...|...|...|...|...|...|...|...|147|-146|...|...|...|...|...|...|...| |186|...|...|...|...|...|...|...|...|...|146|148|...|...|...|...|...|...|...| |38|59|38|59| |---|---|---|---| |39|1|...|39|60|1|...|60| |40|1|...|40|61|1|...|61| Table 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 \( n t - n_1 t \) \[ x = c \lambda' - \varpi \quad \ldots \quad c n t - \varpi \\ z = c_1 \lambda' - \varpi_1 \quad \ldots \quad c_1 n t - \varpi_1 \\ \text{and } y = g \lambda' - \nu \quad \ldots \quad g n t - \nu \] The notation throughout is the same as that used Phil. Trans. 1830, p. 328, with the exception of the indices of the arguments. In the elliptic movement; \[ a^5 r^{-5} = 1 + 5 e^2 \left( 1 + \frac{21}{8} e^2 \right) + 5 e \left( 1 + \frac{27}{8} e^2 \right) \cos x + 10 e^2 \left( 1 + \frac{31}{12} e^2 \right) \cos 2 x \\ + \frac{145}{8} e^3 \cos 3 x + \frac{745}{48} e^4 \cos 4 x \] \[ a^4 r^{-4} = 1 + 3 e^2 + 4 e \cos x + 7 e^2 \cos 2 x \] \[ a^3 r^{-3} = 1 + \frac{3}{2} e^2 \left( 1 + \frac{5}{4} e^2 \right) + 3 e \left( 1 + \frac{9}{8} e^2 \right) \cos x + \frac{9}{2} e^2 \left( 1 + \frac{7}{9} e^2 \right) \cos 2 x \\ + \frac{53}{8} e^3 \cos 3 x + \frac{77}{8} e^4 \cos 4 x \] \[ a^2 r^{-2} = 1 + \frac{e^2}{2} \left( 1 + \frac{3}{4} e^2 \right) + 2 e \left( 1 + \frac{3}{8} e^2 \right) \cos x + \frac{5}{2} e^2 \left( 1 + \frac{2}{15} e^2 \right) \cos 2 x \\ + \frac{13}{4} e^3 \cos 3 x + \frac{103}{24} e^4 \cos 4 x \] \[ a r^{-1} = 1 + e \left( 1 - \frac{e^2}{8} \right) \cos x + e^2 \left( 1 - \frac{e^2}{3} \right) \cos 2 x + \frac{9}{8} e^3 \cos 3 x + \frac{4}{3} e^4 \cos 4 x \] \[ \frac{r}{a} = 1 + \frac{e^2}{2} - e \left( 1 - \frac{3 e^2}{8} \right) \cos x - \frac{e^2}{2} \left( 1 - \frac{2 e^2}{3} \right) \cos 2 x - \frac{3 e^3}{8} \cos 3 x - \frac{e^4}{3} \cos 4 x \] \[ \frac{r^2}{a^2} = 1 + \frac{3 e^2}{2} - 2 e \left( 1 - \frac{e^2}{8} \right) \cos x - \frac{e^2}{2} \left( 1 - \frac{e^2}{3} \right) \cos 2 x - \frac{e^3}{4} \cos 3 x - \frac{e^4}{6} \cos 4 x \] \[ \frac{r^3}{a^3} = 1 + 3 e^2 \left( 1 + \frac{e^2}{8} \right) - 3 e \left( 1 + \frac{3}{8} e^2 \right) \cos x - \frac{5}{8} e^4 \cos 2 x + \frac{e^3}{8} \cos 3 x + \frac{e^4}{8} \cos 4 x \] \[ \frac{r^4}{a^4} = 1 + 5 e^2 - 4 e \cos x + e^2 \cos 2 x \] \[ \frac{a}{r} = r_0 \\ + r_1 \cos 2 t \\ + e r_2 \cos x \\ + e r_3 \cos (2 t - x) \\ + e r_4 \cos (2 t + x) \\ + e r_5 \cos z \\ + e r_6 \cos (2 t - z) + &c. &c. \] λ = n t + λ₁ cos 2 t + e λ₂ cos x + e λ₃ cos (2 t - x) + e λ₄ cos (2 t + x) + e λ₅ cos z &c. &c. The quantities λ correspond to the quantities b in M. Damoiseau's notation. s = γ s₁₄₆ sin y + γ s₁₄₇ sin (2 t - y) + γ s₁₄₈ sin (2 t + y) + eγ s₁₄₉ sin (x - y) &c. &c. γ = tan t \[ R = m_i \left\{ \frac{r^3 r_i \cos (\lambda - \lambda_i)}{r_i^3} - \frac{1}{\{r^2 - 2 r^3 r_i \cos (\lambda' - \lambda_i) + r_i^2\}^{3/2}} \right\} \] \[ = m_i \left\{ -\frac{1}{r_i} + \frac{r^2}{2 r_i^3} - \frac{3}{8} \left\{ 2 r^3 r_i \cos (\lambda' - \lambda_i) - r_i^2 \right\}^2 - \frac{15}{48} \left\{ 2 r^3 r_i \cos (\lambda' - \lambda_i) - r_i^2 \right\}^3 \right\} \] \[ = m_i \left\{ -\frac{1}{r_i} + \frac{r^2}{2 r_i^3} - \frac{3}{2} \frac{r^3 r_i^2}{r_i^3} \cos (\lambda' - \lambda_i)^2 + \frac{3}{2} \frac{r^3 r_i^2}{r_i^5} \cos (\lambda' - \lambda_i) - \frac{5}{2} \frac{r^3 r_i^2}{r_i^7} \cos (\lambda' - \lambda_i)^3 \right\} \] \[ = m_i \left\{ -\frac{1}{r_i} - \frac{r^2}{4 r_i^3} \left\{ 1 + 3 \cos (2 \lambda' - 2 \lambda_i) - 2 s^2 \right\} \right\} \] \[ - \frac{r^3}{8 r_i^4} \left\{ 3 (1 - 4 s^2) \cos (\lambda' - \lambda_i) + 5 \cos (3 \lambda' - 3 \lambda_i) \right\} \] \[ r^3 r_i \cos (\lambda' - \lambda_i) = rr_i \left\{ \cos^2 \frac{\lambda'}{2} \cos (\lambda - \lambda_i) + \sin^2 \frac{\lambda'}{2} \cos (\lambda + \lambda_i - 2 \nu) \right\} \] \[ = * a a_i \cos^2 \frac{\lambda'}{2} \left( 1 - \frac{e^2}{2} - \frac{e^4}{64} \right) \left( 1 - \frac{e^2}{2} - \frac{e^4}{64} \right) \cos t - \frac{3}{2} e \left( 1 - \frac{e^2}{2} \right) \cos (t - x) \] \[ + \frac{e}{2} \left( 1 - \frac{3}{4} e^2 \right) \left( 1 - \frac{e^2}{2} \right) \cos (t + x) + \frac{3}{8} e^2 \left( 1 - e^2 \right) \left( 1 - \frac{e^2}{2} \right) \cos (t + 2 x) \] \[ + \frac{e^3}{3} \cos (t + 3 x) + \frac{125}{384} e^4 \cos (t + 4 x) + \frac{e^2}{8} \left( 1 + \frac{e^2}{3} \right) \left( 1 - \frac{e^2}{2} \right) \cos (t - 2 x) \] \[ + \frac{e^3}{24} \cos (t - 3 x) + \frac{3}{128} e^4 \cos (t - 4 x) - \frac{3}{2} e_i \left( 1 - \frac{e^2}{2} \right) \cos (t + z) \] \[ + \frac{9}{4} e e_i \cos (t - x + z) - \frac{3}{4} e e_i \left( 1 - \frac{3}{4} e^2 \right) \cos (t + x + z) \] \[ - \frac{9}{16} e^2 e_i \cos (t + 2 x + z) - \frac{e^3 e_i}{2} \cos (t + 3 x + z) - \frac{3}{16} e^2 e_i \cos (t - 2 x + z) \] * See Phil. Trans. 1830, p. 343. \[ -\frac{e^3}{16} e_i \cos(t - 3x + z) + \frac{e_i}{2} \left(1 - \frac{3}{4} e_i^2\right) \left(1 - \frac{e^2}{2}\right) \cos(t - z) \] \[ - \frac{3}{4} e e_i \left(1 - \frac{3}{4} e_i^2\right) \cos(t - x - z) \] \[ + \frac{e e_i}{4} \left(1 - \frac{3}{4} e^2\right) \left(1 - \frac{3}{4} e_i^2\right) \cos(t + x - z) + \frac{3}{16} e^2 e_i \cos(t + 2x - z) \] \[ + \frac{e^3}{6} e_i \cos(t + 3x - z) + \frac{e^2}{16} e_i \cos(t - 2x - z) + \frac{e^3}{48} e_i \cos(t - 3x - z) \] \[ + \frac{3}{8} e_i^2 \left(1 - e_i^2\right) \left(1 - \frac{e^2}{2}\right) \cos(t - 2z) - \frac{9}{16} e e_i^2 \cos(t - x - 2z) \] \[ + \frac{3}{16} e e_i^2 \cos(t + x - 2z) + \frac{9}{64} e^2 e_i^2 \cos(t + 2x - 2z) \] \[ + \frac{3}{64} e^2 e_i^2 \cos(t - 2x - 2z) + \frac{e^3}{3} \cos(t - 3z) - \frac{e e_i^3}{2} \cos(t - x - 3z) \] \[ + \frac{e e_i^3}{6} \cos(t + x - 3z) + \frac{125}{384} e_i^4 \cos(t - 4z) \] \[ + \frac{e_i^3}{8} \left(1 + \frac{e^2}{3}\right) \left(1 - \frac{e^2}{2}\right) \cos(t + 2z) - \frac{3}{16} e e_i^2 \cos(t - x + 2z) \] \[ + \frac{e e_i^2}{16} \cos(t + x + 2z) + \frac{3}{64} e^2 e_i^2 \cos(t + 2x + 2z) + \frac{e^2 e_i^2}{64} \cos(t - 2x + 2z) \] \[ + \frac{e_i^3}{24} \cos(t + 3z) - \frac{e e_i^3}{16} \cos(t - x + 3z) + \frac{e e_i^3}{48} \cos(t + x + 3z) \] \[ + \frac{3}{128} e_i^4 \cos(t + 4z) \] \[+ a a_i \sin^2 \frac{t}{2} \left\{ \left(1 - \frac{e^2 + e_i^2}{2}\right) \cos(t - 2y) - \frac{3}{2} e \cos(t + x - 2y) + \frac{e}{2} \cos(t - x - 2y)\right.\] \[ + \frac{3}{8} e^2 \cos(t - 2x - 2y) + \frac{e^2}{8} \cos(t + 2x - 2y) - \frac{3}{2} e_i \cos(t + z - 2y) \] \[ + \frac{9}{4} e e_i \cos(t + x + z - 2y) - \frac{3}{4} e e_i \cos(t - x + z - 2y) \] \[ + \frac{e_i}{2} \cos(t - z - 2y) - \frac{3}{4} e e_i \cos(t + x - z - 2y) + \frac{e e_i}{4} \cos(t - x - z - 2y) \] \[r^2 r_i^2 \cos(\lambda' - \lambda)^2 = a^2 a_i^2 \cos^4 \frac{t}{2} \left\{ \frac{1}{2} + \left(-\frac{1}{2} + \frac{9}{8} + \frac{1}{8}\right)(e^2 + e_i^2) + \left(\frac{1}{2} - \frac{9}{8} - \frac{1}{8} - \frac{9}{8} + \frac{81}{32} + \frac{9}{32}\right)\right.\] \[ - \frac{1}{8} + \frac{9}{32} + \frac{1}{32}\right\} e^2 e_i^2 + \left\{\frac{7}{64} - \frac{3}{16} + \frac{9}{128} + \frac{1}{128}\right\}(e^4 + e_i^4)\] [0] \[ + \left\{ \frac{1}{2} + \left( -\frac{1}{2} - \frac{3}{4} \right) (e^2 + e_i^2) + \left( \frac{1}{2} + \frac{3}{4} + \frac{3}{4} + \frac{9}{16} + \frac{9}{16} \right) e^2 e_i^2 \right\} \cos 2t \] \[ + \left\{ \frac{7}{64} + \frac{9}{16} + \frac{3}{64} \right\} (e^4 + e_i^4) \] \[ + \left\{ -\frac{3}{2} + \frac{1}{2} + \left( \frac{3}{4} - \frac{5}{8} - \frac{3}{16} + \frac{3}{16} \right) e^2 \right\} e \cos x \] \[ + \left\{ \frac{3}{2} - \frac{1}{2} - \frac{27}{8} + \frac{9}{8} - \frac{3}{8} + \frac{1}{8} \right\} e_i^2 \] \[ + \left\{ -\frac{3}{2} + \left( \frac{3}{4} + \frac{1}{16} \right) e^2 + \left( \frac{3}{2} + \frac{9}{8} + \frac{9}{8} \right) e_i^2 \right\} e \cos (2t - x) \] \[ + \left\{ -\frac{1}{2} + \left( \frac{5}{8} - \frac{9}{16} \right) e^2 + \left( -\frac{1}{2} - \frac{3}{8} - \frac{3}{8} \right) e_i^2 \right\} e \cos (2t + x) \] \[ + \left\{ \frac{3}{8} + \frac{1}{8} - \frac{3}{4} + \left( -\frac{9}{16} - \frac{1}{48} + \frac{9}{16} - \frac{1}{16} + \frac{1}{6} \right) e^2 \right\} e^2 \cos 2x \] \[ + \left\{ -\frac{3}{8} - \frac{1}{8} + \frac{3}{4} + \frac{27}{32} + \frac{9}{32} - \frac{27}{16} + \frac{3}{32} + \frac{1}{32} - \frac{3}{16} \right\} e_i^2 \] \[ + \left\{ \frac{9}{8} + \frac{1}{8} + \left( -\frac{1}{48} + \frac{1}{48} \right) e^2 \right\} e^2 \cos (2t - 2x) \] \[ + \left\{ \frac{1}{8} + \frac{3}{8} + \left( -\frac{3}{16} - \frac{9}{16} - \frac{1}{2} \right) e^2 \right\} e^2 \cos (2t + 2x) \] \[ + \left\{ -\frac{3}{4} - \frac{3}{4} + \frac{9}{4} + \frac{1}{4} + \left( \frac{15}{16} + \frac{3}{8} - \frac{9}{8} - \frac{3}{32} - \frac{9}{32} - \frac{5}{16} \right) \right\} e e_i \cos (x + z) \] * 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. \[ \begin{align*} &+ \left\{ -\frac{3}{4} - \frac{3}{4} + \left\{ \frac{3}{8} + \frac{3}{8} + \frac{1}{32} + \frac{1}{32} \right\} e^2 \right. \\ &\quad + \left\{ \frac{15}{16} + \frac{15}{16} + \frac{27}{32} + \frac{27}{32} \right\} e^2 e_i \cos (2t - x - z) \\ &\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. \\ &\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. \\ &\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. \\ &\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. \\ &\left. \quad + \left\{ \frac{1}{3} + \frac{1}{24} - \frac{9}{16} + \frac{1}{16} \right\} e \cos 3x \right. \\ &\left. \quad + \left\{ \frac{1}{24} - \frac{3}{16} \right\} e^3 \cos (2t - 3x) \right. \\ &\left. \quad + \left\{ \frac{1}{3} + \frac{3}{16} \right\} e^3 \cos (2t + 3x) \right. \\ &\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. \\ &\left. \quad + \left\{ \frac{1}{16} + \frac{9}{8} + \frac{1}{16} \right\} e^2 e_i \cos (2t - 2x - z) \right. \\ &\left. \quad + \left\{ -\frac{9}{16} - \frac{3}{8} - \frac{9}{16} \right\} e^2 e_i \cos (2t + 2x + z) \right. \\ &\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. \\ &\left. \quad + \left\{ -\frac{3}{16} - \frac{27}{8} - \frac{3}{16} \right\} e^2 e_i \cos (2t - 2x + z) \right. \\ &\left. \quad + \left\{ \frac{3}{16} + \frac{1}{8} + \frac{3}{16} \right\} e^2 e_i \cos (2t + 2x - z) \right. \\ &\left. \quad + \left\{ \frac{125}{384} + \frac{3}{128} - \frac{1}{2} + \frac{1}{48} + \frac{3}{64} \right\} e^4 \cos 4x \right. \end{align*} \] \[ + \left\{ \frac{1}{128} + \frac{3}{128} - \frac{1}{16} \right\} e^4 \cos (2t - 4x) \] \[ + \left\{ \frac{9}{128} + \frac{125}{384} + \frac{1}{6} \right\} e^4 \cos (2t + 4x) \] \[ + \left\{ -\frac{1}{2} + \frac{1}{48} + \frac{27}{32} + \frac{1}{32} - \frac{9}{32} + \frac{1}{6} - \frac{3}{32} - \frac{1}{16} \right\} e^3 \cos (3x + z) \] \[ + \left\{ \frac{1}{48} - \frac{3}{32} - \frac{3}{32} + \frac{1}{48} \right\} e^3 e_i \cos (2t - 3x - z) \] \[ + \left\{ -\frac{1}{2} - \frac{9}{32} - \frac{9}{32} - \frac{1}{2} \right\} e^3 e_i \cos (2t + 3x + z) \] \[ + \left\{ -\frac{1}{16} + \frac{1}{6} - \frac{9}{32} - \frac{3}{32} + \frac{27}{32} - \frac{1}{2} + \frac{1}{32} + \frac{1}{48} \right\} e^3 e_i \cos (3x - z) \] \[ + \left\{ -\frac{1}{16} + \frac{9}{32} + \frac{9}{32} - \frac{1}{16} \right\} e^3 e_i \cos (2t - 3x + z) \] \[ + \left\{ \frac{1}{6} + \frac{3}{32} + \frac{3}{32} + \frac{1}{6} \right\} e^3 e_i \cos (2t + 3x - z) \] \[ + \left\{ \frac{3}{64} + \frac{3}{64} - \frac{3}{32} - \frac{9}{32} + \frac{9}{64} + \frac{1}{64} - \frac{3}{32} + \frac{9}{16} - \frac{9}{32} \right\} e^3 e_i^3 \cos (2x + 2z) \] \[ + \left\{ \frac{9}{32} + \frac{3}{64} + \frac{27}{32} + \frac{3}{64} + \frac{1}{32} \right\} e^3 e_i^3 \cos (2t - 2x + 2z) \] \[ + \left\{ \frac{9}{32} + \frac{3}{64} + \frac{1}{32} + \frac{3}{64} + \frac{27}{32} \right\} e^3 e_i^3 \cos (2t + 2x + 2z) \] \[ + \left\{ \frac{9}{64} + \frac{1}{64} - \frac{9}{32} - \frac{3}{32} + \frac{3}{64} + \frac{3}{64} - \frac{9}{32} + \frac{9}{16} - \frac{3}{32} \right\} e^3 e_i^3 \cos (2x - 2z) \] \[ + \left\{ \frac{81}{32} + \frac{1}{64} + \frac{9}{32} + \frac{1}{64} + \frac{9}{32} \right\} e^3 e_i^3 \cos (2t - 2x + 2z) \] \[ + \left\{ \frac{1}{32} + \frac{9}{64} + \frac{3}{32} + \frac{9}{64} + \frac{3}{32} \right\} e^3 e_i^3 \cos (2t + 2x - 2z) \] \[ + a^2 a_i^3 \sin^3 \frac{t}{2} \cos^2 \frac{t}{2} \left\{ 1 + \left\{ -1 - \frac{3}{4} - \frac{3}{4} \right\} e^3 + \left\{ -1 + \frac{9}{4} + \frac{1}{4} \right\} e_i^3 \right\} \cos 2y \] \[ + \left\{ 1 + \left\{ -1 + \frac{9}{4} + \frac{1}{4} \right\} e^3 + \left\{ -1 - \frac{3}{4} - \frac{3}{4} \right\} e_i^3 \right\} \cos (2t - 2y) \] \[ + \left\{ -\frac{3}{2} - \frac{3}{2} \right\} e \cos (x - 2y) + \left\{ \frac{1}{2} + \frac{1}{2} \right\} e \cos (x + 2y) \] \[ + \left\{ \frac{1}{2} - \frac{3}{2} \right\} e \cos (2t - x - 2y) \] \[ + \left\{ -\frac{3}{2} + \frac{1}{2} \right\} e \cos (2t + x - 2y) + \left\{ -\frac{3}{2} + \frac{1}{2} \right\} e_t \cos (z - 2y) \] \[ + \left\{ \frac{1}{2} - \frac{3}{2} \right\} e_t \cos (z + 2y) + \left\{ \frac{1}{2} + \frac{1}{2} \right\} e_t \cos (2t - z - 2y) \] \[ + \left\{ -\frac{3}{2} - \frac{3}{2} \right\} e_t \cos (2t + z - 2y) + \left\{ \frac{1}{8} + \frac{9}{4} + \frac{1}{8} \right\} e^2 \cos (2x - 2y) \] \[ + \left\{ \frac{3}{8} + \frac{1}{4} + \frac{3}{8} \right\} e^2 \cos (2x + 2y) \] \[ + \left\{ \frac{3}{8} - \frac{3}{4} + \frac{1}{8} \right\} e^2 \cos (2t - 2x - 2y) + \left\{ \frac{1}{8} - \frac{3}{4} + \frac{3}{8} \right\} e^2 \cos (2t + 2x - 2y) \] \[ + \left\{ \frac{9}{4} + \frac{9}{4} - \frac{3}{4} - \frac{3}{4} \right\} ee_t \cos (x + z - 2y) \] \[ + \left\{ \frac{1}{4} + \frac{1}{4} - \frac{3}{4} - \frac{3}{4} \right\} ee_t \cos (x + z + 2y) \] \[ + \left\{ \frac{1}{4} - \frac{3}{4} + \frac{1}{4} - \frac{3}{4} \right\} ee_t \cos (2t - x - z - 2y) \] \[ + \left\{ \frac{9}{4} - \frac{3}{4} + \frac{9}{4} - \frac{3}{4} \right\} ee_t \cos (2t + x + z + 2y) \] \[ + \left\{ -\frac{3}{4} - \frac{3}{4} + \frac{9}{4} + \frac{9}{4} \right\} ee_t \cos (x - z - 2y) \] \[ + \left\{ -\frac{3}{4} - \frac{3}{4} + \frac{1}{4} + \frac{1}{4} \right\} ee_t \cos (x - z + 2y) \] \[ + \left\{ -\frac{3}{4} + \frac{9}{4} - \frac{3}{4} + \frac{9}{4} \right\} ee_t \cos (2t - x + z - 2y) \] \[ + \left\{ -\frac{3}{4} + \frac{1}{4} - \frac{3}{4} + \frac{1}{4} \right\} ee_t \cos (2t + x - z - 2y) \] \[ r^2 r^2 \cos (\lambda' - \lambda)^2 = a^2 e_i^2 \cos^4 \frac{t}{2} \left\{ \frac{1}{2} + \frac{3}{4} (e^2 + e_i^2) + \frac{9}{8} e^2 e_i^2 + \left\{ \frac{1}{2} - \frac{5}{4} (e^2 + e_i^2) \right\} \right. \] \[ + \frac{23}{32} (e^4 + e_i^4) + \frac{25}{8} e^2 e_i^2 \} \cos 2t + \left\{ -1 + \frac{e^2}{8} - \frac{3}{2} e_i^2 \right\} e \cos x \] \[ + \left\{ -\frac{3}{2} + \frac{13}{16} e^2 + \frac{15}{4} e_i^2 \right\} e \cos (2t - x) + \left\{ \frac{1}{2} - \frac{19}{16} e^2 - \frac{5}{4} e_i^2 \right\} e \cos (2t + x) \] \[ + \left\{ -\frac{1}{4} + \frac{1}{12} e^2 - \frac{3}{8} e_i^2 \right\} e^2 \cos 2x \] \[ + \left\{ \frac{5}{4} - \frac{25}{8} e_i^2 \right\} e^2 \cos (2t - 2x) + \left\{ \frac{1}{2} - \frac{5}{4} e^2 - \frac{5}{4} e_i^2 \right\} e \cos (2t + 2x) \] \[ + \left\{ 1 - \frac{1}{8} (e^2 + e_i^2) \right\} ee_i \cos (x + z) \] \[ + \left\{ -\frac{3}{2} + \frac{13}{16} e^2 + \frac{57}{16} e_i^2 \right\} ee_i \cos (2t - x - z) \] \[ + \left\{ 1 - \frac{(e^2 + e_i^2)}{8} \right\} ee_i \cos (x - z) \] \[ + \left\{ \frac{9}{2} - \frac{39}{16} (e^2 + e_i^2) \right\} ee_i \cos (2t - x + z) \] \[ + \left\{ \frac{1}{2} - \frac{19}{16} (e^2 + e_i^2) \right\} ee_i \cos (2t + x - z) - \frac{e^3}{8} \cos 3x \] \[ - \frac{7}{48} e^3 \cos (2t - 3x) + \frac{25}{48} e^3 \cos (2t + 3x) + \frac{e^2 e_i}{4} \cos (2x + z) \] \[ \begin{align*} &+ \frac{5}{4} e^2 e_i \cos (2t - 2x - z) - \frac{25}{16} e^2 e_i \cos (2t + 2x + z) + \frac{e^2 e_i}{4} \cos (2x - z) \\ &[24] [31] & [25] [30] & [26] [32] \\ &- \frac{15}{4} e^2 e_i \cos (2t - 2x + z) + \frac{e^2 e_i}{2} \cos (2t + 2x - z) \\ &[27] [33] & [28] [34] \\ &- \frac{e^4}{12} \cos 4x - \frac{e^4}{32} \cos (2t - 4x) + \frac{9}{16} e^4 \cos (2t + 4x) + \frac{e^3 e_i}{8} \cos (3x + z) \\ &[38] [59] & [39] [61] & [40] [60] & [41] [53] \\ &- \frac{7}{48} e^3 e_i \cos (2t - 3x - z) - \frac{25}{16} e^3 e_i \cos (2t + 3x + z) \\ &[42] [55] & [43] [54] \\ &+ \frac{e^3 e_i}{8} \cos (3x - z) + \frac{7}{16} e^3 e_i \cos (2t - 3x + z) + \frac{25}{48} e^3 e_i \cos (2t + 3x - z) \\ &[44] [56] & [45] [57] & [46] [58] \\ &+ \frac{e^2 e_i}{16} \cos (2x + 2z) + \frac{5}{4} e^2 e_i \cos (2t - 2x - 2z) \\ &[47] & [48] \\ &+ \frac{5}{4} e^2 e_i \cos (2t + 2x + 2z) + \frac{e^2 e_i}{16} \cos (2x - 2z) \\ &[49] & [50] \\ &+ \frac{25}{8} e^2 e_i \cos (2t - 2x + 2z) + \frac{e^2 e_i}{2} \cos (2t + 2x - 2z) \\ &[51] & [52] \\ &+ a^3 a_i^3 \cos^3 \frac{\epsilon}{2} \sin^2 \frac{\epsilon}{2} \left\{ \left[ 1 - \frac{5}{2} e^2 + \frac{3}{2} e_i^2 \right] \cos 2y + \left[ 1 + \frac{3}{2} e^2 - \frac{5}{2} e_i^2 \right] \cos (2t - 2y) \right\} \\ &[62] & [63] \\ &- 3e \cos (x - 2y) + e \cos (x + 2y) - e \cos (2t - x - 2y) \\ &[65] & [66] & [67] \\ &- e \cos (2t + x - 2y) - e_i \cos (z - 2y) - e_i \cos (z + 2y) \\ &[69] & [71] & [72] \\ &+ e_i \cos (2t - z - 2y) - 3e_i \cos (2t + z - 2y) \\ &[73] & [75] \\ &+ \frac{5}{2} e^2 \cos (2x - 2y) + e^2 \cos (2x + 2y) \\ &[77] & [78] \\ &- \frac{e^2}{4} \cos (2t - 2x - 2y) - \frac{e^2}{4} \cos (2t + 2x - 2y) \\ &[79] & [81] \end{align*} \] \[ + 3 e e_i \cos (x + z - 2y) - ee_i \cos (x + z + 2y) \] \[ - ee_i \cos (2t - x - z - 2y) + 3 ee_i \cos (2t + x + z - 2y) \] \[ + 3 ee_i \cos (x - z - 2y) - ee_i \cos (x - z + 2y) \] \[ + 3 ee_i \cos (2t - x + z - 2y) - ee_i \cos (2t + x - z - 2y) \] \[ - \frac{3}{8} e_i^2 \cos (2z - 2y) - \frac{5}{8} e_i^2 \cos (2z + 2y) \] \[ + \frac{5}{8} e_i^2 \cos (2t - 2z - 2y) + \frac{19}{8} e_i^2 \cos (2t + 2z - 2y) \} \[ + a^2 a_i^2 \sin^4 \frac{t}{2} \left\{ \frac{1}{2} + \frac{1}{2} \cos (2t - 2y) \right\} \] \[ \frac{r^2}{2r_i^3} = \frac{a^2}{a_i^3} \left[ \frac{1}{2} + \frac{3}{4} e^2 + \frac{3}{4} e_i^2 + \frac{9}{8} e^2 e_i^2 + \frac{15}{16} e_i^4 - e \left\{ 1 - \frac{e^2}{8} + \frac{3}{2} e_i^2 \right\} \cos x \right] \] \[ + \frac{3}{2} e_i \left\{ 1 + \frac{3}{2} e^2 + \frac{9}{8} e_i^2 \right\} \cos z - \frac{e^2}{4} \left\{ 1 - \frac{e^2}{3} + \frac{3}{2} e_i^2 \right\} \cos 2x \] \[ - \frac{3}{2} ee_i \left\{ 1 - \frac{e^2}{8} + \frac{9}{8} e_i^2 \right\} \cos (x + z) \] \[ - \frac{3}{2} ee_i \left\{ 1 - \frac{e^2}{8} + \frac{9}{8} e_i^2 \right\} \cos (x - z) \] \[ + \frac{9}{4} e_i^2 \left\{ 1 + \frac{7}{9} e_i^2 + \frac{3}{2} e^2 \right\} \cos 2z - \frac{e^3}{8} \cos 3x - \frac{3}{8} e^3 e_i \cos (2x + z) \] \[ - \frac{3}{8} e^3 e_i \cos (2x - z) - \frac{9}{4} ee_i^3 \cos (x + 2z) - \frac{9}{4} ee_i^3 \cos (x - 2z) \] \[ + \frac{53}{16} e_i^3 \cos 3z - \frac{e^4}{12} \cos 4x - \frac{3}{16} e^3 e_i \cos (3x + z) \] \[ - \frac{3}{16} e^3 e_i \cos (3x - z) - \frac{9}{16} e^2 e_i^2 \cos (2x + 2z) - \frac{9}{16} e^2 e_i \cos (2x - 2z) \] \[ - \frac{53}{16} e e_i^3 \cos (x + 3z) - \frac{53}{16} e e_i^3 \cos (x - 3z) + \frac{77}{16} e^4 \cos 4z \] Terms in \( R \) multiplied by \(- \frac{3}{2} \cos^4 \frac{t}{2} \frac{a^2}{a_i^3}\) \[ = \left\{ \frac{1}{2} + \frac{3}{4} (e^2 + e_i^2) + \frac{9}{8} e^2 e_i^2 \right\} \left\{ 1 + 5 e_i^2 + \frac{105}{8} e_i^4 \right\} \] \[ + \left\{ -1 + \frac{e_i^2}{8} - \frac{3}{2} e \right\} \left\{ \frac{5}{2} e_i^2 + \frac{135}{16} e_i^4 \right\} - \frac{5}{4} e_i^4 \] \[ + \left\{ \frac{1}{2} - \frac{5}{4} (e^2 + e_i^2) + \frac{23}{32} (e^4 + e_i^4) + \frac{25}{8} e^2 e_i^2 \right\} \left\{ 1 + 5 e_i^2 + \frac{105}{8} e_i^4 \right\} \] \[ + \left\{ \frac{1}{2} - \frac{19}{16} e_i^2 - \frac{5}{4} e^2 - \frac{3}{2} + \frac{13}{16} e_i^2 + \frac{15}{4} e^2 \right\} \left\{ \frac{5}{2} e_i^2 + \frac{135}{16} e_i^4 \right\} \] \[ + \left\{ \frac{5}{2} + \frac{25}{4} \right\} e_i^4 \cos 2t \] \[ + \left\{ -1 + \frac{e_i^2}{8} - \frac{3}{2} e_i^2 - 5 e_i^2 + \frac{5}{2} e_i^2 + \frac{5}{2} e_i^2 \right\} e \cos x \] \[ + \left\{ -\frac{3}{2} + \frac{13}{16} e_i^2 + \frac{15}{4} e_i^2 - \frac{15}{2} e_i^2 - \frac{15}{4} e_i^2 + \frac{45}{4} e_i^2 \right\} e \cos (2t - x) \] \[ + \left\{ \frac{1}{2} - \frac{19}{16} e_i^2 - \frac{5}{4} e_i^2 + \frac{5}{2} e_i^2 + \frac{5}{4} e_i^2 - \frac{15}{4} e_i^2 \right\} e \cos (2t + x) \] \[ + \left\{ -1 + \frac{e_i^2}{8} - \frac{3}{2} e_i^2 - 5 e_i^2 + \frac{5}{2} + \frac{15}{4} e_i^2 + \frac{15}{4} e_i^2 + \frac{135}{16} e_i^2 - \frac{5}{8} e_i^2 - 5 e_i^2 \right\} e_i \cos z \] \[ + \left\{ \frac{1}{2} - \frac{19}{16} e_i^2 - \frac{5}{4} e_i^2 + \frac{5}{2} e_i^2 + \frac{5}{4} e_i^2 + \frac{5}{4} - \frac{25}{8} e_i^2 - \frac{25}{8} e_i^2 + \frac{135}{32} e_i^2 - \frac{15}{2} e_i^2 \right\} e_i \cos (2t - z) \] * 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. \[ + \left\{ -\frac{3}{2} + \frac{13}{16} e_i^2 + \frac{15}{4} e_i^3 - \frac{15}{2} e_i^4 + \frac{5}{4} - \frac{25}{8} e_i^2 + \frac{135}{32} e_i^3 + \frac{25}{8} e_i^4 + \frac{5}{2} e_i^5 \right\} e_i \cos(2t + z) \] \[ + \left\{ -\frac{1}{4} + \frac{e_i^2}{12} - \frac{3}{8} e_i^3 - \frac{5}{4} e_i^4 + \frac{5}{8} e_i^5 + \frac{5}{8} e_i^6 \right\} e_i^2 \cos 2x \] \[ + \left\{ \frac{5}{4} - \frac{25}{8} e_i^2 + \frac{25}{4} e_i^3 + \frac{25}{8} e_i^4 - \frac{75}{8} e_i^5 \right\} e_i^3 \cos (2t - 2x) \] \[ + \left\{ \frac{1}{2} - \frac{5}{4} e_i^2 - \frac{5}{4} e_i^3 + \frac{5}{2} e_i^4 + \frac{5}{4} e_i^5 - \frac{15}{4} e_i^6 \right\} e_i^2 \cos (2t + 2x) \] \[ + \left\{ 1 - \frac{e_i^2}{8} - \frac{e_i^3}{8} + 5 e_i^4 - \frac{5}{2} + \frac{5}{16} - \frac{15}{4} e_i^2 - \frac{135}{16} e_i^3 + \frac{5}{8} e_i^4 + 5 e_i^5 \right\} e_i e_i \cos (x + z) \] \[ + \left\{ -\frac{3}{2} + \frac{13}{16} e_i^2 + \frac{57}{16} e_i^3 - \frac{15}{2} e_i^4 - \frac{15}{4} e_i^5 - \frac{15}{4} + \frac{65}{32} e_i^2 + \frac{75}{8} e_i^3 \right. \] \[ - \frac{405}{32} e_i^4 + \frac{45}{2} e_i^5 \right\} e_i e_i \cos (2t - x - z) + \left\{ -\frac{3}{2} + \frac{13}{16} e_i^2 + \frac{57}{16} e_i^3 - \frac{15}{2} e_i^4 + \frac{5}{4} \right. \] \[ - \frac{95}{32} e_i^2 - \frac{25}{8} e_i^3 + \frac{135}{32} e_i^4 + \frac{25}{8} e_i^5 + \frac{5}{2} e_i^6 \right\} e_i e_i \cos (2t + x + z) \] \[ + \left\{ 1 - \frac{e_i^2}{8} - \frac{e_i^3}{8} + 5 e_i^4 - \frac{5}{2} + \frac{5}{16} e_i^2 - \frac{15}{4} e_i^3 - \frac{135}{16} e_i^4 + 5 e_i^5 \right\} e_i e_i \cos (x - z) \] \[ + \left\{ \frac{9}{2} - \frac{39}{16} e_i^2 - \frac{39}{16} e_i^3 + \frac{45}{2} e_i^4 - \frac{15}{4} + \frac{65}{32} e_i^2 + \frac{75}{8} e_i^3 - \frac{405}{32} e_i^4 \right. \] \[ - \frac{75}{8} e_i^5 - \frac{15}{2} e_i^6 \right\} e_i e_i \cos (2t - x + z) \] \[ + \left\{ \frac{1}{2} - \frac{19}{16} e_i^2 - \frac{19}{16} e_i^3 + \frac{5}{2} e_i^4 + \frac{5}{4} e_i^5 + \frac{5}{4} - \frac{95}{32} e_i^2 - \frac{25}{8} e_i^3 + \frac{135}{32} e_i^4 - \frac{15}{2} e_i^5 \right\} e_i e_i \cos (2t + x - z) \] \[ + \left\{ -\frac{1}{4} + \frac{e_i^2}{12} - \frac{3}{8} e_i^3 - \frac{5}{4} e_i^4 - \frac{5}{2} + \frac{5}{16} e_i^2 - \frac{15}{4} e_i^3 - \frac{135}{16} e_i^4 - \frac{5}{16} e_i^5 + 5 \right. \] \[ + \frac{15}{2} e_i^2 + \frac{15}{2} e_i^3 + \frac{155}{12} e_i^4 - \frac{145}{16} e_i^5 \right\} e_i^3 \cos 2z \] \[ + \left\{ \frac{1}{2} - \frac{5}{4} e_i^2 - \frac{5}{4} e_i^3 + \frac{5}{2} e_i^4 + \frac{125}{96} e_i^5 + \frac{5}{4} - \frac{95}{32} e_i^2 - \frac{25}{8} e_i^3 + \frac{135}{32} e_i^4 + \frac{5}{2} - \frac{25}{4} e_i^2 - \frac{25}{4} e_i^3 \right. \] \[ + \frac{155}{24} e_i^4 - \frac{435}{32} e_i^5 \right\} e_i^3 \cos (2t - 2z) \] \[ \frac{5}{4} - \frac{25}{8} e^3 + \frac{25}{4} e_i^2 - \frac{15}{4} + \frac{65}{32} e_i^3 + \frac{75}{8} e^2 - \frac{405}{32} e_i^2 - \frac{35}{96} e_i^2 + \frac{5}{2} - \frac{25}{4} e^2 - \frac{25}{4} e_i^2 \] \[ + \frac{155}{24} e_i^2 + \frac{145}{32} e_i^2 \] \( e_i^2 \cos (2t + 2z) \) \[ [19] \] \[ - \frac{e^3}{8} \cos 3x - \frac{7}{48} e^3 \cos (2t - 3x) + \frac{25}{48} e^3 \cos (2t + 3x) + \left\{ \frac{1}{4} - \frac{5}{8} \right\} e^3 e_i \cos (2x + z) \] \[ [20] \quad [21] \quad [22] \quad [23] \] \[ + \left\{ \frac{5}{4} + \frac{25}{8} \right\} e^3 e_i \cos (2t - 2x - z) - \left\{ -\frac{3}{2} + \frac{5}{4} \right\} e^3 e_i \cos (2t + 2x + z) \] \[ [24] \quad [25] \] \[ + \left\{ \frac{1}{4} - \frac{5}{8} \right\} e^3 e_i \cos (2x - z) + \left\{ -\frac{15}{4} + \frac{25}{8} \right\} e^3 e_i \cos (2t - 2x + z) \] \[ [26] \quad [27] \] \[ + \left\{ \frac{1}{2} + \frac{5}{4} \right\} e^3 e_i \cos (2t + 2x - z) + \left\{ \frac{1}{4} + \frac{5}{2} - 5 \right\} e^3 e_i \cos (x + 2z) \] \[ [28] \quad [29] \] \[ + \left\{ -\frac{3}{2} - \frac{15}{4} - \frac{15}{2} \right\} e^3 e_i \cos (2t - x - 2z) \] \[ [30] \] \[ + \left\{ \frac{5}{4} - \frac{15}{4} + \frac{5}{2} \right\} e^3 e_i \cos (2t + x + 2z) + \left\{ \frac{1}{4} + \frac{5}{2} - 5 \right\} e^3 e_i \cos (x - 2z) \] \[ [31] \quad [32] \] \[ + \left\{ -\frac{15}{4} + \frac{45}{4} - \frac{15}{2} \right\} e^3 e_i \cos (2t - x + 2z) + \left\{ \frac{1}{2} + \frac{5}{4} + \frac{5}{2} \right\} e^3 e_i \cos (2t + x - 2z) \] \[ [33] \quad [34] \] \[ + \left\{ -\frac{1}{8} - \frac{5}{8} - 5 + \frac{145}{16} \right\} e^3 \cos 3z + \left\{ \frac{25}{48} + \frac{5}{4} + \frac{5}{2} + \frac{145}{32} \right\} e^3 \cos (2t - 3z) \] \[ [35] \quad [36] \] \[ + \left\{ -\frac{7}{48} + \frac{25}{8} - \frac{15}{2} + \frac{145}{32} \right\} e^3 \cos (2t + 3z) \] \[ [37] \] \[ - \frac{e^4}{12} \cos 4x - \frac{e^4}{32} \cos (2t - 4x) + \frac{9}{16} e^4 \cos (2t + 4x) \] \[ [38] \quad [39] \quad [40] \] \[ + \left\{ \frac{1}{8} - \frac{5}{16} \right\} e^3 e_i \cos (3x + z) + \left\{ -\frac{7}{48} - \frac{35}{96} \right\} e^3 e_i \cos (2t - 3x - z) \] \[ [41] \quad [42] \] \[ + \left\{ -\frac{25}{16} + \frac{125}{96} \right\} e^3 e_i \cos (2t - 3x - z) + \left\{ \frac{1}{8} - \frac{5}{16} \right\} e^3 e_i \cos (3x - z) \] \[ [43] \quad [44] \] \[ + \left\{ \frac{7}{16} - \frac{35}{96} \right\} e^3 e_i \cos (2t - 3x + z) \] \[ + \left\{ \frac{25}{48} + \frac{125}{96} \right\} e^3 e_i \cos (2t + 3x - z) + \left\{ \frac{1}{16} + \frac{5}{8} - \frac{5}{4} \right\} e^2 e_i^2 \cos (2x + 2z) \] \[ + \left\{ \frac{5}{4} + \frac{25}{8} + \frac{25}{4} \right\} e^2 e_i^2 \cos (2t - 2x - 2z) + \left\{ \frac{5}{4} - \frac{15}{4} + \frac{5}{2} \right\} e^2 e_i^2 \cos (2t + 2x + 2z) \] \[ + \left\{ \frac{1}{16} + \frac{5}{8} - \frac{5}{4} \right\} e^2 e_i^2 \cos (2x - 2z) + \left\{ \frac{25}{8} - \frac{75}{8} + \frac{25}{4} \right\} e^2 e_i^2 \cos (2t - 2x + 2z) \] \[ + \left\{ \frac{1}{2} + \frac{5}{4} + \frac{5}{2} \right\} e^2 e_i^2 \cos (2t + 2x - 2z) + \left\{ \frac{1}{8} + \frac{5}{8} + 5 - \frac{145}{16} \right\} ee_i^3 \cos (x + 3z) \] \[ + \left\{ -\frac{25}{16} - \frac{15}{4} - \frac{15}{2} - \frac{435}{32} \right\} ee_i^3 \cos (2t - x - 3z) \] \[ + \left\{ -\frac{7}{48} + \frac{25}{8} - \frac{15}{2} + \frac{145}{32} \right\} ee_i^3 \cos (2t + x + 3z) \] \[ + \left\{ \frac{1}{8} + \frac{5}{8} + 5 - \frac{145}{16} \right\} ee_i^3 \cos (x - 3z) + \left\{ \frac{7}{16} - \frac{75}{8} + \frac{45}{2} - \frac{435}{32} \right\} ee_i^3 \cos (2t - x + 3z) \] \[ + \left\{ \frac{25}{48} + \frac{5}{4} + \frac{5}{2} - \frac{145}{32} \right\} ee_i^3 \cos (2t + x - 3z) - \left\{ -\frac{1}{12} - \frac{5}{16} - \frac{5}{4} - \frac{145}{16} + \frac{745}{96} \right\} e_i^4 \cos 4z \] \[ + \left\{ \frac{9}{16} + \frac{125}{96} + \frac{5}{2} + \frac{145}{32} + \frac{745}{192} \right\} e_i^4 \cos (2t - 4z) \] \[ + \left\{ -\frac{1}{32} - \frac{35}{96} + \frac{25}{4} - \frac{435}{32} + \frac{745}{192} \right\} e_i^4 \cos (2t + 4z) \] Terms in \( R \) multiplied by \(-\frac{3}{2} \sin^2 \frac{t}{2} \cos^2 \frac{t}{2} \frac{a^2}{a_i^3}\) \[ = \left\{ 1 - \frac{5}{2} e^2 + \frac{3}{2} e_i^2 + 5 e_i^2 - \frac{5}{2} e_i^2 - \frac{5}{2} e_i^2 \right\} \cos 2y \] \[ + \left\{ 1 + \frac{3}{2} e^2 - \frac{5}{2} e_i^2 + 5 e_i^2 + \frac{5}{2} e_i^2 - \frac{15}{2} e_i^2 \right\} \cos (2t - 2y) \] \[ - 3e \cos (x - 2y) + e \cos (x + 2y) - e \cos (2t - x - 2y) - e \cos (2t + x - 2y) \] \[ + \left\{ -1 + \frac{5}{2} \right\} e_i \cos (z - 2y) + \left\{ -1 + \frac{5}{2} \right\} e_i \cos (z + 2y) + \left\{ 1 + \frac{5}{2} \right\} e_i \cos (2t - z - 2y) \] \[ + \left\{ -3 + \frac{5}{2} \right\} e_i \cos (2t + z - 2y) + \frac{5}{2} e^2 \cos (2x - 2y) + e^2 \cos (2x + 2y) \] \[ - \frac{e^2}{4} \cos (2t - 2x - 2y) - \frac{e^2}{4} \cos (2t + 2x - 2y) \] \[ + \left\{ 3 - \frac{15}{2} \right\} ee_i \cos (x + z - 2y) + \left\{ -1 + \frac{5}{2} \right\} ee_i \cos (x + z + 2y) \] \[ + \left\{ -1 - \frac{5}{2} \right\} ee_i \cos (2t - x - z - 2y) \] \[ + \left\{ 3 - \frac{5}{2} \right\} ee_i \cos (2t + x + z - 2y) + \left\{ 3 - \frac{15}{2} \right\} ee_i \cos (2t - z - 2y) \] \[ + \left\{ -1 + \frac{5}{2} \right\} ee_i \cos (x - z + 2y) + \left\{ 3 - \frac{5}{2} \right\} ee_i \cos (2t - x + z - 2y) \] \[ + \left\{ -1 - \frac{5}{2} \right\} ee_i \cos (2t + x - z - 2y) + \left\{ -\frac{3}{8} - \frac{5}{2} + 5 \right\} e_i^2 \cos (2z - 2y) \] \[ + \left\{ -\frac{5}{8} - \frac{5}{2} + 5 \right\} e_i^2 \cos (2z + 2y) + \left\{ \frac{5}{8} + \frac{5}{2} + 5 \right\} e_i^2 \cos (2t - 2z - 2y) \] \[ + \left\{ \frac{19}{8} - \frac{15}{2} + 5 \right\} e_i^2 \cos (2t + 2z - 2y) \] Terms in \( R \) multiplied by \(-\frac{3}{2} \cos^4 \frac{t}{2} a_i^3\) \[ = \frac{1}{2} + \frac{3}{4} e^2 + \frac{3}{4} e_i^2 + \frac{15}{16} e^4 + \frac{9}{8} e^2 e_i^2 + \left\{ \frac{1}{2} - \frac{5}{4} e^2 - \frac{5}{4} e_i^2 + \frac{23}{32} e^4 + \frac{13}{32} e_i^4 + \frac{25}{8} e^2 e_i^2 \right\} \cos 2t \] \[ + \left\{ 1 + \frac{e^2}{8} - \frac{3}{2} e_i^2 \right\} e \cos x + \left\{ -\frac{3}{2} + \frac{13}{16} e^2 + \frac{15}{4} e_i^2 \right\} e \cos (2t - x) \] \[ + \left\{ \frac{1}{2} - \frac{19}{16} e^2 - \frac{5}{4} e_i^2 \right\} e \cos (2t + x) + \left\{ \frac{3}{2} + \frac{9}{4} e^2 + \frac{27}{16} e_i^2 \right\} e_i \cos z \] \[ + \left\{ \frac{7}{4} - \frac{35}{8} e^2 - \frac{123}{32} e_i^2 \right\} e_i \cos (2t - z) \] \[ + \left\{ -\frac{1}{4} + \frac{5}{8} e^2 + e_i^2 \right\} e_i \cos (2t + z) + \left\{ -\frac{1}{4} + \frac{e^2}{12} - \frac{3}{8} e_i^2 \right\} e^2 \cos 2x \] \[ + \left\{ \frac{5}{4} - \frac{25}{8} e_i^2 \right\} e^2 \cos (2t - 2x) + \left\{ \frac{1}{2} - \frac{5}{4} e^2 - \frac{5}{4} e_i^2 \right\} e^2 \cos (2t + 2x) \] \[ + \left\{ -\frac{3}{2} + \frac{3}{16} e^2 - \frac{27}{16} e_i^2 \right\} ee_i \cos (x + z) + \left\{ -\frac{21}{4} + \frac{91}{32} e^2 + \frac{369}{32} e_i^2 \right\} ee_i \cos (2t - x - z) \] \[ + \left\{ -\frac{1}{4} + \frac{19}{32} e^2 + \frac{e_i^2}{32} \right\} ee_i \cos (2t + x + z) + \left\{ -\frac{3}{2} + \frac{3}{16} e^2 - \frac{27}{16} e_i^2 \right\} ee_i \cos (x - z) \] \[ + \left\{ \frac{3}{4} - \frac{13}{32} e^2 - \frac{3}{32} e_i^2 \right\} ee_i \cos (2t - x + z) + \left\{ \frac{7}{4} - \frac{133}{32} e^2 - \frac{123}{32} e_i^2 \right\} ee_i \cos (2t + x - z) \] \[ + \left\{ \frac{9}{4} + \frac{27}{8} e^2 + \frac{21}{12} e_i^2 \right\} e_i^2 \cos 2z + \left\{ \frac{17}{4} - \frac{85}{8} e^2 - \frac{115}{12} e_i^2 \right\} e_i^2 \cos (2t - 2z) - \frac{e^3}{8} \cos 3x \] \[ - \frac{7}{48} e^3 \cos (2t - 3x) + \frac{25}{48} e^3 \cos (2t + 3x) - \frac{3}{8} e^2 e_i \cos (2x + z) \] \[ + \frac{35}{8} e^2 e_i \cos (2t - 2x - z) - \frac{e^2 e_i}{4} \cos (2t + 2x + z) - \frac{3}{8} e^2 e_i \cos (2x - z) \] \[ - \frac{5}{8} e^2 e_i \cos (2t - 2x + z) + \frac{7}{4} e^2 e_i \cos (2t + 2x - z) \] \[ - \frac{9}{4} ee_i^2 \cos (x + 2z) - \frac{51}{4} ee_i^2 \cos (2t - x - 2z) \] \[ - \frac{9}{4} ee_i^2 \cos (x - 2z) + \frac{17}{4} ee_i^2 \cos (2t + x - 2z) \] \[ + \frac{53}{16} ee_i^3 \cos 3z + \frac{845}{96} ee_i^3 \cos (2t - 3z) + \frac{e^3 e_i}{96} \cos (2t + 3z) - \frac{e^4}{12} \cos 4x - \frac{e^4}{32} \cos (2t - 4x) \] * It is remarkable that the coefficient of argument 19 equals zero. \[ \frac{9}{16} e^t \cos (2t + 4x) - \frac{3}{16} e^3 e_i \cos (3x + z) \] \[ - \frac{49}{96} e^3 e_i \cos (2t + 3x - z) - \frac{25}{96} e^3 e_i \cos (2t - 3x - z) - \frac{3}{16} e^3 e_i \cos (3x - z) \] \[ + \frac{7}{96} e^3 e_i \cos (2t - 3x + z) + \frac{175}{96} e^3 e_i \cos (2t + 3x - z) - \frac{9}{16} e^2 e_i^2 \cos (2x + 2z) \] \[ + \frac{85}{8} e^2 e_i^2 \cos (2t - 2x - 2z) - \frac{9}{16} e^2 e_i^2 \cos (2x - 2z) + \frac{17}{4} e^2 e_i^2 \cos (2t + 2x - 2z) \] \[ - \frac{53}{16} e e_i^3 \cos (x + 3z) - \frac{845}{32} e e_i^3 \cos (2t - x - 3z) + \frac{ee_i^3}{96} \cos (2t + x - 3z) \] \[ - \frac{53}{16} e e_i^3 \cos (x - 3z) - \frac{ee_i^3}{32} \cos (2t - x + 3z) - \frac{25}{96} e e_i^3 \cos (2t + x - 3z) \] \[ - \frac{283}{96} e_i^4 \cos 4z + \frac{2453}{192} e_i^4 \cos (2t - 4z) - \frac{741}{192} e_i^4 \cos (2t + 4z) \] Terms in \( R \) multiplied by \( -\frac{3}{2} \sin^2 \frac{t}{2} \cos^2 \frac{t}{2} \frac{a^2}{a_i^3} \) or \( -\frac{3}{8} \sin^2 \frac{t}{2} \frac{a^3}{a_i^3} \) \[ = \left\{ 1 - \frac{5}{2} e^2 + \frac{3}{2} e_i^2 \right\} \cos 2y + \left\{ 1 + \frac{3}{2} e^2 - \frac{5}{2} e_i^2 \right\} \cos (2t - 2y) - 3e \cos (x - 2y) \] \[ + e \cos (x + 2y) - e \cos (2t - x - 2y) - e \cos (2t + x - 2y) + \frac{3}{2} e_i \cos (z - 2y) + \frac{3}{2} e_i \cos (z + 2y) \] \[ + \frac{7}{2} e_i \cos (2t - z - 2y) - \frac{e_i}{2} \cos (2t + z - 2y) + \frac{5}{2} e^2 \cos (2x - 2y) + e^2 \cos (2x + 2y) \] \[ - \frac{e^2}{4} \cos (2t - 2x - 2y) - \frac{e^2}{4} \cos (2t + 2x - 2y) - \frac{9}{2} ee_i \cos (x + z - 2y) \] \[ + \frac{3}{2} ee_i \cos (x + z + 2y) - \frac{7}{2} ee_i \cos (2t - x - z - 2y) + \frac{ee_i}{2} \cos (2t + x + z - 2y) \] \[ R = m_i \left\{ -\frac{1}{r_i} - \frac{1}{4} \left[ 1 + \frac{3}{2} e^2 + \frac{3}{2} e_i^2 + \frac{9}{4} e^2 e_i^2 + \frac{15}{8} e_i^4 - \frac{3}{2} \gamma^2 - \frac{9}{4} \gamma^2 e^2 - \frac{9}{4} \gamma^2 e_i^2 + \frac{39}{8} \gamma^4 \right] \frac{a^2}{a_i^3} \right\} \] \[ - \frac{3}{4} \left[ 1 - \frac{5}{2} e^2 - \frac{5}{2} e_i^2 + \frac{23}{16} e^4 + \frac{25}{4} e^2 e_i^2 + \frac{13}{16} e_i^4 \right] \cos^4 \frac{t}{2} \frac{a^2}{a_i^3} \cos 2t \] \[ + \frac{1}{2} \left[ 1 - \frac{e^2}{8} - \frac{3}{2} e_i^2 - \frac{3}{2} \gamma^2 \right] \frac{a^2}{a_i^3} e \cos x \] \[ + \frac{9}{4} \left[ 1 - \frac{13}{24} e^2 - \frac{5}{2} e_i^2 \right] \cos^4 \frac{t}{2} \frac{a^2}{a_i^3} e \cos (2t - x) \] \[ - \frac{3}{4} \left[ 1 - \frac{19}{8} e^2 - \frac{5}{2} e_i^2 \right] \cos^4 \frac{t}{2} \frac{a^2}{a_i^3} e \cos (2t + x) \] \[ - \frac{3}{4} \left[ 1 + \frac{3}{2} e^2 + \frac{9}{8} e_i^2 - \frac{3}{2} \gamma^2 \right] \frac{a^2}{a_i^3} e_i \cos z \] \[ - \frac{21}{8} \left[ 1 - \frac{5}{2} e^2 - \frac{123}{56} e_i^2 \right] \cos^4 \frac{t}{2} \frac{a^2}{a_i^3} e_i \cos (2t - z) \] \[ + \frac{3}{8} \left[ 1 - \frac{5}{2} e^2 - 4 e_i^2 \right] \cos^4 \frac{t}{2} \frac{a^2}{a_i^3} e_i \cos (2t + z) \] \[ + \frac{1}{8} \left[ 1 - \frac{e^2}{3} + \frac{3}{2} e_i^2 - \frac{3}{2} \gamma^2 \right] \frac{a^2}{a_i^3} e^2 \cos 2x \] \[ - \frac{15}{8} \left[ 1 - \frac{5}{2} e_i^2 \right] \cos^4 \frac{t}{2} \frac{a^2}{a_i^3} e^2 \cos (2t - 2x) \] \[ - \frac{3}{4} \left[ 1 - \frac{5}{2} e^2 - \frac{5}{2} e_i^2 \right] \cos^4 \frac{t}{2} \frac{a^2}{a_i^3} e^2 \cos (2t + 2x) \] Development of $R$. \[ \begin{align*} &+ \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) \\ &+ \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) \\ &+ \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) \\ &+ \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) \\ &- \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) \\ &- \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) \\ &- \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 \\ &- \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) \\ &+ \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) \\ &+ \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) \\ &+ \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) \\ &+ \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) \\ &+ \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 \\ &- \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 \\ &+ \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) \end{align*} \] \[ \begin{align*} &+ \frac{49}{64} \frac{a^2}{a_i^3} e^3 e_i \cos (2t - 3x - z) + \frac{25}{64} \frac{a^2}{a_i^3} e^3 e_i \cos (2t - 3x - z) + \frac{3}{32} \frac{a^2}{a_i^3} e^3 e_i \cos (3x - z) \\ &- \frac{7}{64} \frac{a^2}{a_i^3} e^3 e_i \cos (2t - 3x + z) - \frac{175}{64} \frac{a^2}{a_i^3} e^3 e_i \cos (2t + 3x - z) \\ &+ \frac{9}{32} \frac{a^2}{a_i^3} e^2 e_i^2 \cos (2x + 2z) - \frac{255}{16} \frac{a^2}{a_i^3} e^2 e_i^2 \cos (2t - 2x - 2z) \\ &+ \frac{9}{32} \frac{a^2}{a_i^3} e^2 e_i^2 \cos (2x - 2z) - \frac{51}{8} \frac{a^2}{a_i^3} e^2 e_i^2 \cos (2t + 2x - 2z) + \frac{53}{32} \frac{a^2}{a_i^3} e e_i^3 \cos (x + 3z) \\ &+ \frac{2535}{64} \frac{a^2}{a_i^3} e e_i^3 \cos (2t - x - 3z) - \frac{1}{64} \frac{a^2}{a_i^3} e e_i^3 \cos (2t + x + 3z) \\ &+ \frac{53}{32} \frac{a^2}{a_i^3} e e_i^3 \cos (x - 3z) + \frac{3}{64} \frac{a^2}{a_i^3} e e_i^3 \cos (2t - x + 3z) \\ &+ \frac{45}{64} \frac{a^2}{a_i^3} e e_i^3 \cos (2t + x - 3z) + \frac{591}{64} \frac{a^2}{a_i^3} e_i^4 \cos 4z \\ &- \frac{2453}{128} \frac{a^2}{a_i^3} e_i^4 \cos (2t - 4z) + \frac{741}{128} \frac{a^2}{a_i^3} e_i^4 \cos (2t + 4z) \\ &- \frac{3}{8} \left\{ 1 - \frac{5}{2} e^2 + \frac{3}{2} e_i^2 \right\} \frac{a^2}{a_i^3} \gamma^2 \cos 2y - \frac{3}{8} \left\{ 1 + \frac{3}{2} e^2 - \frac{5}{2} e_i^2 + \frac{\gamma^2}{8} \right\} \frac{a^2}{a_i^3} \gamma^2 \cos (2t - 2y) \\ &+ \frac{9}{8} \frac{a^2}{a_i^3} \gamma^2 e \cos (x - 2y) \\ &- \frac{3}{8} \frac{a^2}{a_i^3} \gamma^2 e \cos (x + 2y) + \frac{3}{8} \frac{a^2}{a_i^3} \gamma^2 e \cos (2t - x - 2y) \\ &+ \frac{3}{8} \frac{a^2}{a_i^3} \gamma^2 e \cos (2t + x - 2y) - \frac{9}{16} \frac{a^2}{a_i^3} \gamma^2 e_i \cos (z - 2y) \\ &- \frac{9}{16} \frac{a^2}{a_i^3} \gamma^2 e_i \cos (z + 2y) - \frac{21}{16} \frac{a^2}{a_i^3} \gamma^2 e_i \cos (2t - z - 2y) \\ &+ \frac{3}{16} \frac{a^2}{a_i^3} \gamma^2 e_i \cos (2t + z - 2y) - \frac{15}{16} \frac{a^2}{a_i^3} \gamma^2 e^2 \cos (2x - 2y) \end{align*} \] Development of $R$. \[- \frac{3}{8} \frac{a^2}{a_i^3} \gamma^2 e^3 \cos (2x + 2y) + \frac{3}{32} \frac{a^2}{a_i^3} \gamma^2 e^3 \cos (2t - 2x - 2y) \] \[78\] \[79\] \[+ \frac{3}{32} \frac{a^2}{a_i^3} \gamma^2 e^3 \cos (2t + 2x - 2y) + \frac{27}{16} \frac{a^2}{a_i^3} \gamma^2 e e_i \cos (x + z - 2y) \] \[81\] \[83\] \[- \frac{9}{16} \frac{a^2}{a_i^3} \gamma^2 e e_i \cos (x + z + 2y) + \frac{21}{16} \frac{a^2}{a_i^3} \gamma^2 e e_i \cos (2t - x - z - 2y) \] \[84\] \[85\] \[- \frac{3}{16} \frac{a^2}{a_i^3} \gamma^2 e e_i \cos (2t + x + z - 2y) + \frac{27}{16} \frac{a^2}{a_i^3} \gamma^2 e e_i \cos (x - z - 2y) \] \[87\] \[89\] \[- \frac{9}{16} \frac{a^2}{a_i^3} \gamma^2 e e_i \cos (x - z + 2y) - \frac{3}{16} \frac{a^2}{a_i^3} \gamma^2 e e_i \cos (2t - x + z - 2y) \] \[90\] \[91\] \[+ \frac{21}{16} \frac{a^2}{a_i^3} \gamma^2 e e_i \cos (2t + x - z - 2y) \] \[93\] \[- \frac{51}{64} \frac{a^2}{a_i^3} \gamma^2 e_i^2 \cos (2z - 2y) - \frac{45}{64} \frac{a^2}{a_i^3} \gamma^2 e_i^2 \cos (2z + 2y) \] \[95\] \[96\] \[- \frac{195}{64} \frac{a^2}{a_i^3} \gamma^2 e_i^2 \cos (2t - 2z - 2y) + \frac{3}{64} \frac{a^2}{a_i^3} \gamma^2 e_i^2 \cos (2t + 2z - 2y) \] \[97\] \[99\] \[- \frac{3}{8} \left\{ 1 + 3e^2 + 3e_i^2 - \frac{11}{4} \gamma^2 \right\} \frac{a^3}{a_i^4} * \cos t + \frac{15}{16} \frac{a^3}{a_i^4} e \cos (t - x) \] \[101\] \[102\] \[+ \frac{3}{16} \frac{a^3}{a_i^4} e \cos (t + x) - \frac{9}{8} \frac{a^3}{a_i^4} e_i \cos (t - z) - \frac{3}{8} \frac{a^3}{a_i^4} e_i \cos (t + x) \] \[103\] \[104\] \[105\] \[- \frac{33}{64} \frac{a^3}{a_i^4} e^2 \cos (t - 2x) + \frac{9}{64} \frac{a^3}{a_i^4} e^2 \cos (t + 2x) + \frac{45}{16} \frac{a^3}{a_i^4} e e_i \cos (t - x - z) \] \[106\] \[107\] \[108\] \[+ \frac{3}{16} \frac{a^3}{a_i^4} e e_i \cos (t + x + z) + \frac{15}{16} \frac{a^3}{a_i^4} e e_i \cos (t - x + z) + \frac{9}{16} \frac{a^3}{a_i^4} e e_i \cos (t + x - z) \] \[109\] \[110\] \[111\] \[- \frac{159}{64} \frac{a^3}{a_i^4} e_i^2 \cos (t - 2z) - \frac{33}{64} \frac{a^3}{a_i^4} e_i^2 \cos (t + 2z) - \frac{9}{16} \frac{a^3}{a_i^4} \gamma^2 \cos (t - 2y) \] \[112\] \[113\] \[114\] \[- \frac{15}{32} \frac{a^3}{a_i^4} \sin^2 \frac{t}{2} \cos (t + 2y) - \frac{5}{8} \left\{ 1 - 6e^2 - 6e_i^2 - \frac{3}{4} \gamma^2 \right\} \frac{a^3}{a_i^4} \cos 3t \] \[115\] \[116\] * For the coefficients of the terms multiplied by $\frac{a^3}{a_i^4}$ see p. 39. \[ + \frac{45}{16} \frac{a^3}{a_i^4} e \cos (3t - x) - \frac{15}{16} \frac{a^3}{a_i^4} e \cos (3t + x) - \frac{25}{8} \frac{a^3}{a_i^4} e_i \cos (3t - z) \] \[ [117] \quad [118] \quad [119] \] \[ + \frac{5}{8} \frac{a^3}{a_i^4} e_i \cos (3t + z) - \frac{285}{64} \frac{a^3}{a_i^4} e^2 \cos (3t - 2x) - \frac{75}{64} \frac{a^3}{a_i^4} e^2 \cos (3t + 2x) \] \[ [120] \quad [121] \quad [122] \] \[ - \frac{225}{16} \frac{a^3}{a_i^4} ee_i \cos (3t - x - z) + \frac{15}{16} \frac{a^3}{a_i^4} ee_i \cos (3t + x + z) \] \[ [123] \quad [124] \] \[ - \frac{45}{16} \frac{a^3}{a_i^4} ee_i \cos (3t - x + z) - \frac{75}{16} \frac{a^3}{a_i^4} ee_i \cos (3t + x - z) \] \[ [125] \quad [126] \] \[ - \frac{635}{64} \frac{a^3}{a_i^4} e_i^2 \cos (3t - 2z) - \frac{5}{64} \frac{a^3}{a_i^4} e_i^2 \cos (3t + 2z) - \frac{15}{32} \frac{a^3}{a_i^4} \gamma^2 \cos (3t - 2y) \] \[ [127] \quad [128] \quad [129] \] In the elliptic movement; \[ s = \gamma \sin (g \lambda - \nu) \] \[ \lambda = n t + 2e \sin x + \frac{5}{4} e^2 \sin 2x \] \[ s = \gamma (1 - e^2) \sin y + \gamma e \sin (x - y) + \gamma e \sin (x + y) + \gamma e^2 \sin (2x - y) + \frac{9}{8} \gamma e^2 \sin (2x + y) \] \[ [146] \quad [149] \quad [150] \quad [161] \quad [162] \] \[ s^2 = \frac{\gamma^2}{2} - \frac{\gamma^2}{2} (1 - 4e^2) \cos 2y + \gamma^2 e \cos (x - 2y) - \gamma^2 e \cos (x + 2y) \] \[ [62] \quad [65] \quad [66] \] \[ + \frac{5}{8} \gamma^2 e^2 \cos (2x - 2y) - \frac{5}{8} \gamma^2 e^2 \cos (2x + 2y) \] \[ [77] \quad [78] \] \[ z^* = a \gamma \left(1 - \frac{e^2}{2}\right) \sin y + \frac{3a \gamma e}{2} \sin (x - y) + \frac{a \gamma e}{2} \sin (x + y) \] \[ [146] \quad [149] \quad [150] \] \[ - \frac{a \gamma e^2}{8} \sin (2x - y) + \frac{3a \gamma e^2}{8} \sin (2x + y) \] \[ [161] \quad [162] \] \[ \frac{s}{r} = \frac{\gamma}{a} (1 - e^2) \sin y + \frac{\gamma e}{2a} \sin (x - y) + \frac{3 \gamma e}{2a} \sin (x + y) \] \[ [146] \quad [149] \quad [150] \] * 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. \[ \frac{s}{r} \delta \cdot \frac{1}{r} = \left\{ (1 - e^2) r_0 + \frac{e^2}{2} r_2 \right\} \frac{\gamma}{a^2} \sin y - \left\{ (1 - e^2) \frac{r_1}{2} + \frac{e^2}{4} r_3 - \frac{3 e^2}{4} r_4 \right\} \frac{\gamma}{a^2} \sin (2t - y) \] \[ + \left\{ (1 - e^2) \frac{r_1}{2} - \frac{e^2}{4} r_4 + \frac{3 e^2}{4} r_3 \right\} \frac{e \gamma}{a^2} \sin (2t + y) + \frac{e \gamma r_0}{2 a^2} \sin (x - y) \] \[ + \frac{3 r_0}{2 a^2} e \gamma \sin (x + y) + \left\{ - \frac{r_3}{2} - \frac{3 r_1}{4} \right\} \frac{e \gamma}{a^2} \sin (2t - x - y) \] \[ + \left\{ \frac{r_3}{2} - \frac{r_1}{4} \right\} \frac{e \gamma}{a^2} \sin (2t - x + y) + \left\{ - \frac{r_4}{2} - \frac{r_1}{4} \right\} \frac{e \gamma}{a^2} \sin (2t + x - y) \] \[ + \left\{ \frac{r_4}{2} + \frac{3 r_1}{4} \right\} \frac{e \gamma}{a^2} \sin (2t + x + y) + \frac{r_5 e \gamma}{2 a^2} \sin (z - y) + \frac{r_5 e \gamma}{2 a^2} \sin (z + y) \] \[ - \frac{r_6 e \gamma}{2 a^2} \sin (2t - z - y) + \frac{r_6 e \gamma}{2 a^2} \sin (2t - z + y) - \frac{r_7 e \gamma}{2 a^2} \sin (2t + z - y) \] \[ + \left\{ - \frac{r_9}{2} - \frac{3 r_3}{4} - \frac{17 r_1}{16} \right\} \frac{e^2 \gamma}{a^2} \sin (2t - 2x - y) + \left\{ \frac{r_9}{2} - \frac{r_3}{4} - \frac{r_1}{16} \right\} \frac{e^2 \gamma}{a^2} \sin (2t - 2x + y) \] \[ + \left\{ - \frac{r_{10}}{2} + \frac{r_4}{4} + \frac{r_1}{16} \right\} \frac{e^2 \gamma}{a^2} \sin (2t + 2x - y) + \left\{ \frac{r_{10}}{2} + \frac{3 r_4}{4} + \frac{17 r_1}{16} \right\} \frac{e^2 \gamma}{a^2} \sin (2t + 2x + y) \] \[ + \left\{ - \frac{r_{11}}{2} + \frac{r_5}{4} \right\} \frac{e e \gamma}{a^2} \sin (x + z - y) + \left\{ \frac{r_{11}}{2} + \frac{3 r_5}{4} \right\} \frac{e e \gamma}{a^2} \sin (x + z + y) \] \[ + \left\{ - \frac{r_{12}}{2} + \frac{r_6}{4} - \frac{3 r_6}{4} \right\} \frac{e e \gamma}{a^2} \sin (2t - x - z - y) + \left\{ \frac{r_{12}}{2} + \frac{r_6}{4} \right\} \frac{e e \gamma}{a^2} \sin (2t - x - z + y) \] \[ + \left\{ - \frac{r_{13}}{2} + \frac{r_7}{4} \right\} \frac{e e \gamma}{a^2} \sin (2t + x + z - y) + \left\{ \frac{r_{13}}{2} + \frac{3 r_7}{4} \right\} \frac{e e \gamma}{a^2} \sin (2t + x + z + y) \] \[ + \left\{ - \frac{r_{14}}{2} + \frac{r_8}{2} \right\} \frac{e e \gamma}{a^2} \sin (x - z - y) + \left\{ \frac{r_{14}}{2} + \frac{3 r_8}{4} \right\} \frac{e e \gamma}{a^2} \sin (x - z + y) \] \[ + \left\{ - \frac{r_{15}}{2} - \frac{3 r_7}{4} \right\} \frac{e e \gamma}{a^2} \sin (2t - x + z - y) + \left\{ \frac{r_{15}}{2} - \frac{r_7}{2} \right\} \frac{e e \gamma}{a^2} \sin (2t - x + z + y) \] \[ -\frac{r_{16}e_e\gamma}{2a^2}\sin(2t+x-z-y) + \left\{\frac{r_{16}}{2} + \frac{3}{4}r_6\right\}\frac{ee\gamma}{a^2}\sin(2t+x-z+y) \] \[ -\frac{r_{17}e_e^2\gamma}{2a^2}\sin(2z-y) + \frac{r_{17}e_e^2\gamma}{2a^2}\sin(2z+y) - \frac{r_{18}e_e^2\gamma}{2a^2}\sin(2t-2z-y) \] \[ +\frac{r_{18}e_e^2\gamma}{2a^2}\sin(2t-2z+y) - \frac{r_{19}e_e^2\gamma}{2a^2}\sin(2t+2z-y) + \frac{r_{19}e_e^2\gamma}{2a^2}\sin(2t+2z+y) \] \[ \frac{m_i z}{r^3} = \frac{m_i a\gamma}{a_i^3}\left(1 + \frac{3}{2}e_i^2 - \frac{e_i^2}{2}\right)\sin y + \frac{3m_i a\gamma e_i}{2a_i^3}\sin(x-y) + \frac{m_i a\gamma}{2a_i^3}\sin(x+y) \] \[ -\frac{3m_i a\gamma e_i}{2a_i^3}\sin(z-y) + \frac{3m_i a\gamma e_i}{2a_i^3}\sin(z+y) - \frac{m_i a\gamma e_i^2}{8a_i^3}\sin(2x-y) \] \[ +\frac{3m_i a\gamma e_i^2}{8a_i^3}\sin(2x+y) + \frac{9m_i a\gamma ee_i}{4a_i^3}\sin(x+z-y) + \frac{3m_i a\gamma ee_i}{4a_i^3}\sin(x+z+y) \] \[ +\frac{9m_i a\gamma ee_i}{4a_i^3}\sin(x-z-y) + \frac{3m_i a\gamma ee_i}{4a_i^3}\sin(x-z+y) - \frac{9m_i a\gamma ee_i}{4a_i^3}\sin(2z-y) \] \[ +\frac{9m_i a\gamma ee_i}{4a_i^3}\sin(2z+y) \] \[ \frac{a^3}{r^3} = 1 + \frac{3}{2}e^2 + 3e\cos x + \frac{9}{2}e^2\cos 2x \] \(r\) being the elliptic value of \(r\). If \(z = a\gamma z_{146}\sin y + a\gamma z_{147}\sin(2t-y) + a\gamma z_{148}\sin(2t+y)\) &c. \[ \frac{z^*}{r^3} = \left\{(1 + \frac{3}{2}e^2)z_{146} + \frac{3}{2}e^2z_{150} - \frac{3}{2}e^2z_{149}\right\}\frac{\gamma}{a^2}\sin y \] \[ +\left\{(1 + \frac{3}{2}e^2)z_{147} + \frac{3}{2}e^2z_{151} + \frac{3}{2}e^2z_{153}\right\}\frac{\gamma}{a^2}\sin(2t-y) \] \[ +\left\{(1 + \frac{3}{2}e^2)z_{148} + \frac{3}{2}e^2z_{152} + \frac{3}{2}e^2z_{154}\right\}\frac{\gamma}{a^2}\sin(2t+y) \] *This multiplication of \(z\) by \(r^{-3}\) may be effected at once by means of Table II.* \[ + \left\{ z_{149} - \frac{3}{2} z_{146} \right\} \frac{e \gamma}{a^2} \sin (x - y) + \left\{ z_{150} + \frac{3}{2} z_{146} \right\} \frac{e \gamma}{a^2} \sin (x + y) \] \[ + \left\{ z_{151} + \frac{3}{2} z_{147} \right\} \frac{e \gamma}{a^2} \sin (2t - x - y) + \left\{ z_{152} + \frac{3}{2} z_{148} \right\} \frac{e \gamma}{a^2} \sin (2t - x + y) \] \[ + \left\{ z_{153} + \frac{3}{2} z_{147} \right\} \frac{e \gamma}{a^2} \sin (2t + x - y) \] \[ + \left\{ z_{154} + \frac{3}{2} z_{148} \right\} \frac{e \gamma}{a^2} \sin (2t + x + y) + z_{155} \frac{e \gamma}{a^2} \sin (z - y) + z_{156} \frac{e \gamma}{a^2} \sin (z + y) \] \[ + \left\{ z_{161} + \frac{3}{2} z_{149} - \frac{9}{4} z_{146} \right\} \frac{e^2 \gamma}{a^2} \sin (2x - y) + \left\{ z_{162} + \frac{3}{2} z_{150} + \frac{9}{4} z_{146} \right\} \frac{e^2 \gamma}{a^2} \sin (2x + y) \] \[ + \left\{ z_{163} + \frac{3}{2} z_{151} + \frac{9}{4} z_{147} \right\} \frac{e^2 \gamma}{a^2} \sin (2t - 2x - y) \] \[ + \left\{ z_{164} + \frac{3}{2} z_{152} + \frac{9}{4} z_{148} \right\} \frac{e^2 \gamma}{a^2} \sin (2t - 2x + y) \] \[ + \left\{ z_{165} + \frac{3}{2} z_{153} + \frac{9}{4} z_{147} \right\} \frac{e^2 \gamma}{a^2} \sin (2t + 2x - y) \] \[ + \left\{ z_{166} + \frac{3}{2} z_{154} + \frac{9}{4} z_{148} \right\} \frac{e^2 \gamma}{a^2} \sin (2t + 2x + y) + \text{&c.} \] \[ + \left\{ z_{167} + \frac{3}{2} z_{155} \right\} \frac{ee_1 \gamma}{a^2} \sin (x + z - y) + \left\{ z_{168} + \frac{3}{2} z_{156} \right\} \frac{ee_1 \gamma}{a^2} \sin (x + z + y) \] \[ + \left\{ z_{169} + \frac{3}{2} z_{157} \right\} \frac{ee_1 \gamma}{a^2} \sin (2t - x - z - y) + \left\{ z_{170} + \frac{3}{2} z_{158} \right\} \frac{ee_1 \gamma}{a^2} \sin (2t - x - z + y) \] \[ + \left\{ z_{171} + \frac{3}{2} z_{159} \right\} \frac{ee_1 \gamma}{a^2} \sin (2t + x + z - y) + \left\{ z_{172} + \frac{3}{2} z_{160} \right\} \frac{ee_1 \gamma}{a^2} \sin (2t + x + z + y) \] \[ + \left\{ z_{173} - \frac{3}{2} z_{156} \right\} \frac{ee_1 \gamma}{a^2} \sin (x - z - y) + \left\{ z_{174} - \frac{3}{2} z_{155} \right\} \frac{ee_1 \gamma}{a^2} \sin (x - z + y) \] \[ + \left\{ z_{175} + \frac{3}{2} z_{159} \right\} \frac{ee_1 \gamma}{a^2} \sin (2t - x + z - y) + \left\{ z_{176} + \frac{3}{2} z_{160} \right\} \frac{ee_1 \gamma}{a^2} \sin (2t - x + z + y) \] \[ + \left\{ z_{177} + \frac{3}{2} z_{157} \right\} \frac{ee_1 \gamma}{a^2} \sin (2t + x - z - y) + \left\{ z_{178} + \frac{3}{2} z_{158} \right\} \frac{ee_1 \gamma}{a^2} \sin (2t + x - z + y) \] \[ s = \frac{z}{r} \text{ nearly,} \] \[ = \left\{ z_{146} + \frac{e^2}{2} z_{150} + \frac{e^2}{2} z_{149} \right\} \gamma \sin y \] \[ + \left\{ z_{147} + \frac{e^2}{2} z_{151} + \frac{e^2}{2} z_{153} \right\} \gamma \sin (2t - y) \] \[ + \left\{ z_{148} + \frac{e^2}{2} z_{152} + \frac{e^2}{2} z_{154} \right\} \gamma \sin (2t + y) + \&c. \] \[ \frac{d^2 r^2}{2 \cdot d t^2} - \frac{\mu}{r} + \frac{\mu}{a} + 2 \int d R + r \left( \frac{d R}{d r} \right) = 0 \] \[ \frac{d^2 z}{d t^2} + \frac{\mu z}{r^3} + \frac{m_i z}{r_i^3} \left\{ r^2 - 2rr' \cos (\lambda - \lambda') + r_i^2 \right\}^{\frac{3}{2}} \] \[ r^4 \cdot \frac{d \lambda'^2}{d t^2} = h^2 - 2 \int r^2 \left( \frac{d R}{d \lambda'} \right) d \lambda' \] Neglecting the square of the disturbing force \[ - \frac{d^2 r^2}{d t^2} \cdot \frac{1}{r} - \mu \delta \cdot \frac{1}{r} + 2 \int d R + r \left( \frac{d R}{d r} \right) = 0 \] \[ \frac{d^2 z}{d t^2} + \frac{\mu z}{r^3} + \frac{m_i z}{r_i^3} + \frac{3 m_i z r' r \cos (\lambda' - \lambda)}{r_i^5} = 0 \] \[ \frac{d^2 \delta z}{d t^2} + \frac{3 \mu s \delta}{r} + \frac{\mu \delta z}{r^3} + \frac{m_i z}{r_i^3} + \frac{3 \mu_i z r' r \cos (\lambda' - \lambda)}{r_i^5} = 0 \] \[ \frac{d \lambda'}{d t} = \frac{h(1 + s^2)}{r^2} - \frac{(1 + s^2)}{r^2} \int \left( \frac{d R}{d \lambda'} \right) d t \] \[ r \left( \frac{d R}{d r} \right) = a \left( \frac{d R}{d a} \right), \quad \frac{d R}{d \lambda'} = \frac{d R}{d t}, \quad (t \text{ being used for } n_i - n_t). \] Integrating 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; \[ - r_0 - \frac{m_i a^3}{2 \mu a_i^3} \left\{ 1 + \frac{3}{2} e^2 + \frac{3}{2} e_i^2 - \frac{3}{2} \gamma^2 \right\} = 0 \] \[ 4 (1 - m)^2 \left\{ (1 + 3 e^2) r_1 - \frac{3 e^2}{2} \{ r_3 + r_i \} \right\} - r_1 \] \[- \frac{3 m_i a^3}{2 \mu a^3} \left\{ 1 - \frac{5}{2} e^2 - \frac{5}{2} e_i^2 - \frac{\gamma^2}{2} \right\} \left\{ \frac{1}{1-m} + 1 \right\} = 0\] \[c^2 \left\{ 1 - 3 r_0 \right\} - 1 + \frac{2 m_i a^3}{\mu a^3} = 0\] \[(2 - 2m - c)^2 \left\{ r_3 - \frac{3}{2} r_1 \right\} - r_3 + \frac{9}{2} \frac{m_i a^3}{\mu a^3} \left\{ \frac{2 - c}{2 - 2m - c} + 1 \right\} = 0\] \[(2 - 2m + c)^2 \left\{ r_4 - \frac{3}{2} r_1 \right\} - r_4 - \frac{3}{2} \frac{m_i a^3}{\mu a^3} \left\{ \frac{2 + c}{2 - 2m + c} + 1 \right\} = 0\] \[m^3 r_5 - r_5 - \frac{3}{2} \frac{m_i a^3}{\mu a^3} = 0\] \[(2 - 3m)^2 r_6 - r_6 - \frac{21}{4} \frac{m_i a^3}{\mu a^3} \left\{ \frac{2}{2 - 3m} + 1 \right\} = 0\] \[(2 - m)^2 r_7 - r_7 + \frac{3}{4} \frac{m_i a^3}{\mu a^3} \left\{ \frac{2}{2 - m} + 1 \right\} = 0\] \[4 c^2 \left\{ (1 - 3r_0) r_8 - \frac{3}{4} + 3 r_0 \right\} - r_8 + \frac{m_i a^3}{2 \mu a^3} = 0\] \[(2 - 2m - 2c)^2 \left\{ r_9 - \frac{3}{2} r_3 \right\} - r_9 - \frac{15}{4} \frac{m_i a^3}{\mu a^3} \left\{ \frac{2 - 2c}{2 - 2m - 2c} + 1 \right\} = 0\] \[(2 - 2m + 2c)^2 \left\{ r_{10} - \frac{3}{2} r_4 \right\} - r_{10} - \frac{3}{2} \frac{m_i a^3}{\mu a^3} \left\{ \frac{2 + 2c}{2 - 2m + 2c} + 1 \right\} = 0\] \[(c + m)^2 \left\{ r_{11} - \frac{3}{2} r_5 \right\} - r_{11} + \frac{3}{2} \frac{m_i a^3}{\mu a^3} \left\{ \frac{c}{c + m} + 1 \right\} = 0\] \[(2 - 3m - c)^2 \left\{ r_{12} - \frac{3}{2} r_6 \right\} - r_{12} + \frac{63}{4} \frac{m_i a^3}{\mu a^3} \left\{ \frac{c}{2 - 3m - c} + 1 \right\} = 0\] \[(2 - m + c)^2 \left\{ r_{13} - \frac{3}{2} r_7 \right\} - r_{13} + \frac{3}{4} \frac{m_i a^3}{\mu a^3} \left\{ \frac{2 + c}{2 - m + c} + 1 \right\} = 0\] \[(c - m)^2 \left\{ r_{14} - \frac{3}{2} r_5 \right\} - r_{14} + \frac{3}{2} \frac{m_i a^3}{\mu a^3} \left\{ \frac{c}{c - m} + 1 \right\} = 0\] \[(2 - m - c)^2 \left\{ r_{15} - \frac{3}{2} r_7 \right\} - r_{15} - \frac{9}{4} \frac{m_i a^3}{\mu a^3} \left\{ \frac{2 - c}{2 - m - c} + 1 \right\} = 0\] * 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. \[(2 - 3m + c)^2 \left\{ r_{16} - \frac{3}{2} r_6 \right\} - r_{16} - \frac{21}{4} \frac{m_i}{\mu} \frac{a^3}{a_i^3} \left\{ \frac{2 + c}{2 - 3m + c} + 1 \right\} = 0\] \[4m^2 r_{17} - r_{17} - \frac{9}{4} \frac{m_i}{\mu} \frac{a^3}{a_i^3} = 0\] \[(2 - 4m)^2 r_{18} - r_{18} - \frac{51}{4} \frac{m_i}{\mu} \frac{a^3}{a_i^3} \left\{ \frac{2}{2 - 4m} + 1 \right\}\] \[4r_{19} - r_{19} = 0\] The equation for determining \(z\) may be integrated in the same way. \[-g^2 z_{146} + 3r_0 + z_{146} + \frac{m_i}{\mu} \frac{a^3}{a_i^3} = 0\] \[-\left\{ 2(1-m) - g \right\} z_{147} - \frac{3}{2} r_1 + z_{147} = 0\] \[-\left\{ 2(1-m) + g \right\} z_{148} + \frac{3}{2} r_1 + z_{148} = 0\] \[-\left\{ c - g \right\} z_{149} + \frac{3}{2} r_6 + z_{149} - \frac{3}{2} z_{146} + \frac{3m_i}{2\mu} \frac{a^3}{a_i^3} = 0\] \[-\left\{ c + g \right\} z_{150} + \frac{9}{2} r_6 + z_{150} + \frac{3}{2} z_{146} + \frac{m_i}{\mu} \frac{a^3}{a_i^3} = 0\] \[-\left\{ 2(1-m) - c - g \right\} z_{147} + 3\left\{ -\frac{3}{4} r_1 - \frac{r_3}{2} \right\} + z_{151} + \frac{3}{2} z_{147} = 0\] \[-\left\{ 2(1-m) - c + g \right\} z_{148} + 3\left\{ -\frac{r_1}{4} + \frac{r_3}{2} \right\} + z_{152} + \frac{3}{2} z_{148} = 0\] \[-\left\{ 2(1-m) + c - g \right\} z_{149} + 3\left\{ \frac{r_1}{4} - \frac{r_4}{2} \right\} + z_{153} + \frac{3}{2} z_{147} = 0\] \[-\left\{ 2(1-m) + c + g \right\} z_{150} + 3\left\{ \frac{3}{4} r_1 + \frac{r_4}{2} \right\} + z_{154} + \frac{3}{2} z_{146} = 0\] \[-\left\{ m - g \right\} z_{151} + \frac{3}{2} r_5 + z_{155} - \frac{3m_i}{2\mu} \frac{a^3}{a_i^3} = 0\] \[-\left\{ m + g \right\} z_{152} + \frac{3}{2} r_5 + z_{156} + \frac{3m_i}{2\mu} \frac{a^3}{a_i^3} = 0\] \[-\left\{ 2(1-m) - m - g \right\} z_{153} - \frac{3}{2} r_6 + z_{157} = 0\] \[-\left\{ 2(1-m) - m + g \right\} z_{154} + \frac{3}{2} r_6 + z_{158} = 0\] \[-\left\{ 2(1-m) + m - g \right\} z_{155} - \frac{3}{2} r_7 + z_{159} = 0\] \[- \left\{ 2(1-m) + m + g \right\}^2 z_{156} + \frac{3}{2} r_7 + z_{160} = 0\] \[\frac{d\lambda}{dt} = \frac{h}{r^2} + \frac{2h}{r} \cdot \frac{1}{r} + \frac{h^2}{r^4} - \frac{(1+s^2)}{r^2} \int \left( \frac{dR}{d\lambda} \right) dt\] \[\lambda' = \frac{h}{a^2} \left\{ 1 + \frac{e^2}{2} + \frac{\gamma^2}{2} + 2r_0 \right\} t + \frac{2e(1+r_0)}{c} \sin x + \frac{5e^2(1+r_0)}{4c} \sin 2x\] \[+ \left\{ 2r_1 + e^2(r_3 + r_4) - \left\{ -(1 - \frac{5}{2} e^2 - \frac{5}{2} e_i^2 - \frac{\gamma^2}{2}) \right\} \frac{3}{4(1-m)} + \frac{9e^2}{2(2-2m-c)} \right\}\] \[- \frac{3e^2}{2(2-2m+c)} \frac{m_i a^3}{\mu a_i^3} \frac{1}{2(1-m)} \sin 2t\] \[+ \left\{ 2r_3 + e^2 r_1 - \left\{ \frac{9}{2(2-m-c)} - \frac{3}{2(2-m)} \right\} \frac{m_i a^3}{\mu a_i^3} \right\} \frac{e}{(2-2m-c)} \sin (2t-x)\] \[+ \left\{ 2r_4 + e^2 r_1 - \left\{ -\frac{3}{2(2-m+c)} - \frac{3}{2(2-m)} \right\} \frac{m_i a^3}{\mu a_i^3} \right\} \frac{e}{(2-m+c)} \sin (2t+x)\] \[+ \frac{2r_5}{m} \sin z\] \[+ \left\{ 2r_6 + \frac{21}{4(2-3m)} \frac{m_i a^3}{\mu a_i^3} \right\} \frac{e_i}{(2-3m)} \sin (2t-z)\] \[+ \left\{ 2r_7 - \frac{3}{4(2-m)} \frac{m_i a^3}{\mu a_i^3} \right\} \frac{e_i}{(2-m)} \sin (2t+z)\] \[+ \left\{ 2r_9 + r_3 - \left\{ -\frac{15}{4(2-2m-2c)} + \frac{9}{2(2-2m-c)} \right\} \frac{m_i a^3}{\mu a_i^3} \right\} \frac{e^2}{2(1-m-c)} \sin (2t-2x)\] \[+ \left\{ 2r_{10} + r_4 - \left\{ -\frac{3}{2(2-2m+2c)} - \frac{3}{2(2-m+c)} \right\} \frac{m_i a^3}{\mu a_i^3} \right\} \frac{e^2}{2(1-m+c)} \sin (2t+2x)\] \[+ \left\{ 2r_{11} + r_5 \right\} \frac{ee_i}{(c+m)} \sin (x+z)\] \[+ \left\{ 2r_{12} + r_6 - \left\{ \frac{63}{4(2-3m-c)} - \frac{21}{4(2-3m)} \right\} \frac{m_i a^3}{\mu a_i^3} \right\} \frac{ee_i}{(2-3m-c)} \sin (2t-x-z)\] \[+ \left\{ 2r_{13} + r_7 - \left\{ \frac{3}{4(2-m+c)} + \frac{3}{4(2-m)} \right\} \frac{m_i a^3}{\mu a_i^3} \right\} \frac{ee_i}{(2-m+c)} \sin (2t+x+z)\] Considering the terms which depend on the square of the disturbing force \[\frac{d^2}{dt^2} \frac{r^2}{2} - \frac{\mu}{r} + \frac{\mu}{a} + 2 \int dR + r \left( \frac{dR}{dr} \right) = 0\] \[ \frac{d^2 r}{dt^2} = \frac{d^2 r^3 \delta}{dt^2} + \frac{3 d^2 r^4 (\delta \cdot \frac{1}{r})^2}{2 dt^2} - \frac{\mu}{r} + \frac{\mu}{a} + 2 \int dR + r \left( \frac{dR}{dr} \right) = 0 \] \[ \frac{dz}{dt^2} + \frac{\mu z}{r^3} + \frac{m_i z}{r_i^3} - \frac{3 m_i z r r \cos (\lambda' - \lambda)}{r_i^5} = 0. \] \[ \frac{d\lambda'}{dt} = \frac{h}{r^2} \left[ 1 - \frac{1}{h} \int \left( \frac{dR}{d\lambda'} \right) dt \left\{ 1 - \frac{1}{h} \int \left( \frac{dR}{d\lambda'} \right) dt \right\} - \frac{1}{2 h^2} \int \left( \frac{dR}{d\lambda'} \right) dt \right]^2 \] \[ = \frac{h (1 + s^2)}{r^2} - \frac{(1 + s^2)}{r^2} \int \left( \frac{dR}{d\lambda'} \right) dt + \frac{(1 + s^2)}{2 r^2 h} \left\{ \int \left( \frac{dR}{d\lambda'} \right) dt \right\}^2 \] \( dR \) = 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 \( dR \) = the differential of \( R \), supposing only \( nt \) variable + \( \frac{dR}{da} \cdot d.\delta r + \frac{dR}{dt} \cdot d.\delta \lambda' \) + \( \frac{dR}{dr} \cdot d.\delta s; \ d.\delta r', \ d.\delta \lambda, \) and \( d.\delta s, \) being restrained to mean the differentials of those quantities with regard to \( n_t \) only. \[ \delta R = \left( \frac{dR}{dr} \right) \delta r + \left( \frac{dR}{d\lambda'} \right) \delta \lambda' + \left( \frac{dR}{ds} \right) \delta s = -a \left( \frac{dR}{da} \right) r \delta \cdot \frac{1}{r} + \left( \frac{dR}{dt} \right) \delta \lambda' + \left( \frac{dR}{ds} \right) \delta s, \] \( t \) being used in the sense \( nt - n_t. \) \[ \left( \frac{dR}{ds} \right) \delta s = \frac{r^2}{2 r_i^3} s \delta s \text{ nearly.} \] \[ \left( \frac{dR}{dr} \right) d.\delta r + \left( \frac{dR}{d\lambda'} \right) d.\delta \lambda' + \left( \frac{dR}{ds} \right) d.\delta s = -a \left( \frac{dR}{da} \right) d.r \delta \cdot \frac{1}{r} + \left( \frac{dR}{dt} \right) d.\delta \lambda' + \left( \frac{dR}{ds} \right) d.\delta s \] \( d.r \delta \frac{1}{r}, \ d.\delta \lambda' \) and \( d.\delta s \) being restrained to mean the differentials of those quantities with regard to \( n_t \) only. \[ \delta r \left( \frac{dR}{dr} \right) = d.\frac{r}{dr} \left( \frac{dR}{dr} \right) \delta r + d.\frac{r}{d\lambda'} \left( \frac{dR}{dr} \right) \delta \lambda' + d.\frac{r}{ds} \left( \frac{dR}{dr} \right) \delta s \] \[ = -a d.\frac{r}{da} \left( \frac{dR}{dr} \right) r \delta \cdot \frac{1}{r} + d.\frac{r}{dt} \left( \frac{dR}{dr} \right) \delta \lambda' + d.\frac{r}{ds} \left( \frac{dR}{dr} \right) \delta s \] \[ \delta \left( \frac{dR}{d\lambda'} \right) = -a d.\frac{dR}{da} \left( \frac{dR}{d\lambda'} \right) r \delta \cdot \frac{1}{r} + d.\frac{dR}{dt} \left( \frac{dR}{d\lambda'} \right) \delta \lambda' + \left( \frac{dR}{ds} \right) \delta s \] A 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. Similarly, if $\delta'$ denote the variation due to the disturbance of the earth by the moon, $$\delta' R = -a_i d \cdot \left( \frac{dR}{da_i} \right) r_i \delta \cdot \frac{1}{r_i} - d \cdot \left( \frac{dR}{dt} \right) \delta \lambda_i$$ In $dR$ the terms which arise from $$-a \left( \frac{dR}{da} \right) d \cdot r \delta \cdot \frac{1}{r} + \left( \frac{dR}{dt} \right) d \cdot \delta \lambda + \left( \frac{dR}{ds} \right) d \cdot \delta s$$ are multiplied by the small quantity $m$. Considering in $r \left( \frac{dR}{dr} \right)$ and $R$ the terms multiplied by $\frac{a^2}{a_i^3}$, $$r \left( \frac{dR}{dr} \right) = 2R, \quad \delta \cdot r \left( \frac{dR}{dr} \right) = 2\delta R;$$ considering the terms multiplied by $\frac{a^3}{a_i^4}$, $$r \left( \frac{dR}{dr} \right) = 3R, \quad \delta \cdot r \left( \frac{dR}{dr} \right) = 3\delta R$$ Hence 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 $\delta R$. I 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. ### Addition to Table I. | | 146 | 149 | 150 | | 146 | 149 | 150 | | 146 | 149 | 150 | |---|-----|-----|-----|---|-----|-----|-----|---|-----|-----|-----| | 1 | 148 | 153 | 154 | 1 | 147 | 63 | 69 | 4 | 147 | 5 | 83 | | | 147 | 152 | 151 | | 147 | 63 | 67 | 147 | 155 | 71 | 14 | | 2 | 150 | 161 | 162 | 2 | 148 | 64 | 4 | 70 | 148 | 72 | 11 | | | 149 | 162 | -146 | | 148 | 64 | 68 | 3 | 148 | 72 | 11 | | 3 | 152 | 147 | 148 | 3 | 149 | 2 | 77 | 8 | 149 | 6 | 93 | | | 151 | 164 | 163 | | 149 | 2 | 65 | 0 | 149 | 73 | 12 | | 4 | 154 | 165 | 166 | 4 | 150 | 66 | 8 | 78 | 150 | 74 | 16 | | | 153 | 148 | 147 | | 150 | 66 | 62 | 0 | 150 | 74 | 16 | | 5 | 156 | 167 | 168 | 5 | 151 | 3 | 63 | 1 | 151 | 7 | 87 | | | 155 | -1/3 | -174 | | 151 | 3 | 67 | 9 | 151 | 75 | 15 | | 6 | 158 | 169 | 178 | 6 | 152 | 68 | 1 | 64 | 152 | 76 | 13 | | | 157 | 170 | 169 | | 152 | 68 | 80 | 9 | 152 | 76 | 13 | | 7 | 160 | 171 | 172 | 7 | 153 | 4 | 81 | 10 | 153 | 7 | 88 | | | 159 | 176 | 175 | | 153 | 4 | 69 | 1 | 153 | 7 | 88 | | 146 | 62 | 2 | 66 | 146 | 154 | 70 | 10 | 82 | 154 | ### Addition to Table II. | | 146 | 149 | 150 | | 146 | 149 | 150 | | 146 | 149 | 150 | |---|-----|-----|-----|---|-----|-----|-----|---|-----|-----|-----| | 1 | 147 | 152 | 151 | 1 | 10 | 165 | 154 | 153 | 10 | 64 | 148 | | | 148 | 153 | 154 | | 10 | 165 | 154 | 153 | 10 | 64 | 148 | | 2 | 149 | 146 | ..... | 2 | 11 | 167 | 156 | 155 | 11 | 65 | 149 | | | 150 | ..... | -146 | | 11 | 167 | 156 | 155 | 11 | 65 | 149 | | 3 | 151 | ..... | ..... | 3 | 12 | 169 | ..... | ..... | 12 | 66 | 150 | | | 152 | ..... | ..... | | 12 | 169 | ..... | ..... | 12 | 66 | 150 | | 4 | 153 | 148 | 147 | 4 | 13 | 171 | 160 | 159 | 13 | 67 | 151 | | | 154 | ..... | ..... | | 13 | 171 | 160 | 159 | 13 | 67 | 151 | | 5 | 155 | ..... | ..... | 5 | 14 | 173 | ..... | ..... | 14 | 68 | 152 | | | 156 | ..... | ..... | | 14 | 173 | ..... | ..... | 14 | 68 | 152 | | 6 | 157 | ..... | ..... | 6 | 15 | 175 | ..... | ..... | 15 | 69 | 153 | | | 158 | ..... | ..... | | 15 | 175 | ..... | ..... | 15 | 69 | 153 | | 7 | 159 | ..... | ..... | 7 | 16 | 177 | 158 | 157 | 16 | 70 | 154 | | | 160 | ..... | ..... | | 16 | 177 | 158 | 157 | 16 | 70 | 154 | | 8 | 161 | 150 | 149 | 8 | 62 | 146 | ..... | ..... | 62 | 71 | 155 | | | 162 | ..... | ..... | | 62 | 146 | ..... | ..... | 62 | 71 | 155 | | 9 | 163 | ..... | ..... | 9 | 63 | 147 | ..... | ..... | 63 | 72 | 156 | | | 146 | 149 | 150 | | 146 | 149 | 150 | | 146 | 149 | 150 | |---|-----|-----|-----|---|-----|-----|-----|---|-----|-----|-----| | 73 | 157 | | | 73 | 178 | 158 | | 94 | 10 | | 4 | | 74 | 158 | | | 74 | 146 | 62 | | 146 | 11 | | 5 | | 75 | 159 | | | 75 | 147 | 63 | | 147 | 11 | | 5 | | 76 | 160 | | | 76 | 148 | 64 | | 148 | 12 | | 6 | | 77 | 161 | 149 | | 77 | 149 | 2 | | 149 | 12 | | 6 | | 78 | 162 | 150 | | 78 | 150 | 2 | | 150 | 13 | | 7 | | 79 | 163 | 151 | | 79 | 151 | 3 | | 151 | 13 | | 7 | | 80 | 164 | 152 | | 80 | 152 | 3 | | 152 | 14 | | 5 | | 81 | 165 | 153 | | 81 | 153 | 4 | | 153 | 14 | | 5 | | 82 | 166 | 154 | | 82 | 154 | 4 | | 154 | 15 | | 7 | | 83 | 167 | 155 | | 83 | 155 | 5 | | 155 | 15 | | 7 | | 84 | 168 | 156 | | 84 | 156 | 5 | | 156 | 16 | | | | 85 | 169 | 157 | | 85 | 157 | 6 | | 157 | 16 | | | | 86 | 170 | 158 | | 86 | 158 | 6 | | 158 | 17 | | | | 87 | 171 | 159 | | 87 | 159 | 7 | | 159 | 17 | | | | 88 | 172 | 160 | | 88 | 160 | 7 | | 160 | 18 | | | | 89 | 173 | -156 | | 89 | 161 | 8 | | 161 | 18 | | | | 90 | 174 | -155 | | 90 | 162 | 8 | | 162 | 19 | | | | 91 | 175 | 159 | | 91 | 163 | 9 | | 163 | 19 | | | | 92 | 176 | 160 | | 92 | 164 | 9 | | 164 | | | | | 93 | 177 | 157 | | 93 | 165 | 10 | | 165 | | | | | | 161 | 162 | | 161 | 162 | | 161 | 162 | | 161 | 162 | |---|-----|-----|---|-----|-----|---|-----|-----|---|-----|-----| | 8 | 146 | -146 | 8 | 78 | 146 | 78 | 81 | 147 | 81 | 164 | 1 | | 9 | 147 | 148 | 9 | 79 | 147 | 79 | 82 | 148 | 82 | 165 | 1 | | 10| 148 | 147 | 10| 80 | 148 | 80 | 163 | 1 | 163 | 166 | On the Precession of the Equinoxes, supposing the Earth to revolve in a resisting medium. In 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. If 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. This 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. In 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. in 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. Let $x'$, $y'$, $z'$ 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. Let 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. Suppose 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. Now, 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. The equation to a plane perpendicular to the axis of rotation, and passing through the centre of gravity of the body, is \( px + qy + rz = 0 \). Let the body revolving be a spheroid of which the equation is \[ x^2 + y^2 + z^2 (1 + e^2) = a^2 (1 + e^2) \] The equation to the tangent plane to the spheroid at the point \( x, y, z \) is \[ xx' + yy' + zz' (1 + e^2) = a^2 (1 + e^2) \] The equations to the planes from whose intersection the line PB results, are \[ * z(qz' - ry') + y(rx' - pz') + z(py' - qx') = 0 \\ px + qy + rz = D \] \( D \) being a constant. The equations to the line PC are \[ x\{r(qz' - ry') - p(py' - qx')\} + y\{r(rx' - pz') - q(py' - qx')\} = 0 \\ x\{q(qz' - ry') - p(rx' - pz')\} + z\{q(py' - qx') - r(rx' - pz')\} = 0 \] and neglecting \( p^2, q^2, pq \), \[ x(qz' - ry') = y(pz' - rx') \\ x(qy' + px') = z(pz' - rx') \] The equations to the direction of motion of the point P are \[ x(pz' - rx') = y(ry' - qx') \\ x(qx' - py') = z(ry' - qz') \] Cos. angle, which the direction of motion of P makes with the normal to the surface or cos A PC \[ = \frac{x'(ry' - qz') + y'(pz' - rx') + z'(1 + e^2)(qx' - py')}{\sqrt{(ry' - qz')^2 + (pz' - rx')^2 + (qy' - px')^2}} \sqrt{x'^2 + y'^2 + z'^2 (1 + 2e^2)} \] * The notation is the same as p. 20, except that the accents at foot of \( x_p, y_p, z_p \) are omitted. \[ \frac{e^2 z' (q x' - p y')}{r \sqrt{x'^2 + y'^2} \sqrt{x'^2 + y'^2 + z'^2}} \text{ nearly.} \] The resistance acting in the direction of the normal, and since the velocity \[ = \sqrt{x'^2 + y'^2} \sqrt{(p^2 + q^2 + r^2)} \text{ nearly;} \] \[ Cdr = 0 \] \[ B dq + (A - C) rp dt = dt \int \left\{ \frac{z' x' - x' z' (1 + e^2)}{r \{ x'^2 + y'^2 + z'^2 \}} \right\} e^2 z' (q x' - p y') \sqrt{x'^2 + y'^2} ds (p^2 + q^2 + r^2) \] \[ A dp + (C - B) qr dt = dt \int \left\{ \frac{y' z' (1 + e^2) - z' y'}{r \{ x'^2 + y'^2 + z'^2 \}} \right\} e^2 z' (q x' - p y') \sqrt{x'^2 + y'^2} ds (p^2 + q^2 + r^2) \] \[ \sin \frac{C - A}{A} (nt + \gamma) dc + c \frac{(C - A)}{A} \cos \frac{C - A}{A} (nt + \gamma) d\gamma \] \[ = - \frac{n dt e^t}{A} \int \frac{x' z'^2 (q x' - p y') \sqrt{x'^2 + y'^2}}{\{ x'^2 + y'^2 + z'^2 \}} ds \] \[ \cos \frac{C - A}{A} (nt + \gamma) dc - c \frac{(C - A)}{A} \sin \frac{C - A}{A} (nt + \gamma) d\gamma \] \[ = \frac{n dt e^t}{A} \int \frac{y' z'^2 (q x' - p y') \sqrt{x'^2 + y'^2}}{\{ x'^2 + y'^2 + z'^2 \}} ds \] since \( \int x'^2 z'^2 ds = \int y'^2 z'^2 ds \) \[ dc = - \frac{n dt e^t c}{A} \int \frac{x'^2 z'^2 \sqrt{x'^2 + y'^2}}{\{ x'^2 + y'^2 + z'^2 \}} ds + \frac{n dt e^t}{2A} \sin 2 \left( \frac{C - A}{A} (nt + \gamma) \right) \int \frac{x' y' z'^2 \sqrt{x'^2 + y'^2}}{\{ x'^2 + y'^2 + z'^2 \}} ds \] neglecting the term which is periodic, \[ dc = - nc \frac{e^t dt}{A} \int \frac{x'^2 z'^2 \sqrt{x'^2 + y'^2}}{\{ x'^2 + y'^2 + z'^2 \}} ds \] Let \( \int \frac{x'^2 z'^2 \sqrt{x'^2 + y'^2}}{\{ x'^2 + y'^2 + z'^2 \}} ds = D \) \( D \) being a positive quantity. \[ dc = - \frac{n D e^t dt}{A} \quad e^c = \frac{n De^t}{A}, \quad e \text{ being the base of Naperian logarithms.} \] When \( t \) is infinite \( c = 0 \); hence the latitude of the axis of instantaneous rotation increases until it reaches 90°, which is its limit. Having 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 \[ p dt = \sin \phi \sin \theta d\psi - \cos \phi d\theta \] \[ q dt = \cos \phi \sin \theta d\psi + \sin \phi d\theta \] \[ r dt = d\phi - \cos \theta d\psi \]