Researches in Physical Astronomy [Continued]

I propose in this paper to extend the equations I have already given for determining the planetary inequalities, as far as the terms depending on the squares and products of the eccentricities, to the terms depending on the cubes of the eccentricities and quantities of that order, which is done very easily by a Table similar to Table II. in my Lunar Theory; and particularly to the determination of the great inequality of Jupiter, or at least such part of it as depends on the first power of the disturbing force. That part which depends on the square of the disturbing force may I think be most easily calculated by the methods given in my Lunar Theory; but not without great care and attention can accurate numerical results be expected. I have however given the analytical form of the coefficients of the arguments in the development of $R$, upon which that inequality principally depends. It is I think particularly convenient to designate the arguments of the planetary disturbances by indices. The system of indices adopted in this paper is given as appearing better adapted for the purpose than that used in my former paper on the Planetary Theory; but it is not advisable to make use of the same indices in this as in the Lunar Theory. I have also given analytical expressions for the development of $R$ to the terms multiplied by the squares and products of the eccentricities inclusive, and for the terms in $r \left( \frac{dR}{dr} \right)$ multiplied by the first power of the eccentricities, which are I believe the simplest that can be proposed. The following are the arguments which occur in the Planetary Theory. Column 1 contains the index. —— 2 contains the index of the argument, which is symmetrical. —— 3 contains the index used Phil. Trans. Part II. 1830, p. 349. | Column 1 | Column 2 | Column 3 | |----------|----------|----------| | 0 | 0 | | | 1 | t = n t - n_i t | | | 2 | 2 t = 2 n t - 2 n_i t | | | 3 | 3 t = 3 n t - 3 n_i t | | | 4 | 4 t = 4 n t - 4 n_i t | | | 10 | x = n t - ω | | | 11 | t - x = -n_i t + ω | | | 12 | 2 t - x = n t - 2 n_i t + ω | | | 13 | 3 t - x = 2 n t - 3 n_i t + ω | | | 14 | 4 t - x = 3 n t - 4 n_i t + ω | | | 21 | t + x = 2 n t - n_i t - ω | | | 22 | 2 t + x = 3 n t - 2 n_i t - ω | | | 23 | 3 t + x = 4 n t - 3 n_i t - ω | | | 30 | z = n_j t - ω_i | | | 31 | t - z = n t - 2 n_i t + ω_i | | | 32 | 2 t - z = 2 n t - 3 n_i t + ω_i | | | 33 | 3 t - z = 3 n t - 4 n_i t + ω_i | | | 41 | t + z = n t - ω_i | | | 42 | 2 t + z = 2 n t - n_i t - ω_i | | | 43 | 3 t + z = 3 n t - 2 n_i t - ω_i | | | 44 | 4 t + z = 4 n t - 3 n_i t - ω_i | | | 50 | 2 x = 2 n t - 2 ω | | | 51 | t - 2 x = -n_i t + 2 ω | | | 52 | 2 t - 2 x = -2 n_i t + 2 ω | | | 53 | 3 t - 2 x = n t - 3 n_i t - 2 ω | | | 61 | t + 2 x = 3 n t - n_i t - 2 ω | | | 62 | 2 t + 2 x = 4 n t - 2 n_i t - 2 ω | | | 63 | 3 t + 2 x = 5 n t - 3 n_i t - 2 ω | | | 70 | x + z = n t + n_i t - ω_i | | | 71 | t - x - z = -2 n_i t + ω + ω_i | | | 72 | 2 t - x - z = -3 n_i t + ω + ω_i | | | 73 | 3 t - x - z = 2 n t - 4 n_i t + ω + ω_i | | | 74 | 4 t - x - z = 3 n t - 5 n_i t + ω + ω_i | | | 81 | t + x + z = 2 n t - ω - ω_i | | | 82 | 2 t + x + z = 3 n t - n_i t - ω - ω_i | | | 83 | 3 t + x + z = 4 n t - 2 n_i t - ω - ω_i | | | 84 | 4 t + x + z = 5 n t - 3 n_i t - ω - ω_i | | | 90 | x - z = n t - n_i t - ω + ω_i | | | 91 | t - x + z = ω - ω_i | | | 92 | 2 t - x + z = n t - n_i t + ω - ω_i | | | 93 | 3 t - x + z = 2 n t - 2 n_i t + ω - ω_i | | | 94 | 4 t - x + z = 3 n t - 3 n_i t + ω - ω_i | | | 101 | t + x - z = 2 n t - 2 n_i t - ω + ω_i | | | 102 | 2 t + x - z = 3 n t - 3 n_i t - ω + ω_i | | | 103 | 3 t + x - z = 4 n t - 4 n_i t - ω + ω_i | | | 104 | 4 t + x - z = 5 n t - 5 n_i t - ω + ω_i | | | 110 | 2 z = 2 n_i t - 2 ω_i | | | 111 | t - 2 z = n t - 3 n_i t + 2 ω_i | | | 112 | 2 t - 2 z = 2 n t - 4 n_i t + 2 ω_i | | | 113 | 3 t - 2 z = 3 n t - 5 n_i t + 2 ω_i | | | 114 | 4 t - 2 z = 4 n t - 6 n_i t + 2 ω_i | | | 121 | t + 2 z = n t + n_i t - 2 ω_i | | | 122 | 2 t + 2 z = 2 n t - 2 ω_i | | | 123 | 3 t + 2 z = 3 n t - n_i t - 2 ω_i | | | 124 | 4 t + 2 z = 4 n t - 2 n_i t - 2 ω_i | | | 130 | 2 y_i = 2 n_i t - 2 ω_i | | | 131 | t - 2 y = n t - 3 n_i t + 2 ω_i | | | 132 | 2 t - 2 y = 2 n t - 4 n_i t + 2 ω_i | | | 133 | 3 t - 2 y = 3 n t - 5 n_i t + 2 ω_i | | | 134 | 4 t - 2 y = 4 n t - 6 n_i t + 2 ω_i | | | 141 | t + 2 y = n t + n_i t - 2 ω_i | | | 142 | 2 t + 2 y = 2 n t - 2 ω_i | | | 143 | 3 t + 2 y = 3 n t - n_i t - 2 ω_i | | | 144 | 4 t + 2 y = 4 n t - 2 n_i t - 2 ω_i | | | 150 | 3 x = 3 n t - 3 ω | | | 151 | t - 3 x = 2 n t - n_i t + 3 ω | | | 152 | 2 t - 3 x = -n t - 2 n_i t + 3 ω | | | 153 | 3 t - 3 x = -3 n_i t + 3 ω | | | 154 | 4 t - 3 x = n t - 4 n_i t + 3 ω | | | 161 | t + 3 x = 4 n t - n_i t - 3 ω | | | 162 | 2 t + 3 x = 5 n t - 2 n_i t - 3 ω | | | 163 | 3 t + 3 x = 6 n t - 3 n_i t - 3 ω | | | 164 | 4 t + 3 x = 7 n t - 4 n_i t - 3 ω | | | 170 | 2 x + z = 2 n t + n_i t - 2 ω - ω_i | | | 171 | t - 2 x - z = -n t - 2 n_i t + 2 ω + ω_i | | | 172 | 2 t - 2 x - z = -3 n_i t + 2 ω + ω_i | | | 173 | 3 t - 2 x - z = n t - 4 n_i t + 2 ω + ω_i | | | 174 | 4 t - 2 x - z = 2 n t - 5 n_i t + 2 ω + ω_i | | | 181 | t + 2 x + z = 3 n t - 2 ω - ω_i | | | 182 | 2 t + 2 x + z = 4 n t - n_i t - 2 ω - ω_i | | | 183 | 3 t + 2 x + z = 5 n t - 2 n_i t - 2 ω - ω_i | | | 184 | 4 t + 2 x + z = 6 n t - 3 n_i t - 2 ω - ω_i | | | 190 | 2 x - z = 2 n t - n_i t - 2 ω + ω_i | | | 191 | t - 2 x + z = -n_i t + 2 ω - ω_i | | | 192 | 2 t - 2 x + z = -n_i t + 2 ω - ω_i | | | 193 | 3 t - 2 x + z = n t - 2 n_i t + 2 ω - ω_i | | | 194 | 4 t - 2 x + z = 2 n t - 3 n_i t + 2 ω - ω_i | | | 201 | t + 2 x - z = 3 n t - 2 n_i t - 2 ω + ω_i | | | 202 | 2 t + 2 x - z = 4 n t - 3 n_i t - 2 ω + ω_i | | | 203 | 3 t + 2 x - z = 5 n t - 4 n_i t - 2 ω + ω_i | | | 204 | 4 t + 2 x - z = 6 n t - 5 n_i t - 2 ω + ω_i | | | 210 | x + 2 z = n t + 2 n_i t - 2 ω_i | | | 211 | t - x - 2 z = -3 n_i t + ω + 2 ω_i | | | 212 | 2 t - x - 2 z = n t - 4 n_i t + ω + 2 ω_i | | | | | | | |---|---|---|---| |213|224|3t−x−2z=2nt−5nt+ω+2ωi|282|2t+x+2y=3nt−ω−2νi| |214|225|4t−x−2z=3nt−6nt+ω+2ωi|283|3t+x+2y=4nt−nt−ω−2νi| |221|171|t+x+2z=2nt+nt−ω−2ωi|284|4t+x+2y=5nt−2nt−ω−2νi| |222|172|2t+x+2z=3nt−ω−2ωi|290|x−2y=nt−2nt−ω+2νi| |223|173|3t+x+2z=4nt−nt−ω−2ωi|291|t−x+2y=nit+ω−2νi| |224|174|4t+x+2z=5nt−nt−ω−2ωi|292|2t−x+2y=nt+ω−2νi| |230|190|x−2z=nt−2nt−ω+2ωi|293|3t−x+2y=2nt−nt−ω−2νi| |231|191|t−x+2z=nit+ω−2ωi|294|4t−x+2y=3nt−2nt+ω−2νi| |232|192|2t−x+2z=nt+ω−2ωi|301|t+x−2y=2nt−3nt−ω+2νi| |233|193|3t−x+2z=2nt−nt+ω−2ωi|302|2t+x−2y=3nt−4nt+ω+2νi| |234|194|4t−x+2z=3nt−2nt+ω−2ωi|303|3t+x−2y=4nt−5nt−ω+2νi| |241|201|t+x−2z=2nt−3nt−ω+2ωi|304|4t+x−2y=5nt−6nt−ω+2νi| |242|202|2t+x−2z=3nt−4nt−ω+2ωi|310|z+2y=3nt−ω−2νi| |243|203|3t+x−2z=4nt−5nt−ω+2ωi|311|t−z−2y=nt−4nt+ωi+2νi| |244|204|4t+x−2z=5nt−6nt−ω+2ωi|312|2t−z−2y=2nt−5nt+ωi+2νi| |250|150|3z=3nt−3ωi|313|3t−z−2y=3nt−6nt+ωi+2νi| |251|161|t−3z=nt−4nt+3ωi|314|4t−z−2y=4nt−7nt+ωi+2νi| |252|162|2t−3z=2nt−5nt+3ωi|321|t+z+2y=nt+2nt−ωi−2νi| |253|163|3t−3z=3nt−6nt+3ωi|322|2t+z+2y=2nt+nt−ωi−2νi| |254|164|4t−3z=4nt−7nt+3ωi|323|3t+z+2y=3nt−ωi−2νi| |261|151|t+3z=nt+2nt−3ωi|324|4t+z+2y=4nt−nt−ωi−2νi| |262|152|2t+3z=2nt+nt−3ωi|330|z−2y=−nt−ωi+2νi| |263|153|3t+3z=3nt−3ωi|331|t−z+2y=nt+ωi−2νi| |264|154|4t+3z=4nt−nt−3ωi|332|2t−z+2y=2nt−nt+ωi−2νi| |270|..|x+2y=nt+2nt−ω−2νi|333|3t−z+2y=3nt−2nt+ωi+2νi| |271|..|t−x−2y=−nt+ω+2νi|334|4t−z+2y=4nt−3nt+ωi−2νi| |272|..|2t−x−2y=nt−4nt+ω+2νi|341|t+z−2y=nt−2nt−ωi+2νi| |273|..|3t−x−2y=2nt−5nt+ω+2νi|342|2t+z−2y=2nt−3nt−ωi+2νi| |274|..|4t−x−2y=3nt−6nt+ω+2νi|343|3t+z−2y=3nt−4nt−ωi+2νi| |281|..|t+x+2y=2nt+nt−ω−2νi|344|4t+z−2y=4nt−5nt−ωi+2νi| Table I. Showing the arguments which result from the combination of the arguments 10, 50 and 150 with the arguments in the first or left-hand column, by addition and subtraction. | | 10 | 50 | 150 | | 10 | 50 | 150 | | 10 | 50 | 150 | |---|----|----|-----|---|----|----|-----|---|----|----|-----| | 1 | 21 | 61 | 161 | 1 | 51 | 11 | | 51 | 102| 202| | | | 11 | 51 | 151 | | 52 | 12 | | 52 | 103| 203| | | 2 | 22 | 62 | 162 | 2 | 53 | 13 | | 53 | 104| 204| | | | 12 | 52 | 152 | | 54 | 14 | | 54 | 110| 210| | | 3 | 23 | 63 | 163 | 3 | 55 | 15 | | 55 | 111| 211| | | | 13 | 53 | 153 | | 56 | 16 | | 56 | 112| 212| | | 4 | 24 | 64 | 164 | 4 | 57 | 17 | | 57 | 113| 213| | | | 14 | 54 | 154 | | 58 | 18 | | 58 | 114| 214| | | 10| 50 | 150| | 10| 61 | 161| | 61 | 111| 241| | | | 0 | -10| | | 62 | 162| | 62 | 112| 242| | | 11| 1 | 21 | | 11| 63 | 163| | 63 | 113| 243| | | | 51 | 151| | | 64 | 164| | 64 | 114| 244| | | 12| 2 | 22 | | 12| 65 | 165| | 65 | 115| 245| | | | 52 | 152| | | 66 | 166| | 66 | 116| 246| | | 13| 3 | 23 | | 13| 67 | 167| | 67 | 117| 247| | | | 53 | 153| | | 68 | 168| | 68 | 118| 248| | | 14| 4 | 24 | | 14| 69 | 169| | 69 | 119| 249| | | | 54 | 154| | | 70 | 170| | 70 | 120| 250| | | 21| 61 | 161| | 21| 71 | 171| | 71 | 121| 251| | | | 1 | 11 | | | 72 | 172| | 72 | 122| 252| | | 22| 62 | 162| | 22| 73 | 173| | 73 | 123| 253| | | | 2 | 12 | | | 74 | 174| | 74 | 124| 254| | | 23| 63 | 163| | 23| 75 | 175| | 75 | 125| 255| | | | 3 | 13 | | | 76 | 176| | 76 | 126| 256| | | 24| 64 | 164| | 24| 77 | 177| | 77 | 127| 257| | | | 4 | 14 | | | 78 | 178| | 78 | 128| 258| | | 30| 70 | 170| | 30| 79 | 179| | 79 | 129| 259| | | | -90| -190| | | 80 | 180| | 80 | 130| 260| | | 31| 101| 201| | 31| 81 | 181| | 81 | 131| 301| | | | 71 | 171| | | 82 | 182| | 82 | 132| 302| | | 32| 102| 202| | 32| 83 | 183| | 83 | 133| 303| | | | 72 | 172| | | 84 | 184| | 84 | 134| 304| | | 33| 103| 203| | 33| 85 | 185| | 85 | 135| 305| | | | 73 | 173| | | 86 | 186| | 86 | 136| 306| | | 34| 104| 204| | 34| 87 | 187| | 87 | 137| 307| | | | 74 | 174| | | 88 | 188| | 88 | 138| 308| | | 41| 81 | 181| | 41| 89 | 189| | 89 | 139| 309| | | | 91 | 191| | | 90 | 190| | 90 | 140| 310| | | 42| 82 | 182| | 42| 91 | 191| | 91 | 141| 311| | | | 92 | 192| | | 92 | 192| | 92 | 142| 312| | | 43| 83 | 183| | 43| 93 | 193| | 93 | 143| 313| | | | 93 | 193| | | 94 | 194| | 94 | 144| 314| | | 44| 84 | 184| | 44| 95 | 195| | 95 | 145| 315| | | | 94 | 194| | | 96 | 196| | 96 | 146| 316| | | 50| 150| | 50| 101| 201| | 101| 147| 317| | Table II. Showing the arguments which, by their combination with the arguments 10, 50, and 150, by addition and subtraction, produce the arguments in the first or left-hand column. | | 10 | 50 | 150 | |---|----|----|-----| | 1 | 11 | 21 | | | 2 | 12 | 22 | | | 3 | 13 | 23 | | | 4 | 14 | 24 | | | 10| 0 | 50 | -10 | | 11| 51 | 1 | 21 | | 12| 52 | 2 | 22 | | 13| 53 | 3 | 23 | | 14| 54 | 4 | 24 | | 21| 61 | 1 | 11 | | 22| 62 | 2 | 12 | | 23| 63 | 3 | 13 | | 24| 64 | 4 | 14 | | 30| 70 | -90| | | 31| 71 | 101| | | 32| 72 | 102| | | 33| 73 | 103| | | 34| 74 | 104| | | 41| 91 | 81 | | | 42| 92 | 82 | | | | 10 | 50 | 150 | |---|----|----|-----| | 1 | 43 | 93 | 83 | | 2 | 44 | 94 | 84 | | 3 | 50 | 10 | 0 | | 4 | 51 | 11 | 1 | | 10| 52 | 12 | 2 | | 11| 53 | 13 | 3 | | 12| 54 | 14 | 4 | | 13| 61 | 21 | 1 | | 14| 62 | 22 | 2 | | 21| 63 | 23 | 3 | | 22| 64 | 24 | 4 | | 23| 70 | 30 | | | 24| 71 | 31 | | | 30| 72 | 32 | | | 31| 73 | 33 | | | 32| 74 | 34 | | | 33| 81 | 41 | | | 34| 82 | 42 | | | 41| 83 | 43 | | | 42| 84 | 44 | | | | 10 | 50 | 150 | |---|----|----|-----| | 90| -30| | | | 91| 41 | | | | 92| 42 | | | | 93| 43 | | | | 94| 44 | | | | 101| 31 | | | | 102| 32 | | | | 103| 33 | | | | 104| 34 | | | | 150| 50 | 10 | 0 | | 151| 51 | 11 | 1 | | 152| 52 | 12 | 2 | | 153| 53 | 13 | 3 | | 154| 54 | 14 | 4 | | 161| 61 | 21 | 1 | | 162| 62 | 22 | 2 | | 163| 63 | 23 | 3 | | 164| 64 | 24 | 4 | | 170| 70 | 30 | | | 171| 71 | 31 | | MDCCCXXXI. The following examples will show the use of the preceding Table, in forming the equations of condition which serve to determine the coefficients of the inequalities of the reciprocal of the radius vector and of the longitude. $$-\frac{d^3 r^3}{dt^2} \frac{1}{r} - \mu \frac{1}{r} + 2 \int dR + r \left( \frac{dR}{dr} \right) = 0$$ \[ r^3 = a^3 \left\{ 1 + 3 e^2 \left( 1 + \frac{e^2}{8} \right) - 3 e \left( 1 + \frac{3}{8} e^2 \right) \cos (nt + \varepsilon - \varpi) + \frac{e^3}{8} \cos (3nt + 3\varepsilon - 3\varpi) \right\} \] \[ \frac{(n-n_i)^2}{n^2} \left\{ (1 + 3 e^2) r_1 - \frac{3 e^2}{2} (r_{11} + r_{21}) \right\} - r_1 + \frac{m_i}{a} q_1 = 0 \] \[ \frac{4(n-n_i)^2}{n^2} \left\{ (1 + 3 e^2) r_2 - \frac{3}{2} e^2 (r_{12} + r_{22}) \right\} - r_2 + \frac{m_i}{a} q_2 = 0 \] \[ \frac{d\lambda}{dt} = \frac{h}{r^2} + \frac{2h}{r} \delta \cdot \frac{1}{r} - \frac{1}{r^3} \int \frac{dR}{d\lambda} dt \] \[ \frac{a^3}{r^2} = 1 + \frac{e^2}{2} + 2 e \left( 1 + \frac{3 e^2}{8} \right) \cos (nt + e - \varpi) + \frac{5 e^3}{2} \cos (2nt + 2\varepsilon - 2\varpi) \] \[ + \frac{13}{4} e^3 \cos (3nt + 3\varepsilon - 3\varpi) \] \[ \frac{a}{r} = 1 + e \left( 1 - \frac{e^2}{8} \right) \cos (nt + \varepsilon - \varpi) + e^2 \cos (2nt + 2\varepsilon - 2\varpi) + \frac{9}{8} e^3 \cos (3nt + 3\varepsilon - 3\varpi) \] \[ \lambda = n \left\{ 1 + 2 r_0 \right\} t + \varepsilon \] \[ + \left\{ 2 \left\{ r_1 + \frac{e^2}{2} (r_{11} + r_{21}) \right\} \right. \] \[ - \frac{m_i}{\mu} \left\{ \left( 1 + \frac{e^2}{2} \right) \frac{an R_1}{(n-n_i)} + \frac{e^2}{n_i} an R_{11} + \frac{e^2 an R_{21}}{(2n-2n_i)} \right\} \frac{n}{(n-n_i)} \sin (nt - n_i t + \varepsilon - \varepsilon_i) \] \[ + \left\{ 2 \left\{ r_2 + \frac{e^2}{2} (r_{12} + r_{22}) \right\} \right. \] \[ - \frac{m_i}{\mu} \left\{ \left( 1 + \frac{e^2}{2} \right) \frac{an R_2}{(n-n_i)} + \frac{2 e^2 an R_{12}}{(n-2n_i)} + \frac{2 e^2 an R_{22}}{(3n-2n_i)} \right\} \frac{n}{2(n-n_i)} \sin (2nt - 2n_i t + \varepsilon - \varepsilon_i) \] In the same way, by means of the Table, all the other coefficients may be found. The great inequality of Jupiter consists of the arguments 155, 174, 213, 273, and 312, the variable part of which is \(2n - 5n_i\), and arises, as is well known, from the introduction of the square of this quantity, which is small, by successive integrations in the denominators of the coefficients of the sines in the expression for the longitude, of which the above named are the arguments. The following are the equations which have reference to these arguments, and which may be found at once by Table II. \[ \frac{(2n-5n_i)^2}{n^2} \left\{ r_{155} - \frac{3}{2} r_{54} + \frac{1}{16} r_4 \right\} - r_{155} + \frac{m_i a}{\mu} q_{155} = 0 \] \[ \frac{(2n-5n_i)^2}{n^2} \left\{ r_{174} - \frac{3}{2} r_{74} \right\} - r_{174} + \frac{m_i a}{\mu} q_{174} = 0 \] \[ \frac{(2n-5n_i)^2}{n^2} \left\{ r_{213} - \frac{3}{2} r_{113} \right\} - r_{213} + \frac{m_i a}{\mu} q_{213} = 0 \] \[ \frac{(2n - 5n_i)^2}{n^3} \left\{ r_{273} - \frac{3}{2} r_{133} \right\} - r_{273} + \frac{m_i a}{\mu} q_{273} = 0 \] \[ \frac{(2n - 5n_i)^2}{n^3} \left\{ r_{312} - r_{312} \right\} + \frac{m_i a}{\mu} q_{312} = 0 \] \[ \delta \lambda = \left\{ 2 \left\{ r_{155} + \frac{1}{2} \left( r_{55} + r_{15} + \frac{9}{8} r_4 \right) \right\} \right. \] \[ - \frac{m_i}{\mu} \left\{ \frac{5naR_{155}}{(2n - 5n_i)} + \frac{5naR_{55}}{(3n - 5n_i)} + \frac{5.5naR_{15}}{4(3n - 4n_i)} + \frac{13.5naR_5}{8.5(n - n_i)} \right\} \frac{n e^3}{(2n - 5n_i)} \sin (2nt - 5n_t + 3\varphi) \] \[ + \left\{ 2 \left\{ r_{174} + \frac{1}{2} (r_{74} + r_{34}) \right\} \right. \] \[ - \frac{m_i}{\mu} \left\{ \frac{4naR_{174}}{(2n - 5n_i)} + \frac{4naR_{74}}{(3n - 5n_i)} + \frac{5.4naR_{34}}{4(4n - 5n_i)} \right\} \frac{n e^2 e_i}{(2n - 5n_i)} \sin (2nt - 5n_t + 2\varphi + \varphi_i) \] \[ + \left\{ 2 \left\{ r_{213} + \frac{1}{2} r_{113} \right\} - \frac{m_i}{\mu} \left\{ \frac{3naR_{213}}{(2n - 5n_i)} + \frac{3naR_{113}}{(3n - 5n_i)} \right\} \right\} \frac{n ee^2}{(2n - 5n_i)} \sin (2nt - 5n_t + \varphi + 2\varphi_i) \] \[ + \left\{ 2 \left\{ r_{273} + \frac{1}{2} r_{133} \right\} - \frac{m_i}{\mu} \left\{ \frac{3naR_{273}}{(2n - 5n_i)} + \frac{3naR_{133}}{(3n - 5n_i)} \right\} \right\} \frac{n e \sin^2 \frac{t_i}{2}}{(2n - 5n_i)} \sin (2nt - 5n_t + \varphi + 2\varphi_i) \] \[ + \left\{ 2 r_{312} - \frac{2m_i na R_{312}}{\mu(2n - 5n_i)} \right\} \frac{n e \sin^2 \frac{t_i}{2}}{(2n - 5n_i)} \sin (2nt - 5n_t + \varphi_i + 2\varphi_i) \] The quantities \( r_{55}, r_{74}, r_{113} \) and \( r_{133} \) have the quantity \( 2n - 5n_i \) in the denominator, rejecting those quantities in the value of \( \delta \lambda \) which have not \( (2n - 5n_i)^2 \) in the denominator. \[ r_{155} = - \frac{4m_i n^3 a R_{155} e^3}{\mu (n - 5n_i)(3n - 5n_i)(2n - 5n_i)} \] \[ r_{174} = - \frac{4m_i n^3 a R_{174} e^2 e_i}{\mu (n - 5n_i)(3n - 5n_i)(2n - 5n_i)} \] \[ r_{213} = - \frac{4m_i n^3 a R_{213} e e_i^2}{\mu (n - 5n_i)(3n - 5n_i)(2n - 5n_i)} \] \[ r_{273} = - \frac{4m_i n^3 a R_{273} e \sin^2 \frac{t_i}{2}}{\mu (n - 5n_i)(3n - 5n_i)(2n - 5n_i)} \] \[ r_{312} = - \frac{4m_i n^3 a R_{312} e_i \sin^2 \frac{t_i}{2}}{\mu (n - 5n_i)(3n - 5n_i)(2n - 5n_i)} \] \[ \delta \lambda = \left\{ 2 r_{155} + r_{55} - \frac{5m_i na R_{155}}{\mu(2n - 5n_i)} \right\} \frac{n e^3}{(2n - 5n_i)} \sin (2nt - 5n_t + 3\varphi) \] \[ + \left\{ 2 r_{174} + r_{74} - \frac{4m_i na R_{174}}{\mu(2n - 5n_i)} \right\} \frac{n e^2 e_i}{(2n - 5n_i)} \sin (2nt - 5n_t + 2\varphi + \varphi_i) \] \[ + \left\{ 2 r_{213} + r_{113} - \frac{3 m_i a n R_{213}}{\mu (2n-5n_i)} \right\} \frac{n e e_i^2}{(2n-5n_i)} \sin (2nt - 5n_i t + \varpi + 2\nu_i) \] \[ + \left\{ 2 r_{273} + r_{193} - \frac{3 m_i a n R_{373}}{\mu (2n-5n_i)} \right\} \frac{n e \sin^2 \frac{t_i}{2}}{(2n-5n_i)} \sin (2nt - 5n_i t + \varpi + 2\nu_i) \] \[ + \left\{ 2 r_{312} - \frac{2 m_i a n R_{312}}{\mu (2n-5n_i)} \right\} \frac{n e \sin^3 \frac{t_i}{2}}{(2n-5n_i)} \sin (2nt - 5n_i t + \varpi_i + 2\nu_i) \] The coefficients of the terms in the development of \( R \) multiplied by the cubes of the eccentricities, as regards the quantities \( b_5 \) and \( b_7 \), (they also contain the quantities \( b_3 \)) may be found by changing \( b_3 \) into \( b_5 \), in the terms in \( R \) multiplied by the eccentricities, and multiplying the result by \[ - \frac{9}{8} \frac{(a^2 e^2 + a_i^2 e_i^2)}{a_i^3} + \frac{3}{8} \frac{a^2}{a_i^3} e^2 \cos 2x - \frac{3}{4} \frac{a}{a_i} \left( e^2 + e_i^2 + 2 \sin^2 \frac{t_i}{2} \right) \cos t + \frac{9}{16} \frac{a}{a_i} e^2 \cos (t + 2x) \] and changing \( b_5 \) into \( b_7 \), in the terms in \( R \) multiplied by the squares and products of the eccentricities, and multiplying the result by \[ - \frac{5}{6} \text{ and } - \frac{2 a^2}{a_i^3} e \cos x + \frac{3 a}{a_i} e \cos (t - x) + \frac{3 a}{a_i} e_i \cos (t + z) - \frac{a}{a_i} e \cos (t + x) \] and changing \( b_3 \) into \( b_5 \) in the terms in \( R \) multiplied by the squares and products of the eccentricities, and multiplying the result by \(- \frac{3}{4}\) and the same quantity. Thus \( R_{155} \) results from the combination of the arguments \( 51 \times 14, 50 \times 15, 61 \times 16, 10 \times 55, \) and \( 11 \times 54. \) \( 51 \times 14 \) gives \( + \frac{3}{32} \frac{a}{a_i} \left\{ \frac{3 a}{4 a_i^2} b_{5,3} - \frac{a^2}{2 a_i^3} b_{5,4} - \frac{a}{4 a_i^2} b_{5,5} \right\} \) 50 \times 15 \text{ gives } + \frac{3}{16} \frac{a^2}{a_i^3} \left\{ \frac{3a}{4a_i^3} b_{5,4} - \frac{a^2}{2a_i^3} b_{5,5} - \frac{a}{4a_i^3} b_{5,6} \right\} 61 \times 16 \text{ gives } + \frac{9}{32} \frac{a}{a_i} \left\{ \frac{3a}{4a_i^3} b_{5,5} - \frac{a^2}{2a_i^3} b_{5,6} - \frac{a}{4a_i^3} b_{5,7} \right\} R_{55} = - \frac{a}{16a_i^2} b_{5,4} - \frac{a^2}{8a_i^3} b_{5,5} - \frac{3a}{16a_i^2} b_{5,6} - \frac{3.9}{2.4.4a_i^3} b_{5,9} + \frac{3.3}{2.4a_i^4} b_{5,3} - \frac{3a^2}{2.4.2a_i^5} (2a^2 - 3a_i^2) b_{5,5} - \frac{3}{2.4a_i^4} b_{5,6} - \frac{3a^2}{2.4.4a_i^3} b_{5,7} changing \(b_3\) into \(-\frac{3}{4} b_5\), and \(b_5\) into \(-\frac{5}{6} b_7\), we have \[ \begin{align*} &\frac{3a}{64a_i^3} b_{5,4} + \frac{3a^2}{32a_i^3} b_{5,5} + \frac{9}{64a_i^2} b_{5,6} + \frac{3.9.5}{2.4.4.6a_i^3} a^2 b_{7,3} - \frac{3.3.5}{2.4.6a_i^4} a^3 b_{7,4} \\ &+ \frac{3.5}{2.4.2} \frac{a^2}{a_i^3} (2a^2 - 3a_i^2) b_{7,5} + \frac{3.5}{2.4.6a_i^4} a^3 b_{7,6} + \frac{3.5}{2.4.4.6a_i^3} a^2 b_{7,7} \end{align*} \] \[ = \frac{3}{64a_i^3} b_{5,4} + \frac{3a^2}{32a_i^3} b_{5,5} + \frac{9}{64a_i^2} b_{5,6} + \frac{3.5}{8.6a_i^3} \left\{ \frac{a^2 + a_i^2}{a_i^2} b_{7,3} - \frac{a}{a_i} b_{7,4} - \frac{a}{a_i} b_{7,6} \right\} \] \[ + \frac{3.9.5}{8.4.6a_i^3} \left\{ b_{7,3} - b_{7,5} \right\} - \frac{3.5}{4.6a_i^4} \left\{ b_{7,4} - b_{7,6} \right\} - \frac{3.5}{32.6a_i^3} \left\{ b_{7,5} - b_{7,7} \right\} \] and since \(b_{5,5} = \frac{a^2 + a_i^2}{a_i^3} b_{7,5} - \frac{a}{a_i} b_{7,4} - \frac{a}{a_i} b_{7,6}\) \[ 4b_{5,4} = \frac{5}{2} \frac{a}{a_i} \left\{ b_{7,3} - b_{7,5} \right\} \quad 5b_{5,5} = \frac{5}{2} \frac{a}{a_i} \left\{ b_{7,4} - b_{7,6} \right\} \quad 6b_{5,5} = \frac{5}{2} \frac{a}{a_i} \left\{ b_{7,5} - b_{7,7} \right\} \] \[ = \frac{3}{64a_i^3} b_{5,4} + \frac{3a^2}{32a_i^3} b_{5,5} + \frac{9}{64a_i^2} b_{5,6} + \frac{15}{48a_i^3} b_{5,5} + \frac{27}{24a_i^3} b_{5,4} - \frac{15}{12a_i^3} b_{5,5} - \frac{3}{16a_i^2} b_{5,6} \] \[ = \frac{75}{64a_i^3} b_{5,4} - \frac{27}{32a_i^3} b_{5,5} - \frac{3}{64a_i^3} b_{5,6} \] \(R_{54} = - \frac{a}{16a_i^2} b_{5,3} - \frac{a^2}{8a_i^3} b_{5,4} - \frac{3a}{16a_i^2} b_{5,5} - \frac{3.9a^2}{2.4.4a_i^3} b_{5,2} + \frac{3.3a^3}{2.4a_i^4} b_{5,3} - \frac{3a^2}{2.4.2a_i^5} (2a^2 - 3a_i^2) b_{5,4} - \frac{3a^3}{2.4a_i^4} b_{5,6} - \frac{3a^2}{2.4.4a_i^3} b_{5,6}\) Similar changes and reductions give \[ \frac{57}{64a_i^3} b_{5,3} - \frac{19}{32a_i^3} b_{5,4} - \frac{a}{64a_i^3} b_{5,5} \] \[ R_{155} = \frac{3}{32} \frac{a}{a_i} \left\{ \frac{3}{4} \frac{a}{a_i^3} b_{3,3} - \frac{a^2}{2a_i^3} b_{5,4} - \frac{a}{4a_i^3} b_{5,5} \right\} + \frac{3}{16} \frac{a^2}{a_i^3} \left\{ \frac{3}{4} \frac{a}{a_i^3} b_{5,4} - \frac{a^2}{2a_i^3} b_{5,5} - \frac{a}{4a_i^3} b_{5,6} \right\} \\ + \frac{9}{32} \frac{a}{a_i} \left\{ \frac{3}{4} \frac{a}{a_i^2} b_{5,5} - \frac{a^2}{2a_i^3} b_{5,6} - \frac{a}{4a_i^2} b_{5,7} \right\} - \frac{a^2}{a_i^3} \left\{ \frac{75}{64} \frac{a}{a_i^3} b_{5,4} - \frac{27}{32} \frac{a^2}{a_i^3} b_{5,5} - \frac{3}{64} \frac{a}{a_i^2} b_{5,6} \right\} \\ + \frac{3}{2} \frac{a}{a_i} \left\{ \frac{57}{64} \frac{a}{a_i^2} b_{5,5} - \frac{19}{32} \frac{a^2}{a_i^3} b_{5,4} - \frac{a}{64} \frac{a}{a_i^2} b_{5,5} \right\} \] and adding the terms which depend upon \( b_3 \), \[ R_{155} = \frac{a}{96a_i^3} b_{3,4} - \frac{a^2}{16a_i^3} b_{3,5} + \frac{a}{12a_i^2} b_{3,6} + \frac{45}{32} \frac{a^2}{a_i^3} b_{5,3} - \frac{63}{32} \frac{a^3}{a_i^4} b_{5,4} + \frac{(21a_i^2 + 96a^2)}{128a_i^5} a^2 b_{5,5} \\ - \frac{9}{64} \frac{a^3}{a_i^4} b_{5,6} - \frac{9}{128} \frac{a^2}{a_i^3} b_{5,7} \] which may be still further reduced. \( R_{174}, R_{213}, R_{273}, \) and \( R_{312} \) may be obtained in a similar manner. The following Table shows the arguments which, by their combination with the arguments 1, 2, 3, 12, 13, 31, 32, 64, 65, 73, 74, 112, and 113, by addition and subtraction produce the arguments 155, 174, 213, 273, and 312. | | 1 | 2 | 3 | 12 | 13 | 31 | 32 | 64 | 65 | 73 | 74 | 112 | 113 | |---|----|----|----|----|----|----|----|----|----|----|----|-----|-----| | 155 | 154 | 153 | 152 | 53 | 52 | ..... | ..... | 11 | ..... | 192 | 191 | ..... | ..... | | 174 | 173 | 172 | 171 | 72 | 71 | 53 | 52 | ..... | ..... | 11 | ..... | 192 | 191 | | 213 | 212 | 211 | -210 | 111 | ..... | 72 | 71 | ..... | ..... | 11 | ..... | -10 | | 273 | 272 | 271 | -270 | 131 | ..... | ..... | ..... | -291 | -292 | 330 | ..... | ..... | | 312 | 311 | 310 | -311 | ..... | ..... | 131 | ..... | ..... | ..... | 330 | ..... | ..... | If \[ r \delta \cdot \frac{1}{r} = r'_1 \cos (nt - n_t t) + r'_2 \cos (2nt - 2n_t t) + r'_3 \cos (3nt - 3n_t t) + er'_1 \cos (nt - 2n_t t + \varpi) \\ + er'_2 \cos (2nt - 3n_t t + \varpi) + \&c. \] \[ r_t \delta \cdot \frac{1}{r_t} = r'_1 \cos (nt - n_t t) + r'_2 \cos (2nt - 2n_t t) + r'_3 \cos (3nt - 3n_t t) + er'_1 \cos (nt - 2n_t t + \varpi) \\ + er'_2 \cos (2nt - 3n_t t + \varpi) + \&c. \] \[ \delta \lambda = \lambda_1 \sin (nt - n_t t) + \lambda_2 \sin (2nt - 2n_t t) + \lambda_3 \sin (3nt - 3n_t t) + e\lambda'_1 \sin (nt - 2n_t t + \varpi) \\ + e\lambda'_2 \sin (2nt - 3n_t t + \varpi) + \&c. \] \[ \delta \lambda_i = \lambda_{i1} \sin (nt - n_i t) + \lambda_{i2} \sin (2nt - 2n_i t) + \lambda_{i3} \sin (3nt - 3n_i t) + e \lambda_{i12} \sin (nt - 2n_i t + \varphi) + e \lambda_{i13} \sin (2nt - 3n_i t + \varphi) + \&c. \] Supposing that the arguments 1, 2, 3, 12, 13, 31, 32, 64, 65, 73, 74, 112, 113, 155, 174, 213, 273, and 312 are alone sensible in \( \delta \frac{1}{r} \), \( \delta \lambda \), \( \delta \frac{1}{r_i} \) and \( \delta \lambda_i \) the coefficient of \( e^3 \cos (2nt - 5n_i t + 3\varphi) \) in the expression for \( \delta R \) or \( \delta R_{155} \) \[ = -\frac{1}{2} \left\{ \frac{a d R_{154}}{da} + \frac{a d R_{156}}{da} \right\} r'_1 + \left\{ 2R_{154} - 3R_{156} \right\} \left\{ \lambda_1 - \lambda_{11} \right\} - \frac{1}{2} \left\{ \frac{a d R_{153}}{da} + \frac{a d R_{157}}{da} \right\} r'_2 \\ + \frac{1}{2} \left\{ 3R_{153} - 7R_{157} \right\} \left\{ \lambda_2 - \lambda_{12} \right\} - \frac{1}{2} \left\{ \frac{a d R_{159}}{da} + \frac{a d R_{158}}{da} \right\} r'_3 \\ + \left\{ R_{152} - 4R_{158} \right\} \left\{ \lambda_3 - \lambda_{13} \right\} - \frac{a d R_{53}}{2da} r'_{12} + \frac{3}{2} R_{53} \left\{ \lambda_{12} - \lambda_{112} \right\} \\ - \frac{a d R_{59}}{2da} r'_{13} + R_{52} \left\{ \lambda_{13} - \lambda_{113} \right\} - \frac{a d R_{64}}{2da} r'_{11} + 2R_{64} \left\{ \lambda_{11} - \lambda_{111} \right\} \\ - \frac{a d R_{65}}{2da} r'_{10} - \frac{5}{2} R_{65} \left\{ \lambda_{10} - \lambda_{110} \right\} - \frac{a d R_{102}}{2da} r'_{73} - R_{102} \left\{ \lambda_{73} - \lambda_{173} \right\} - \frac{a d R_{105}}{2da} r'_{74} \\ - \frac{1}{2} R_{109} \left\{ \lambda_{74} - \lambda_{774} \right\} - \frac{a d R_{0}}{da} r'_{155} - \frac{1}{2} \left\{ \frac{a d R_{154}}{da} + \frac{a d R_{156}}{da} \right\} r'_1 \\ - \frac{1}{2} \left\{ \frac{a d R_{153}}{da} + \frac{a d R_{157}}{da} \right\} r'_2 - \frac{1}{2} \left\{ \frac{a d R_{159}}{da} + \frac{a d R_{158}}{da} \right\} r'_3 - \frac{a d R_{53}}{2da} r'_{12} \\ - \frac{a d R_{59}}{2da} r'_{13} - \frac{a d R_{64}}{2da} r'_{11} - \frac{a d R_{65}}{2da} r'_{10} - \frac{a d R_{102}}{2da} r'_{73} - \frac{a d R_{105}}{2da} r'_{74} - \frac{a d R_{0}}{da} r'_{155} \] In the same way the expression for \( \delta R_{174}, \delta R_{213}, \delta R_{273}, \) and \( \delta R_{312} \) may be found from the preceding Table. If \( a < a_i \) and \[ \left\{ 1 - \frac{a}{a_i} \cos \theta + \frac{a^2}{a_i^2} \right\}^{-\frac{1}{2}} = \frac{1}{2} b_{1,0} + b_{1,1} \cos \theta + b_{1,2} \cos 2\theta + \&c. \] \[ \left\{ 1 - \frac{a}{a_i} \cos \theta + \frac{a^2}{a_i^2} \right\}^{-\frac{5}{2}} = \frac{1}{2} b_{3,0} + b_{3,1} \cos \theta + b_{3,2} \cos 2\theta + \&c. \] \[ R = m_i \left\{ \frac{a}{a_i^2} \left( \cos^2 \frac{t}{2} - \frac{e^2 + e_i^2}{2} \right) \right\} \cos (nt - n_i t) + \frac{3m_i a}{2a_i^3} e \cos (nt - n_i t - \varphi) + \frac{m_i a}{a_i^2} e \cos (2nt - n_i t - \varphi) + \frac{2m_i a}{2a_i^2} e_i \cos (nt - 2n_i t + \varphi) \] * The notation is slightly changed from that used before. † \( e \) and \( e_i \) which accompany \( nt \) and \( n_i t \) are omitted for convenience. \[ \begin{align*} &+ \frac{m_i}{8a_i^3} e^2 \cos (nt + n_i t - 2\varpi) + \frac{3m_i}{8a_i^3} e^2 \cos (3nt - n_i t - 2\varpi) - \frac{3m_i}{a_i^3} ee_i \cos (2n_i t - \varpi - \varpi_i) \\ &+ \frac{m_i}{a_i^3} ee_i \cos (2nt - 2n_i t - \varpi + \varpi_i) + \frac{27}{8} \frac{m_i}{a_i^3} e_i^2 \cos (nt - 3n_i t + 2\varpi) \\ &+ \frac{m_i}{8a_i^3} e_i^2 \cos (nt + n_i t - 2\varpi) + \frac{m_i}{a_i^3} \sin^2 \frac{\theta_i}{2} \cos (nt + n_i t - 2\varphi_i) \\ &+ m_i \left\{ -\frac{b_{1,i}}{2a_i} + \frac{a}{4a_i^2} \sin^2 \frac{\theta_i}{2} \left( b_{3,i} - 1 + b_{3,i} + 1 \right) \right\} \cos i(nt - n_i t) \\ &+ \frac{a(e^2 + e_i^2)}{16a_i^3} \left( (3i - 1)b_{3,i} - 1 - (3i + 1)b_{3,i} + 1 \right) \cos i(nt - n_i t) \\ &+ m_i \left\{ -\frac{a}{4a_i^2} b_{3,i} - 1 - \frac{a^2}{2a_i^3} b_{3,i} + \frac{3a}{4a_i^2} b_{3,i} + 1 \right\} e \cos \left( i(nt - n_i t) + nt - \varpi \right) \\ &+ m_i \left\{ \frac{3}{4} \frac{a}{a_i^2} b_{3,i} - 1 - \frac{1}{2a_i} b_{3,i} - \frac{a}{4a_i^2} b_{3,i} + 1 \right\} e_i \cos \left( i(nt - n_i t) + n_i t - \varpi \right) \\ &+ m_i \left\{ -\frac{(2+i)}{16} \frac{a}{a_i^2} b_{3,i} - 1 - \frac{(1+i)}{2} \frac{a^2}{a_i^3} b_{3,i} \right. \\ &\quad \left. + \frac{(8+9i)}{16} \frac{a}{a_i^2} b_{3,i} + 1 \right\} e^2 \cos \left( i(nt - n_i t) + 2nt - 2\varpi \right) \\ &+ m_i \left\{ \frac{(3+9i)}{8} \frac{a}{a_i^2} b_{3,i} - 1 - \frac{i}{a_i} b_{3,i} \right. \\ &\quad \left. - \frac{(1+i)}{8} \frac{a}{a_i^2} b_{3,i} + 1 \right\} ee_i \cos \left( i(nt - n_i t) + nt + n_i t - \varpi - \varpi_i \right) \\ &+ m_i \left\{ -\frac{(1+3i)}{8} \frac{a}{a_i^2} b_{3,i} - 1 \right. \\ &\quad \left. + \frac{3(1+i)}{8} \frac{a}{a_i^2} b_{3,i} + 1 \right\} ee_i \cos \left( i(nt - n_i t) + nt - n_i t - \varpi + \varpi_i \right) \\ &+ m_i \left\{ \frac{(8-9i)}{16} \frac{a}{a_i^2} b_{3,i} - 1 + \frac{(1-i)}{2a_i} b_{3,i} \right. \\ &\quad \left. - \frac{(2-i)}{16} \frac{a}{a_i^2} b_{3,i} + 1 \right\} e_i^2 \cos \left( i(nt - n_i t) + 2n_i t - 2\varpi_i \right) \\ &- m_i \left\{ \frac{a}{2a_i^2} b_{3,i} - 1 \sin^2 \frac{\theta_i}{2} \cos \left( i(nt - n_i t) + 2n_i t - 2\varphi_i \right) \right\} \end{align*} \] \(i\) being every whole number, positive and negative and zero, and observing that \(b_{m,n} = b_{m,-n}\). Considering only the terms multiplied by \(e\) and \(e_i\), \[ r \left( \frac{dR}{dr} \right) = -\frac{3m_i}{2a_i^2} e \cos (nt - \varpi) + \frac{m_i}{2a_i^2} e \cos (2nt - n_i t - \varpi) \] \[ \frac{a}{r} = -\frac{m_i}{\mu} \left( \frac{n^2}{(3n-n_i)(n-n_i)} \right) \left\{ \frac{2n}{2n-n_i} + \frac{1}{2} \right\} \frac{a^2}{a_i^2} e \cos (2nt-n_i t - \varpi) \] \[ -\frac{m_i}{\mu} \frac{3n^2}{2(n-n_i)(n+n_i)} \frac{a^2}{a_i^2} e \cos (n_i t - \varpi) \] \[ + \frac{m_i}{\mu} \frac{n^2}{n_i(2n-2n_i)} \left\{ \frac{2n}{(n-2n_i)} + 1 \right\} \frac{a^2}{a_i^2} e_i \cos (nt-2n_i t + \varpi_i) \] \[ + \sum \frac{n^2}{(i(n-n_i)+2n)} i(n-n_i) \left\{ \frac{3(i(n-n_i)+n)}{2n^2} \right\} 2r_i^* \] \[ -\frac{m_i}{\mu} \left\{ \frac{2(1+i)n}{i(n-n_i)+n} \left[ -\frac{a^2}{4a_i^2} b_{3,i-1} - \frac{a^3}{2a_i^3} b_{3,i} + \frac{3a^2}{4a_i^2} b_{3,i+1} \right] \right\} \] \[ -\frac{i}{4} \frac{a^2}{a_i^2} b_{3,i-1} + \frac{(1+2i)}{2} \frac{a^3}{a_i^3} b_{3,i} - \frac{3i}{4} \frac{a^2}{a_i^2} b_{3,i+1} \right\} e \cos \left( i(nt-n_i t) + nt - \varpi \right) \] \[ + \frac{m_i}{\mu} \sum \frac{n^2}{(1-i)(n-n_i)((i+1)(n-n_i)+2n)} \left\{ \frac{2in}{i(n-n_i)+n_i} \left[ \frac{3a^2}{4a_i^2} b_{3,i-1} \right] \right\} \] \[ -\frac{a}{2a_i} b_{3,i} - \frac{a^2}{4a_i^2} b_{3,i+1} \right\} - \frac{3(1+i)}{4} \frac{a^2}{a_i^2} b_{3,i-1} \] \[ + \frac{ia}{a_i} b_{3,i} + \frac{(1-i)}{4} \frac{a^3}{a_i^3} b_{3,i+1} \right\} e_i \cos \left( i(nt-n_i t) + n_i t - \varpi_i \right) \] \[ \lambda = -\left\{ \frac{3n^2}{2n_i^2} + \frac{n^2}{n_i(n-n_i)} \frac{m_i}{\mu} \right\} \frac{a^2}{a_i^2} e \sin (n_i t - \varpi) \] \[ -\left\{ \frac{n^2}{(2n-n_i)^2} + \frac{n^2}{(2n-n_i)(n-n_i)} \right\} \frac{m_i}{\mu} \frac{a^2}{a_i^2} e \sin (2nt-n_i t - \varpi) \] \[ -\frac{2n^2}{(n-2n_i)^2} \frac{m_i}{\mu} \frac{a^2}{a_i^2} e_i \sin (nt-2n_i t + \varpi_i) \] \[ * r_i being the coefficient of \cos \left( i(nt-n_i t) \right) in the expression for \frac{a}{r}. \] \[ + \sum_{i} \frac{n}{(n-n_i) + n} \left\{ 2 \left( r^* + \frac{r_i}{2} \right) - \frac{m_i n_i}{\mu (i(n-n_i) + n)} \left( -\frac{a^2}{4a_i^3} b_{3,i-1} - \frac{a^3}{2a_i^5} b_{3,i} \right) \right. \\ + \frac{3a^2}{4a_i^3} b_{3,i+1} \right\} e \sin \left( i(n t - n_i t) + n t - \varpi \right) \\ + \sum_{i} \frac{n}{(n-n_i) + n_i} \left\{ 2 r^* - \frac{m_i n_i}{\mu (i(n-n_i) + n)} \left( \frac{3}{4} \frac{a^2}{a_i^2} b_{3,i-1} - \frac{a}{2a_i} b_{3,i} \right) \right. \\ - \frac{a^2}{4a_i^3} b_{3,i+1} \right\} e_i \sin \left( i(n t - n_i t) + n_i t - \varpi_i \right) \] If \( a > a_p \), and \[ \begin{align*} \left\{ 1 - \frac{a_i}{a} \cos \theta + \frac{a_i^2}{a^2} \right\}^{-\frac{1}{2}} &= \frac{1}{2} b_{1,0} + b_{1,1} \cos \theta + b_{1,2} \cos 2 \theta + &c. \\ \left\{ 1 - \frac{a_i}{a} \cos \theta + \frac{a_i^2}{a^2} \right\}^{-\frac{3}{2}} &= \frac{1}{2} b_{3,0} + b_{3,1} \cos \theta + b_{3,2} \cos 2 \theta + &c. \end{align*} \] the value of \( R \) may be easily inferred from the value which it has in the former case. Considering only the terms multiplied by the eccentricities \[ \begin{align*} r \left( \frac{dR}{dr} \right) &= -\frac{3m_i}{2} \frac{a}{a_i^2} e \cos (nt - \varpi) + \frac{m_i}{2} \frac{a}{a_i^2} e \cos (2nt - n_i t - \varpi) \\ &+ \frac{m_i}{2} \frac{a}{a_i^2} e_i \cos (nt - 2n_i t + \varpi_i) \\ &+ m_i \sum \left\{ -\frac{i}{4} \frac{a_i}{a^2} b_{3,i-1} + \frac{(1+2i)}{2a} b_{3,i} \right. \\ &\quad - \frac{3i}{4} \frac{a_i}{a^2} b_{3,i+1} \right\} e \cos \left( i(n t - n_i t) + nt - \varpi \right) \\ &+ m_i \sum \left\{ -\frac{3}{4} \frac{(1+i)}{a^2} b_{3,i-1} + \frac{i a_i^2}{a^3} b_{3,i} \right. \\ &\quad + \frac{(1-i)}{4} \frac{a_i}{a^2} b_{3,i+1} \right\} e_i \cos \left( i(n t - n_i t) + n_i t - \varpi_i \right) \end{align*} \] All these expressions are to a certain extent arbitrary, on account of the equation which connects \( b_{3,i-1} \), \( b_{3,i} \), and \( b_{3,i+1} \) \[ \frac{(2i+1)}{2} \frac{a}{a_i} b_{3,i+1} = \frac{i(a^2 + a_i^2)}{a_i^3} b_{3,i} - \frac{(2i-1)}{2} \frac{a}{a_i} b_{3,i-1} \] † \( r^* \) being the coefficient of the cosine of the same argument in the expression for \( \frac{a}{r} \) and excluding the case of \( i = 0 \). The reader is requested to make the following corrections. Page 50, line 4, read \( q_6 = -\frac{3a}{2a_i^2} + \frac{3}{2} \frac{a}{a_i^2} b_{3,0} - \frac{a^3}{2a_i^3} b_{3,1} + \frac{a}{4a_i^3} b_{3,2} \) Page 53, line 3, read \( m_i \left\{ \frac{2a^3}{a_i^3} b_{3,0} - \frac{5}{4} \frac{a^2}{a_i^2} b_{3,1} \right\} \) Page 247, line 1, read \( \lambda = n t \) \[ + \lambda_1 \sin 2t \] \[ + e \lambda_2 \sin x \] \[ + e \lambda_3 \sin (2t - x) \] \[ + e \lambda_4 \sin (2t + x) \] \[ + e_i \lambda_5 \sin z \quad \text{&c., &c.} \] for \( \lambda = n t \) \[ + \lambda_1 \cos 2t \] \[ + e \lambda_2 \cos x \quad \text{&c., &c.} \) Page 254, line 1, read \( -\frac{3}{2} e^2 e_i \cos (2t + 2x + z) \) Page 260, line 6, read \( + \left\{ 3 - \frac{15}{2} \right\} ee_i \cos (x - z - 2y) \) Page 262, line 6, read \( -\frac{15}{32} ee_i^3 \cos (2t + x - 3z) \) Page 265, line 1, read \( + \frac{25}{64} \frac{a^2}{a_i^3} e^3 e_i \cos (2t + 3x + z) + \frac{3}{32} \frac{a^2}{a_i^3} e^3 e_i \cos (3x - z) \) Page 274, line 6, read \( + \left\{ 2r_3 + r_1 - \frac{9}{2(2 - m - c)} \right\} \text{&c.} \) Page 274, line 7, read \( + \left\{ 2r_4 + r_1 - \frac{3}{2(2 - m + c)} \right\} \text{&c.} \) Page 291, line 9, read \( + \frac{3}{16} \frac{a}{a_i^2} e_i^3 \cos (t + 2z) \) Page 294, line 20, read \( + \frac{m_i a}{2a_i^3} \cos (2nt - n_i t - \varpi) + \frac{2m_i a}{a_i^3} e_i \cos (nt - 2n_i t + \varpi_i) \)