JUPITER. 



Explanation of the Talks of Jupiter, 



The firft Table contains the epochs of the mean lonoritude 

 •of the aphelion and node of Jupiter; together with the 

 Ai -^'umeiitsof the equations expreiled by dividing the circle 

 10,000 eqi;al parts for Arguments II, III, IV, V ; and 

 izoo equal parts for Arguments VI, VII, VIII, 

 ' V Table II. cohtains the mean motions of the fame for 

 \ jars. Tabic III. contains the mean motions for months ; 

 that is, for the beginning of each month reckoned from the 

 beginning of the year; Table IV. contains the mean mo- 

 tions for days. Table V. contains the me.ui motions for 

 hours and minutes. Table VI. contains the groat equa- 

 tion of Jupiter for every to years. Table VII. contains 

 the equation of Jupiter's orbit for the year 17^3, with 

 the fecular variation. Table VIII. contains the equa- 

 tion of the orbit corrjfponding to Argument II. Table 

 IX. contains the equation anfwering to Argument III. 

 Table X. contains the equation anfwering toArgument IV. 

 Table XI. contains the equation anfwering to Argument V. 

 Table XII. contains the equation anfwering to Argument 

 VI. Table XIII. contains the equation anfwering to Ar- 

 gument VII. Table XIV. contains the equation anfwering 

 to Argument VIII. Table XV. contains the equation an- 

 fwering to Argument IX. Table XVI. contains the ccjua- 

 tion anfwering to Argument VII. — Argument VIII. Table 

 XVII. contains the equation anfwering to Argument VII. + 

 Argument VIII. Table XVIII. contains the radius veftor 

 i)f the orbit of Jupiter for 1750, with the fecular variation; 

 the mean diiance of the earth from the fun being unity. 

 Table XIX. contains the equation of the radius veftor an- 

 fwering to Argument II. Table XX. contains the equation 

 anfwering to Argument IV. Table XXI. contains the 

 equation anfwering to Argument V. Table XXII. con- 

 tains the hehocentric latitude of Jupiter for the year 1750, 

 with the fecular variation. Table XXIII. contains the re- 

 duAion to the ecliptic ; with the log. cofine of the heliocen- 

 tric latitude. 



All the above mentioned equations of the motion of Ju- 

 piter, the equations of the orbit e.xcepted, arife from the at- 

 traction of Saturn, and depend upon the relative Situation of 

 Jupiter and Saturn. The theory of thcfe equations was 

 given by M. de la Place in the Memoirs of 1786, from the 

 theory of gravity, and they were computed by M. de 

 JLambre. 



Let a exprefs the number of years from 1750, S the 

 mean longitude of Saturn, / the mean longitude of Jupiter : 

 then the great equation of Jupiter is, 



- (20' 49", J -ax. o".o42733) x fm. (ji' - 2/ 

 + 5' .U'«" - " X 58 '.88). 



The great equation of Saturn arifing from the attraftiou 

 cf Jupiter, is alfo found to be (Mem. 1785), 



- (48-44" -rtx o".i) X fin. (5.?- 2/ +5:" 34' 8"- 



a X 58".8S). 

 The period of thefe two equations is 918.76 years. By 

 way of diilinction, thcfe are called ihe great equations of Ju- 

 piter and Saturn : thefe two equations of Jupiter and Sa- 

 turn are very nearly in the ratio of 3 to 7. Now it is mani- 

 fell, that when 55" — 1I — ax 58'. 88 = 360 , the equation 

 riiuft then begin again, and therefore this will determine the 

 period; now 55 — 2I — a x jS '.88 uicreafes i4lo".6 in a 



, . j6o'=: 1206000'' „ , 



common year; therctore " ■ — ; = 01070 



^ 1410.6 



years, the period in which thefe equations return again the 

 lame. 



Thefe equations (hew the rcafon, why the pretnt mean 



motions of .lupiter at^d Saturn differ from the true mean mO' 

 tion?, as the latter cannot be determined but by taking in 

 the above period. We have the apparent mean motion of 

 Saturn, in a common year, by addiiig to its mean annual 

 motion, the quantity by which the great equation varies in 

 that time, and this quantity is very nearly rr — (48'44" — 

 a X o".i) X fin. 2,5'3l" X cof. (56' — 2 / -)- J ' 34' 8'' 

 — ax 5S".S8) ; and tlie apparent mean motion of Jupiter is 

 this quantity with a contrary fign, diminidied in the ratio oj 

 7 to 3, and added to his mean annual motion. 



Now the fluxion of the cofine of any quantity = fine x 

 flux, arc, therefore when cof. = a maximum, fine x flux, 

 arc = o, or the fine of the arc =: o, or the arc = o ; there- 

 fore when the above quantity =3 a maximum, j5 — 2/ -(- 

 5 34'8" — rt X 58 '.88 =0, which happened in the year 

 I yf)o. At that time, the apparent annual motion of Saturn 

 was iefs than the true, by 20". i, and that of Jupiter greater 

 by 8".6 ; fince that time their apparent mean motions ha'.o 

 been approaching to their true mean motions, and in I78<j 

 their apparent and trwe mean motions were equal. Tliefe rc- 

 fults fliew, why, in comparing the modern with the ancient 

 obfervations, the mean motion of Saturn appears to be re- 

 tarded, and that of Jupiter accelerated ; and that from a com- 

 parifoh of the modern obfervations, the mean motion of Sa- 

 turn appears to be accelerated, and that of Jupiter to be re- 

 tarded. 



Now if H be the heliocentric mean longitude of Jupiter; 

 h that of Saturn, corrected by the above equations rcfpec- 

 tively, computed from the equinox of the j'car 1750; the 

 other equations of the motion of Jupiter from the action of 

 Saturn, are, 



- I 22 '.7 fin. (H- h) + V24".?fin. 2 [H-Ij) + 17" 



fin. 3 // - /•) -!- 3 .9 fin. 4 ^'H - h) 

 + 2'i8'.4fin. ( //- 2h - \y 33' 7" + fl x I3".7) 

 -f I 27.4 fin. {2H — T,h — 61 59 48 4- a x 21.0) 

 + 2 47.0 fin. (3// - 5/j 4- ))- J9 21 + a x 43 .0) 



[6.0 r.u. (3// 



I 19) 



+ 12.S fin. (3// — 2/) - 8 30 15) 



— 13.0 fin. (^A — //— 5831 o; 

 f II 6 fin. (/ji- 4 J 4') 



-I- lo.o fin. (4// — 5/j 4. 4,- 16 32 ) 



— 5.4 fin. [2H — A f 16 I 27 ) 



Now the fum of the maxima of all thefe equations, is 

 II' 56 '.3; and in the Tables computed for thefe equa- 

 tions, each equation is increafed by its maximum, i[i order 

 to render ail the equations additive ; this quantity, there- 

 fore, mud always be fubtraftcd from the fum of the 

 equations taken from the Tables. 



To Jintl the heliocentric Latitude and Longitude of Jupiter. 



From Table I. of the epochs, take out the epochs of the 

 mean longitude of the aphelion and node, with the Argu- 

 ments II, III. IV, V, VI, VII, VIII, IX, and place 

 them in an horizontal line. But if the given year be not 

 found in that Table, take the nearefl year preceding the- 

 given year as an epoch, and take out as before ; under 

 which, from Table II, place the mean motion in longitude, 

 of the aphelion and node, with the Arguments, anfwering 

 to the number of years elapfed fince the epoch to the given 

 year. 



Under thefe, write down (Table III.) the mean motions 

 of the fame, for Uie given month. 



Under thefe, write down (Table IV.) the mean motions 

 of tlie fame, for the given day of the month. 



ITndcr thefe, write down (Table V.) the mean motions 



of the fame, for the given hours and minutes. The aryu- 



mi nts fov hours and minutes arc fo fmatl that they are here 



6 omitted-; 



