Oct. 4, 1877] 



NATURE 



479 



The equations of condition in latitude are treated 

 in a similar manner, being grouped according to the dis- 

 tances of the planet from its ascending node. From 

 these equations the corrections of the inclination of 

 the orbit and longitude of the node are found separately 

 from the ancient and from the modern observations. The 

 results differ very little, but the second solution is 

 employed in the construction of the tables. After the 

 application of these corrections to the elements, the 

 agreement between theory and observation may be con- 

 sidered perfect : so that the action of the minor planets 

 on Jupiter appears to be insensible, and there is no indi- 

 cation of any unknown disturbing causes. 



There are some peculiarities in the mode of tabulating 

 the perturbations caused by the action of Saturn. The 

 perturbations of longitude and of radius vector are not, 

 as usual, exhibited directly, but instead of them !\I. 

 Leverrier gives the perturbations, both secular and 

 periodic, of the mean longitude, of the longitude of the 

 perihelion, of the e,\cenlricity, and of the semi-axis major 

 of the orbit, and then from the elements corrected by 

 these perturbations he derives the disturbed longitude and 

 radius vector by the ordinary formula; of elliptic motion. 



Where the perturbations are large M. Leverrier con- 

 siders this preferable to the ordinary method of proceeding. 

 The perturbations of latitude being small, he applies to 

 the inclination and longitude of the node their secular 

 variations alone, and then determines directly the periodic 

 inequalities of latitude. 



All these perturbations, whether of the elements or of 

 the latitude, are developed in a series of sines and cosines 

 of multiples of the mean longitude of Saturn, including a 

 constant term, the coefficients multiplying these several 

 terms being functions of the mean elongation of Saturn 

 from Jupiter, which for a given elongation are developed 

 in powers of the time reckoned from the epoch 1S50. 

 These coefficients only are tabulated with the mean 

 elongation as the argument, and the perturbations are 

 thence calculated by means of the ordinary trit;ono- 

 mctrical tables. The intervals of the argument are so 

 small, that the requisite interpolations ai every simple, and 

 the coefficients which relate to the four elements, and 

 depend on the same argument, are given at the same 

 opening cf the tables. 



The tables have been calculated specially for the 500 

 years included between the years iS5oand 2350. Never- 

 theless they may be applied to epochs anterior to 1S50, by 

 simply changing the sign of the time reckoned from 1850. 

 For one or two centuries before 1850 this extension will 

 have all the rigour of modem ob-ervations, while for still 

 earlier times the accuracy of the tables will greatly surpass 

 that of the observations which we have to compare with 

 them. 



M. Leverrier's Tables of Jupiter are now employed in 

 the computations of the Nautical Almanac, beginning 

 with the year 187S. 



The thirteenth volurrc of the Annals is devoted to 

 the theories of Uranus and Neptune. Tl ese theories are 

 not unattended with difficulties. In the first place, these 

 planets are disturbed by the actior s of the two great 

 masses Jupiter and Saturn, interior to their orbits, and 

 these actions are modified by the great inequaliiies cf 

 Jupiter and Saturn defending on five times the mean 

 motion of Saturn niinus twice the mean motion ol Jupiter. 

 In the next place twice the mean rro ion of Neptune 

 differs very little from the mean motion of Uranus, and 

 thus arise inequalities of long period in the elements of 

 their orbits which are laige enough to produce very 

 sensible teims of the second ordsr. Lastly, the mean 

 elliptic elements of the two planets are not \ et sufficiently 

 well known. 



In a preliminary chapter, the 24th, M. Leverrier investi- 

 gates formula; which are specially applicable to the 

 case of a planet disturbed by another which is consider- 



ably nearer to the sun. In this case it is easily seen that, 

 by the direct action of the disturbing planet on the sun, 

 perturbations of large amount may be produced in the 

 elements of the orbit of the disturbed planet, while the 

 corresponding perturbations of the co-ordinates of the 

 planet are comparatively small. Hence arises the advan- 

 tage of considering this case apart. 



We have seen how closely the theories of Jupiter and 

 Saturn are related to each other. In a similar manner 

 the theories of Uranus and Neptune are also closely 

 related in consequence of the great perturbations intro- 

 duced into the elements of their orbits by the near 

 approach to commensurability in their mean motions. 

 Hence, before entering upon the separate theories, M. 

 Leverrier devotes Chapter 25 of his researches to the 

 determination of the mutual actions of Uranus and Nep- 

 tune, and this forms the base of the theories of both 

 planets. The method employed is similar to that adopted 

 in the case of Jupiter and Saturn, and the results are 

 exhibited in the same general form. 



It is important to remark that f.he elements of Uranus 

 and Neptune as determined from observations severally 

 differ from their mean elliptic values by the amount of 

 their perturbations of long period corresponding to the 

 mean epoch of the observations. The apparent elements 

 of Uranus and Neptune for the epoch 1850 have been 

 carefully determined by Prof. Newcomb in his excellent 

 work on the theory of those planets which obtained the 

 Society's medal in 1S74. By the application of his own 

 general formula?, M. Leverrier deduces from these 

 elements the values of the mean elliptic elements corre- 

 sponding to the same epoch. It maybe remarked that 

 the mean elements thus determined will depend on the 

 assumed masses of the two planets, and will therefore 

 lequire small corrections when more accurate values of 

 the masses have been obtained. 



When the secular variations of Uranus and Neptune 

 given in Chapter 19 were found, the elements were less 

 accurately known, and M. Leverrier has therefore recal- 

 culated the values of the excentricities and longitudes of 

 the perihelia of the two planets for the same five epochs 

 as before, starting from the mean elliptic values of the 

 elements above referred to. 



Chapter 26 contains the completion of the theory of 

 Uranus. The last chapter, which contains the comple- 

 tion of the theory of Neptune, is not yet printed. 



The twenty-third chapter also, vhich contains the 

 comparison of the theory of Saturn with observations, 

 together with the tables of the planet, and which will 

 form the latter part of the twelfth volume of the Annals, 

 is not yet printed. The results of this comparison of the 

 theory with observations have, however, been fully pub- 

 lished in the G?;«/A-j- yvV;/(/;cj-, ;nd 1 understand that the 

 tables will be used for computing the place of Saturn in 

 tt.e foithcoming volume of the Nautical Ahnanac. 



Although the comparison of the theory ol Saturn with 

 observations shows in general a satisfactory accordance, 

 there occur some discrepancies in individual years which 

 are larger than might be desired. 



During the thirty-two years ever which the modern 

 observations extend, viz., from 1S37 to 1869, the dis- 

 crepancy between theory and observation, however, 

 remains constantly less than 2"5 of arc, excepting in two 

 instances, viz., in the years 1S39 and 1844, when the 

 differences amount to 4"'5 of arc. 



In the ancient observations only, made in the time of 

 Maskelyne, rather largtr differences occur, amounting in 

 t«o instances to nearly 9" of arc. 



In Older to test whether these discrepancies could be 

 due to any imperfections in the theory, M. Leverrier has 

 not shrunk from the immense labour of forming a second 

 theory of the planet independent of the former, employing 

 methods of interpolation instead of the analytical develop- 

 ments. I learn directly from M. Leverrier that this 



