478 



NATURE 



\Pct.i,, 1877 



These orbits are in very satisfactory confirmation of 

 each other. 



May we hope that at no distant period Ur. Doberck 

 may find he has sufficient material to induce him to investi- 

 gate the elements of a Centauri ; a fair approximation to 

 the true orbit might be expected from his experienced 

 hand. 



PROF. ADAMS ON LEVERRIER'S PLANETARY 



THEORIES ' 



II. 



THE nineteenth chapter of M. Leverrier's researches, 

 which forms the first part of the eleventh volume of 

 the Annals of ilic Paris Obs.'ivatory, contains the deter- 

 mination of the secular variations of the elements of the 

 orbits of the four planets, Jupiter, Saturn, Uranus, and 

 Neptune. 



In the first place are collected the differential formulas 

 which are established in the previous chapter, and which 

 give the rates of secular change of the various elements at 

 any epoch in terms of the elements themselves, which by 

 the previous operations have been cleared of all periodic 

 inequililies. 



The terms of different orders which enter into these 

 formula; are carefully distinguished. 



If we were to confine our attention to the terms of the 

 first degree with respect to the exccntricities and inclina- 

 tions of the orbits, and of the first order with respect to 

 the masses, the differential equations which determine 

 the secular variations would become linear, and their 

 general integrals might be found, so as to give the values 

 of the several elements for an ind-finite period. 



In the present case, however, the terms of higher orders 

 are far too important to be neglected, and when these are 

 taken into account the equations b;come so complicated 

 as to render it hopeless to attempt to determine their 

 general integrals. 



Fortunately, however, these are not needed for the 

 actual requirements of astronomy, and for any definite 

 period the simultaneous integrals m?y be determined with 

 any degree of accuracy that may be desired by the method 

 of quadratures. 



In this way M. Leverrier has determined the values 

 of the elements for a period of 2,000 years, starting from 

 1850, at successive intervals of 500 years. The first steps 

 in this integration were attended with some difficulties, 

 because the determination of the numerical values of the 

 rates of change of the several elements at the various 

 epochs depends on the ele.Tients themselves which are to 

 be determined. Hence several approximations were 

 necessary in order to obtain the requisite precision. 



After this work of M. Leverrier, however, the extension 

 of the investigation to other epochs, past or future, is no 

 longer attended with the same difficulties. In fact, from 

 his results we may at once find, by the method of differ- 

 ences, very approximate values of the elements at an epoch 

 500 years earlier or later than those which he has con- 

 sidered. His general formukc will then give the rates of 

 change of the several elements at the epoch in question, 

 and having these we can determme by a direct calculation 

 the small corrections which should be applied to the 

 approximate values of the elements first found. 



This process may evidently be repeated as often as we 

 choose. 



It is important to remark that in the formula: which 

 give the rates of change of each of the elements at the five 

 principal epochs considered, as well as in those which give 

 the total variations of the elements at the same epochs, 

 the masses of the several planets appear in an indetermi- 

 nate form, so that it may be at once seen what part of the 

 variation of any element is due to the action of each of 

 the planets, and what changes would be produced in the 



■ Coininued from p. 464. 



value of any element at any epoch by any changes in the 

 assumed values of the masses. 



Consequently, when the astronomer of the future, say 

 of 2,000 years hence, has determined the values of the 

 elements of the planetary orbits correspanding to that 

 epoch, it will be easy for him, by comparing those values 

 with the general expressions given by M. Leverrier, to 

 determine with the greatest precision the actual values of 

 the masses, provided that all the disturbing bodies are 

 known ; and should there be any unknown disturbing 

 causes, their existence would be indicated by the in- 

 consistency of the values of the masses which would be 

 found from the different equations of condition. 



By means of the work which has just b^en described, 

 ever) thing has been prepared which is required for the 

 treatment of the theories of the several planets. 



The remainder of the eleventh volume of the Annals 

 is accordingly occupied by the complete theories of Jupiter 

 and Saturn, the former theory being given in Chapter 20, 

 and the latter in Chapter 21 of M. Leverrier's researches. 



The coefficients of the periodic inequalities of the 

 mean longitudes and of the elements of the orbits are 

 not only exhibited in a general form, but are also calcu- 

 lated numerically for the five principal epochs considered 

 in Chapter 19 of these researches, viz, for 1850, 2350, 

 2850, 3350, and 3850. 



The long mequalities of the second order with respect 

 to the masses, depending on twice the mean motion of 

 Jupiter plus three times the mean motion of Uranus 

 minus six times the mean motion of Saturn, are also 

 determined in a similar form. 



Chapter 22 of M. Leverrier's researches, forming the 

 first part of the 12th volume of the " Annals," contains the 

 comparison of the theory of Jupiter with the observations, 

 the deduction of the definitive corrections of the elements 

 therefrom, and finally the resulting tables of the motion of 

 Jupiter. The observations employed are the (ireenwich 

 observations from 1750 to 1830 and from 1836 to 1869 

 together with the Paris observations from 1837 to 1867. 



To the results given in the Astronomer-Royal's 

 " Reduction of the Greenwich Observations of Planets 

 from 1750 to 1830," M. Leverrier has applied the cor- 

 rections which he has found to be required by his own 

 reduction of Bradley's observations of stars and his rede- 

 termination of the Right Ascensions of the fundamental 

 stars^ published in the second volume of the " Annals " 

 (Chapter 10). 



The equations of condition in longitude, for finding 

 the corrections of the elements and of the assumed 

 mass of Saturn, are divided into two series corre- 

 sponding to the observations made from 1750 to 1830, 

 and into two other series corresponding to the obser- 

 vations made from 1836 to 1869. Moreover in each, 

 of these series the equations are subdivided into eight 

 groups, corresponding to the distances of the planet from 

 Its perihelion, 0' to 45°, 45" to 90", and so on. From these 

 are formed four final equations, the solution of which 

 gives the corrections of the epoch, of the mean motion, of 

 the excentricity, and of the longitude of the perihelion, in 

 terms of the correction required by the mass of Saturn, 

 which is left in an indeterminate form. The substitution 

 of these expressions in the thirty-two normal equations 

 corresponding to the several groups above-mentioned, 

 gives the residual differences between theory and observa- 

 tion in terms of the correction of the mass of Saturn. No 

 conclusion can be drawn from the ancient observations ; 

 but from the modern observations M. Leverrier finds 

 that the mass of Saturn assumed— which is that of 

 Bouvard- -should be diminished by about its ~.\rX^ part. 

 This correction is very small, but M. Leverrier regards it 

 as well established. 



On the other hand, Bessel's value of the mass ol 

 Saturn, founded on his observations of the Huyghenian 

 satellite, exceeds Bouvard's by about its issuth pait. 



