JUPITER AND SATURN. IO5 



Hansen's methods and no one had a more extensive practical know 

 ledge of all methods of computing perturbations. The aim of the 

 earlier perturbational astronomers such as Lagrange and Laplace, 

 was to develop theories which would furnish a unique set of tables 

 based on a hteral development which would be available for thou- 

 sands of years. That such a result is not impossible is proved by 

 their theory of Jupiter's Galilean satellites. The lunar theory 

 may also be cited as a sufficient solution of the same problem. 

 Le Verrier's work on Jupiter and Saturn had the same end in view, 

 but it was not reached and it is safe to say never will be reached 

 by his road. The circumstances which favour the solution in 

 the cases of Jupiter's satellites and in the lunar theory are absent 

 in the theories of the eight major planets: in the first case the 

 eccentricities and mutual inclinations are exceedingly small ; 

 in the second, the eccentricity and inclination do not vary, and 

 the ratio of distances is very small. All this is well-known ; it 

 is recognised that the attack must be directed differently. So 

 far as merely obtaining an ephemeris goes, it is probable that 

 the method of special perturbations would have given one for 300 

 years or so with less labour than was involved in either the theories 

 of Hill or Le Verrier. But a mere numerical exhibit is not what 

 is wanted. A theory, good for ages, in which t alone has to be 

 substituted will be the aim of the astronomer-mathematician. 

 Le Verrier's attempt to arrive at this end by means of variation 

 of elements failed, but this failure does not account for the errors 

 of his tables. The mean distances in the planetary theories are 

 so nearly constants that as early as possible their numerical ratios 

 are introduced into the formulae, and thus each ratio becomes, as 

 it were, the backbone of the theory. It is in this fundamental 

 point that Le Verrier has made a considerable error. The value 

 of the ratio (which as usual will be designated by a) for Jupiter 

 and Saturn adopted by Le Verrier is too large and in consequence 

 all the perturbating coefficients are too big. To a certain extent, 

 this error can be compensated by reducing the masses of these 

 two planets. In the introduction to his theory. Hill says : 



" The motions assigned to the excentricity and perihelion of Saturn in 

 Le Verrier's tables are considerably greater than those given by my theory. 

 In the case of Jupiter, Le Verrier's values of the coefficients of the large terms 

 are quite as large as mine, although they profess to correspond to the 

 value I/3S29-6 of the mass of Saturn, while mine have been computed with 

 the mass 1/3501-6. This, perhaps, explains why Le Verrier's discussion led 

 him to the too small mass of Saturn."* 



Unfortunately, the relation between o, m and in' , is not linear, 

 so that a good adjustment is impossible and the resulting theory 

 must remain faulty. 



This paper deals with the value of a which should be used and 

 gives the values of the b functions appearing in the expansion 

 of the perturbing function ; then some of the secular perturbations 

 are computed and compared with Le Verrier's results ; lastly, 

 the great inequality of Jupiter by Saturn and some other large 

 inequalities are computed to the first power of the masses. The 



* Papers of Amer. Eph. IV., p. 18. 



