Sept. 27, 1877] 



NA TURE 



463 



On January 12, 1874, he presented his tables of 

 Jupiter, founded on the theory which has just been men- 

 tioned, as compared with obst^rvations made at Greenwich 

 from 1750 to 1830, and from 1S36 to 1S69, and with obser- 

 vations made at Paris from 1S37 to 1867. 



Again, on November 9, 1874, he presented to th; 

 Academy a complete theory of Uranus. Already in 1S46, 

 in his researches which led to the discovery of Neptune, 

 M. Leverrier had given a very full investigation of the 

 perturbations of Uranus by the action of Jupiter and 

 Saturn. In the memoir just mentioned he gives a fresh 

 investigation, including a full treatment of the perturba- 

 tions of Uranus by the action of Neptune. 



On December, 14, 1874, he presented a new theory of 

 the planet Neptune, thus completing the theoretical part 

 of the immense labours which he had undertaken with 

 respect to the planetary system. 



Finally, on August 23, 1875, he presented to the 

 Academy the comparison of the theory of Saturn with 

 observations. 



Such is a bare enumeration of the various labours for 

 which our science is already indebted to our illustrious 

 Associate. 



That any one man should have had the power and 

 perseverance required thus to traverse the entire solar 

 system with a firm step, and to determine with the utmost 

 accuracy the mutual disturbances of all the primary 

 planets which appear to hive any sensible influence on 

 each other's motions, might well have appeared incredible 

 if we had not seen it actually accomplished. 



I will now proceed to give a brief outline of the 

 investigations relating to the motions of the four larger 

 planets, with which we are now more particularly con- 

 cerned. The most important parts of these investigations 

 are printed in full detail in the volumes of Memoi7-s\s\aOt\ 

 form part of the Annals of the Observatory of Paris. 



As in his former researches, M. Leverrier here also 

 exclusively employs the method of variation of elements, 

 and the investigations are based on the development of 

 the disturbing function given by him, in the first volume 

 of the Annals of the Paris Observatory, with greater 

 accuracy and to a far greater extent than had ever been 

 done before. 



The eighteenth chapter of M. Leverrier's researches, 

 which forms nearly the whole of the tenth volume of the 

 Memoirs^ is devoted to the determination of the mutual 

 action of Jupiter and Saturn, which forms the foundation 

 of the theories of these two planets. 



These theories are extremely complicated, and I shall 

 endeavour briefly to paint out, and to explain as far as I 

 can without the introduction of algebraical symbols, the 

 nature of the peculiar difficulties which M. Leverrier has 

 had to encounter in their treatment, and which he has so 

 successfully overcome. These difficulties either do not 

 present themselves at all, or do so in a very minor degree 

 in the theories of the smaller planets. 



First, then, the masses of Jupiter and Saturn are far 

 larger than those of the inferior planets, the mass of 

 Jupiter being more than 300 times and that of Saturn 

 being nearly 100 times greater than the mass of the earth. 

 For this reason it is necessary to develop the infinite 

 series in which the perturbations are expressed to a much 

 greater extent when we are dealing with Jupiter and 

 Saturn than when we are concerned with the mutual dis- 

 turbances of the inferior planets. Also Jupiter and 

 Saturn arc so far removed from these latter planets that 

 the disturbances which they produce in the motion of 

 these planets are extremely small, in spite of the large 

 misses of the disturbing bodies. 



But the great magnitude of the disturbing masses is 

 far from being the only reason why the theory of the 

 mutual disturbances of Jupiter and Saturn is so compli- 

 cated. 



Another cause wh'ch aggravates the effect of the 



former is the near approach to commensurability in the 

 mean motions. 



Twice the mean motion of Jupiter differs very little 

 from five times that of Saturn. In other words, five 

 periods of Jupiter occupy nearly the same time as two of 

 Saturn, so that if at a given time the planets were in con- 

 junction at certain points in their orbits, then after three 

 synodic periods they would be again in conjunction at 

 points not far removed from their positions at starting. 

 Hence, whatever uncompensated perturbations may have 

 been produced in the motions of the two planets during 

 these three synodic periods will be very nearly repeated 

 in the next three synodic periods, and again in the next 

 three, and so on. 



Hence the disturbances will go on accumulating in 

 the same direction during many revolutions of the two 

 planets, and will become very important. The inequalities 

 of Jong period thus arising will affect all the elements of 

 the orbits of the two planets ; but the most important 

 are those which affect the mean longitudes of the bodies, 

 since these are proportional to the square of the period 

 of the inequalities, whereas the inequalities affecting the 

 other elements are proportional to the period itself. 



The principal terms of the inequalities of mean longi- 

 tude are of the third order, if we consider the excen- 

 tricities of the orbits and their mutual inclination to be 

 small quantities of the first order. 



Terms of the same period, however, and those far 

 more numerous and more complicated in expression, 

 occur among those of the fifth and of the seventh order 

 of small quantities, and I\L Leverrier has included these 

 terms also in his approximations. 



But the circumstance which contributes in the highest 

 degree to cause the superior complexity of the theories of 

 the larger planets is the necessity, in their case, of taking 

 into account the terms which depend on the squares and 

 higher powers of the disturbing forces. 



I will endeavour to point out the nature of these 

 terms and the manner in which they arise. 



By the theory of the variation of elements we are 

 able to express at any given time the rate of variation of 

 any one of the elements in terms of the mean longitudes 

 and the elements of the orbits of the disturbed and the 

 several disturbing bodies. If this rate of variation were 

 given in terms of the time and known quantities, we 

 should at once find the value of the element for any 

 given time by a simple integration. But this is not the 

 case. 



The method of variation of elements gives us not a 

 solution, but merely a transformation of our original 

 differential equations of motion. The rates of variation 

 are given in terms of the unknown elements themselves ; 

 and in order to find the elements from the equations so 

 formed, we must employ repeated approximations. 



Let us consider this matter a little more particularly. 



The terms which express the rate of variation of any 

 element may be divided into two classes — 



1. Those which involve the mean longitudes of one or 

 both of the planets concerned, as well as the elements of 

 their orbits. 



2. Those which involve the elements only. 



The first are called panodic terms, since they pass 

 from positive to negative, and vice versa, in periods com- 

 parable with those of the planets themselves. The second 

 are called secular terms, and vary very slowly, since the 

 elements on which they depend do so. Each of the terms 

 in the expression of the rate of variation of any element 

 will involve the miss of one of the disturbing bodies as a 

 fcictor. Hence, if all these masses be very small, all the 

 periodic inequalities of the elements will be likewise very 

 small, and we shall obtain a value of the rate of variation 

 which is very near the truth if we substitute for the com- 

 plete value of any element its value when cleared of 

 periodic inequalities. Then the periodic inequalities in 



