364 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [46 



terms of the first order, in order that their expressions may hold good for 

 any epoch whatever. The formulse relating to these terms are necessarily 

 very complicated. The coefficient belonging to a given argument depends, in 

 general, on a great number of terms which are classed methodically. 



Next are determined the terms of the second order in the secular 

 variations of the elements of the orbits. 



Afterwards, M. Le Verrier takes into account the influence of the secular 

 inequalities on the values of the integrals on which the periodic inequalities 

 depend. 



The last part of this chapter is devoted to the completion of the 

 differential expressions of the secular inequalities by the determination of 

 certain secular terms in the rates of variation of the eccentricities and the 

 longitudes of the perihelia, which are of the third and fourth orders with 

 respect to the masses. 



The 19th Chapter of M. Le Verrier's researches, which forms the first 

 part of the llth Volume of the Annals of the Paris Observatory, contains 

 the determination 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 formulae which are esta- 

 blished 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 in- 

 equalities. 



The terms of different orders which enter into these formulae are carefully 

 distinguished. 



If we were to confine our attention to the terms of the first degree with 

 respect to the eccentricities and inclinations 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 indefinite 

 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 become so complicated as to render it hopeless to attempt to 

 determine their general integrals. 



