362 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [46 



In order to find more correct values, we must substitute for the elements 

 in the second member of the equation their secular parts augmented by the 

 approximate periodic inequalities before found. 



Now, if in any periodic term we increase any element by a periodic 

 inequality depending on a different argument, that is involving different 

 multiples of the mean longitudes, the result will evidently be to introduce 

 new periodic terms which will involve the square of one of the masses or 

 the product of two of them as a factor. 



Similarly, if in any periodic term any element be increased by a periodic 

 inequality depending on the same argument, the result will also introduce 

 new terms of the second order which do not involve the mean longitudes, 

 and which therefore constitute new secular terms. 



These will be particularly important if the inequality in question be one 

 of long period. 



Also in the secular terms the result of increasing any element by a 

 periodic inequality will be to introduce a new periodic term depending on 

 the same argument. 



Lastly, it should be remarked that in finding the periodic inequalities of 

 any element by integration of the corresponding differential equation, we 

 must take into account the secular variations of the elements which were 

 neglected in the first approximation. The new terms thus introduced, like 

 the others which we have just described, will evidently be of the second 

 order with respect to the masses. 



If the disturbing masses be large, as in the case of the mutual disturb- 

 ances of Jupiter and Saturn, it may be necessary to proceed to a further 

 approximation, and thus to obtain new terms, both periodic and secular, 

 which involve the cubes and products of three dimensions of the masses. 



The number of combinations of terms which give rise to these terms of 

 the second and third orders is practically unlimited, and the art of the 

 calculator consists in selecting those combinations only which lead to sensible 

 results. 



This is the chief cause of the great complexity of the theories of the 

 larger planets, and more especially of those of Jupiter and Saturn. 



M. Le Verrier lays it down as the indispensable condition of all progress 

 that we should be able to compare the whole of the observations of a planet 



