46] EOYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 369 



five times the mean motion of Saturn minus twice the mean motion of 

 Jupiter. 



In the next place, twice the mean motion of Neptune differs very little 

 from the mean motion of Uranus, and thus arise inequalities of long period 

 in the elements of their orbits which are large enough to produce very 

 sensible terms of the second order. 



Lastly, the mean elliptic elements of the two planets are not yet suf- 

 ficiently well known. 



In a preliminary chapter, the 24th, M. Le Verrier investigates formula; 

 which are specially applicable to the case of a planet disturbed by another 

 which is considerably nearer to the Sun. 



In this case it is easily seen that, by the direct action of the disturbing 

 planet on the Sun, perturbations of large amount may be produced in the 

 elements of the orbit of the disturbed planet, while the corresponding pertur- 

 bations of the coordinates of the planet are comparatively small. Hence 

 arises the advantage of considering this case apart. 



We have seen how closely the theories of Jupiter and Saturn are related 

 to each other. In a similar manner the theories of Uranus and Neptune are 

 also closely related in consequence of the great perturbations introduced into 

 the elements of their orbits by the near approach to commensurability in 

 their mean motions. 



Hence, before entering upon the separate theories, M. Le Verrier devotes 

 Chapter 25 of his researches to the determination of the mutual actions of 

 Uranus and Neptune, and this forms the base of the theories of both planets. 



The method employed is similar to that adopted in the case of Jupiter 

 and Saturn, and the results are exhibited in the same general form. 



It is important to remark that the elements of Uranus and Neptune as 

 determined from observations severally differ from their mean elliptic values 

 by the amount of their perturbations of long period corresponding to the 

 mean epoch of the observations. 



The apparent elements of Uranus and Neptune for the epoch 1850 

 have been carefully determined by Professor Newcomb in his excellent work 

 on the theory of those planets which obtained the Society's Medal in 1874. 



A. 47 



