THEORBITOPURANUS 55 



The term of Long Period. 



From the expressions for the perturbations of Uranus, subsequently given, it will 

 be seen that several of the terms have very large coefficients, that of sin ('21' — </) 

 being nearly an entire degree. The magnitude of most of the terms in which i is 

 even arises from the near approach to commensurability in the mean motions of 

 the two planets. Twice tlie mean annual motion of Neptune exceeds that of 

 Uranus by only 303".8. The elements ef the orbits of both planets will there- 

 fore, in consequence of their mutual action, be affected with a slow oscillation, 

 having a period of about 4266 years. The employment of these large terms and 

 the great inconveniences to which they will give rise, especially in the corrections 

 of the elements of Uranus, may be avoided by the device employed in the theory 

 of Neptune. The following are the essential features of this method: 



First, all the perturbations arising from tliat portion of the perturbative func- 

 tion in which tlie coefficient of the time is 2n' — n or its multiples are considered 

 and developed as perturbations of the elements. 



Secondly, the arbitrary constants to be added to the integrals of these perturba- 

 tions are so taken that the perturbations shall vanish at the epoch 18-50.0. 



In other words, the perturbations in question will be treated as producing secular 

 variations of the elements of the orbit, only, instead of being developed in powers 

 of the time, these variations will be retained in their rigorous form. 



The formula; for the computations of the perturbations in question, are as 

 follows : 



Let 



— h cos {!• I' + i I +/cy +ycj) = — cos N 



be any term of the perturbative function, h being a function of a, e, and a. 



sin a|/ = e 

 g =^ cos 4> tan 2 4' 



n 



III -j- t/i 



For each such term, compute 



A= 2 Ih 



ij = — o II h — 2 -J 



drj ' de 



eW^ cos »J/ 



E=-h(;g+jcoti) 



da 



