THE ORBIT OF URANUS. 



45 



~^ I +(iJ.?2— iV7i) (I'-i — 1-2) ' *• ^ Ji~r ^ \ J)s 



I -)- etc. etc. 



A law of the factors of h^ cos {N-\- u(j) -\- l\ sin (N'-\- vg) which will be noticed 

 in the above expression, is this: Representing this factor by A'^, we have 



the index i representing the coefficient of g in y, so that only half the values of 

 A' need be separately computed. 



As the computation of )\^Sp from these formulae can be arranged in such a way 

 as to be very simple, the computation of the terms in which the index i' is — 1 is 

 here presented quite fully. The logarithms only are omitted, being used only in 

 the cases in which they are more convenient than a table of products. In prac- 

 tice I find it convenient to write them in red ink immediately under the numbers 

 which they represent. 



m'a 



First, to find M, it will be noticed that in the expression 



, the a in the 



rti(l + rn) 



numerator represents the mean distance of the <Jisti(rhei7 Yilnnet, as deduced from tlie 

 observed mean motion by the equation a^ir^u (l-|-7») while r/, represents the mean 

 motion of the outer planet. AVhen the outer planet is the disturbed one, the ratio 



— would be unity, but that, to avoid a large class of second order terms, Oj has 



been corrected for perturbations in the beginning (p. 32). In the case of Uranus 

 disturbed by Saturn, we have in consequence 



Whence 



log — =9.999803. 



M= 285.44 



in units of the sixth place of decimals. 



Computing the values of jh and g^ from (16)' we find, for Uranus, 



1 31 p,q, = + 142.56 



1 M p^: = + 0.0784 



iM 2Mi = - l<^-<^-t^ 



i^^Hp.q.+2hqd = + 3.3433 



ii)/(i>,73+P372)= + 0.0028 



i M poq-, = — 0.2358 



I3f(p,q,-^p,qi)=^+ 0.118 



i 31 p,q, = - 0.008 



I31{p,q,-^p,q,)=+ 0.005 



I31{p,q,-p,q,)- + 0.003 



In the computation the first three lines are copied from previous pages. 



In units of the sixth 

 place of decimals. 



