Til K i) K i: I T OF U II AM US. 



45 



-f- 



etc. 



etc. 



| *. cos (N - 30) + Ar. sin (N- 30) | 



A law of the factors of k\ cos (X-\- u<j) -\- k,sin (iV-f- </) which will be noticed 

 in the above expression, is this: BepCWentiBg this factor by A' B , we have 



A"' A' ( ' + "' 

 j\. j\. _ 



the index / representing the coefficient of g in N, so that only half the values of 

 A' need be separately computed. 



As the computation of r,*fy from these formulae can be arranged in such a way 

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

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

 the eases 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. 



First, to find J/, it will be noticed that in the expression , the a in the 



numerator represents the mean distance of the (Iwlurberl planet, as deduced from the 

 observed mean motion by the equation a*n* = [t (l-\-m) whiles, represents the mean 

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



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



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

 disturbed by Saturn, we have in consequence 



log- =9.999803. 



Whence 



M = 285.44 



in units of the sixth place of decimals. 



Computing the values of p t and q { from (16J we find, for Uranus, 



| M pfli = + 142.56 



= -f 0.0784 



= _ 10.044 



= -f 3.3433 



= 4- 0.0028 



= 0.2358 



= -f 0.118 



= 0.008 



= -+- 0.005 



- + 0.003 



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



M 



In units of the sixth 

 place of decimals. 



