8 



THE ORBIT OF URANUS. 



C being the arbitrary constant added to the integral. -Adding this equation to the 

 first of equations (1) we have 



Let us now represent by r that elliptic value of r which satisfies the equation 



<^(r 2 ) p(l+Q 

 ~ ~ 



Subtracting this equation from the last we have 



- (9p - 



\ 



> 



in which no constant is to be added to the integral, and both sides of the equation 

 are of the order of the disturbing forces. As there is a decided advantage in taking 

 the logarithm of the radius vector as the variable instead of r itself, we substitute 

 for the latter its value 



and put 

 Then 



= p p . 



=c 



= r c = 



etc 



.) 



1 (r 2 - r a ) = ?yfy + 7- 2 V -f etc. 



1 1 Sp fy 2 . 



- = -- '- 4- etc. 



r r r 2r Q ' 



Substituting these values in the above equation, carrying the development only to 

 terms of the second order, and transposing those terms to the right hand side of the 

 equation, and putting p ^ (1 -j- TO), we find 



an equation which gives the perturbations of radius vector. 



The general mode of solving this equation by successive approximation is familiar. 

 The principles on which the successive approximations are made being the same, 

 we shall begin by assuming that we have obtained first approximations to the values 

 of 5v, 5V, $p, <5p', 5y, and that from these we wish to pass to a second approximation. 

 We must first carry this approximation into the functions of R in the second mem- 

 ber of (4). To effect this we must show how, from the development of R in terms 

 of the elements and the time, we may form its successive derivatives with respect 

 to the quantities which enter into it. H, while originally a function of v, V, p, p', 

 and y, is, in its developed form, a function of a, X', o, u', e, e\ , ,' and y, the 

 development being effected by substituting for the first set of quantities their 

 values in terms of the second. The substitution is as follows : 





(5) 



