THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 17 



2V=- ~' , m which a is very nearly equal to a', and the computation of these 



quantities cannot be readily effected by logarithms with sufficient precision to give 

 their difference correct to more than three or four significant figures; and, in all 

 these cases, the other formula for the same quantity gives a value which is free 

 from this source of error. 



The computation of the successive approximations to the values of the required 

 quantities is then arranged as follows: — 



We first find the roots of equation (53), on the supposition that £ is equal to 

 nothing. We shall designate these roots by g, g lt g 2 , and g 3 , and the correspond- 

 ing values of the second member by £, £ l5 ^ 2 , and £ 3 . The roots of equation (54) 

 are also designated by g^ g b , g 6 , and g 7 , and the corresponding values of the second 

 member by £ 4 , £ 6 , £„, and ^ 7 . When g, g x , g 2 , &c. have been determined, we must 

 transform equation (53) into others whose roots shall be smaller by the values 

 $■> 9\i U*> an d #3- Then if we denote the corrections to be applied to g, g u &c, in 

 order to obtain their correct values by &g, hg x , bg 2 , and <§<7 3 , we shall have a system 

 of equations for the determination of bg, bg x , hg 2 , and bg 3 , of the following form: — 



bg 4 +« bg 3 +Z> bg 2 +c hg =z ; 

 '\'/ 1 4 +«' &g*+V <tyi 2 +c' ^i=%i ; 

 bg^+a" bg.?+b" } g 2 ~-\-c" bg 2 =% 2 ; 



bg 3 i +a"'bg 3 3 -\-b"'bg 3 2 -\-c"'bg 3 =x^ I 

 $0+501+^2+^8+^4+^5+^6+^7=0. ^ j- Equation of cond i tioil .] 



(65) 



The equations for the determination of 8g±, bg b , bg c , and bg 7 arc entirely similar. 

 Then, having determined the approximate values of g, g x , g 2 , &c, we must substi- 

 tute them in succession in equations (31 — 42) inclusive, and we shall obtain the 

 values of A, A', A", &c, D, D', D", &c, which are to be substituted in equations 

 (57 — 59), and we shall obtain the approximate values of N', N", and N'". These 

 quantities are then to be substituted in equations (28), and we shall get the values 

 of b x , b 2 , b 3 , and 5 4 ; which quantities, together with the value of g, are to be sub- 

 stituted in equations (47 — 50) inclusive, and we obtain B x , B 2 , B 3 , G x , C 2 , &c, &c. 

 Then equation (52) will give f x , f 2 , &c, which are to be substituted in equations 

 (61—64), and we shall obtain N'\ N\ N", and N T ". Equations (27) will then 

 give b, b', b", and &'", which are to be substituted along with g in equations (43 — 46). 

 Then equations (51) will give/, /',/", &c, which being substituted in equations 

 (54) will give the value of £, on which the value of bg depends in the first of 

 equations (65). When bg has been determined we must add it to the approximate 

 value of g, and repeat the whole computation with the corrected value of g, and by 

 this means we shall obtain the correct values of g, N', N", N'", &c. In like manner 

 we shall obtain g x , N x , N x , jV/", &c. ; g 2 , iV 2 ', N 2 , &c. 



12. We shall now reduce the preceding formula? to numbers, and illustrate by a 

 numerical example the extreme simplicity of the formulas (which appear so unwieldy 

 in their algebraic form), and the comparative facility with which the required 

 quantities can be obtained. 



3 August, 1371. 



