THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 



Therefore when we have computed the quantities (0,1), (, 2 ), (, 3 ), & c -> or the 

 coefficient for an interior planet, depending on the action of an exterior planet, we 

 shall obtain the corresponding coefficient for an exterior planet, depending on the 

 action of an interior planet, by means of equations (6), (7), and (8). 



Equations (6) and (7) may be written as follows : 



, . in f , m 

 (1,0) = (0,1) 



n'a' na 



, m" , > TO 



I 2 , 0) = (0,2) 



' ri'a" ' na 



&c. 



We shall also similarly have 



&c. 

 We shall therefore have 



(3,0)(0, 2 )( 2 ,3) = (0,3)(3,2)(2,0), 



&c. 

 We shall also have the following products of four factors 



&c 

 And of five factors we have 



, O) = (l, 0)(0,4)(4,S)(S, 



(9) 



(10) 



(H) 



(12) 



j. . (13) 



( 1 ,2)(2,4)(4,3)(3, 5 )( 5 , 1 ) = (2, 1 )( 1 ,a)(5,3)(3,4)(4,2), 



&c. . 



And in like manner we may form the products of six, or any number of factors. 

 Equations (11), (12), and (13) are very useful in reducing two terms to a single 

 one. We may in like manner form the following equations : 



(14) 



