THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 101 



One of the roots of this equation will evidently be equal to nothing, since equations 

 (E') will be satisfied by supposing #=0, and N=N'=N", &c. 



'2. We shall now suppose, 



(366) 



(2,2)=, / +(2,o)- H 2, 1 )-j-(2,3).f&C., 



&C.J 



b =o, 



-f(S , 



I, =(4 , 0) JV -f (4 , !)#' +(4 , 2)#" -|-(4 , 

 ^=(,0)# -f(5,l)^'+(5,2)JVr" + (3,3) 

 63 =(6 , 0) JV +(6 , l) #' -j_( 6 , 2)^" _j_(. , 3) 



(367) 



(368) 



If we now substitute these quantities in equations (E'), they will become 



=0, 



(E") 



(3,0)JV (3,1)^' (3,5 



( 5 , ,)2V' ._ ( 8 , 4 ) N ir (* , *)N " ( , 7) 

 (e,6)JV' w ^(^4)^-^^9,)jf^^(,,r) 



(7 , 7)iV m (7 , 4) JV /r (7 , 5) JV r (7 , G) 



2 =0, 

 j=0, 



4 =0. 



(E'") 



These equations are similar to equations (B") and (B'") of 9; and we may make 

 use of equations (31-64) for their solution, if we suppose [Q,O]=(O,Q), [i,i|=(i.i), 



&c.; |,ij^^ (".O) |. 2 I= (. 2 ), &c.; [TTol^ ( 1 ') i 1 - 2 ]^ 0' 2 ) ^ c - i n these 

 equations. We have given the values of (o,i) (' 2 )> ^ C M O-'Oi C 1 * 2 )? ^ c - ^ n ^- '^ ne 

 values of (0,0), (0,1), (2,2), &c., are given by means of the corresponding values |Q.Q|, 

 |i.i|, [2,2), &c., in equations (67), by simply changing the sign of the numerical 

 terms of the second member. 



3. We shall now reduce equations (31-64) to numbers. The values of the pro- 

 ducts (o,o)(i,i)i (.)( 2 < 2 ), &c. are given by means of equations (68-79) by simply 

 changing the sign of the coefficients of y. 



