THE ORBITS OF THE EIGUT 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, 



(O.o)=0+(o,l) + (o, a )+(o,8)+&c., 



( 1 , 1 )= J7 +( 1 ,0) + ( 1 ,2) + (1 ,3) + &C., 

 ( 2 , ,)=,,+ (., ) + (.,, l) + ( 2 ,3) + &C., 



&c.; 

 — b = (o,4)^ r -f(o,:,) i V r +(o, ( 0iV r/ +(o, 7 )iV r ' 



— Z>'=(l,4)iV /F +(l, 5)N •'+(!, «)iV r/ +(l.7)iV r/ 



— &" = ( 2 ,4)iV"'+(2,:,)^ r +(2,0)iV r/ +(2,7)iVr'-' 



— Z>'"=(3,4)^ r/ +(3, 5 )^ r +0,e)i^ r/ +(3,7)iV n 



— 6 1= (4,o)JV -f (4,i)iV'+(4,2)iVr" _|_( 4l s)iV w , 

 -b z =( 6 ,o)N -f( 5 ,i)JV+( 6 ,s)JV"" +(«,3)i^ w , 



— £> 3 =(6,0)iV +(6,l)i^'+(6,2) i V" +(6,3)^"'", 



— Z) 4 =(7,0)iV +(7,l)N'-\-(7,*)N" ■f(7,3)JV". 



(366) 



(367) 



(368) 



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



(0,o)i\T _ (O.I)JV' — (0 , 2)i\T" _( ,3)iV'" +6 =0, 



(l.l)iV' — (1,0) iV — (1,2)JV"_ (l, S )JV"» +6'=0, 



( 2 ,2)iV" — (*,0)ir — (2,l)iV' — (2,3)iV" +&"=0, 



(3,3)lV" — (s,o)JV — (S.I)JV' — (S,2)JV" +&'"=(), 



}• (E") 



(4,4)A r/r — (4,S)JV' 

 (6,5)iV r — («,4)iV J 



( 6 , a )ivr"_( G ,4)iV" 

 (7,7)JV r "— (r.4)^' 



-(4 , 6)i\T "—(4 , 7)N r "-|- J x =0, 



.( e , 6 )N r -(e,7)N r "-\-b s =0, 



.(7 , 5 )JV T —(7 , «>) tf "'-f& 4 =0. 



(E'") 



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

 use of equations (31-61) for their solution, if we suppose |q,o| =(o,q), |i,i| =Q ,i), 

 &C; | o , 1 1 = — (o,i), |o,a| = — (0,2), &c; |T7o] = — (1,0), |i,2| = — (}>*), &c., in these 

 equations. We have given the values of (0 , 1), (0 , 2), &c., (1 , 0), (1 , 2), &c., in § 7. The 

 values of (0,0), (o. 1 )? ( 2 ' 2 )> & c -> are given by means of the corresponding values | q,»1 , 

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

 changing the sign of the coefficients of g. 



