12 



SECULAR VARIATIONS OF THE ELEMENTS OF 



The equation of the eighth degree when completely developed is therefore com- 

 posed of 80640 distinct monomial terms, each of which contains eight factors. The 

 actual formation of this equation could therefore with difficulty be brought within 

 the compass of an ordinary lifetime ; and we must, therefore, seek for a shorter and 

 more expeditious method of attaining results which seem to necessarily involve such 

 an immense expenditure of labor. 



9. For this purpose we shall resume equations 



\HH=g—\(}> 6 . 



E3=<7-i( 2 



r*t*i=.<7— 1(«.° 



nZ}=g—\( e ' 6 



) + (C2) + (0 



)+( 2 .0+(s 



)+(3,!) + (3 

 )+0,l) + (4 



)+(«,!)+(« 



■)+(6,l) + (6 



') + ('• + 



B) of § 2, and shall suppose 



+ &c.<; 

 + &c. 

 + &c. 

 + &c. 

 + &c; 

 rf.&c.; 

 + &c; 

 + &c.i 



(26) 



By this means equations (B) will be reduced to the following : — 



\m\n" +r^^+FT^^ , +[^^ ,,, +r^^ jr +F^^ r +r^A /F/ +F^iiV TO -o, 



\TJ]N'" +F7^iY + f^W' + r^W" -\-{W^]N I7 -\-\TJ\N 7 -\-[T^]N TI -\-[^]N rir =0, 

 [TJ]N ir + f^UV + F~J]N'-\-[I^\N"-\-[I^}N'" -\-[^]N v -\-U^]N rz -\-m\N T "=0, 



[7Tiiv"+r^^+FTriiV'+[T^jv''+|T^^ w +r^^ jr +r^^ F +[^-^" r =o, 



Now since the coefficients of N' r , N\ N ri , and N T " of the first four of the 

 equations (B') are independent of g, and also the coefficients of N, N', N" and N'", 

 in the last four of equations (B'), we may divide them into two distinct groups, and 

 determine the values of g, N\ N", 2V"', &c, by successive approximations. We shall 

 therefore suppose 



b"=[J^]N ir -\- r ^]N r -\-l^\N ri -\-\^J]N TI 



(27) 





}■ (28) 



Substituting these quantities in equations (B'), they will become 

 {^]N +[oT|iV [TJ]N" ±\oj\N»-\-b =0, 

 \ry\N' +[JZ]N \J^]N" +[TJ]N'"-\-b' =0, 



(B") 



(B) 



