280 



NEW FORMUKE RELATIVE TO COMETS. 



Values analogous to these, though very diflferent in their form and 

 object, are made the basis of Lagrange's method of determining the 

 orbits of Comets. They may be seen in his Mecanique Analytique. 



In consequence of the supposed proximity of the extreme positions 

 C, C", to the mean position C of the comet, it is easily conceived that 

 the denominator D of (10) is a small quantity, liable to be affected by 

 unavoidable errors of observations. We shall therefore eliminate it, 

 and employ, instead of (10), the following values: 



w 



• N ' 







I 



(12) 



in which we have assumed 



N = AH- B>7, N' = A'e -f- B'>7, N" = A"| + BV (13) 



These quantities are the numerators of (10) ; they give, in virtue of the 

 conditional equations (9) and (11) 



Na H- NV + N"a" = D^, N/3 + N'/3' + N"/?" = D)?. (14) 



Let now (12) be substituted in the general expressions (6), and let 

 us attend to the preceding forms for D^, D>7. We shall then obtain 

 the following values : 



N \v V J v-V-v 



// 5 





(■ (1«) 



^' = i^(?NV->V")' 



u u 

 in which we have put w (v'4-d"") = — v" r, v. 



