282 NEW FORMULA RELATIYE TO COMETS. 



and by substitution, the last term of the component velocity x, becomes 



v'-\-v" 



— s Kf t Jv, • 



Values similar to these will evidently result for >? and the last term 

 of y, ; and if, for the sake of brevity, we put 



^, = X-^(t'-t")X, y!, = Y^^(t'-t")Yr, (17) 



so that the quantities |, yj may take the form 



^ = iA:(^ + ne, y! = hk{f-\-t")yj,; (IS) 



we shall immediately perceive that after substitution in formulae (15), 

 the factors of ^„ yj, will disappear; and that all the terms but the last of 

 x„ y,, will be independent of the small quantity ^ k. To eliminate 

 ^ k from these terms, we must have recourse to the first of the equa- 

 tions (10), which in virtue of the values (4) becomes 



t'-^t" 

 ft" 



n ill\ — A^ + ^^ 

 ^ "" 3rV "~ D • 



From this and the preceding values of ^, >7 we get \ht't" = 



(1 — —-A . -TT -^ — ; so that if for conciseness we put the fac- 



t' t" 

 tor 1 — gp- = v^ and 



M = AeH-B,7, , M' = A'e,-|-B',7, , M^ = A"|, + B'V, , 

 the general expressions (15) will take the form 



