NEW FORMULJE RELATIVE TO COMETS. 281 



These formulae are well adapted to the proposed determination of 

 the comet's component velocities. The generality which we have in- 

 tentionally observed in presenting them, has imposed on their last terms 

 a complex appearance. But this will immediately vanish, when they 

 are limited to the particular state in which approximate results are the 

 objects of computation. 



For the purpose here intimated, let us employ the expressions (4) 

 and their similar forms with double accents. If the intervals t\ t" be 

 regarded as small quantities, so that the terms which involve the dif- 

 ferentials r,, R, of the distances r, R, may be rejected, we shall have 

 for the coefficients of N', N" in (15), the values 



v' ~" /' ^ 3r') ' v" ~~ 't" ^ 3rV' 



in which we shall assume for brevity 



and to a near approximation r may be supposed equal to unity. 



The factor D in the fourth terms of the components x„ ?/,, we have 

 already noticed as being a small quantity ; and we shall presently see 

 that ^, y] depend on factors which never exceed the radius R. We 



may therefore to great accuracy take w = —p-pr (1 — "Ft)' *^^ 



second factor of which, analogously to (16), we shall denote by ii. 

 Substitute the expressions (4) just employed in the assumed 



quantities (5) ; and put the factor -3 ^^ = k. We shall obtain, 



in consequence of the minuteness of k and the intervals /', t% the two 

 values 



e = hk{krX, — f X), r=: ifea^'X, 4-^"X); 



whence there results ^" — ^', or 



