the Orl'its of Comets. 89 



we shall varv by a very small quantity the instant of the 

 perihelion passage : take in this hypothesis, 



a-r=p, u'-F' = p', U"-F" = p". 



This being done, with the perihelion distance and the 

 instant of the passage of the comet by this point, found in 

 the first hypothesis, we shall calculate the angle v and the 

 vector radius r, in the supposition of a very eccentric el- 

 lipsis ; so that by calling e the relation of the eccentricity 

 of this ellipsis to its half great axis, the difference 1 — e 

 is ecjual to a very small quantity, for instance to -jL-. In 

 order to have the angle v, in this supposition, it will be 

 sufficient to add to the anomaly v, calculated in the para- 

 bola, a small angle, the sine of which is 



■jL (1 — £').tang ^ ?;.(4— S.cos^ iy — 6.cos' i?') ; 

 The new anomaly v bemg thus known, we shall substi- 

 tute it in the equation 



r = — -—■ (1 tang^ I y ) 



which is the expression of the vector radius in a very ec- 

 centric ellipsis : bv this means we shall have the corre- 

 sponding vector radius r. We shall calculate in the same 

 way v', r' ; v", r" ; 7>"', r'"; from which we shall extract 

 U,'U'j U", V, V, V ; in this case take 



u- r= q, U'- V'= q', U"-r"= q". 



Finally, let us call u the number by which we ought to 

 multiply the supposed variatioi) in the perihelion distance, 

 in order to have the true one; t the number by which we 

 ought to multiply the supposed variation in the instant of 

 the perihelion j)assage ; and ^ the number by which we 

 ought to multiply the value supposed fori— e; we shall 

 form the three equations 



u.{7i —m ) -f- t.{p — m ) + ^.{q —m ) + w = 0, 

 u.{n'—m') + t.{p'—m) + S.(q' ~m) + m — 0, 

 ■ii.{n" — m")- -{■ t.[p" — m') + S.{q'' — m") + m"=, 0. 

 We shall have, by means of these equations, the values 

 of 71, i and (J, from which we shall extract the true perihe- 

 lion distance, the true instant of the passage by this point, 

 and tlie true value of 1—e. Let D be the perihelion di- 

 stance, and a the semi great axis; we shall have a = ; 



from which it is easy to conclude, that the time of the re- 

 volution of the comet will be equal to the number of sidereal 



years expressed bv j. We shall have as n\ page 



(l-e)^ 



85 J 



