48 General Method for determining 



of the comet will make known the value of CST, of which 

 ve ought to make use, particularly if these two angles 

 are verv different; for ihen one of them will place the 

 comet further than the other from the earth, and it will be 

 easy to ascertain by the apparent motion of the comet, at 

 the instant of the observation, which of the two ought to 

 be preferred. In a great number of cases, one of them 

 will be negative, and must consequently be rejected ; but 

 if any uncertainty remains on this head, we might always 

 determine the true values of /3, /3', /3'', by observing to take 

 for |3 and jS', the two angles which render V very little dif- 

 ferent from U, and to take ibr jS and /S" the two angles 

 which render F' very little different from U'. 



We shall afterwards form a second hypothesis, in which, 

 by preserving the same [instant of perihelion passage with 

 the above, we shall vary the perihelion distance by a small 

 quantity, for example, by the fiftieth part of its value, and 

 we shall find out ni this hypothesis the values of U— V, 

 and of U'— F' ; thus, 



7«'= U- F, 71' ~ U' - F'; 

 finallv, we shall form a third hypothesis, ill which, by pre- 

 serving the same perihelion distance as in the first, we shall 

 vary by half a day, or a whole day (more or less) the instant 

 of the perihelion passage. We shall find out in this new 

 hypothesis, the values of U— F and of U'— F'; thus, 



m"= U- F, ??"= U'- F'-, 

 this being done, if we call u the number by which we ought 

 to multiply the supposed variation in the perihelion distance 

 in order to have the true one, and t the number by which 

 we ought to multiply the supposed variation in the instant 

 of the perihelion passage in order to have the true instant, 

 we shall have the two equations, 



u {m—m) -j- t {in— 'in!') =w, 

 u{n — n') -{-tln — n") =?i, 

 from which we shall extract u and I ; and consequently the 

 perihelion distance corrected and the true instant of the 

 passage of the comet by this point. 



The foregoing correction supposes that these elements 

 determined bv the first approximation, arc sufficiently exact 

 to treat as infinitely small their differences from the true; 

 but if the second approximation did not still appear suffi- 

 cient, we might have recourse to a third by operating on 

 the elements already corrected, as has been done upon the 

 first : we must only take care to make them undergo 

 •mailer variations. But in most cases this third approxi- 

 mation 



