288 NEW FORMULAE RELATIVE TO COMETS. 



E = sin (/"-/'> E' = sin {l-l"), E" sin (/'— /), s" = ^"', 



or when these longitudes vary from each other less than the corre- 

 sponding latitudes, we may more suitably adopt the values 



E r= a'y" — ay, E' =: a"y — ay", E" = a/ — a>/ ; e" = y% 



the three first of which are the values of B, B', B" (8) taken with a 

 change of signs. In both cases here implied, the values of a, a', &c. 

 will be as in formulae (A) ; and we have taken d = t' -h t\ 



The formulae (E) are regular in the composition of their terms, and 

 but little more complex than (A), from which they chiefly diflfer in 

 this respect, that the factor k cannot be eliminated, but must corre- 

 spond to a determinate value of the comet's distance r. Any change 

 in the value of this distance will however be attended only with small 

 additional computation ; since the calculated values of the diflferent 

 terms in (E) will, from the form which we have adopted, continue in- 

 variable. 



When the intervals f, t" happen to be equal, we shall have K = 

 Rfef, and the more simple values 



X, = X, + ^ (EW - E''a») + K (f - i + ^)? 



> (E') 



*»/ 



z,=^(Ey-EV) + K^ 



the quantities E, E', E", e" being as in (E), and the values of aja'j&c. 

 the same as in the particular forms (A'). 



The investigation of the formulae now given, we reserve for another 

 paper, which will soon appear. At present we shall briefly show how 

 the expressions given by Mr Pontecoulant may be readily deduced 

 from our expressions (19). 



If in these we omit the last terms depending on the earth's veloci- 



