NEW FORMULA RELATIVE TO COMETS. 287 



In the formulae hitherto given for this particular case, the factor (i 

 has not been introduced or noticed; and on this account, and the last 

 terms involving the earth's velocities X,, Y„ they are less exact than 



DY, 



the preceding expressions (A'), in which the term — ^ at least should 



be computed as no negligible, minute quantity. 



The general formulae, of which we -have now completed the inves- 

 tigation, and also (A'), fail in giving accurate results, when the quan- 

 tities A, A', A" and D are minute; and they become indeterminate, 

 when these quantities vanish. If we conceive the apparent path of 

 the comet during the interval /' -f- t" to be a great circle, passing 

 through a point in the ecliptic of which the longitude is less than that 

 of the sun by the small arc a, the tangent of the inclination of the orbit 



y y' y" 



will be evidently expressed by either of the three relations -^, — „ -^; 



and we shall have then A = 0, A' = 0, A" = 0, and also D = 0, to 

 which the formulae (A) and (A') are inapplicable. The case in which 

 these quantities become small, is now obvious without further reflec- 

 tion ; and should it be inconvenient or impossible to take new observa- 

 tions of the comet at other places in its orbit, we must abandon for- 

 mulae (A), and have recourse to different expressions, capable of deter- 

 mining the component velocities x,,y,, z,. 



Such expressions are the following : they have not to our knowledge 

 been hitherto presented. 





HE) 



There are in fact two sets of formulae, comprehended in these ex- 

 pressions. We may take in terms of the geocentric longitudes ; 

 VI. — 3 w 



