Brinkley on Cornets. 167 



July 14, and probably were the first elements published, and 

 even when compared with the above, they must be considered 

 as having some claim to exactness. 



In correcting the first approximations, Dr. Brinkley employed 

 a method, which, it is beUeved, has not been used before. 

 Instead of changing the approximate perihelion distance, and 

 approximate time of passage through perihelion by small quan- 

 tities, as in M. Laplace's method, he obtained two equations 

 in which the unknown quantities were the corrections of the 

 perihelion distance, and of time of passage through perihelion. 



This was done by investigating the fluxions of the anomalies, 

 heliocentric longitudes, and latitudes, computed by help of the 

 approximate perihelion distance, and approximate time of pe- 

 rihelion, and of three observations. This, at first sight, might be 

 supposed to lead to intricate formulae ; but it is by no means the 

 case. The operations will be found very considerably shorter 

 than by M. Laplace's method, when great exactness is required. 

 This method is particularly applicable in cases where it is ne- 

 cessary to investigate the elliptic orbit. Also, in M. Laplace's 

 method, there is nothing by which the degree of exactness, re- 

 quired in the first approximate elements, to apply with advan- 

 tage his method of corrections, is easily shown. The want of 

 this may sometimes lead to very tedious calculations. Thus, in 

 the present instance, considerable exactness is required in the 

 first elements, when the observations of July 4th, 13th, and 

 20th, are used ; because, on the first day, the angle at the 

 comet was nearly a right angle, and the difference between the 

 heliocentric longitudes of the comet and earth only amounted 

 to a few degrees ; and, on the second, the angle at the sun 

 was nearly a right angle, and the difference of the heliocentric 

 longitudes nearly 80°. It was on this account that Dr. Brink- 

 ley found it convenient to re-compute the first approximation 

 with all the accuracy of which the observations were sus- 

 ceptible. 



