Astronoirdciil and Nautical Collections. 381 



servations, the tli=;tance of the perihelium, and the time from the 

 periheUum to the Hist and third observations ; which gives us 

 the time that ought to ehipse between these observations, accord- 

 ing to each of the hypotheses : and hence we have the first com- 

 parison. We then add, for each of the three orbits, to the time 

 between the perihelium and the first observation, the observed 

 time from the first to a second observation, sufficiently remote 

 from both the others, and compute the respective geocentric lon- 

 gitude ; or the latitude, if the latitudes vary more rapidly than 

 the longitudes : and the place so computed gives, together with 

 the observation, the second comparison. 



§. 77. 

 The whole proceeding will therefore stand thus, 



1 Hyp. 2 Hyp. 3 Hyp. True Oibi(, 



Curtate dist. I Obs. a' a'+w* a' a'+j- 



3 Obs. a'" a'" a"'+w a"'-|-?/ 



Time between 1 and 3 obs. t r+p T+q t" obs. 



Longitude in 2 obs. a a.+ -j a.+s a" obs. 



We have then J^ + ^11= t" -r, and — + !l = a." - 

 in n VI n 



a. ; whence we obtain the value of x and y in the same manner 



as in §. 75. If now ?« and n have been assumed not too great, 



and X and y are still smaller, or not conbiderably larger, the whole 



of the elements ofthe orbit may be readily found by interpolation, 



§. 78. 



It must, however, be remembered, that three complete obser- 

 vations are more than sufficient to determine the orbit of a comet 

 considered as a parabola : that is, if the orbit were not correctly 

 parabolic, or if the observations were erroneous, we could only 

 represent by a parabola three longitudes and two latitudes, 

 or two longitudes and three latitudes : and for this reason 

 I have employed, in the method of correction here ex- 

 plained, pnly the longitude or the latitude of the middle ob- 

 servation ; while in the methods of Lambert and LAPL-ict, 

 as well as in that of Newton here applied to the parabola, we 

 appear to satisfy three complete observations. But this appear. 



Vot. XIIL 2 D 



