Gibers* Essaj/ on Comets. 



421 



the proportion of the true or of the curtate distances of the 

 comet from the earth may be found for the first and third ob- 

 servations, as will appear in the next secUon. Now a fourth 

 and a fifth observation may again be combined with the third, 

 and we may obtain, by means of these, the proportion of the 

 first, third, and fifth, distances ; and having this proportion, we 

 may determine the distances themselves, from the condition 

 that the sun must be in the plane of the orbit. 



§ 26. 



In order to show this, we need only to find an equation 

 between x, y, z, and the longitude of the node and the inclina- 

 tion of the orbit of the comet. Let S be the centre of the sun, 

 S «V> a line pointing to the vernal equinox, S 9, the line of th« 



nodes ; let S A be ar, A B n y, B C, perpendicular to the ecliptic 

 = z, and C the place of the comet. Now if B D be perpendi- 

 cular to S gx , B D C will be the inclination of the orbit, and 

 calling gi Scyo, the longitude of the node, A, and BDC = i, we 

 have AE = x tang. h\ BE =: y — x tang. A; BD zr BE cos. 

 h^iy cos. h — X sin. A, and B C zz « zz B D tang, t =: y cos. h 

 tang, i — X sin. h tang. i. We shall therefore obtain, frbm 

 three observations, three equations of the form z'=.y cos. h 

 tang. % — X sin. h tang, i, each containing, when the proportions 

 of the curtate distances are given, only three unknown quanti- 

 ties, 5, A, and iy which may therefore be deduced from tlicm ; 

 for X, y, and z, are all dependent on ^ . (§ 7). 



