determining the Orbits of Comets. 13 



equation with one unknown quantity, but of a very high de- 

 gree, which would have to be solved by trials. It is therefore 

 preferred to solve the system of those equations by trials. 



If we denote by x 9 y 9 #, the coordinates of a point referred 

 to the centre of the sun as the beginning of the coordinates, 

 we have the following three equations for three points in the 

 same plane passing through the centre of the sun: 



Ax + By + C * =0 



Ax 1 + By + C*' =0 



Ax" + By" + Cz"=:0. 



The equation of condition for these nine coordinates, after 



B C 



the ratios— r- and — r- have been eliminated, may be repre- 



A A. 



sented in different forms useful for future application. In or- 

 der to find the factors by which each of these equations must 

 be multiplied with a view to eliminate B and C, by which means 

 A likewise disappears, we have to combine these two equa- 

 tions : 



W + PW + p n y ,h = 

 pz + ptz* + p"z u = 0. 

 We obtain from them 



p proportional to y n z' — %f z u 



p' y z" — y u z 



pii yi z — yz*, 



or as these latter values are the double areas of the triangles 

 between the beginning of the coordinates and the points y #', 

 y *", &c, respectively in the plane of the y ?, and as these 

 again are proportional to the areas of the triangles in the plane 

 of the orbit, these factors become proportional to [Vr"], 

 — [*" ^"]j \? ^]y if these symbols stand for the double areas of 

 the three triangles, viz. sun, second and third places of the co- 

 met ; sun, first and third places of the comet ; sun, first and 

 second places of the comets. The negative sign is introduced 

 in order to avoid negative areas, in as much as all areas are sup- 

 posed to originate from the turning of the first radius vector 

 always in the same sense. 



Hence the equation of condition has three forms : 

 [V yff] x _ [ r jgirj x \ + [ r ^ x \\ = 



(12) [r 1 r"] y - [r /•"] y' + [r rf] y" = 



[ r t r"] x _ [ r r >q % ' + [ r r t] % n = . 



Let us now designate 

 The three curtate distances of the comet froml , „ 



the earth, by J" ?> § > ? > 



