86 RELATIONS PERTAINING SIMPLY [BoOK 1. 



72. 



Certain problems, which are in frequent use in the determination of the orbits 

 of planets and comets, will bring this section to a close. And first, we will revert 

 to the parallax, from which, in article 70, we showed how to free the observed 

 place. Such a reduction to the centre of the earth, since it supposes the distance 

 of the planet from the earth to be at least approximately known, cannot be made 

 when the orbit of the planet is wholly unknown. But, even in this case, it is pos 

 sible to reach the object on account of which the reduction to the centre of the 

 earth is made, since several formulas acquire greater simplicity and neatness 

 from this centre lying, or being supposed to lie, in the plane of the ecliptic, 

 than they would have if the observation should be referred to a point out of the 

 plane of the ecliptic. In this regard, it is of no importance whether the obser 

 vation be reduced to the centre of the earth, or to any other point in the plane 

 of the ecliptic. Now it is apparent, that if the point of intersection of the 

 plane of the ecliptic with a straight line drawn from the planet through the true 

 place of observation be chosen, the observation requires no reduction whatever, 

 since the planet may be seen in the same way from all points of this line : * where 

 fore, it will be admissible to substitute this point as a fictitious place of observa 

 tion instead of the true place. We determine the situation of this point in the 

 following manner : 



Let X be the longitude of the heavenly body, /? the latitude, // the distance, 

 all referred to the true place of observation on the surface of the earth, to 

 the zenith of which corresponds the longitude /, and the latitude b ; let, more 

 over, n be the semidiameter of the earth, L the heliocentric longitude of the cen 

 tre of the earth, B its latitude, II its distance from the sun ; lastly, let L be the 

 heliocentric longitude of the fictitious place, 1? its distance from the sun, A -j- d 



* If the nicest accuracy should be wanted, it would be necessary to add to or subtract from the given 

 time, the interval of time in which light passes from the true place of observation to the fictitious, or from 

 the latter to the former, if we are treating of places affected by aberration : but this difference can 

 scarcely be of any importance unless the latitude should be very small. 



