Astronomical Society. 137 



A paper was also read, entitled " On the computation of the geo- 

 centric places of the planets for ephemerides :" by J. F. Littrow. 



The usual mode of computing the geocentric places of the planets 

 for ephemerides consists in instituting a first computation of their 

 heliocentric positions, which are then reduced to geocentric by well 

 known formulae. To avoid the very laborious calculations entailed 

 by this process, Prof. Gauss devised a method much more expeditious, 

 and which gives at once the geocentric right ascensions and declina- 

 tions, as well as the distance of the planet from the earth, in terms of 

 the co-ordinates of the earth and planets at the instant of observation. 

 The equations expressive of these relations are stated by M. Littrow, 

 and are extremely simple. By them, the right ascension (a), the de- 

 clination (o), and mutual distance (§ ) are given directly as follows : 



tan a = 



X + * 



tan o = — cos a 



X-f-* 



(I) 



t 



z + * 



sin 3 



X, Y, Z being the co-ordinates of the earth, and x, y, z of the planet. 



Now it is obvious, that if we neglect the perturbations (on which 

 neglect it will be observed the application of this method mainly de- 

 pends), all these co-ordinates are, ultimately, functions respectively of 

 the mean longitudes of the earth and planet, and their values may 

 therefore be tabulated as such ; and the tables, once constructed, will 

 last till the secular variations of the elements of the planetary orbits 

 shall have so changed them as to produce an error surpassing that 

 which can be tolerated in an ephemeris. — These tables once com- 

 puted, the foregoing equations reduce the annual computations to the 

 utmost brevity. The only point then to consider is the construction 

 of the tables themselves. 



The expressions of the co-ordinates in terms of the mean longitude 

 are made to depend on the true longitude as the independent variable, 

 by means of six constant quantities for each planet. These are, 1st, 

 the respective inclinations of the plane of the planet's orbit to the 

 solstitial and equinoctial colures and the earth's equator ; and, 2ndly, 

 the plane angles made by the line of intersection of the planet's orbit, 

 and the ecliptic respectively with its line of intersection with the sol- 

 stitial and equinoctial colures and equator. The author states the 

 method of computing these constants from the known elements of 

 the orbit, and gives their actual numerical values for all the planets 

 with their secular variations. — These once known, and the true lon- 

 gitude, as a preliminary step, computed for every mean longitude, he 

 states the equations by which the co-ordinates are formed from them 

 for every given value of the longitudes ; and, finally, embodies the 

 whole result of his computations in a table stating their values for each 

 of the planets, and for intervals of 4° for Mercury and Venus, 3° for 

 Mars, 2° for Jupiter, Saturn, and Uranus, and for each day in the year 

 for the Sun. 



New Series. Vol. 3. No. 14. Feb. 1828. T The 



