68 CORRECTION OF ELEMENTS. [SECT. vm. 



can only be regarded as approximate, because they are so 

 connected, that the estimation of any one independently, 

 will induce errors in the others. The excentricity depends 

 upon the longitude of the perihelion, the mean motion de- 

 pends upon the major axis, the longitude of the node upon 

 the inclination of the orbit, and vice versd. Consequently, 

 the place of a planet computed with the approximate data 

 will differ from its observed place. Then the difficulty is to 

 ascertain what elements are most in fault, since the differ- 

 ence in question is the error of all ; that is obviated by 

 finding the errors of some thousands of observations, and 

 combining them, so as to correct the elements simulta- 

 neously, and to make the sum of the squares of the errors 

 a minimum with regard to each element (N. 136). The 

 method of accomplishing this depends upon the Theory 

 of Probabilities ; a subject fertile in most important results 

 in the various departments of science and of civil life, and 

 quite indispensable in the determination of astronomical 

 data. A series of observations continued for some years 

 will give approximate values of the secular and periodic 

 inequalities, which must be corrected from time to time, 

 till theory and observation agree. And these again will 

 give values of the masses of the bodies forming the solar 

 system, which are important data in computing their mo- 

 tions. The periodic inequalities derived from a great 

 number of observations are employed for the determination 

 of the values of the masses till such time as the secular in- 

 equalities shall be perfectly known, which will then give 

 them with all the necessary precision. When all these 

 quantities are determined in numbers, the longitude, lati- 

 tude, and distance of the planet from the sun are computed 

 for stated intervals, and formed into tables, arranged accord- 

 ing to the time estimated from a given epoch, so that the 

 place of the body may be determined from them by in- 

 spection alone, at any instant for perhaps a thousand years 

 before and after that epoch. By this tedious process, tables 



