Scr. VHI. CORRECTION OF ELEMENTS. 59 



direction, but that some are in excess and others in de- 

 fect, they will compensate each other when combined. 



However, the values of the elements determined sep- 

 arately, 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 

 eccentricity depends upon the longitude of the perihe- 

 lion, the mean motion depends upon the major axis, the 

 longitude of the node upon the inclination of the orbit, 

 and vice versa. Consequently, the place of a planet com- 

 puted with the approximate data will differ from its ob- 

 served place. Then the difficulty is to ascertain what 

 elements are most in fault, since the difference in ques- 

 tion 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 simultaneously, 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 Proba- 

 bilities ; 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 ine- 

 qualities, 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 

 motions. The periodic inequalities derived from a great 

 number of observations are employed for the determina- 

 tion of the values of the masses till such time as the 

 secular inequalities shall be perfectly known, which will 

 then give them with all the necessary precision. When 

 all these quantities are determined in numbers, the lon- 

 gitude, latitude, and distance of the planet from the 

 sun are computed for stated intervals, and formed into 

 tables, arranged according to the time estimated from a 

 given epoch, so that the place of the body may be deter- 

 mined from them by inspection alone, at any instant, for 

 perhaps a thousand years before and after that epoch. 

 By this tedious process, tables have been computed for 

 eleven planets, besides the moon and the satellites of 



