60 CORRECTION OF ELEMENTS. SECT. VI 11. 



with the approximate data will differ from its observed place. 

 Then the difficulty is to ascertain what elements are most in 

 fault, since the difference 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 simultaneously, 

 and to make the sum of the squares of the errors a minimum 

 with regard to each element (N. 1.38). The method of accom- 

 plishing 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 deter- 

 mination of astronomical data. A series of observations con- 

 tinued 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 motions. 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 inequalities shall be perfectly 

 known, which will then give them with all the necessary pre- 

 cision. When all these quantities are determined in numbers, 

 the longitude, latitude, and distance of the planet from the sun 

 are computed for stated intervals, and formed into tables, ar- 

 ranged according to- the time estimated from a given epoch, so 

 that the place of the body may be determined from them by 

 inspection alone r at any instant for perhaps a thousand years 

 before arid after that epoch. By this tedious process, tables have 

 been computed for all the great planets, and several of the 

 small, besides the moon and the satellites of Jupiter. In 

 the present state of astronomy the masses and elements of the 

 orbits are pretty well known, so that the tables only require 

 to be corrected from time to time as observations become 

 more accurate. Those containing the motions of Jupiter, 

 Saturn, and Uranus have already been twice constructed within 

 the last thirty years, and the tables of Jupiter and Saturn 

 agree almost perfectly with modern observation. The following 

 prediction will be found in the sixth edition of this book, pub- 

 lished in the year 1842 : " Those of Uranus, however, are already 

 defective, probably because the discovery of that planet in 1781 

 is too recent to admit of much precision in the determination of 



