THE 



PHILOSOPHICAL MAGAZINE 

 AND JOURNAL. 



31'' MARCH 1825. 



XXV. On the Method of the Least Squares. By J. Ivory, Esq. 

 M.J. F.B.S. 



[Concluded from p. 88.] 



4. A DMITTING the expression 4> {e^) for the probability 

 -^^*- of an error, we may both demonstrate the rule of the 



least squares, and determine the form of the function, by the 



following short process: 



Let P stand for the product, 



<p{e').<p{/^).^{e"^)8ic.; 



then the most probable system of the errors will be found by 

 the equations which make P a maximum, viz. 



( -r— = 0, — — = 0, &C. 



' dx ^ dy ^ 



Now, n being the number of the errors, we have 



log 



l-t log.il = log.l^ + log.^^ + &c. 



Again, by expanding !p(f'), we get 



^ = 1 -¥e''-ge*-^c.; 

 wherefore, log. -^^ = — h^e"- — (g - -^)^— &C' 

 Hence log. ^ = - h'^ii.e'- (^g - J^'^S.e*- &c.: 



and, by substituting this value in the foregoing differential 

 equations, we get, 



,, d.fi.e' / hi \ d.S.e* „ 



''=^ '-77- + {s-~)-jr +«'^-' 



d^ \o 2 / dy ^ 



But as we are not now in search of a mere mathematical 

 Vol. as. No. 323. March 1825. X Theory, 



