358 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 



The greater the number of observations^ the narrower will be 

 the Urn its between which t will lie with equal probability. 

 Therefore, if the observations are sufficiently numerous, deve- 

 loping v! and m" according to Taylor's theorem, we may be 

 permitted to consider only the first power of 8. Whence, 



?«'= r\tdt - SvJ/ (r) ... = i-8rKr); 



»^ 



and similarly, 



«" = i + SvI/(r). 



This form, as well as the combination of u and 1 — m in the 

 integral, shows that a still moi-e convenient form may be ob- 

 tained by bringing in another variable for u ; and this may be 

 best done by the equation 



M = i + 



2 ^/» 



consequently. 



1 — M = A — 



= K'-v„). 



the limits in relation to s being found by 



^ ' 2 j^/ n 

 According to this the integral becomes 



( ^ _ *'\" ; 

 V^ n ) '^^' 



or, because s in the differential contains only even powers. 



/2 S V" 4 {r) 

 \ n ) 



1.2.3 . . , 



a s. 



(1.2.3 



Now let 8 v' ^ be a finite magnitude = y ; thus the limit 8 

 decreasing with the increase of »/ n, s remains finite within the 

 assumed limits, however much n may increase. But if n is 

 large, we may, according to the develojament of logarithms in 

 Euler's Introductio, put 



(>-0'=- 



