= ?\/- 



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



[A"] = «" + /3" + y" + 8" . . . = w A:("), 



squaring, we may write, 



the probable deviation of a single datum is 



and consequently that of the arithmetical mean of m data 



2 (yfc2«)_^(,0 ^(«)) ^ 



m ' 



consequently it is an equal chance that k"- lies between 



[A"] / / 2 (/^(^'O-y^W/fcWj X 



m ^ V \ m /' 



and I^i- , ./ ("-("'■'-'■> -k('n^'))\ 

 m V \ ni / 



or that 



where the bracket refers to the limiting values, between which 

 the probabiHty = . 



In the apphcation to the law ^ A found above for ■^ (A), we 



require each time the value of aV A:^"). Thus, if we designate 

 generally nf r^„-] 



V = ^M 



* m 



and extract on both sides the Mth root, neglecting the higher 

 powers of the limiting values, then 



^^'■'-{■±^\/In/(-I^,-0}- 



This formula requires besides only the determination of the 

 values of A;(") for any n that may be taken. For the function 

 4i A which here obtains, we have 



h ")"°° 



;t(«)=_-_ r A"e-^'^'</A; 



or if, in order to be able to bring into calculation the uneven 

 powers of the errors (which else must always destroy each other), 

 we regard all errors as positive 



