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



and combining with the expression of log W 



m , rn 



4- 1 _ A3— ' — A44- Slc 



If the value of h is here to become the most probable (conse- 



W 



quently if W is to be an absolute maximum, and log ^^ is on 



that account always to have a negative value) the coefficient 

 of A must be made = 0. For the maximum of W there 

 will be 



?w 1 



J -2 mil s.,^ = 0, or ^ = ^2 >/2, (16.) 



and if we substitute this most probable value in the remaining 

 members, every other value of W, so far as it depends on another 

 h, will be given by the formula 



We may here make the series contained as factor in the ex- 

 ponent = 1. For if we introduce the value of the most probable 

 h, it becomes 



A^ A . 



which series must still be multiplied by m y| . If — is a small 



fraction the series will deviate little from unity, and, stiU more, 



the difference of the complete rigorous value from the ap- 



£fi , ... .A 



proximate e "' ifi wiU be quite insensible. But if -r were to 



have a greater value, W would be very small compared to W, 

 and for that reason the whole accurate expression would have 



no material interest. Hence the probabihty that h =* — — or 



1 ' 



W, is to the probability that the value of h = — y + A, or W ^ 



1 : e ' or 1 : e n- 



