LAPLACE. 585 



I'abscisse qui i^epond k la plus grande ordonnee de la courbe ; mais le 

 milieu que nous adoptons, est evidemment iudique par la theorie des 

 probabilites. 



This extract illustrates a remark which we have already made 

 in Art. 1008, namely that strictly speaking Laplace's method does 

 not profess to give the 7nost probable result but one which he con- 

 siders the most advantageous. 



lOl-i. Laplace gives an investigation in his pages 335 — 340 

 which amounts to solving the following problem : if we take the 

 average of the results furnished by observations as the most 2:)ro- 

 bable result, and assume that positive and negative errors are 

 equally hkely and that the function of the facility of error is the 

 same at every observation, what function of the facihty of error is 

 implicitly assumed ? 



Let the function of the facility of an error z be denoted by 

 e-'/'(z')^ which involves only the assumption that positive and nega- 

 tive errors are equally likely. Hence the value of y in the pre- 

 ceding Article becomes 



He-^, 



where (7 = '^{x — a^Y + y)r{x — a^Y + yjr {x — a^Y + • • • 



To obtain the most probable result we must determine x so 

 that a shall be a minimum ; this gives the equation 



{x — aj yjr' {x — ay + {x— aj i^' (^ ~ ^2)^ 



+ {x — a^) yjr' {x-ay+ ...^ 0. 



Now let us assume that the average result is always the most 

 probable result ; suppose that out of s observations ^ coincide in 

 giving the result a^, and s — i coincide in giving the result a^ ; the 

 preceding equation becomes 



I {x — aj -v/r' (x — a^ + (s — i) {x — a^ yfr' (x — a^ = 0. 



The average value in this case is 



ia^ + (g — i) a^ 

 s 



Substitute this value of x in the equation, and we obtain 

 (s-{ V {i ^* 



