LAPLACE. 587 



Here SJ^ stands for ; J.X., , and tlie value of B, will therefore 



depend on the values of the factors y^^, %, ... which we employ; for 

 example we may take each of these factors equal to unity, which 

 amounts to adopting the average of the results of observation ; or 

 we may take for these factors the system of values which we have 

 called the most advantageous system : if we adopt the latter we 



Similarly the probability that the error of the result deduced 

 from the second set of observations will lie between x — a^ and 



And so on for the other sets of observations. 



Hence we shall find, in the manner of Art. 1013, that the pro- 

 bability that X is the true value of the unknown quantity is pro- 

 portional to 



x-\-dx-a^ is -^ e-^2Mx-a2)Vic, 



e-^ 



where (t = ^^ (x-a^^ + ^^ (x- a^"" + I3^\x- a^' + ... 



Now determine x so that this probability shall have its 

 greatest value ; a must be a minimum, and we find that 



^ /3>^ + ff /g, + /g>., + . . . 



We may say then that Laplace obtains this result by deducing 

 a value of the unknown quantity from each set of observations, 

 and then seeking for the most p7'ohahle inference. If a^, a^, a^, ... 

 are determined by the most advantageous method, this result is 

 similar in form to that which is given in Art. 1007, if we suppose 

 that positive and negative errors are equally likely, and that one 

 function of facility of error applies to the first set of observations, 

 another function to the second set, and so on. For the numerator 



of the value of x just given corresponds with the S-y^, and the 



h^ 



denominator with the Sti> of Art. 1007. 



