Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/168

taneous existence of the errors $e$, $e'$, $e''$, etc., will be then proportional to the product of these divers functions, a product which will be a function of $x$. This being granted, if one conceives a curve whose abscissa is $x$, and whose corresponding ordinate is this product, this curve will represent the probability of the divers values of $x$, whose limits will be determined by the limits of the errors $e$, $e'$, $e''$, etc. Now let us designate by $X$ the abscissa which it is necessary to choose; $X$diminished by $x$ will be the error which would be committed if the abscissa $x$ were the true correction. This error, multiplied by the probability of $x$ or by the corresponding ordinate of the curve, will be the product of the loss by its probability, regarding, as one should, this error as a loss attached to the choice $X$. Multiplying this product by the differential of $x$ the integral taken from the first extremity of the curve to $X$ will be the disadvantage of $X$ resulting from the values of $x$ inferior to $X$. For the values of $x$ superior to $X$, $x$ less $X$ would be the error of $X$ if $x$ were the true correction; the integral of the product of $x$ by the corresponding ordinate of the curve and by the differential of $X$ will be then the disadvantage of $X$ resulting from the values $x$ superior to $x$, this integral being taken from $x$ equal to $X$ up to the last extremity of the curve. Adding this disadvantage to the preceding one, the sum will be the disadvantage attached to the choice of $X$. This choice ought to be determined by the condition that this disadvantage be a minimum; and a very simple calculation shows that for this, $X$ ought to be the abscissa whose ordinate divides the curve into two equal parts, so that it is thus probable