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

158 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