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