If one wishes to make a life loan it will be observed that the tables of life annuities give the capital required to constitute a life annuity at any age, a simple proportion will give the rent which one ought to pay to the individual from whom the capital is borrowed. From these principles all the possible kinds of loans may be calculated. The principles which we have just expounded concerning the benefits and the losses of institutions may serve to determine the mean result of any number of observations already made, when one wishes to regard the deviations of the results corresponding to divers observations. Let us designate by _x_ the correction of the least result and by _x_ augmented successively by _q_, _q´_, _q´´_, etc., the corrections of the following results. Let us name _e_, _e´_, _e´´_, etc., the errors of the observations whose law of probability we will suppose known. Each observation being a function of the result, it is easy to see that by supposing the correction _x_ of this result to be very small, the error _e_ of the first observation will be equal to the product of _x_ by a determined coefficient. Likewise the error _e´_ of the second observation will be the product of the sum _q_ plus _x_, by a determined coefficient, and so on. The probability of the error _e_ being given by a known function, it will be expressed by the same function of the first of the preceding products. The probability of _e´_ will be expressed by the same function of the second of these products, and so on of the others. The probability of the simultaneous 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_