Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/167

Rh acquired would be formed by these funds and from this annuity the State contributes annually to the sinking fund.

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