92 M. Poisson's Memoir en the 



true value by a quantity greater than some assigned limit. To 

 solve it, he supposes known the law of probability of error in 

 the observations, or the values which they may give for the 

 thing which it is sought to determine ; an hypothesis which 

 prevents the formulae deduced from it being applicable and of 

 any use in practice. It is to Laplace we are indebted for hav- 

 ing rendered the probability of the mean result independent of 

 this law in the cases where the observations are numerous ; so 

 that from the sole numerical data of the observations the pro- 

 bability may be calculated of a determinate limit of error to 

 be apprehended in taking this result for the value of the un- 

 known quantity. I hope the details on this subject, into which 

 I have entered in my Memoir, will be well calculated to dissi- 

 pate the doubts which might still remain as to the degree of 

 approximation of this probability*. 



To form a precise and general idea of the limit to which the 

 mean result of observations approaches indefinitely in propor- 

 tion as the number of them increases, we must suppose the 

 construction of a curve, the ordinates of which are proportional 

 to the probabilities of the values of the unknown quantity, 

 which last is expressed by the corresponding abscissae. If the 

 law of probability change from one observation to another, this 

 curve will change also; and another will be constructed, the 

 ordinates of which will be means, for each abscissa, between 

 those of all the particular curves. This being so, the limit in 

 question, in every case, is the abscissa which corresponds to 

 the centre of gravity of the area of the mean curve. This 

 limit to which the mean result of the observations converges, 

 is not necessarily one of the values of the unknown quantity 

 which have the most probability, and are given most frequently 

 by isolated observations. It may even happen that the proba- 

 bility is altogether none, and that it cannot be given by any 

 single observation ; which, in fact, will be the case if the ordi- 

 nates of all the curves of probability are null for the same 

 abscissa, and symmetrical on each side of it. In the general 

 case where the curve of probability varies from one observation 

 to another, it may also happen that the areas of all the curves 



* Supplement to Theorie Analytique des Probability, p. 1. 



