Mean Results of Observations, 95 



this determination. It will therefore be important to ascertain 

 by the observations themselves, if they are incompatible with 

 the supposition that has been made, that is to say, with the 

 symmetry of all the curves of probability on each side of a 

 point corresponding to the same abscissa : now, in fact, condi- 

 tions do exist which the observations should satisfy if this sym- 

 metry really have place. 



Let us suppose, for the sake of illustration, that the mean 

 result of a great number of observations be successively taken 

 away from the particular results of each of them, which will 

 make known their differences each way from the mean, which 

 will be in general very small quantities, positive or negative, 

 the sum of which will be nothing. If we take the sum of any 

 powers of these errors, neglecting the signs, and the sum of 

 the double of those powers, it is evident that the ratio of the 

 first sum to the second will be inversely as the greatness of the 

 errors, and consequently a very considerable quantity. Also, 

 if the square root of the second sum be taken, the ratio of the 

 first to this root will also be a large number of the order of the 

 square root of the number of observations ; that is, if there be, 

 for example, a million of observations, the ratio in question will 

 be comparable to one or to several thousands ; but it will not 

 be the same where the first sum is composed of uneven powers, 

 and that their changes of sign are considered. This circumstance 

 will diminish this sum ; and the calculus shews that in the hy- 

 pothesis of an equal probability of equal errors, plus or minus, 

 the ratio of the sum of their uneven powers to the square root 

 of the sum of the double powers, must be an inconsiperable 

 fraction : we find, forexample, that there is one against one to 

 wager that the observations are incompatible with this suppo- 

 sition, when the ratio shall equal a fraction not differing much 

 from | ; thus, in comparing the sum of the cubes of the errors 

 to the square root of the sum of the sixth powers, or the sum 

 of the fifth powers to the square root of the sum of the tenth 

 powers, &c. when we find for one of these ratios a fraction 

 which is not much below ■§■, that will suffice to shew the hypo- 

 thesis in question to be improbable, and, consequently, that 

 the observations ought to be rejected, as was stated above. 



In a great number of cases, and particularly in questions of 



