30i LAGEANGE. 



560. In the third problem it is supposed that there are a 

 cases at each observation in which no error is made, h cases in 

 which an error equal to — 1 is made, and c cases in which an error 

 equal to r is made ; the probability is required that the error of 

 the mean of n observations shall be contained within given 

 limits. 



In the fourth problem the suppositions are the same as in the 

 third problem ; and it is required to find the most probable error 

 in the mean of n observations ; this is a particular case of the 

 fifth problem. 



561. In the fifth problem it is supposed that every observation 

 is subject to given errors which can each occur in a given number 

 of cases ; thus let the errors be p, q^ r, s, ... ^ and the numbers of 

 cases in which they can occur be a, h, Cyd, ... respectively. Then 

 we require to find the most probable error in the mean of n ob- 

 servations. 



In the expansion of {ax^ + hx^ + cic** + ...)" let M be the coeffi- 

 cient of iC* ; then the probability that the sum of the errors is yit, 



and therefore that the error in the mean is — is 



n 



M 



Hence we have to find the value of //. for which M is greatest. 



Suppose that the error p occurs a times, the error q occurs 

 /3 times, the error r occurs 7 times, and so on. Then 



a + /3 + 7+ =n, 



pOL + ql3 + ry -\- = fi. 



It appears from common Algebra that the greatest value of fjL 



is when 



a_/3_7_ _ n 



a h c a4-& + c+..'' 



^T , Lb pa-^- qh + rc-\- ... 



so that - —^ S • 



n a + 6 + c+ ... 



This therefore is the most probable error in the mean result. 



562. With the notation of Art. 561, suppose that a, h, c, ... 



