562 LAPLACE. 



greater than p, so that a — ^ is positive. The chance of an as- 

 siofned error will not be assumed the same at each observation. 

 If 71 be any integer comprised between a and /3 we shall denote 

 the chance of an error 71(0 at the first observation by N^, at the 

 second observation by N^, at the third observation by N^, and 

 so on. Let -ot be a factor such that all the products -377^, -5772 , 

 'S773, ... -5773 are integers ; such a factor can always be found either 

 exactly or to any required degree of approximation. Let 



where S denotes a summation with respect to n for all values 

 from P to a, both inclusive ; and let 



J- ^^ V1V2 * • * ^s * 



then the probability that 'utE will be exactly equal to mo), where 

 m is a given integer, is the coefficient of t^'^ in the development 

 of T according to powers of t; or, which is the same thing, the 

 probability is equal to the term independent of t in the develop- 

 ment of Tr^". 



For f^ put e^^^ and denote by X what T becomes ; then the 

 required probability is equal to 



Let X and fi be two given integers, such that \ — yit is positive ; 

 then the probability that tjjE will lie between \co and /xo), both 

 inclusive, may be derived from the last expression by putting for 

 m in succession the values jx, //- + !, ^6 + 2, ... X, and adding the re- 

 sults. Since the sum of the values of ^"'"^^"^ is 



7~i 



2sini6> 



2 



,-(x+i)0V-i _ p-{i^-i)esl-i 



the required probability is equal to 



sm - ^ 



we shall denote this probability by P. 



