ERRORS OP OBSERVATION. 416 



\lfiC.ai^dx, (series) x f -^x.w^dx, &c. 



Hence, when these last integrals are diminished without limit, we obtain -y^ce""^: -v/tt as the 

 final modulus. 



There is a want about the preceding investigation, and also about that of Laplace, which 

 has never been complained of, and for a sufficient reason. It requires very high principle to 

 scrutinise the accounts of a debtor who is always making mistakes in our favour, and always 

 accompanies his statement by a cheque. All the methods in which e~"* is employed give 

 much better approximations than could have been expected from the demonstration, for even 

 rather small values of er. It would have been no matter of surprise, judging by the rejections 

 of the process, if every decimal place of correct result had demanded a cipher in the numerical 

 value of (T. Nevertheless, we get three places when we hardly want tlie second, and do not 

 deserve the first. The reason must be sought at the beginning of the process : or rather 

 presumption of the fact ; for I can give no account of the matter which sufficiently explains 



the phenomenon. If we take any even function (piv which gives j <pxdx = 1, and if 



/ 



(px .x'^dx'='Ai, we have, far more closely than we could expect, 



2 jc-.-JiAi 



(bxdx = — y— I e~^ dt. 



Let all values be supposed equally probable, the most extreme case of a theory of errors. 

 This supposes (px such a discontinuous function as 



(-co)o(-^).(+l)o(+o=), 

 and gives A2 = (I2a')"', and 



2ax = — — / €"' dt. 



Let a = 1. Then x = -01 gives '02 = "027; and -J? = ^ gives 1 = -Q. Now try the case 

 of uniformly descending facility, the limits being - 1 and + 1. This gives 



(- cc ) (- 1) 1 + J? (0) 1 - 0? (+ 1) (+ CO ) ; 

 also y^j = 65 and 



Here x = -Ol gives -02 = -OlS ; -05 gives -10 = '10 ; •! gives -2 = "193 ; '5 gives 76 = -78 ; 

 and 1 gives 1 = '99. 



fX 1 Vc.(it+6a:?+...) 



If we assume <px dx = — r— / e~^'dt; 



and if / (px . a^dx = J^. : the first approximation, in which b, &c. are rejected, gives 



c = (2^2)"'; the second gives 



5^2 - ^/(25A^'' - 7J4) ^ c (1 - 20.^2) 



c = 6 = . 



2A^ 3 



Vol. X. Part IL 53 



