422 Mr de morgan, on the theory of 



produced upon our probable results by the supposition of an extraneous and more accurate 

 mode of estimating errors. For this purpose it will be sufficient to take the simple average 

 of observations of one quantity. In this case the functions are a^— x, a-^— x,...a, — x: and 

 the first approximation, obtained by making '2.{-^"o(a — xf\ a minimum, gives for oo the 

 weighted average 2(v/'"o , a) -r- 2\|/"o. Let T be a value which we see reason to prefer to 

 this average for determination of errors, so that «[ = T + e, , &c. : whence we get 



w=T + 2(x|."0 . e) (2>/,"0)-'. 



Take S{x//"'o (a - xf\, or 2f v|/"'o (T - a? + e)'}, and substitute for T - w. We get for 

 the next equation of approximation 2(\|/"o . a — at) + ^ V = 0, where 



r.<.ro..,-.cfv.,i^).,.(ro.«,(S0^)--.ro(?<^))-. 



When we can only obtain reputed errors, we have 2(\|/-"o, e) = 0, and T = x. The first 

 correction of the average is 



S(>/."0 . a) 1 2(^/."'0■e^) 

 ^~ 2x|/"0 ^6 ^f"0 ' 



I have verified the value of V by application of both Taylor's theorem and the rever- 

 sion of series to 2\fA'(y + e) = 0. Supposing the observations made under one law of facility, 

 and turning back to our second approximation, in which the variable part of yj/x is 

 — ex' + log {p + qsc^), we find for the correction of the average the average cube of the 

 reputed errors multiplied by (q : p)" -~ c. This is a very small quantity, having the sign 

 of the average cube : whence we infer, as we might have expected, that a positive average cube 

 of error indicates a presumption that the average of observations is less than the truth; and 

 vice versa. 



The only law of facility under which the average is necessarily the most probable result 

 is that in which 2v//-'(a -x) = gives '2c{a — x) = independently of the values of letters 

 and of the number of the functions. Let y^'u = xi^'"')^ ^^^ ^^ ^^^ ^^^^ ^X"" ~ ^ ^'^^ 2a? = 

 must be true together for all values of x : and we know that -^x must be an odd function. 

 Hence S^-* + ^v = gives 2a; + « = 0, or 2x<» + x(~^^) = ^' °'' x(^) = ^X^» which admits 

 of no solution except j^a? = ax. Hence we deduce Jce''^ : Jir as the only modulus which 

 absolutely gives the average as in all cases the most probable result. 



The greatest mathematical difficulty of the subject, the connexion of the sum of an 

 unlimited number of errors with its modulus, may receive the following illustration of its 

 demonstration, though of a character requiring much consideration, and at first of a repulsive 

 aspect. It must be premised that an observation., as yet so called, need not be simply a result 

 of perception, but may combine both thought and sense. It is enough that, being subject to 

 error, positive and negative errors should be equally likely. Thus the average of a number 

 of similarly situated observations may itself be considered as an observation, in anything 

 hitherto laid down. 



