566 LAPLACE. 



but no serious error will arise from this circumstance, because 

 the true value of Y and the approximate value are both very- 

 small when X is sensibly different from zero. We may put (3) in 

 the form 



F = — j < I cos (Ix — CX + XV) dv j- 6""^'^'^ dx ; 



then by changing the order of integration, and using a result 

 given in Art. 958, we obtain 



^=2^\f^^^' (^)- 



This is therefore approximately the probability that E will lie 

 between c — rj and c-\-7j. 



It is necessary to shew that the quantity which we have 

 denoted by k^ is really positive; this is the case since hi is really 

 positive, as we shall now shew. From the definition of hi in con- 



junction with the equation I fi [z) dz =^ 1, we have 



J 6 



U^ = f " z'f, {z) dz ("f, (z) dz' - r zf, {z) dz ("z'f, (z') &■ 



J h J b J b J b 



= jj\z''-zz')f{z)f,{z')dzdz'. 



And so also 



2A? = JJ 1^ (/2 - z^')f, {z)fi {£) dz dz'. 

 Hence, by addition, 



^h^=n\z-z'ffi{z)f,{z')dzdz'. 



J b J h 



Thus ^hi is essentially a positive quantity which cannot be zero, 

 for every element in the double integral is positive. 



It is usual to call^ {z) the function which gives the facility of 

 error at the i^'^ observation ; this means that ^^ (2;) dz expresses the 

 chance that the error will lie between z and z + dz. 



If the function of the facility of error be the same at every 



