380 Mr. F. Y. Edgeworth's Problems in Probabilities. 



And c (the real Modulus) 



= 2-096 ... x OQ (the real probable-error) 

 = 2-096 (0 r Q' + 2-?). 



The probability of the (n + l)th observation exceeding P is 

 therefore proportional to 



l+ O'Q' I 



'Q' J 



This expression is to be multiplied by 



1 _?_ 2 1 _£ 

 -y^e * 2 X —yz.e x' 2 dzd%, 



VTT V IT 



and integrated through the whole range of z and ?. The 

 first term of the expanded expression is 



| [1-00477*)], 



which is the uncorrected value of the sought probability. 

 The second term vanishes. For the third term, the correction, 

 I find (putting, as before, /3 for *477 a) 



e-t* (ff-B) V^ . ^ 2 (ff 3 + 2-4/3) x ^ 



KH H) 2xl-7x-227xn *9n 



The second portion of this addendum is the correction due 

 to the error of putting the apparent Mean for the real Mean. 

 The first part of the expression corresponds to the correction 

 which we obtained when we supposed the real Mean known. 

 The present result differs from the former one in that its sign 

 is negative, and its absolute quantity less by half; as it ought 

 to do in view of the different enunciation of the problem. 

 It should be added that in so far as z and J* tend to vary in 

 the same direction, the correction is not accurate but of the 

 nature of a superior limit. 



I have applied this correction to the question discussed 

 under Problem I., namely, what was the probability that the 

 limit fixed in 1844 for the uncovered Notes would not be 

 passed in the proximate future by the Notes in the hands of 

 the public. The value found by Problem I., viz. *98, is 

 reduced by Problem II. to -95. 



III. A third problem is suggested by the procedure of the 



* Above, p. 375. 



