and the Elimination of Chance. 321 



not exceeded. With regard to the range or extent of diver- 

 gence with which we are here more particularly concerned, 

 w^hen this is large it will he found necessary to desert alto- 

 gether the neighbourhood of the received formula, and revert 

 to Poisson's earlier formula *, where the argument of the 

 probability-curve is the unreduced k (written above, p. 310), 

 and the expression outside the sign of integration is the unre- 

 duced one written on p. 312. This incorrectness and corri- 

 gibility may be illustrated by the following example. Let 

 fx — 999, p = -jLy, m = 299, ri = 100. Here the current, as con- 

 trasted with the unreduced formula, exaggerates by trillions 

 of times the improbability of the observed divergence being 

 due to accident — the odds in favour of a disturbing cause. 

 For the t 2 of the former formula is 222*4, giving as the value 

 of the probability of mere chance about 10 -98 . Whereas P, 

 which measures the order of the sought probability according 

 to the unreduced formula, is 153" 7. And the corresponding 

 value of the qusesitum is 10 -68 , which is correct to its last 

 decimal. 



lhave verified this calculation by the following simple pro- 

 cess, which has the advantage of being most efficacious just 

 when the received methods are least so : when the asymmetry 

 and the divergence are great. The sought probability may be 

 written 



rq n x iL n + V x n f «(«-!) -i 



11 mn[_ q m + 1 q- (m -f-l)(?;i + 2) J 



If p is small and n less than fiq, m greater than /Lip, as in 

 the case which presents difficulty, that of the longer limb, 

 the second term within the brackets is fractional, say =r. 

 The third term is less than r 2 . Hence we have a convergent 

 series, which converges the more rapidly the less p and n are, 

 the greater the asymmetry and divergence. For a superior 

 limit we have the geometric series 1+r + r 2 &c. multiplied by 

 the term outside the brackets ; for an inferior limit that term 

 alone. 



The formula given in the text-books is quite inadequate to 

 deal with such extreme cases. And not only is it accurate 

 only for a short range, but also wdien it ceases to be accurate, 

 it ceases to be secure. For, as before, compare the extreme 

 term with the corresponding ordinate of the probability-curve. 

 The former is now p*. The (Napierian) logarithm of the 

 latter is 



~|^~* l0g2 ^~* l0g7r - 

 * Ibid. (15) 

 Phil Mag. S. 5. Vol. 21. No. 131. April 1886. 2 A 



