and the Elimination of Chance. 319 



neglect of terms of the order -, but terms of the order 



1 P 



. For that is the order of the first term* of &'s ex- 



y/2pq/i 



pansion, which vanished in the symmetrical case, but now 

 makes itself felt. Among derived authorities the only one 

 known to me who has called attention to this point is Prof. 

 Lazarus, in a paper " On some Questions . . ."in the 15 th vol. of 

 the ' Assurance Magazine ' f . In short for other than small di- 

 vergences the formulae in the text-books are apt to be unsuited 

 to our purpose J, the determination of the probability that a 

 given divergence in an assigned direction would have occurred 

 by mere chance. 



The conclusion that asymmetry diminishes the range and 

 the degree of accuracy may be verified by that method of 

 approximation which proceeds by way of a partial differential 

 equation §. 



Let the odds be u : |3 where a and /3 are integers. Let 

 yLt = s(a+j3); and put u sx for the .n'th term from that which 

 is the greatest. Consider the formation of u s+l)X ; it is thus 

 made up: 



tt S HF« v -«x«^ + (a + fltt M - M X|3a (a " 1)+ ^&c. 



+ («+l3K, + ( M )X#+^- ] ) + w M+ ^. 



Expanding as before, we have 



where B is the mean error of the binomial (a + j8) a +0 about 

 its centre of gravity, and C is the mean square of error about 

 the same point. The first coefficient vanishes || , and the second 



* Poisson, art. 78. 



t The correction which, he derives from Poisson is accurate only for a 

 very limited extent of divergence. 



X I have treated the case of unsymmetrical Binomials more fully in the 

 1 Journal for Psychical Research/ part x. 



§ Above, p. 313. 



|| These propositions, if they are not already known, may thus he 

 proved. Put 



p= ^h' ?= db' M=a+/3 - 



The centre of gravity of the binomial locus (p+qY, measured from one 

 extremity is 



^ I/* 



2, T—r±= p>»q».-m x m, r 



m |/ x— m ' [Over. 



