322 Mr. F. Y. Edge worth on the Law of Error 



1 - ■* 7) 



If we compare ?, A 4 w ^h P^geP) we see ^ na ^ as P 



P 

 diminishes from J ; the first exponent soon becomes, and con- 

 stantly continues, of a lower order than the second. Hence 

 at the extremity of the longer limb of the binomial the proba- 

 bility-curve does not escape free from the binomial. It is 

 otherwise at the extremity of the shorter limb. These con- 

 clusions may be verified by examining the explicit general 

 terms of the compared loci. It will be found now that, as 

 regards the longer limb, it is not in general possible to expand 



in terms of— (where p is the divergence from the central 



*P. 

 point, the position of the largest term) . For p may be greater 



than fip. And if we examine* the differentials of 97, we shall 

 find that -^ is apt to be continually negative and large. 



afi 7. The inaccuracy and insecurity which have been proved 

 for the binomial species of asymmetry may be extended by 

 parity of reasoning to the other species, multinomials and con- 

 tinuous facility-curves. But the attribute of corrigibility is less 

 easily generalized. First, it is to be observed that in case of 

 asymmetry the modulus of the ancillary probability-curve is 

 apt to be small. This may be shown by induction from par- 

 ticular hypotheses, like that which is indicated by the accom- 

 panying figure, where is the centre of gravity, and the sum 



e a 



of the rectangles erected on P and Q is unity. The 

 modulus squared = twice sum of squares of error 



= i(OQ 3 Q? + OPP P ), 



where iOQ 2 Q?=40P 2 Pp and OQx Q? + OP x Pp = l. Also 

 PQ, the total range, is finite, say of the order unity. Whence 

 if OP is small, the modulus is small. This proof may be thus 

 generalized: — The curve being huddled asymmetrically, the 

 mean error for the short limbs must be small. But it is equal 

 to the mean error for the other limb. Therefore the total 



* See above, p. 314. 



