150 Prof. F. Y. Edgeworth on the 



uniformities of statistics cannot be explained by other kinds- 

 of uniformity. 



Of course, other kinds of uniformity, propositions other 

 than those proper to Probabilities, may be employed to 

 ascertain whether and how far the conditions necessary for 

 the genesis of the law-of-error are fulfilled in any concrete 

 case. The gambler may reasonably inquire whether a 

 roulette table is faked, or the dice loaded. The student of 

 Probabilities may properly expect that a long set of figures 

 forming the period of a recurring decimal will approximately 

 conform to the law-of-error : and that the same will be true 

 of pairs formed at random from these figures. And yet he 

 may learn from theory or observation that pairs selected in 

 a particular manner from certain periods — e. g. by adding the 

 first place to the (?i + l)th, the second place to the (n+2)th r . 

 and so on in a period of 2n decimals — will not fulfil the 

 conditions necessary for the genesis of the law of error *.. 

 The principal conditions are three. (6) First, the number of 

 constituents must be large — how large depends on the degree 

 in which the other conditions are fulfilled. Secondly, the 

 variation of the constituents must be independent, a condition 

 which is, perhaps, never perfectly filled in concrete nature, 

 not even in g;imes of chance f. Thirdly, the constituents are 

 to be aggregated by simple summation. This condition is 

 sometimes perfectly fulfilled, as in games of chance ; but 

 often only approximately, as where the compound is some 

 function other than the sum of the constituents, but such 

 that, when expanded in ascending powers of the constituents, 

 it is equateable approximately to their sum. (7) According 

 as these conditions are more or less perfectly satisfied the 

 law-of-error is more or less perfectly fulfilled. So the path 

 of a projectile is more nearly a parabola the less dense the 

 resisting medium ±. 



There is a fourth condition which is, indeed, essential,, 

 but hardly requires to be stated since it is always fulfilled 

 in concrete nature. (8) The standard deviation and generally 

 the mean powers of deviation for each and all of the 

 constituent variables must not be infinite. The definition 

 does not exclude the possibility of an infinite deviation 

 provided that it is sufficiently rar.e. Thus a group fulfilling 

 the law-of-error is qualified to be a constituent in the genesis 



* The property is illustrated by the recurring decimal representing 

 1/19, viz. -052631578. 947368421. 1/1861 furnishes a better instance.. 

 See " Law of Error." Transactions of Cambridge Philosophical Society,, 

 p. 129 (1905). 



t See Venn, ' Logic of Chance/ ed. 3, p. 77 et seq. 



+ Cp. Phil. Mag. vol. xxxv. p. 422 (1918). 



