388 Dr. Dorothy Wrinch and Dr. H. Jeffreys on Certain 



with regard to fair sampling *, in which we showed that 

 the proportional composition of an aggregate is probably 

 very near that of a large sample taken from it, the prior 

 probability of a particular composition being practically 

 irrelevant as long as it lies within certain very wide 

 limits. 



It may be pointed out that in the well-ordering of the 

 series of functional relations admissible in physics, laws 

 identical in all features, except that corresponding con- 

 stants have slightly different values, are likely to be widely 

 separated. Evidently the arrangement of such laws, re- 

 garded as a subset of the whole sequence of possible laws, 

 cannot be in the order or inverse order of magnitude of 

 the values of these constants. For a subset of a well- 

 ordered sequence is necessarily itself well-ordered, while 

 such a series as this must be of the type of the series 

 of the possible constants — namely, the rational numbers — 

 in order of magnitude, and cannot be well ordered. On 

 the other hand, we cannot specify completely what the 

 actual order must be, for if a sequence can be well ordered 

 in one way it can be in many. One possible and apparently 

 quite satisfactory way of well-ordering this subset is to 

 express each fraction in its lowest terms, and to adopt the 

 rule that where the sum of the numerator and denominator 

 in one is less than the corresponding sum in the other 

 the former shall precede the latter in the series. Where 

 the sums are equal in the two cases, the fraction with the 

 smaller numerator should come first. In this arrangement 

 a fraction differing from an integer by 1/w, where n is 

 large, will not be reached till after something of the order 



n 



of £</>(n) terms, where <f>(n) is the number of integers less 

 i 



than n and prime to it. Now, if the probability of the 

 rath term be p m , %p m must be a convergent series, so that 

 p m must decrease with m more rapidly than 1/m. Hence 

 the probability of a fraction differing from an integer 

 by lj?i must decrease as n increases more rapidly than 



n 



l/£i/>(w). Even with different well-ordered arrangements 



from this, there is no way in which such fractions can be 

 made to occur early in the series save in exceptional cases. 

 Thus laws differing from simple ones in having very slightly 

 different values of numerical constants will be expected 



* Loc. cit. p. 728. 



