386 Dr. Dorothy Wrinch and Dr. R. Jeffreys on Certain 



rational coefficients is tf , so that all our data are consistent 

 with the following assumption : 



Every law of physics is expressible as a differential 

 equation of finite order and degree, with rational 

 coefficients. 



It may be objected that some of the arbitrary constants 

 involved in the solutions of physical equations are capable 

 of a continuous series of values within a definite range, and 

 therefore that the true number of solutions is ; so that 

 we are no further forward. It may be replied that the 

 differential form, not the integrated form, is the physically 

 fundamental one ; or, alternatively, that these constants, 

 in so far as they are physically determinable, can have 

 only tf distinguishable values. Either alternative appears 

 tenable, and the two are consistent ■ so that there will 

 be no objection to the adoption of both if that course is 

 considered desirable. 



If this assumption be granted, it follows at once that 

 the series of possible physical laws can be well ordered. 

 If a series can be well ordered in one way, however, it 

 can be in many, and we still need a criterion that will 

 enable us to decide which of these possible arrangements 

 is the one determined by probability. Our suggestion is 

 that a valuable guide is provided by the notion of simplicity, 

 which is a quality easily recognizable when present ; in 

 other words, the simpler the law the greater is its prior 

 probability. On this theory, then, the prevalent practice 

 of adopting the approximate but simple law in preference 

 to the exact but complex one is based directly on the higher 

 prior probability of the former. The probabilities of the 

 various possible laws must form a convergent series, whose 

 sum is unity. It would be interesting to have precise 

 estimates of these, but there seems to be no satisfactory 

 way of evaluating them at present. It is, however, enough 

 for our present purpose to suppose that whatever be the 

 number of theories that have been tested and abandoned, 

 the probability of the most probable remaining law is 

 larger than some fraction and is never infinitesimal. It 

 is not necessary to suppose that the series converges 

 with extreme rapidity, so that the ratio of the proba- 

 bilities of consecutive laws may not be very different from 

 unity. 



With these premises, the problem of the probability to be 

 attached to an inference can be dealt with. If p denote 

 the most probable law at any stage, and q an additional 



