600 Mr. J. H. Jeans on the 



average throw averaged over all possible series of 10 throws) 

 is 3^, but the " expectation " o£ the difference between this 

 and the average lor any single series (in other words, the 

 probable error) is (about) 0*6. I£ we consider a series of 

 1000 throws the expectation of the average throw remains 

 the same, but the probable error is only (about) 0*06. I£ we 

 pass to the limit, and consider a series consisting of an 

 infinite number of throws, the expectation of the average 

 throw remains 3-J, but the probable error becomes nil. 



§ 5. This suggests the definite proposition : " The average 

 value of a throw in an infinite series of throws is 3^.' v This 

 proposition does not stand on the same level of absolute truth 

 as the proposition " 2x2 = 4," but its truth is sufficient for 

 all practical purposes, and, moreover, it represents the highest 

 level of truth which is attainable in the absence of a definite 

 knowledge of the values of individual throws. The pro- 

 position (understood in its proper sense) is not refuted by 

 pointing out that a series such as 



1, 1, 1. . . . ad inf. (series A) 



is a possible series, and that the average value in this series is 

 not 3i but unity. The reply to this criticism is that the 

 probability is infinitely against a series o£ random throws 

 being of the form of series A, or of any other form for 

 which the proposition is not true ; for a series of random 

 throws it is infinitely more probable that the proposition will 

 be true than that it will be untrue. We may conveniently 

 indicate that a proposition is of this type, by prefixing the 

 words " It is infinitely probable that . . ." 



§ 6. The propositions o£ the kinetic theory, founded upon 

 the basis which is now proposed, will be of this type: they 

 will state infinite probabilities, and not certainties. The 

 uncertainty as to the positions and velocities of the individual 

 molecules of the gas will correspond to the uncertainty as to 

 the actual values of the throws in the illustrative problem of 

 the dice: the theoretical uncertainty in the final result will 

 replace the uncertainty which enters, in the usual treatment, 

 at the outset by assuming the gas to be ungeordnet. The 

 assumption o£ an ungeordnet state, as enunciated by Boltz- 

 mann *, was obviously intended to exclude special cases 

 analogous to the special case of series A. That it does not 

 have this result in the form in which it is used will appear 

 later. But with our present understanding it is quite un- 

 necessary to make any assumption or limitation o£ this kind, 

 just as in our question of the dice, it was quite unnecessary 

 * Vorlesungen, i. p. 20. 



