Distribution of Velocities in a Molecular Chaos. 107 



with scientific order. The double aspect o£ law and chance 

 is well illustrated by the sequence of digits in the develop- 

 ment of mathematical constants when not periodical (and 

 not commensurate with the arithmetical radix). For example, 

 below are a set of forty-eight figures whicji were obtained 

 as follows. The seventh digit in the logarithms of 101, 

 102 .... 125 respectively having been noted, there was 

 formed the (algebraic) sum of the differences between each 

 digit and the mean of digits taken at random, viz. 4*5 (=the 

 sum of the observed digits —25x4*5) ; and this aggregate 

 was divided by 5. The result, —1*9, having been recorded, 

 another batch of twenty-five successive logarithms, beginning 

 with that of 126, was similarly treated ; and so on, for forty- 

 eight batches of twenty-five elements, up to the logarithm of 

 1300. It will be seen that the group of compound quantities 

 corresponds approximately to a normal error-curve for which 

 the mean is zero and the mean-square-of-deviation, 8'25. 

 Thus, whereas the modulus is 4*062, there ought to be half 

 the total number of observations between the limits + 1'94 

 ( = •4769 modulus); and four-fifths of the total number, say 

 thirty-eight or thirty-nine, between the limits +3*7 ( = *906 

 modulus). In fact half the observations lie between — 1*9 

 and -f-1'7; forty observations lie between +3*7. 



The negative deviations from the Mean (zero), rearranged in 

 the order of magnitude, are : 



6*9, 6*5, 5*5, 3*5, 2*7, 2-7, 2*5, 2*5, 2*5, 2*3, 2*1, 1*9, 

 1*9, 1*7, 1*7, 1*5, 1-3, 1*3, 1*3, 1*3, 1*1, *9, -3. 



The positive deviations from the Mean (zero) are : 



•1, *1, *1, *3, *5, *5, -7, *7, *7, *9, 1*1, 1*5, 1*5, 



1-7, 1*9, 1*9, 2*3, 2-7, 2*9, 3*5, 4*9, 5*1, 5*3, 5*5, 6*5. 



The character of the group would not have been sensibly 

 altered if, instead of 25, we had used any other (tolerably 

 large) number ??i, and divided the sum of deviations by \/m. 

 The correspondence with the normal law of frequency would 

 be more perfect the larger m was; but the mean would still 

 be zero, and the mean square of deviation 8*25. Nor would 

 these characteristics be affected if we multiply any two 

 assigned elements {e. g. the first and last) in each batch by 

 */2 sin 6 and \/2 cos respectively. It comes to the same if 

 we attach the sine and the cosine not to any assigned pair of 

 elements but to any pair at random in each batch. Instead 

 of sin and cos we might employ sin 20, sin 30 . . . cos 20, 

 cos 30 . . . or their equivalents in terms of sin and cos 0. 



