( 682 ) 



p ' 

 so that tlie uniform distribution will be the most probable. 



We shall next consider the probability for a dislribiition differing 

 a little from the most ])robable one. Let ns ])iit 



1,1 1 



a' = 1- « , V = \-{i , m' = [- n . (34) 



P P P 



and let ns suppose the luimbers a, [i .... n, to l)e so small in com- 



1 

 parison with — , that in the expansion of the quantities in (32) in 



P 

 ascending powers of a, ^j . . . . [j, we mav neglect all jtowers surpassing 

 the second. We have for instance 



ia'q + ^j lop a' =: _(--- + - j lopp^lrj + ^p - q Ion p jr« + -^ P f '/"^pj"' ' 



1 

 where, in the last term, we mav omit the term — )>, because it is 



. 2 



much smaller than (/. If we put 



1 1 

 ^ (/> — 1) lofi (2 rr v) -f — ;) lofi p — lofi P,n 



and keep in mind that, in virtue of (33), 



« + l?-f .... + ;i = 0, (35) 



the equation (32) becomes 



lori r = lofi P,n -jPQ i<r + 15^ + . . . . + in- 



P=P,„e- 2'"/'^' + ''' + -"- + "'\ 



It is seen from this that P,n is the maximum of the probability, 



with ^vhich Ave shall have to do, if « = j? =.... = f* = 0. The 



equation shows also that, conforndy to what has been said above, 



the probability ^vill only be comparable to P;„ so long as «, ,?.... /x 



are far below — . Indeed, if one of these numbers had this last value, 



P 

 Pn would be multiplied by 



e 2/>' 

 which, by our assumptions, is extremely small. 



§ 15. Let 2u be the length of the line, .v the distance along the 



line, reckoned from the end A, and let us take — for the value oi 



p 



