542 LAPLACE. 



the application which Laphxce makes of the result to the subject 

 of the planes of motion of the planets. 



By proceeding as in Art. 148, we find that the probability that 

 after i drawings the sum of the numbers drawn will be s is the 

 coefficient of x^ in the expansion of 



— ^ (1 - ic^^y (1 - xY\ 



Thus we obtain for the required probability 

 1 { \i+s-l i U*+s-n-2 



[n-^Vf Mi-l \s 1 [i:il Lf^i^^ij^l 



i[i-\) h' + g-2;2~3 



1.2 ^-l 5-2^ 



If the balls are marked respectively 0, Q, 2^, 8^, ,.,nd, this 

 expression gives the probability that after i drawings the sum of 

 the numbers drawn will be sQ. 



Now suppose to become indefinitely small, and n and 5 to 

 become indefinitely great. The above expression becomes ulti- 

 mately 



i-\ \\nl \\n 1 ^ 1.2 \n j '" n 



s 1 



Let - be denoted by x, and - by dx, so that we obtain 



n " ' n 



this expression may be regarded as the conclusion of the follow- 

 ing problem. The numerical result at a single trial must lie 

 between and 1, and all fractional values are equally probable : 

 determine the probability that after { trials the sum of the results 

 obtained will lie between x and x + dx^ where dx is indefinitely 

 small. 



Hence if we require the probability that after i trials the sum 

 of the results obtained will lie between x^ and x^, we must inte- 



