176 THEORY OF PROBABILITIES. 



P being the probability that the required sum T shall be 

 precisely equal to 27T+ t. The greatest value of P corresponds 

 to t = ; consequently the most probable value of T is ^K. 



It is to be remarked that the approximate formula (p) is 

 independent of the law of probability expressed by the func- 

 tion <f>x : it depends merely on the two definite integrals 



/* r 



I x<f>xdx and I 



J o J o 



We have stopped the approximation to the value of G at 

 the second power of a. Had we gone farther, and retained 

 only the principal term in the coefficient of each power of a, 

 a similar result, viz. one which may be assumed as coincident 

 with the exponential function, would, there is little doubt, 

 have been obtained; while the coefficient of each power of a 

 in H would be negligible in comparison of the corresponding 

 power in @. Some remarks on this, or at least on a cognate 

 question, will be found in the paper already mentioned. 



As a verification of the approximation we have employed, 

 which is in effect the same as that of Laplace, let us suppose 

 that the functions < 1? < 2 ... are all of the same form <, and that 

 <frx = e"*. Then, as we have seen, the required probability is 

 obtained by integrating 



for all positive values of x lt x 2 ... a? B _ l which do not transgress 

 the limits 



Now 



and thus we have P e T dTf dx l . . . f dx n _ lt 



the limits being given by 



tf 1 + * 2 ...o; n _ 1 = 7 7 . 

 Hence, it is easily seen that 



e -T 



P - T n ~ l JT 



~ T (n) * 



In order to compare this with the approximate expression 

 (p), I remark that T=n l renders P a maximum; assume, 

 therefore, 



T=n-l+t, 



