430 Mr DE MORGAN, ON A QUESTION, &c. 



Addition. 



The formula (1) at the beginning of this paper is unnecessarily 

 complex, seeing that if / + ^ were written instead of / in the limits 

 of the integral, the increment of the latter would differ from the 

 additional term only by a quantity of such an order as was rejected 

 in the approximation. 



If then V and w be the components of n (or o + w), which are as 

 the probabilities of A and ^ in a single trial, the probability that 

 A will happen a number of times between v + 1 and v — I in « 

 trials is 





which is of the same order of exactness as the formula given by 

 Laplace, and is somewhat more symmetrical and less difficult to cal- 

 culate. 



Perhaps it may not be here out of place to notice that the usual 



approximation to the product 1.2.3 x may be made very much 



more exact without being rendered materially more difficult to calcu- 

 late. As follows : instead of 



1.2.3 .r = V^ x'^^e-', 



substitute 



1 .2.3 X = -s/2^ .r'+^c-'+i^ 



this follows immediately from 



1.2.3 x = \/2^ x'^^e-'il +~ + 



\2x 288J- 



since the third term of the series is half the square of tlu; setuud. 

 The approximation is so close that even if we take x - 1, the error is 

 very little more than the five hundredth part of the whole. 



