DE MOIYRE. 161 



Progression, which supposition he says "will very little err from 

 the truth, especially if the proportion of ^ to 1, be not very small." 



Put r for ; thus the equation becomes 



P 



1 / 1 /(/- 1) 1 /(/- 1 ) (/- 2) 2_ . ^1. 

 1?^""^ 1.2 r''' \S r''''^'" 2' 



that is ^1__)=2. 



Hence -■ = 1 _ f _ ] , 



and then n may be found by logarithms. 



De Moivre says in his Preface respecting this problem, 



The 40th Problem is the reverse of the preceding; It contains a 

 very remarkable Method of Solution, the Artifice of which consists 

 in changing an Arithmetic Progression of Numbers into a Geometric 

 one; this being always to be done when the Numbers are large, and 

 their Intervals small. I freely acknowledge that I have been indebted 

 long ago for this useful Idea, to my much respected Friend, That Ex- 

 cellent Mathematician Dr. Halley, Secretary to the Royal Society, 

 whom I have seen practise the thing on another occasion: For this 

 and other Instructive Notions readily imparted to me, during an un- 

 interrupted Friendship of five and Twenty years, I return him my 

 very hearty Thanks. 



Laplace also notices this method of approximation in solving 

 the problem, and he compares its result with that furnished by his 

 own method ; see Theorie ... des Proh. pages 198 — 200. 



293. Problem XLI. is as follows : 



Supposing a regular Prism having a Faces marked i, h Faces 

 marked ii, c Faces marked iii,* d Faces marked iv, kc. what is the 

 Probability that in a certain number of throws n, some of the Faces 

 marked i will be thrown, as also some of the Faces marked ii ? 



This is an extension of Problem xxxix ; it was not in the first 

 edition of the Doctrine of Chances. 



Let a + h ■{- c -{■ d + ...=s\ then the Probability required 

 will be 



1 [," _ {(, _ „). + (, _ j)«j + {s-a- in 



11 



