HAM. 205 



This is Ham's formula, given as we have said without de- 

 monstration. Since we assumed 



we have 7 = Napierian log of 6 = 1-791759 ; thus 



5 = 2c -7= 2c- 1-791759. 



Ham says that this series will determine the value of z in 

 all cases when ^ is greater than 4-1473. This limit is doubtless 



obtained by making 2c - 7 = 0, which leads to (l + - j = V6 ; 



and this can be solved by trial. But Ham seems to be un- 

 necessarily scrupulous here ; for if 2c be less than 7 we shall still 



have - numerically less than unity, so long as 7 — 2c is less than 

 r 



c - -^ , that is so long as c is greater than k + q . 



357. The work finishes with some statements of the nu- 

 merical value of certain chances at Hazard and Backgammon. 



358. We have next to notice a work entitled Calcul du Jeu 

 appellS par les Frangois le trente-et-quarante, et que Von nomme 

 d Florence le trente-et-un.,,. Par Mr D. M. Florence, 1739. 



This is a volume in quarto. The title, notice to the reader, 

 and preface occupy eight pages, and then the text follows on 

 pages 1 — 90. 



The game considered is the following : Take a common pack 

 of cards, and reject the eights, the nines, and the tens, so that 

 forty cards remain. Each of the picture cards counts for ten, and 

 each of the other cards counts for its usual number. 



The cards are turned up singly until the number formed by 

 the sum of the values of the cards falls between 31 and 40, both 

 inclusive. The problem is to determine the chances in favour of 

 each of the numbers between 31 and 40 inclusive. 



The problem is solved by examining all the cases which can 

 occur, and counting up the number of ways. The operation is 

 most laborious, and the work is perhaps^ the most conspicuous 



