DE MOIYRE. 177 



The demonstrations of the formulae are to be found in the Mis- 

 cellanea Analytica, pages 76 — 83, and in the Doctrine of Chances, 

 pages 230 — 234. De Moivre supposes the players to start with the 

 same number of counters ; but he says on page 83 of the Miscel- 

 lanea Analytica, that solutions similar but somewhat more complex 

 could be given for the case in which the original numbers of 

 counters were different. This has been effected by Laplace in his 

 discussion of the whole problem. 



314. De Moivre's own demonstrations depend on his doctrine 

 of Recurring Series ; by this doctrine De Moivre could effect what 

 we should now call the integration of a linear equation in Finite 

 Differences : the equation in this case is that furnished by the first 

 of the two laws which we have explained in Arts. 304, 306. Cer- 

 tain trigonometrical formulae are also required ; see Miscellanea 

 Analytica, page 78. One of these, De Moivre says, constat ex 

 -^quationibus ad circulum vulgo notis ; the following is the pro- 

 perty : in elementary works on Trigonometry we have an expan- 

 sion of cos 7x6 in descending powers of cos 6 ; now cos nO vanishes 



when nO is any odd multiple of -^ , and therefore the equivalent ex- 



pansion must also vanish. The other formulae which De Moivre 

 uses are in fact deductions from the general theorem which is 

 called De Moivre's property of the Circle; they are as follows ; 



TT 



let a = ^7- , then we have 

 2?i 



1 = 2""^ sin a sin 3a sin 5a ... sin (2/ia — a) ; 



also if n be even we have 



cos n(^ = 2""^ [sin^ a — sin^ </>} {sin^ 3a — sin^ 0} . . . 



. . . { sin^ {n — 3) a — sin^ <^} { sin^ (ti — 1) a — sin^ 0} : 



see Plane Trigonometry, Chap, xxiii. 



De Moivre uses the first of these formulae ; and also a formula 

 which may be deduced from the second by differentiating with 

 respect to (f), and after differentiation putting </> equal to a, or 

 3a, or 5a, ... 



315. De Moivre applies his results respecting the Duration 



12 



