201 HAM. 



have frequent use made of De Moivre's results as to the number 

 of trials in which it is an even chance that an event will happen 

 once, or happen twice ; see Art. 264. 



856. There is however an addition given without demon- 

 stration, to De Moivre's results, which deserves notice. 



De Moivre made the problem of finding the number of trials 

 in which it is an even chance that an event will occur twice 

 depend on the following equation : 



(l + ^)' = 2 (1 + z). 



If we suppose q infinite this reduces to 



^ = log 2 + log (1 + ^) ; 



from which De Moivre obtained z = 1-678 approximately. But let 

 us not suppose q infinite ; put ( 1 + -j =6"; so that our equation 



becomes 



6*'^= 2(1+^). 



Assume z=2 —y, thus 



Assume 2c = 7 + s where e*^ = 6. 



1 



Thus, e*-^ = 1 - g y. 



Take the logarithms of both sides, then 



1 1 , 1 3 



that is ''i/ - Yg .V' - gj 3/' - ••• = ^ ; 



where r = c — ^. 



Hence by reversion of series we obtain 



