we ou^lit to read 7- . De Moivre is correct in an example which 



18G I)E MOIVRE. 



is that in the series 1 - a?- aa?^-aV - ... -a^~V instead of a 



a 

 I 



he gives on his page 255. Let j =Cy then according to De Moivre's 



rule corrected we have to expand 



1 a^ , . 1 — ex c? 



a 



\ — X 



\ — cx 



This will be seen to agree with our result remembering that we 

 took a^h = \. 



De Moivre himself on his page 256 practically gives this form 

 to his result by putting 



\ — c^ d^ 



1 — X — i for 1 —x^cx"^ — c^ x^ — . . . — (f ^x^. 



I — ex 



De Moivre gives without demonstration on his page 259 an 

 approximate rule for determining the number of trials which must 

 be made in order to render the chance of a run of _p equal to 

 one half. 



De Moivre's Problem LXXiv. has been extended by Condorcet, 

 Essai...de V Analyse... pages 73 — 86, and by Laplace, Theorie...des 

 Frob. pages 247 — 253. 



326. De Moivre's pages 261 — 328 are devoted to Annuities on 

 Lives; an Appendix finishes the book, occupying pages 329 — 348 : 

 this also relates principally to annuities, but it contains a few notes 

 on the subject of Probability. As we have already stated in 

 Art. 53, we do not profess to give an account of the investigations 

 relating to mortality and life insurance. 



We may remark that there is an Italian translation of De 

 Moivre's treatise on Annuities, with notes and additions ; the title 

 is La Dottrina degli Azzardi...de Ahramo Moivre: Trasportata 

 dalV Idionia Inglese,...dal Padre Don Roberto Gaeta...sotto Vassis- 

 tenza del Padre Don Gregorio Fontana...In Milano 1776. This 

 translation does not discuss the general Theory of Probability, but 

 only annuities on lives and similar subjects. 



