LAPLACE. 521 



du passage du r^el a rimaginaire, dans le 3"^® chapitre du Calcul d'^s 

 P^'ohahilites, et qu'il vient de confirmer par des metliodes rigoureuses 

 dans quelques additions faites a cet ouvrage. 



The additions to which Cauchy refers occupy pages 464 — 484 

 of the Theorie...des Froh., and first appeared in the second edi- 

 tion, which is dated 1814. 



965. An important application which Laplace makes of his 

 method of approximation is to evaluate the coefficients of the 

 terms in the expansion of a high power of a certain polynomial. 



Let the polynomial consist of 2n + l terms and be denoted 

 by 



111 1 



-^ + -iFT + -1F2 + — + - + 1 + «+••• + «""' + «*""' + «" ; 



a a a ^ a 



and suppose the polynomial raised to the power s. 



First, let it be required to find the coefficient of the term 

 independent of a. 



Substitute e^^^^ for a ; then we require the term which is 

 independent of 6 when 



{ 



l + 2cos^ + 2cos2^+ ... + 2cos?i^y 



is expanded and arranged according to cosines of multiples of 0. 

 This term will be found by integrating the above expression with 

 respect to 6 from to tt, and dividing by tt. Sum the series of 

 cosines by the usual formula ; then the required term 



. 2w + l^ 

 t n^ 1 sm — 71 — u 



ir\ J 



TT j„ J • 1 /I 



sm ^ o 



2 



de 



2 C^' /sin m(l) 

 "ttJo Vsin^ 



where (f)=^-9, and 7n = 2n + 1. 





Now the expression ( . ^ j vanishes when 



TT 27r Stt 

 = — or — or — 

 ^ m m m 



