LAPLACE. 4S5 



relate to tlie solution of partial differential equations of the 

 second order, and have no connexion with our subject. 



The formulae which occur at the top of pages 18 and 19 of 

 the Theorie...des Proh. are stated in the memoir to agree with 

 those which had been given in Newton's Methodus differentialis ; 

 this reference is omitted in the TJieorie...des Proh. 



907. Laplace's next memoir on our subject is entitled ^itr les 

 approximations des Formides qui sont fonctions de tres-grands nom- 

 hres; it is published in the volume for 1782 of the Histoire de 

 V Acad... Paris: the date of publication is 1785. The memoir 

 occupies pages 1 — 88 of the volume. 



Laplace refers at the commencement to the evaluation of 

 the middle coefficient of a binomial raised to a high power by 

 the aid of Stirling's Theorem ; Laplace considers this to be one 

 of the most ingenious discoveries which had been made in the 

 theory of Series. His object in the memoir is to effect similar 

 transformations for other functions involving large numbers, in 

 order that it might be practicable to calculate the numerical 

 values of such functions. 



The memoir is reproduced without any important change 

 in the T]ieorie...des Proh., in which it occupies pages 88 — 171-. 

 See Arts. 89-i, 899. 



A mistake occurs at the beginning of page 29 of the memoir, 

 and extends its influence to the end of page 30. Suppose that a 

 function of two independent variables, 6 and 6', is to be exjDanded 

 in powers of these variables: we may denote the terms of the 

 second degree by Md^ + 2N96'+ P6"^ : Laplace's mistake amounts 

 to omitting the term 2X66'. The mistake does not occur in the 

 corresponding passage on page 108 of the Tlieorie...des Proh. 



908. Laplace's next memoii* is the continuation of the pre- 

 ceding; it is entitled, Suite du Memoire sur les approximations 

 des Formides qui sont fonctions de tres-grands Komhres; it is pub- 

 lished in the volume for 1783 of the Histoire de V Acad... Paris: 

 the date of publication is 1786. The memoir occupies pages 

 423 — 467 of the volume. 



909. Laplace gives here some matter which is reproduced in 

 the Theorie.., des Proh. pages 363—365, 394—396. Pages 440—444 



