LAPLACE. 609 



compound interest is neglected we shall form too high an estimate 

 of the stability of insurance companies ; see Cournot's Exposition 

 de la Theorie des Chances... ^Rge 333: se^ also page 143 of the 

 same work for a formula by Bienayme connected with the result 

 given in Ai't. 1038. 



1042. Laplace's tenth Chapter is entitled De Tesperance morale: 

 it occupies pages 432 — 445. This Chapter may be described as 

 mainly a reproduction of the memoir by Daniel Bernoulli, which 

 we have analysed in Arts. 377 — 393 ; Laplace himself names his 

 predecessor. Laplace adds the demonstration to which we have 

 referred in Art. 388 ; see his pages 436, 437. Laplace also applies 

 the theory of moral expectation to an example connected with life 

 annuities ; see his pages 442 — 444. 



The following example in inequalities is involved in Laplace's 

 page 444. If a^, a^, a^, ... and h^, h^, h^, ... are series both in in- 

 creasing or both in decreasing order of magnitude 



is greater than 



a^\ + a^\ + o^\ + . . . + OnK 



afi^ + <^2^2 + ^3^3 + • • • + <^n^n 



a^ + a^ + a^+ ... + Gn ' 



for if we multiiDly up and bring all the terms together, we find 

 that the result follows from the fact that ara^ {cij. — a^ {hr — h^ is 

 positive. 



Hence too if one of the two series is in increasing and one in 

 decreasing order of magnitude the inequality becomes inverted. 



1043. Laplace's eleventh Chapter is entitled De la Prohahilite 

 des Umoig^iages: it occupies pages 446 — 461. 



We have given sufficient indication of the main principle of 

 the Chapter in Art. 735 ; see also Art. 941. 



Laplace's process on his page 457, although it leads to no eiTor 

 in the case he considers, involves an unjustifiable assumption ; see 

 Poisson, Eecherches siir la Pro6. ...page 112. See also pages 

 3 and 364 of Poisson's work for criticisms bearing on Laplace's 

 eleventh Chapter. 



39 



