458 PREVOST AND LHUILIER. 



probabilities of causes are estimated. The principle is quoted as 

 given by Laplace in the Memoires . . .par divers Savans, Yol. VI. : 

 Si un evenement pent etre produit par un nombre n de causes 

 differentes, les probabilites de I'existence de ces causes prises de 

 I'evdnement, sont entre elles comme les probabilites de I'evene- 

 ment prises de ces causes. The memoir considers it useful and 

 necessary to demonstrate this principle ; and accordingly deduces 

 it from a simple hypothesis on which it is conceived that the whole 

 subject rests. Some remarks made by Condorcet are criticised ; 

 and it is asserted that our persuasion of the constancy of the laws 

 of nature is not of the same kind as that which is represented by 

 a fraction in the Theory of Probability. See Dugald Stewart's 

 Works edited hy Hamilton, Yol. i. pages 421, 616. 



The second section of the memoir applies Laplace's principle 

 to some easy examples of the following kind. A die has a certain 

 number of faces ; the markings on these faces are not known, but 

 it is observed that out of ^ + ^ throws p have given ace and q 

 not-ace. Find the probability that there is a certain number of 

 faces marked ace. Also find the probability that in p' + q' more 

 throws there will bej/ aces and q not-aces. 



It is shewn that the result in the last case is 



where 2 denotes a summation taken with respect to m from m = 1 



to m = n', and 7i is the whole number of faces. This is the result 



if the aces and not-aces are to come in a prescribed order \ if they 



I // + q 

 are not we must multiply by — ;- — r- . 



^ *^ *^ p \q 



The memoir states without demonstration what the approxi- 

 mate result is when n is supposed very great ; namely, for the 

 case in which the order is prescribed. 



Li L^ \p + q+p + </' + ! ' 



852. The third memoir is entitled Remarques siir Vutilite et 

 Vetendue dii 2)7'inci2)e par lequel on estime la p^i'ohahilite des causes. 

 This memoir also relates to the principle which we have quoted 



