43i MALFATTI. 



Borda considers him to affirm the first two with equal emphasis, 

 and the last with double emphasis. See Condorcet's Discours 

 Preliminaire, page CLXXVii, Laplace, Theorie . . . des Froh. page 274. 



807. We have next to notice a memoir by Malfatti, entitled 

 Esame Critico di un Prohlema di pjvhahilita del Sig. Baniele 

 Bernoulli, e soluzione d'un cdtro Prohlema analogo al Bermdliano. 

 Del Sig. Gio: Francesco Malfatti Professore di Matematica nell' 

 Universita di Ferrara. 



This memoir is published in the Memorie di Matematica e 

 Fisica delta Societa Italiana, Tomo i. 1782 ; the memoir occupies 

 pages 768 — 824. The problem is that which we have noticed in 

 Art. 416. Malfatti considers the solution of the problem about 

 the balls to be erroneous, and that this problem is essentially 

 different from that about the fluids which Daniel Bernoulli used 

 to illustrate the former ; see Art. 420. Malfatti restricts himself 

 to the case of two urns. 



Malfatti in fact says that the problem ought to be solved by 

 an exact comparison of the numbers of the various cases which 

 can arise, and not by the use of such equations as we have given 

 in Art. 417, which are only probably true ; this of course is quite 

 correct, but it does not invalidate Daniel Bernoulli's process for 

 its own object. 



Let us take a single case. SujDpose that originally there are two 

 white balls in A and two black balls in B ; required the probable 

 state of the urn A after x of Daniel Bernoulli's operations have 

 been performed. Let u^ denote the probability that there are 

 two black balls in A ; v^ the probability that there is one black 

 ball and one white one, and therefore 1-u^-v^ the probability 

 that there are two white balls. 



808. We will first give a Lemma of Malfatti's. Suppose there 

 tiren-p white balls in A, and therefore p black balls ; then there 

 are n —p black balls in B and p white balls. Let one of Daniel 

 Bernoulli's operations be performed, and let us find the number 

 of cases in which each possible event can happen. There are w^ 

 cases altogether, for any ball can be taken from A and any ball 

 from B. Now there are three possible events ; for after the opera- 

 tion A may contain n—p-^\ white balls, or n—p, or n—p — \. 



