EULER. 249 



Now % 13, 'y, ... are all independent of m. Hence we may put 

 in succession for on the values 1, 2, 3, ... ; and we shall thus be 

 able to determine /S, y — 



446. Euler enters into some detail as to the values of /3, 7 . . . ; 

 but he then shews that it is not necessary to find their values for 

 his object. 



For he proposed to find the probable expense which will fall 

 on the managers of the lottery. Now on the first hypothesis 

 it is m ducats, on the second it is m — 1 ducats, on the third it 

 is m — 2 ducats, and so on. Thus the probable expense is 



-r> \am + /3m (m - 1) + jm {m - 1) (m — 2) + . . . L 



= -^ ja + /3(w-l)+7 (771-1) (m-2) + ...L 



The expression in brackets is what we shall get if we change 

 m into m — 1 in the right-hand member of the value of M in 

 Art. 445 ; the expression therefore is what M becomes when Ave 

 change m into m — 1. Thus 



a + y5(m-l) +7(7??- 1) {m-2) + ... 



= [m (771 + 1) . . . (m + n - 1) }'"\ 



Thus finally the probable expense is 



m Y ^ 



m 



/)n + Uj 



Euler then confirms the truth of this simple result by general 



reasoning. 



447. We have next to notice a memoir entitled Eclaircisse- 

 mens sur le memoire de Mr. De La Gra^ige) inserS dans le V'^ 



volume de Melanges de Turin, concernant la methode de prendre le 



milieu entre les residtats de plusieurs observations, <^c. Presente 



a VAcademie le 27 Nov. 1777. This memoir was published in the 



Nova Acta Acad. ... Petrop. Tom. 3, which contains the history 



of the Academy for the year 1785 ; the date of publication 



of the volume is 1788 : the memoir occupies pages 289 — 297. 



