EULEE. 241 



Similarly, at the end of the second year there will be 



(?/i + 2) 

 M —7 — -^ of the persons alive, each of whom is to receive x : 



(w) 



therefore the present worth of the whole sum to be received is 

 -5 M ^ , . . And so on. 



The present worth of all the sums to be received ought to be 

 equal to Ma ; hence dividing by M we get 



_ X ((m + 1) (m + 2) (m + 3) 



Euler gives a numerical table of the values of (1), (2), ... (95), 

 which he says is deduced from the observations of Kerseboom. 



434. Let iV denote the number of infants born in one year, 

 and r'i\^ the number born in the next year ; then we may suppose 

 that the same causes which have changed N" into riV will change 

 rN into rW, so that r^N will be the number born in the year 

 succeeding that in which rN were born. Similarly, r^N will be 

 born in the next succeeding year, and so on. Let us now express 

 the number of the population at the end of 100 years. 



Out of the N infants born in the present year, there will 

 be (100) N alive ; out of the rN born in the next year, there will 

 be (99) rN alive ; and so on. Thus the whole number of persons 

 alive at the end of 100 years will be 



[ ^. ^. ^ 



Therefore the ratio of the population in the 100*'' year to the 

 number of infants born in that year will be 



If we assume that the ratio of the population in any year to the 

 number of infants born in that year is constant, and we know this 

 ratio for any year, we may equate it to the expression just given : 

 then since (1), (2), (3), ... are known by observation, we have 

 an equation for finding r. 



16 



