270 d'alemfsErt. 



iind I (x — «) -v/r' (x) dx = (x — a) ^jr {x) — I yfr (x) dx, 

 therefore I (x — a) yjr' (x) dx = — j yjr (x) dx; 



and \//' (x) dx=: — yjr (a). 



a 



Thus 



I (a? — a) (p (x) dx I yjr (x) dx 



a .' a 



V (.X) dx ^ ^"^ 



a 



This shews that ihe two methods give the same mean duration. 

 In the same way it may be shewn that they give the same pyvhahle 

 duration. 



487. D'Alembert draws attention to an erroneous solution of 

 the problem respecting the advantages of Inoculation, which he 

 says was communicated to him by un savant Geotnetre. D'Alem- 

 bert shews that the solution must be erroneous because it leads to 

 untenable results in two cases to which he applies it. But he does 

 not shew the nature of the error, or explain the principle on which 

 the pretended solution rests ; and as it is rather curious we will 

 now consider it. 



Suppose that N infants are born at the same 

 epoch, and let a table of mortality be formed by 

 recording how many die in each year of all dis- 

 eases excluding small-pox, and also how many die 



of small-pox. Let the table be denoted as here ; 



so that u^ denotes the number who die in the r*'^ year excluding 

 those who die of small-pox, and v,. denotes the number who die of 

 small-pox. Then we can use the table in the following way : sup- 

 pose M any other number, then if u^ die in the r*'' year out of N 



M 

 from all diseases except small-pox, -^^ w,. would die out of M; and 



so for any other proportion. 



Now suppose small-pox eradicated from the list of human dis- 

 eases ; required to construct a new table of mortality from the 

 above data. The savant Geometre proceeds thus. He takes the 



