d'alembeht. 2G7 



attacked, and that 1 died out of every m attacked ; on these 

 hypotheses he solved definitely the problem which he undertook. 

 D'Alembert also gives a mathematical theory of inoculation ; but he 

 does not admit that Daniel Bernoulli's assumptions are established 

 by observations, and as he does not replace them by others, he 

 cannot bring out definite results like Daniel Bernoulli does. 

 There is nothing of special interest in D'Alembert's mathematical 

 investigation; it is rendered tedious by several figures of curves 

 which add nothing to the clearness of the process they are sup- 

 posed to illustrate. 



The follomng is a specimen of the investigations, rejecting the 

 encumbrance of a figure which D'Alembert gives. 



Suppose a large number of infants born nearly at the same 

 epoch ; let y represent the number alive at the end of a certain 

 time ; let ti represent the number who have died during this 

 period of small-pox : let z represent the number who would have 

 been alive if small-pox did not exist : required z in terms of y 

 and u. 



Let dz denote the decrement of ^ in a small time, dy the 

 decrement of y in the same time. If we supposed the z individuals 

 subject to small-pox, we should have 



dz = - dii. 



y ^ 



But we must subtract from this value of dz the decrement 



arising from small-pox, to which the z individuals are by hypo- 



z 

 thesis not liable : this is - du. 



y 



Thus, dz = - dy + - du ; 



y y 



z z 

 we put -^ - du and not du, because z and y diminish while 



/ y y 



u increases. Then 



dz dif du 



^ y y 



therefore log z = \ogy + \ — 



/du 



