218 DANIEL BERNOULLI. 



Let p = — ; — SLiid q= — ; — , where m and n are inteofers : 



then we have to compare 



{a + x- eY^*" with {a + xY a\ 



Wlien a = the right-hand member is the less ; when a is 

 infinite the right-hand member is the greater, provided mx is 

 greater than (m -^ n) [x — e) : we will assume that this is the case. 

 Thus the equation 



{a-^x- e)'""-" = (a + xY oT 



has one positive root. We must examine if it has another. 



Let log {a + x- ef'^'' = y, log (a + xY dr = z\ 



. dy m-\-n dz m n 



then - -f = — , , -7- = — , h - . 



da a + X — e da x + a a 



d z d II 



Thus when a is zero -j- is greater than j- , so that z begins 



by increasing more rapidly tlian y does. If we suppose 



dy dz 

 da da 



* 



, , . nx (x — e) 



we obtam a = - — — ^r '■— . 



(??i -\-n) e — nx 



Now begin with a = 0, and let a gradually increase until we 

 have y = z\ then it is obvious that we have not yet reached the 

 value of a just given. And if by increasing a we could arrive 

 at a second value at which y = z, we should have passed beyond 

 the value of a just given. Then after that value z would increase 

 more slowly than y, and the final value of z would be less than 

 the final value of y, which is impossible. Thus there is only one 

 value of a which makes y = z, and this value is less than 



nx {x — e) 

 {m ■\- n) e — nx' 



If mx is less than (m -\-n) {x — e) the original equation has 

 no positive root; for then we have z always increasing more 

 rapidly than y, and yet the final value of z less than that of y ; 

 so that it is impossible that any value of a can make y = z. 



