358 MEMOIRS OF THE AMERICAN ACADEMY 



As another illustration of the use which may be made of differentiation in logic, let 

 us consider the following problem. In a certain institution all the officers (x) and 

 also all their common friends (/) are privileged persons (y). How shall the class of 

 privileged persons be reduced to a minimum ? Here we have 



dy = dx + df* = dx — /*,(1 —f)dx . 

 When y is at a minimum it is not diminished either by an increase or diminution of 



x. That is, 



[rfy] > ° . 



and when [x] is diminished by one, 



When x is a minimum, then 



E*y] < •- 



[rfx -/*,(i -f)*x] >- o [&-/'->,(i-/yx]-<o 



(A.) [rfx] - [A(l -/)</*] >- [dx] - [/ X - J ,(l -f)dx-] -< . 

 .Now we have by (30) 



A(i- ( /yx=yk-(M),(i-/Wx- 



Hence, 



[/*] -< D**] + [0j0,].[(l -/)rfxj [/*-«] >- [rfx] + [0;0,].[(1 -/)«**]. 



But [0;0,] lies between the limits o and i, and 



(153.) 

 We have, therefore, 



[dx] = I 



l/*]^: i + [(i-/)i] [/*— ]>i 



This is the general solution of the problem. If the event of a person who may be an 

 official in the institution being a friend of a second such person is independent of and 

 equally probable with his being a friend of any third such person, and if we take p, or 



the whole class of such persons, for our universe, we have, 



[/*>] 



p=l; 



_ un — (ins w 



m "am; ' 



[(1 -f)dx] = [1 -f].{dx] = ([p] - [f]).[dx] , 



[/*,(! -f)dx] = (gl) W . ([ P ] - [/]).[rfx] . 



