CHAP. IX.] METHODS OF ABBREVIATION. 143 



Let the function t (1 -p) to be determined, be represented by z ; 

 then the full development of z in respect of t and p is 



z = tp + t (1 - p) + (1 - t) p + (1 - /) (1 - p). 

 Hence, by the last problem, we have 



where E = w (1 - sr) + sr (1 - w) ; 



JE'= {w (1 - s) + s (1 - w)} x w x w = w (I - s); 

 .-. {w (1 - sr) + sr (1 - w?)) z + w (1 - 5) (1 - z) = 0. 



Hence, 



_ w(l-s) 

 2wsr - ws - sr 



= - wsr + ws (l-r) + -w (I- s)r +-w(l - s)(l- r), 

 + (1 - w) sr + ? (1 - ,) , (1 - r) + (1 - w) (1 - *) r 



z = - lvsr + -(l- w ) s (l- r ) + ^(i- w ) (l _ s ), 

 with w (1 - 5) = 0. 



Hence, Things transferable and not productive of pleasure are 

 either wealth (limited in supply and preventive of pain); or things 

 which are not wealth, but limited in supply and not preventive of 

 pain ; or things which are not wealth, and are unlimited in supply. 



The following results, deduced in a similar manner, will be 

 easily verified : 



Things limited in supply and productive of pleasure which are 

 not wealthy are intransferable. 



Wealth that is not productive of pleasure is transferable, limited 

 in supply, and preventive of pain. 



Things limited in supply which are either wealth, or are pro- 

 ductive of pleasure, but not both, are either transferable and pre- 

 ventive of pain, or intransferable. 



1 1 . From the domain of natural history a large number of 

 curious examples might be selected. I do not, however, con- 



