340 MEMOIRS OF THE AMERICAN ACADEMY. 



Formulce relating to the Numbers of Terms 

 We have already seen that 



(77.) If ar-O, then [*] -< [>] . 



(78.) When *,y = 0, then [* -fcy] = [>] -fc [>] . 



(79.) When \_xy\\n*y\ = [>]:[»*], then [>#] = [>]|>] . 

 (80.) When \_xny\ = |>] [l^] [1], then [ats] = [*]M . 



It will be observed that the conditions which the terms must conform to, in order that 

 the arithmetical equations shall hold, increase in complexity as we pass from the more 

 simple relations and processes to the more complex. 

 We have seen that 



(81.) [O] = 0. 



(82.) [/] = ;. 



Most commonly the universe is unlimited, and then 

 (83.) [1] 



and the general properties of 1 correspond with those of infinity. Thus 



* ~b" 1 = 1 corresponds to x -J- °o = oo 



? 



?1 = 1 " « ?00 =00, 



** = 1 « " oo* = oo 



lar = l « « 00*= OO. 



The formulae involving commutative multiplication are derived from the equation 



But if 1 be regarded as infinite, it is not an absolute infinite: for 1 = 0. 



i, = r 



On the other hand, Z 1 = 



It is evident, from the definition of the number of a term, that 



(84.) [>,] = [>]:[1] . 



We have, therefore, if the probability of an individual being • to any y is independ- 

 ent of what other /s it is * to, and if z is independent of* 



(85.) [a*,] = [* ,] M . 



