by a Table of Single Entry. 431 



4«6 = (« + 6)- — (a — S/^ the question of finding the product of 

 two numbers is virtually reduced to a process of addition and 

 subtraction, and of finding the values of two squares out of a 

 table of squares. If the two factors a and b are both even or 

 both odd, the formula ought to be changed into 



-=m-c-^)'-: 



if one of them is odd and the other even, we may employ the 

 formula 



ab 



= \—T-) -\—T-) +«• 



So, again, for the product of three numbers, there exists the 

 analogous formula 



Gabc ={a + b ■{- cf —{a -^-b —cf — {b -\- c — af— {c -\- a~ bf. 



Object of the Paper. 



The object of this brief note is to exhibit and demonstrate the 

 generalization of the above formulae, i. e. to express the product 

 of any n quantities «j, «^ g, . . . «„ under the form of the sum of 

 powers of simple linear functions of «j, «2> • • • ^n- '^^^^ ^^7 ^^ 

 done as follows. 



General Formula. 



Let Pj, 01^, u^, , . . 0,1 



be disjunctively equal to 



1, 2, 3,.../^, 

 then 



(2.4.6...2w)(«, .«2...fl!„) 



= K + «0j + «03+ • • • +«9„r-S(-«9, + «02+ . . . +%J" 



+ {-)"{- %-«e^- ■ ■ ■ -«0„)". 

 which I call the principal equation. 



Demonstration of the principal Equation. 



Ivet ^„ <^2J <^3> • • • ^n-l 



be disjunctively equal to 



1, 2, 3, . . . («-l), 



