DESCRIPTION OF A NOTATION FOR THE LOGIC OF RELATIVES. 340 



Then the negatives of the different terms of the binomial development ai 



M-lf- 11 ^ 1 = * + *" + *" + *"" • 



— Bi&zri3.iy-.*2,«*2 — _ if _ *>'" _ jf* - *V" - z",z"" - j >" 



2 



a» = — WJ'W" 



The 



Now, since this addition is invertible, in the first term, / that is z", is count. 1 over 

 twice, and so with every other pair. The second term subtracts each of these pairs 

 so that it is only counted once. But in the first term the z that is z" that is z" if 

 counted in three times only, while in the second term it is subtracted three times 

 namely, in (x,x") in (x',x'") and in {z",z'"). On the whole, therefore, a triplet 

 not be represented in the sum at all, were it not added by the third term. 

 quartette is included four times in the. first term, is subtracted six times by 

 term, and is added four times in the third term. The fourth term robin 

 and thus in the sum of these negative terms each combination occurs one 

 only ; that is to say the sum is 



j j. x - jf. ** + *-" = z(r + r-t r" -fe r -") = x, . 



t 



I 



for whatever is a to 



If we write (axf for [>].[> — i].[s — a].l 

 any three x% regard being had for the order of the z's; and employ the mode 

 numbers as exponents with this signification generally, then 



I , x. I 



l_ w + >)2-^K + etc 



2V ' 3 



is the development of (1 - a? and consequently it reduces itself to 1 - az. That is, 



(112.) 



• 1 ~2 _l_ -ir 3 — z* -4- etc 



1 



+w ;« whatever is other than every x y so 

 x denotes everything except z, that is, whatever i 



that 6- means ■ not." We shall take log z in such a sense that 



£ log* -s X 



* 



. I, make, another reeercblanee between 1 and infinity that [eg 



1. 



VOL. IX. 



48 



