446 Mr Norbert Wiener, A Contribution to the 
whence we get 
= =~ 
Petr Gh (SiO Ns aay @) ao Cie ener (cue IP OS) e. 
=> 
TEa. YER. Dn, y.-~ePy.~yPxi Dd: Wea. Dy. We p' PKB 
> = 
ory ihe: @, 3 eComstve >) so — pola) — ap nad ne 
wea. ye B.d,,.~aePy.~ ya: Dee | 
By an exactly similar argument, 
j =. = 
Fira, Be Canst(P.D:.a=p'P,.“a. 8 =p'P,,.“B :. 
wea. yeP.o,,.~ePy.~yka: >. eee ( 
Combining these, we get 
_ ~ 
Pat a Ole OC ainsteh sD) ri pole cl) 0 1) ole nea emen 
eet. ye B.dn,.-~ePy.~yPa sD 50 ee | 
This we may write as 
; = = 
eh GL (Ose yD) on eae el (= DP 2, 
(qa, y).cea.yeB.cPy:v:(qv,y).vea.yeR.YPurV sae 
By (1), this becomes 
bana, Se Cmst(P .D:aimst (PG .v. Sinsteeannv aver 
or +. inst*P e connex (4) 
Combining (2), (3), and (4), we get the desired conclusion; | 
namely, 
t. inst““R {R| R,, | RE R} C ser. 
From this we can easily conclude that 
F . inst‘‘cs C ser. 
It will be noticed that two of the three serial properties of 
inst“P—its being contained in diversity and its connexity—are 
independent of the properties of P itself. It is especially notice- 
able that no use is made of P G J in proving inst‘P G J, nor, indeed, ~ 
in deducing any of thé serial properties of inst‘P. inst is a valuable 
tool for what Mr Russell calls “fattening out” a relation: Le. 
deriving from a non-serial relation a relation with many of the — 
properties of series*. 
It is interesting to consider under what conditions inst‘P will 
be compact. If we define esd as follows: 
ax u vu aS 
x02. esd=csn K(k CA R,.| hk. h| RC & |mim_| ieee 
* Since writing this article, I have discovered an operation which will turn any 
relation into a series (though not necessarily an existent one) and will leave un- 
changed the relation-number of any series to which it is applied. It is the 
operation which transforms P into inst‘[(inst‘P),,]. 
