Theory of Relative Position. 445 
| and hence that 
F . inst*‘es C ser. 
This shows us how we can construct a serial relation from any 
relation of the same sort as complete succession ; or, indeed, from 
any relation agreeing with it in only one respect. 
. A | 
#01. | .inst“h {Rh | R,,| R © R} Cser. 
ie Proof. 
It is easy to show that 
Li aS = 
a= pE Ma. B= pe. eB -(qr,y)-vea.yeB.xPy (1) 
from the definitions of ee and tp. From this we can deduce 
iE: pmsi-e 3.) . b= oP“B. (qv,y).vea.yeB.rv~aPyy, 
since, by the definition of P,., ePy and wP,.y are incompatible. 
This reduces to 
= = 
meeomst Pe. D. B=pP,.“8.(qe).vea.~(vep'P,,“*8), 
from which we can deduce 
F:ainst‘P8.3,,2.aJ8 
Ge F.inst‘Pe RIV (2) 
Also, we find from (1) that 
F:ainst‘P8. Binst‘Py.). 
= = = 
eS p Pa B= pPeB “y = p* P.M“ y : 
(qx, y,U,v).vea.yueP.vey.aPy.uPr. 
This implies 
F:ainst‘P@. Binst‘Py.). 
= > 
eel — Pens cy lala, Y) CE Vie cy ble || lee lad 
This, together with (1), gives us 
F.inst“R{R| R,,| RG R} C trans: (3) 
By the definitions of inst and rp, we find that 
= = 
meee Cinst PD .a=pP..“a.8 =p P48. 
By an easy deduction, we can arrive, from this proposition and 
the definition of P,,, at the proposition 
= > 
meee O-mst©P .D:.0—p' Pa. B= p' Pa “Bt 
DEANE > yar UO VO OPEL EN] ate od 
