| Theory of Relative Position. 449 
uU vu —> 
Since Recsd also implies R| R C R| ming| #,,, this becomes 
i : = iy 
fee ccd: D:ainst\PO.D.0= pP,,“a.B=p'P. ‘8 - 
i Se 
2 
(qa, y).c2ea.yeB.xP||minp| P,.|| Py. 
i Se 
ic 
«P|{minp|P,,|| Py says that there are a wu and a v such that 
aPu, ey: and yminp|P,,u. This latter proposition is equivalent 
=> > > 
to ve oni p'P,.“u. We have just seen, moreover, that ue minp‘P,, u, 
land that minp‘P,,we tp. This gives us 
Fi: Pecsd:.3:. nantes De 
= > 72 OS 
cco 2) 5 Pa. s= ae a Hine DERE CUTE Dayan ae lezen ey 
—> 
(ae Y)- cea. yeR.u,ve nob Wh dP 3 QUE D. 
‘From this and *0°1, (1) it is an easy matter to deduce that 
Fs. Pecsd: D: ainst‘PB.D.(qy) - ainst’Py . y inst*PB. 
This gives us immediately 
‘F . inst“csd C comp. 
It will be noticed that it is not true that esd Ccomp. For 
example, if P is the relation of complete succession between one- 
inch stretches on a line, P will be a member of esd, and an inch 
stretch beginning half an inch after the end of another will bear 
the relation P to it, yet there will be no inch stretch to which the 
first bears the relation P and which bears the relation P to the 
second. PG P|P,.|P is a weaker hypothesis than PG P?, which 
implies it if PCJ. 
