| Theory of Relative Position. 447 
we shall find that Recsd is a sufficient condition for the density 
‘of inst‘R. This condition says that (1) R is a relation of complete 
sequence, (2) if « precedes y by the relation Rk, there are two 
members of the field of R neither of which bears the relation R 
to the other, while # precedes the one by R, while the other 
precedes y by R, (8) if a follows by R some R-contemporary of y, 
it follows some initial R-contemporary of y. This latter condition, 
which was first formulated by Mr Russell, ensures that if ve C‘R 
| 
} 
i 
} 
> > 
and Reesd, minp‘R,,.{veTp. This I now wish to prove. 
> > 
et. fF: Peesd.ceC’P. >. minp‘P,,“ve rp. 
Proof. 
— 
It follows from the definition of p‘« and minp‘a@ that 
Pp 
ft 
Hy 
} 
. i > 2S vu => 
ipep F.. minp P,,“« = 7 {ae P.e“| Pa a CS P— P“P.,§2|.3.. yea. 
‘Since it follows from the definition of P,, that +.C‘P,,C C*P, 
this reduces to 
—— > - 2 = vy > 
meeeee.. minp P...2—% lace P..“[ P60 — P“ Pot]. D..¥,€ ah: 
| This becomes by a little manipulation 
. ao ie to v 
Ue: p-P..“minp’P,,°2 = 9 [2P x. %~2— P| Pye ® 1 Dz + YP oe X} (1) 
| : : — 
On the other hand, it follows from the definition of minp‘a that 
= = vy 
Fa minp! Pe = 9 [yPoo ty P| Poe x}. 
Since by definition any & which belongs to esd satisfies the 
3 u u —> 
‘condition, #| R,,€ R| ming| R,., we get 
Ean eed S =) 
(Pi Pecsd.d.minp‘P,e= 7 yPs. @ .y — P| minp| P,, a}. 
From this we may deduce 
> 2 
MeePccsd. >. minp'P,,2=9 {y Px. 
| 2P_0.2-+ P| Pyntd,tyPz.V.y + P2.2+ Py.y,2eCP). 
But when yPz is the correct alternative in the conclusion of the 
_ Second proposition in the brackets, together with yP,, 7, this gives 
us zP|P,,2, which contradicts the hypothesis. Hence, by the 
definition of P,., we have 
: eee eccsd. 2D. 
en ue 
MMM hee OY (Yl 02 2 ge. 2 P|) Pie. D;.yP 2} (2) 
