522 



DK. A. N. WHITEHEAD ON 



(vii). Case II. We have a and a' identical. Now (cf. *31'22) A, B, C are distinct 

 points. Hence [cf. (ii, 8) and *30'1] u Rt , v Rt , w Rt are distinct points, and [cf. (ii, )] 

 they are collinear. Hence, by XVHpE (cf. *2272), they are in some point-order. 

 Hence [cf. (ii)] three interpoints u", v", w" exist on a common objective real x, which 

 is distinct from a and cogredient with it, and also [u" m v" m w" Rt ~\ persp R( [u Rt v Rt iv m ~]. 



(viii). Hence Case II. divides into two subclasses, either (Case II., ) x is not 

 identical with d, or (Case II., ft) x is identical with d. 



(ix). Case II., a. a and a' are identical and distinct from both d and x, and d and 

 x are distinct; also a, d, x are cogredient and therefore (cf. *26'11'22) copunctual. 

 Hence [cf. (ii, 8) and (vii) and *30'3] we have [ABC] persp K( \u" *$' *&/' TU\- Hence 

 [cf. (iii, 8) and *30'3] [u' Rt v' Rt u>' nt ~\ persp R{ [u\ t v" Jtt w" Rt ~]. Hence [cf. *20'41 and (ii, y) 

 and (vii)] Il in ; (u'v'iv't}. Hence (cf. *1'64) Case II., a, cannot hold. 



(x). Case II., ft. a and a' are identical, x and d are identical, and a and d are 

 cogredient and distinct. Since (cf. *20 - 5l) A, B, C are not cogredient points, they 

 are distinct from u' K , v' K , w/ K , since the only point common to the punctual associates 

 of a and d is a cogredient point. Thus none of u' m , v' m , it/ Rt can be cogredient points. 

 Hence (cf. *28 - 33) there is a punctual line joining B and w' m which does not possess 

 A or u' Rt or V (cf. figure annexed). Hence (cf. *28'42) there is at least one other 



V 



If! 



B" 



point, A" say, lying in the punctual line joining A and u' Rt in addition to V and u' 

 and A. Hence (cf. *28'33) there is a punctual line, z say, joining A" to the cogredient 

 point common to the punctual associates of a and d. This punctual line must meet 

 (cf. *28'42) the punctual lines VB and VC (cf. figure annexed) in the points B" and 

 C". Hence 



[A"B"C"]persp m [ABC], 

 and hence (cf. *30'3) we have 



and hence (cf. *30'8) we have 



[A"B"C>ersp m [ W V> 



