514 DR. A. N. WHITEHEAD ON 



*25'13. Proposition. Assuming (II.-VL) HpR, if t be an instant of time, then 

 R( is homalous. In symbols, 



Proof From *T65 and *21'11, O K( is identical with R ; (;---). Hence (cf. 

 *l-3r61'62) every member of O m is doubly secant with O R( . Again (cf. *2M1), 

 every objective real which is doubly secant with R( is a member of it. 



*25'14. Proposition. Assuming (II.-VL, IX-XI.)HpR, if t be an instant of time, 

 then all the special deductions of the theory of dimensions, namely *11 to *16, hold 

 respecting homaloty, that is, with p. }U substituted for <. 



Proof Cf. *21"31 and *22'4r42'43 and *25'13. 



*25'21. Proposition. Assuming I HpR, if I is an instant of time, then O = O R( . 

 In symbols, 



Proof. Cf. *2fll and *22'1. 



Note. The above theorem is not used in any geometrical reasoning. 

 *25'31. Proposition. Assuming (II. -VIII.) HpR, if a be a member of O R( , then 

 the number of points on a is at least three. In symbols, 



|- . '. (II. VIII.) Hp R . D : e O l{( . D . Nc'ass K( Va ^ 3 



Proof.~Cf. *172 and *2M1 and *22'32'33 and *25'1. 



*25'32. Proposition. Assuming, (II. -VII., IX.-XI.)HpR, if u be an interpoint, 

 there is one and only one point containing it. In symbols, 



h .'. (IT. -VII., IX.-XF.) Hp R . D : u t intpntB, . D . P {P e pnt R( . u c P} e 1 

 Proof Cf. *171 and *14'21 and *22'32. 



*2G. On Cogredient Points. 



*26'11. Proposition. Assuming (II.-VL, IX-XL, XIII.) HpR, if a point possesses 

 two members which are cogredient to each other, it is a cogredient point. In 

 symbols, 



h.'. (II.-VL, IX-XL, XIII.) HpR . D : 



A e pnt K( . a, b e A . b e cogrd K( 'a . ^ b . D . A e OO R( 



Proof Cf. *14'21 and *20'42 and *22'61. 



*26'22. Proposition. Assuming (II.-VL, IX. -XL, XIII.) Hp R, if A is a cogredient 

 point and a is a member of A, then A is identical with ,' a u cogrd lu ' a. In symbols, 



|- . (II.-VI. , IX. -XL , XIII. ) Hp R . : A e OO R( . a e A . D . A = t ' a U cogrd R( ' a 

 Proof Cf. *14'21 and *20'42 and *21'41 and *22'61. 



