Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

SECTION B]

THE SERIES OF SEGMENTS

653

*212-601. F:. Pe Ser. a C C'P.:
-4-4
E! maxpa. -. E! max (P;P)'P"a..E! max (s'P)'P" a
*212'602. F:. P e Ser. f! P. a C C'P.: E! naxp'a. -. P"a e P"a
*212-61. F: P e trans n connex.! P. D. limax ('P)P'Pa = Pa
*212-632. F: P e Ser.! P. a C C'P. P"a e PWC'P. D. P"a = It ('P)'P"a
*212 661. F: P e Ser. KC DP,. E! It (SP)tC. D.
It (S'P)'K = It (S'P)'P"cs' = s'K
This shows that every limit in the series of segments is a limit of a class
of what we may call rational segments (i.e. segments of the form P'x), namely
it is the limit of P"s'K.
*212-667. F: P e Ser.:). D'lt ('P) - t'A = (Isgm'P
This shows that the segments (other than A) which are limits of classes
of segments are the segments (other than A) which have no maximum in P.
The number ends with a set of propositions (*212*7 —72) on the relations
of the sections and segments of two correlated series. If S is a correlator of
P with Q, then Se (with its converse domain limited) is a correlator of S'P*
with s'Q*,;'P with s'Q and sgm'P with sgm'Q (*212-71-711'712). Hence
*212'72. F: P smor Q. ). 'P* smor sQ,. s'P smor;'Q. sgm'P smor sgm'Q
This proposition is used in the next number, and also in *271.
*212-01.;'P = Plc, D'P Df
*212-02. sgm'P = Pc D'(Pe n I) Df
*212'1. F: a (s'P)/3. -. a, /? e D'P. -!/3 -a-P"(a - )
[*170102. *37'15]
*212'11.: a (sgm'P)... a,/3e D'(Pe, I). [!,8 -a
Dem.. *170-102. *37'15. D
F: a(sgm'P),. -. a,,leD'(Pe i I).! /3 - a - P"(a - f8) (1). *211-12. ): a e D'(Pe A I).. - a =-P"a.
[*37-2.Transp]. - C - P"(a - 9).
[*22-621] D. - a- P"(a- /3) =- a (2)
F. (1). (2). ) F. Prop
*212-12. F: a(sgm'PO)...a, p/ esect'P. a! - a [*211 13. 212-11]
Thus sgm'P, has the same connection with sect'P as sgm'P has with
D'(Pe I). When P is transitive, sgm'P, also has the same connection