Phytopharmacological Reactions of Normal, Toxic and Atoxic Sera.

As soon as we have established (10) and (11) we can complete the discussion in the customary manner. An m-chain will be exact if its derivative vanishes, it will be derived if it is the derivative of some (m 1)-chain. By (10), every derived chain will be exact; therefore, the group GD' of all derived m-chains will be a subgroup of the group GE of all exact m-chains. The residue group GE mod GD will be the m-th connectivity group of B. Finally, by (11), the product of two exact chains will be exact and the product of an exact chain by a derived chain or of a derived chain by an exact chain will be derived. Therefore, the exact chains will generate a sub-ring RE of R and the derived chains an ideal RD of RE. The ring RE mod RD will be the connectivity ring of the space B. 1 Cf. a previous note under the same title communicated to these PROCEEDINGS on March 30, 1936. A more detailed account of the theory outlined in the present and previous notes is now being prepared for publication. Moreover, an alternative treatment, from quite another angle, will appear in the Ann. Math. for July of this year.