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MATHEMATICS: C. N. MOORE 
Proc. N. a. S. 
the case of the descriptive properties. In the case of a function of n- 
variables, we may project upon an {n — 1) space, an {n — 2) space, etc. 
The proofs of the preceding theorems are contained in a paper that has 
been offered to the Transactions of the American Mathematical Society. 
1 Ann. Math. Princeton, 18, 1917 (147). 
2 For the terminology cf. Denjoy, /. Math., Paris, ser. 7, 1, 1915 (122-125). 
2 Cf . for example, Frechet, Rend. Circ, Math. Palermo, 22, 1907, p. 1 ; and Hausdorff, 
Grundziige der Mengenlehre, 1914, p. 211. 
4 Hausdorff, 1. c, p. 315. 
5 For the case where only measurable sets are admitted cf., for example, de la Vallee 
Poussin, Cours d' Analyse, 2, 1912, p. 114. For the linear case of general (not neces- 
sarily measurable) sets cf. Blumberg, Bull. Amer. Math. Soc, 25, 1919 (350). 
6 Cf. Denjoy, Bull. Soc. Math. France, 43, 1915 (165). 
^ In this connection cf. Gateaux, Ibid., 47, 1919 (47). 
GENERALIZED LIMITS IN GENERAL ANALYSIS 
By Chari,e:s N. Moore 
Department of Mathematics, University of Cincinnati 
Communicated July 15, 1922 
It is well known that to each of the various methods for summing 
divergent series there corresponds an analogous method for summing di- 
vergent integrals. It is readily seen that similar methods may be used for 
obtaining types of generalized derivatives of a function at a point where the 
ordinary derivative fails to exist. Likewise, in any other case in Analysis 
where we wish to associate a limit with a variable that oscillates, we will 
naturally be led to make use of methods that have been tried out in the 
case of divergent series. 
It would be manifestly poor economy of time and thought to elaborate 
for each of these special theories such fundamental results as are common 
to them all, if these results can be obtained in one central theory that in- 
cludes all the others. According to a principle of generalization formu- 
lated by B. H. Moore and stated by him on several occasions,^ the exis- 
tence of such a general theory is implied by the analogies found among the 
various special theories. It is natural to designate this general theory as 
the theory of generalized limits in General Analysis. 
It is the purpose of the present communication to illustrate the nature 
and usefulness of this theory by outlining the proof of a theorem in it which 
is a generalization of one of the important theorems in the theory of di- 
vergent series. This latter theorem is the Knopp-Schnee-Ford theorem^ 
with regard to the equivalence of the Cesaro and Holder means of order k 
