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converge to a function </>(V) analytic in 5 and such that | <f)(z) | <1 for z in S. Since the original sequence {pn(z)} converges by hypothesis in R, the limit function of the subsequence would necessarily agree with ƒ(z) in R. But <j>(z) could not be identical with f(z) in R} for then <j>(z) would have to equal unity at interior points of S, namely, at the boundary points of R which lie interior to S. It would appear from this remark that Theorem 1 (and Theorem 2 as well) might admit of extension to an arbitrary finite simply connected region whose complete boundary is also the boundary of an infinite region, and that a modification of the methods used in the proofs of the present paper would serve to establish such an extension. The writer hopes to answer this question in a later paper.

Note on a simple type of algebra in which the cancellation law of addition does not hold