Recent Progress of Investigations by Symbolical Methods of the Invariants of Bi-Ternary Quantics.

the set T1 + T2 + T3 + . . . + Tn. Let K denote the set of all points X of M such that X can be joined to A by a connected subset of M which contains no point outside of C and let E denote the sum of the sets H and K. Let D denote the complementary domain of E which contains B and let L denote the boundary of D. Since K is2 a continuous curve and each Ti is3 a continuous curve, therefore, E is a continuous curve. Hence L is3 a continuous curve. Now A does not belong to L. For suppose it does. Then since every point of L is' accessible from D there exists an arc AB which, except for A, is a subset of D. Let P denote the first point that the arc AB has in common with C1. The interval AP ofAB has only the point A in common with M and hence AP-A lies in one of the complementary domains D,, D2, D3,... Dn. This, however, is impossible since the interval BP of AB contains no point of the boundary of any one of these domains. Hence A lies in one complementary domain of the continuous curve L and B lies in another one. Hence by a theorem of R. L. Moore4 there exists a simple closed curve which separates A from B and which is a subset of L and, therefore, of M. 1 Schoenflies, A., "Die Entwickelung der Lehre von den Punktmannigfaltigkeiten," Zweite Teil, Leipzig, 1908. 2 Gehman, H. M., Ann. Math., 27, 1925 (29-46), Lemma B. 3Moore, R. L., Math. Zs., 15, 1922 (259). 4Moore, R. L., loc. cit. (260).