1. Presented to the Society, December 29, 1920, and April 23, 1921.
2. When this paper was first offered for publication, the writer believed Theorem I to be new. Professor D. R. Curtiss has kindly pointed out its connection with a theorem due to Grace and has indicated an entirely new proof of Theorem I; the reader will refer to Professor Curtiss’s note which immediately follows this paper. The point of view of the present paper in the proof and application of Theorem I seems to be new, and also the results obtained except where otherwise stated. This paper is the development of a short note published in the Paris Comptes Rendus, March, 1921, to which explicit reference is made later, and which contained in outline the proof of Theorem I. In the interval between the publication of that note and the publication of the present paper, there have appeared a number of other papers dealing with Grace’s Theorem. See Szegö, Mathematische Zeitschrift, vol. 13 (1922), pp. 28–55; Cohn, Mathematische Zeitschrift, vol. 14 (1922), pp. 110-148; Egerváry, Acta Litterarum ac Scientiarum, Regiae Universitatis Hungaricae Fran-cisco-Josephinae, vol. 1 (1922), pp. 39-45; Fekete, same Journal, vol. 1 (1923), pp. 98-100. The present paper has very little in common with these other papers, but with Szegö’s Theorem 9 the reader should compare our Theorem X, and with Szegö’s Theorems 13–15 compare our Theorem IV and also Walsh, American Mathematical Monthly, vol. 29 (1922), pp. 112-114.
3. Professor Curtiss has recently published a very interesting report on this general field, Science, vol. 55 (1922), pp. 189–194.
4. The coefficients of f (z) need not be homogeneous in each of these sets of variables, but each coefficient must be a linear combination of the elementary symmetric functions of each of these sets with coefficients linear combinations of the elementary symmetric functions of the other sets. These linear combinations may, moreover, contain constant terms.
5. This is what actually occurs in the situation of Theorem II if we choose P inside C.