* Presented to Society, December 27, 1928; received by editors in January, 1929. t National Research Fellow. : Bibliography. The following articles are cited at least twice. Each will hereafter be referred to by letter B and a roman numeral. BI: M. Bocher, Proceedings of American Academy of Arts and Sciences, vol. 40 (1904), pp.469-484. BII: J. L. Walsh, Comptes Rendus du Congres International des Mathematiciens, Strasbourg, 1920, pp. 1-4. BIII: J. L. Walsh, these Transactions, vol. 19 (1918), pp. 291-298. BIV: J. L. Walsh, these Transactions, vol. 22 (1921), pp. 101-116. BV: J. L. Walsh, these Transactions, vol. 24 (1922), pp. 31-69. BVI: J. L. Walsh, Proceedings of National Academy of Sciences, vol. 8 (1922), pp. 139-141. BVII: J. L. Walsh, Rendiconti del Circolo Matematico di Palermo, vol. 46 (1922), pp. 1-13. BVIII: A. B. Coble, Bulletin of American Mathematical Society, vol. 27 (1921), pp. 434-437. BIX: M. Marden, Bulletin of American Mathematical Society, vol. 35 (1929), pp. 363-370. ? I wish to express my deep gratitude to Professor Walsh for his many suggestions and criticisms, and, above all, his constant encouragement in course of this work. 11 This theorem was proved geometrically by Walsh (see BIV). Subsequently an analytical treatment of same problem was published by Coble (see BVIII). Although suggestive as to manipulation of difficult algebra of problem, latter article is open to following objections. (1) It quotes Walsh's theorem incorrectly by substituting the interior of a circle for the circular (2) It considers in proof only case of regions C1 being interiors of circles. (3) It implicitly assumes that locus is finite whenever regions C} are finite. This need not be case, as is shown in chapter IV ?2 of present paper. (4) It implicitly assumes without proof that locus is a simply-connected region. Walsh's theorem, as stated by Coble, was later challenged by T. Nakahara (T6hoku Mathematical Journal, vol. 23 (1924), p. 97) as if it were Walsh's own statement. Nakahara's contribution to subject consists of a set of simple conditions for locus of theorem to be finite. The conditions given are, however, only sufficient, not both necessary and sufficient, as announced by Nakahara. They may be deduced more easily by methods of chapter IV ?2 of present paper. ? See BIV, pp. 101-112. In Walsh's statement of theorem, point z is defined by means of real constant cross ratio (z zl Z2 Z3) = X. Our definition is, however, equivalent to his, provided X0, 00. By a circular region is meant in theorem interior and circumference of a circle, exterior and circumference of a circle, a half-plane including its boundary line, or an entire plane.