satisfied by the substitutions which correspond to the unities that are employed.1 These relations between the quaternion unities could also have been obtained directly by means of the corresponding substitutions. As any relation between quaternion unities remains true if we replace all these unities by those which correspond to them in any simple isomorphism of their group to itself, it follows directly that a knowledge of the group of isomorphisms of this group to itself is of great utility in transforming quaternion relations; e. g., from the simple isomorphism it follows that i may be replaced by j, j by k, and k by i at the the following: ). j (−j) ). k (k) ). k (k) ijk. (-i) (—j) (−k) j (−k) (—j) k i (−k) (-i) k The twenty- ij. (—i) (—j). k (— k) i (-i). jk. (j) (−k) i (-i). j (k). (—j) k i (—j). j (—i). k (—k) ik. j (—j). (—i) (—k) i (−k). j (—j). k (−i) when an equation between the quaternion unities admits a of these 1 Cf. Tait's Quaternion, 1890, p. 46. substitutions these substitutions must form a subgroup of this group of isomorphism and the given equation must assume 24 a different forms which are equally true in case it is transformed by all these substitutions, e. g., each of the three equations in the last set given above admits a cyclical subgroup of order 4. Hence each of these equations gives rise to 24 4 6 true equations. In addition to the three that have been given we have (—¿)2= (—j)2= (−k)2: =-1. We have already noticed that the group of cogredient isomorphisms of is the four-group. Hence has only two operators that are commutative to each one of its operators. These are evidently the operators which correspond to 1 and 1 in the quater. nion unities. These two unities are therefore the only ones in the quarternion number system that are commutative to all the numbers of the system. It need scarcely be remarked that any one of the three cyclical subgroups of order 4 contained in Q may correspond to the unities of the ordinary complex number system. RELATION BETWEEN THE QUATERNION GROUP AND THE HAMILTONIAN GROUPS. 1 One of the most remarkable properties of the quaternion group is that each of its subgroups is self-conjugate. Dedekind has called all the groups which have this property Hamiltonian groups and he has pointed out that the quaternion group is of fundamental importance in the study of the Hamiltonian groups. It has recently been proved that every Hamiltonian group is the direct product of an Abelian group of an odd order and a Hamiltonian group of order 2a, and that there is one and only one Hamiltonian group of order 2a for every integer value of a greater than 2." I It is easy to see that the direct product of the quaternion group and the Abelian group of order 2-3 which contains 2-3. operators of order 2 is Hamiltonian. Since there is only one Hamiltonian group of this order it follows that every such Hamiltonian group may be constructed in this manner. Hence we have that every Hamiltonian group whose order is divisible by 2°, but not 2+1 must be the direct product of some Abelian group of an odd order, the Abelian group of order 2-3 which contains 2a-3 operators of order 2, and the quaternion group. 1 Dedekind, loc. cit. 2 Miller, Comptes Rendus, 1898, Vol. cxxvi, p. 1406. I While the direct product of the quaternion group and any Abelian group of an odd order is always a Hamiltonian group, the direct product of the quaternion group and an Abelian group whose order is divisible by a power of a is only Hamiltonian when the latter group contains no operator whose order is divisible 4. This follows directly from the fact that the group generated by the product of an operator of order 4 in the Hamiltonian group and any operator in such an Abelian group must be self-conjugate. We may determine the number of the quaternion groups that are contained in a Hamiltonian group whose order is divisible by 2" without being divisible by 2+1 in the following manner. Such a group contains a single subgroup of order 2°. This subgroup in<cludes 3 times 2a- operators of order 4. Each quaternion subgroup includes two of the operators of order 4 that are included in a subgroup of order 2a which involves only 2°Hence there are 22 group. 6 -2 operators of order 4. quaternion subgroups in the given Hamiltonian have the commutator subgroup of the entire In other words, the commutator subgroup of a Hamiltonian group is the same as that of any one of its quaternion subgroups. All of these group in common. CORNELL UNIVERSITY, June, 1898. Stated Meeting, October 21, 1898. Vice-President SELLERS in the Chair. Present, 12 members. Prof. Lighter Witmer, a newly elected member, was presented to the Chair, and took his seat. The minutes of the last stated meeting were read and approved. Dr. Frazer read a letter from the International Geological Congress in regard to the establishment of an international floating institute, and offered the following resolution : Resolved, That the President of the Society be requested to memorialize Congress in favor of an appropriation in aid of the in1 Sylow, Mathematische Annalen, 1872, Vol. v, p. 584. PROC. AMER. PHILOS. SOC. XXXVII. 158. U. PRINTED FEB. 23, 1899 vestigations proposed at the meeting of the International Geological Congress held at St. Petersburg, Russia, in August, 1897, and that the President be requested to communicate to the Secretary of State what had been done at the St. Petersburg Congress in respect of establishing an international floating institute for the purposes named in the action of that Congress, and to request the Secretary of State to bring the subject to the attention of the proper committees of Congress. which resolution, on his motion, was referred to the Officers and Council. The Librarian presented a list of the donations to the Library, and called special attention to a valuable gift from Mr. Henry Pettit, of five volumes of contemporaneous clippings, illustrating the day-to-day history of the HispanoAmerican War; and of two volumes of L'Illustration, July, 1870-July, 1871, being the numbers issued in Paris during the Commune. Mr. Pettit, by invitation, made some interesting remarks in connection with this donation. Announcement was made of the decease of Prof. Gabriel de Mortillet, of St. Germain-en-Laye, France, who was elected to membership on February 15, 1895. Prof. Albert H. Smyth read a paper on "Thomas Moore in Philadelphia," which was discussed by Messrs. Dickson and Wood. A paper by Dr. Daniel G. Brinton was presented on "Two Unclassified Recent Vocabularies from South America." Pending nominations for membership Nos. 1432, 1464, 1469, 1470, 1471, 1472, and new nominations Nos. 1473 and 1474 were read. The rough minutes were read, and the Society was adjourned by the presiding officer. ON TWO UNCLASSIFIED RECENT VOCABULARIES FROM SOUTH AMERICA. BY DANIEL G. BRINTON, M.D. (Read October 21, 1898.) The time has almost passed when any South American Indian can speak in an unknown tongue. The hundreds and even thousands of "radically distinct" languages which the early travelers and missionaries supposed to exist on that continent have been reduced to about sixty linguistic stocks, with a fair prospect of further diminution when materials for analysis become available. To aid in this work it is important that each vocabulary collected by travelers be scrutinized and referred to its appropriate stock, if known, and, if not, that it be noted for further consideration. In pursuance of this, I shall briefly examine two vocabularies from South America which have been published within the last year, but which have not been referred by the writers who obtained them to any of the leading stocks. The first is furnished by Mr. A. Rimbach, in his "Reise im Gebiet des oberen Amazonas," printed in the Zeitschrift der Gesellschaft für Erdkunde, Berlin, 1897, p. 379. He calls it the "Gay" language, and adds that he obtained it from some Andoas Indians whom he encountered on the lower reaches of the river Pastaza. He gives only five words, which are as follows: This vocabulary belongs to what I have called in my work, The American Race, to the "Zaparo" linguistic stock, as is easily seen by comparing it with the Zaparo vocabulary collected by the Italian traveler, Osculati.1 Although by some writers the Andoas have been said to speak Quichua, this has been refuted by Tyler and others. The name 2 Esplorazione delle Regioni Equatoriali, App. (Milan, 1850). 2 Cf. Tyler, in The Geographical Journal, June, 1894. |