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On Covariant Types of Binary n ‐ICS *
Author(s) -
Young A.
Publication year - 1904
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s2-1.1.202
Subject(s) - citation , binary number , covariant transformation , table (database) , information retrieval , computer science , arithmetic , mathematics , combinatorics , library science , data mining , geometry
1. The " finiteness" o( the complete system of concomitants of a single binary form was first proved by Gordant in the year 1868. Before that time it was believed that quantics of order higher than 4 did not possess a finite system of concomitants. Since then many different proofs of this central fact in the theory of forms have been given by various writers. Its truth has been established for ternary forms and forms having any greater number of variables ; for forms possessing two or more different sets of variables, whether cogredient or otherwise ; for any simultaneous system of forms ; and for types of concomitants when the actual number of base forms becomes infinite, but the order of each is finite. Information, with references, as to the various proofs of the finiteness is to be found in any of the various editions of Meyer's " Bericht iiber den gegenwartigen Stand der Invariantentheorie."! The majority of the proofs, especially of the later ones, keep the attention fixed on the collective idea of finiteness, and give little or no information as to the actual cuiujjosition, formation, or extent of the complete system. There are two exceptions, which stand out pre-eminently from the above general rule : these are the proof given by Gordan and that due to Jordan. Gordan proved the theorem by actually giving a process by means of which the complete system might be established, and by showing that this process yielded only a finite number of irreducible concomitants. In the later editions of his proof the process was considerably modified, but the principle remained the same.