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The geometry of determinantal loci
BOOK
REVIEWS.
T H E GEOMETRY OF DETERMINANTAL LOCI, by T. G. Room. 483 pages, illustrations, I9 X 28 cms. N e w York, The Macmillan Company, 1938. Price $Io. The subject matter of this book holds fascination for many mathematicians, particularly those interested in research in geometry. There is stillmuch unexplored territory and the problems contained therein offer a means for exercise and ingenuity. The author of this book states that it is thc result of 13 years work. It should provide a fair start along lines of this branch of mathematical investigation. The plan of the book is such that general theorems are first established, followed by a discussion of the particular configurations which appear to be of greatest interest. The point made by this method is the saving in time and labor, although thc investigation of the peculiarities of special Iocl and configurations are of prime importance. After an introductory chapter which sets out the suffix and summation conventions adopted in the book and discusses brieflythe meanings of terms in current use in projective geometry, the book is divided into thrce parts with a total of 25 chapters. Part I has the general titlcof manifolds dcfined by matrices of general form. Under this a survey is made of the fundamcntal properties of manifolds, and there are discussed thrce different ways of establishing a birational correspondence between two projectivcly generated manifolds, the application of the degeneration principle, the pairing theorcms, the key manifolds, determinantal primals, and the freedom of a projectively generated manifold. This material is more or less of a general nature and m a y be passed over by some and used only for reference. The second part of the book has to do with manifolds defined by matrices of special forms. In it there is attention given to key-manifolds, manifolds repre sentable by the vanishing of minors of asymmctrical determinant, those defined by skew symmetrical determinants, normal rational scrolls,and representations of spaces by points. Finallythere are given some generalizations of the classictheorem of the "fifth associated line" containing a generalization of the Scgre primal. Determlnantal quartic primals in space of four dimensions is the heading of part Ill of the book. This covers almost ~oo pages and is followed by appendixes on Veroneseans and Grassmannians, the characteristic numbers of certain determinantal manifolds, and the freedoms of manifolds. A n index is in the back. The book provides a course of study that should prove illuminating as well as inspirational to those qualified for its perusal. R. H. OPPERMANN. SIR CHANDRASEKHARA VENKATA RAMAN, I888--7th November--1938. Jubilee volume of original papers presented in commemoration of his fiftiethbirthday. 322 pages, illustrations, plates, 2o X 27 cms. Bangalore, Indian Academy of Sciences, I938. Every scientistis familmr with at least some of the works of Sir C. V. Raman, the best known of which is his discovery in I928 of thc radiation effectknown by 728
REVIEWS.
T H E GEOMETRY OF DETERMINANTAL LOCI, by T. G. Room. 483 pages, illustrations, I9 X 28 cms. N e w York, The Macmillan Company, 1938. Price $Io. The subject matter of this book holds fascination for many mathematicians, particularly those interested in research in geometry. There is stillmuch unexplored territory and the problems contained therein offer a means for exercise and ingenuity. The author of this book states that it is thc result of 13 years work. It should provide a fair start along lines of this branch of mathematical investigation. The plan of the book is such that general theorems are first established, followed by a discussion of the particular configurations which appear to be of greatest interest. The point made by this method is the saving in time and labor, although thc investigation of the peculiarities of special Iocl and configurations are of prime importance. After an introductory chapter which sets out the suffix and summation conventions adopted in the book and discusses brieflythe meanings of terms in current use in projective geometry, the book is divided into thrce parts with a total of 25 chapters. Part I has the general titlcof manifolds dcfined by matrices of general form. Under this a survey is made of the fundamcntal properties of manifolds, and there are discussed thrce different ways of establishing a birational correspondence between two projectivcly generated manifolds, the application of the degeneration principle, the pairing theorcms, the key manifolds, determinantal primals, and the freedom of a projectively generated manifold. This material is more or less of a general nature and m a y be passed over by some and used only for reference. The second part of the book has to do with manifolds defined by matrices of special forms. In it there is attention given to key-manifolds, manifolds repre sentable by the vanishing of minors of asymmctrical determinant, those defined by skew symmetrical determinants, normal rational scrolls,and representations of spaces by points. Finallythere are given some generalizations of the classictheorem of the "fifth associated line" containing a generalization of the Scgre primal. Determlnantal quartic primals in space of four dimensions is the heading of part Ill of the book. This covers almost ~oo pages and is followed by appendixes on Veroneseans and Grassmannians, the characteristic numbers of certain determinantal manifolds, and the freedoms of manifolds. A n index is in the back. The book provides a course of study that should prove illuminating as well as inspirational to those qualified for its perusal. R. H. OPPERMANN. SIR CHANDRASEKHARA VENKATA RAMAN, I888--7th November--1938. Jubilee volume of original papers presented in commemoration of his fiftiethbirthday. 322 pages, illustrations, plates, 2o X 27 cms. Bangalore, Indian Academy of Sciences, I938. Every scientistis familmr with at least some of the works of Sir C. V. Raman, the best known of which is his discovery in I928 of thc radiation effectknown by 728