Science Volume 4, Nos. 79-104 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1896 edition. Excerpt: ...For n 5 it is holoedrically isomorphic to the symmetric group in n letters. It contains a sub-group of collineations permuting amongst themselves certain n-1 fundamental points, and first given by Klein. Prof. Moore's group results from extending that of Prof. Klein by an 'inversion ' having n-2 of the fundamental points as critical points and the remaining one as a fixed point. This paper is intended for the Mathematische Annalen. In Dr. Snyder's paper Lie's haxaspherical coordinates were employed to define, without the use of line-geometry, a Dupin's cyclide. By associating the three simultaneous linear equations of definition with the point-complex and the plane-complex, one obtains determinants the signs of which indicate the reality of spheres common to the four complexes and thus show the presence of nodes. Dr. Snyder's paper will appear in the Annals of Mathematics. On a convex surface of deficiency zero Euler's equation, together with the requirement of numerical regularity, gives three sets of integers for vertices, faces and edges of a polyedron. These by duality become five, corresponding to the five regular polyedra. On a surface of deficiency greater than unity the modified equation of Euler, together with similar limitations, gives again a finite number of sets of integers for vertices, faces and edges. These sets are of two sorts: 'derivative, ' obtained from sets belonging to lower deficiencies; and 'special, ' not so obtainable, but peculiar to the deficiency in question. These sets of characteristic numbers can be realized on concrete models. Prof. White discussed the subject in detail and exhibited models, constructed by Mr. O. H. Basquin, for deficiency 2 and for the 'special' sets of deficiency 3. For deficiency 2 there were 13...