File:The Americana; a universal reference library, comprising the arts and sciences, literature, history, biography, geography, commerce, etc., of the world (1908) (14578691178).jpg

Identifier: americanaunivers08newy (find matches)
Title: The Americana; a universal reference library, comprising the arts and sciences, literature, history, biography, geography, commerce, etc., of the world
Year: 1908 (1900s)
Authors:
Subjects: Encyclopedias and dictionaries
Publisher: New York : Scientific American Compiling Dept.
Contributing Library: University of California Libraries
Digitizing Sponsor: Internet Archive

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greatcircle (equator), form a group. These groupsare of orders 60 (icosahedron, dodecahedron),24 (octahedron, cube), 12 (tetrahedron), 2n(dihedron). n (cyclical). They give all thenon-isomorphic t>-pes of finite groups of lineartransformations of z. The octahedral group isalso isomorphic with the sjTnmetric substitutiongroup of four letters, the tetrahedral and icosa-hedral groups with (alternating) substitutiongroups of four and five letters, respectively. .■\ simple example of a (dihedral) group oforder 6 is generated by the transformationsS: z^= l/z and T: r = i — z above. The linear transformations of z written inhomogeneous form 5, =02,-f/S.jj.y, = 7^,-;-iz, furnish homogeitcous linear groups. Increas-ing the number of variables, we arrive at thegeneral homogeneous linear groups s, • = a,, s, + au j + +aH Sn,r.=aj, 3, -f-djjSj -;- + a^xSH, .... Sn^ ■= OhiS, + an,s,+ + annSn identified, for example, with project-ive geometry. Cur^•es, surfaces, etc, frequently GROUSE.
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1. Prairie Hen (Tympaiuichus Americanus). 2. Blackcock (Tetrao urogallus). 3. Moor-hen or Ptarmigan (Lagopus albus) in winter dress. 4. Scotch Ked Grouse (Lagopus scoticus). 5. Ruffed Grouse (Bonasa umbellus). 6. Hybrid between Blackcock and Capercailzie. GROUSE —GROVE CITY have linear transformations into themselves,these always forming a gronp. Thus a planecubic curve has in general such a group of order432. Linear congruence groups should also bementioned. An example is the simple group oforder lApiP—J) (P>4) composed of the lin-ear transformation z = (a 3-)-p) / (y s+ S)wherio,/3, 7, 5, z, s are integers taken mod./Continuous Groups.—These are groups oftransformations involving continuous parameters,such as the entire group of linear transforma-tions of z, or the entire group of motions in aplane or in space. The theory of these groups,■which has been extensively developed by SophusLie and his followers since 1870, has importantapplications to geometry, and especially t

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