## Classification of the Surfaces of Singularities of the Quadratic Spherical Complex ... |

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annular surface biplanar point bitangent planes Canonical Form class 12 coincident double complex consists complex of planes complex of points cone of revolution consists of spheres counted four cuspidal sphere cyclide counted twice developable of order developables of bitangent director sphere double conic double directrices double elements coincide double line double spheres Dupin cyclide counted envelope of oo2 finite minimum lines focal developable focal line fundamental sphere involution Laguerre surface consists Laguerre transformation line in common linear congruences lines of curvature minimum lines counted minimum lines enveloped oo1 spheres opable order and class parabolic cylinder pencils of spheres plane at infinity plane counted twice point sphere points transform quadrilateral of minimum quartic ruled cyclide simple line single directrix singular spheres skew quadrilateral sphere in common sphere whose points spheres which touch sphero-quartic surface becomes surface of order surface of singularities tacnodal tangent pencils twisted cubic vertex

### Popular passages

Page 248 - ... Spherical Complex. The author completes the classification of the surface of singularities mentioned in the title, discussed by PF Smith (Trans, of the Amer, m-ith. sac., vol. 1 , 1900, p. 371, Rev. sem. IX 2, p. 8), as Weiler has done for the quadratic line complex (Math. Ann. , vol. 7 , p. 145). Since spheres as well as lines can be represented by six homogeneous coordinates, Weiler's symbols and notations for the fundamental complexes are used, but no further use is made of line geometry....

Page 267 - The directrices of these special congruences form two tangent pencils in (2, 1) correspondence having a self-corresponding element. The involution formed by the pair of elements of the first pencil corresponding to the elements of the second pencil, has the common self-corresponding element for double element. The complex is, therefore, composed of those spheres which touch corresponding spheres of two tangent pencils in (2, 1) correspondence having a selfcorresponding element. VI. Sixth Canonical...

Page 253 - P*)2 = tt which defines the general involution [2] . The complex is, therefore, obtained by establishing a general involution between the spheres of a Dupin cyclide and taking all the spheres which touch each pair of corresponding spheres. In the case [(lll)(ll) l] a4 = aB and, therefore, c = 0 and the involution becomes A2 (pi + p*)' = jp- p!p2 , and, therefore, has two pairs of coincident roots, two zero and two infinite.

Page 278 - The complex consists of those spheres which touch paired planes and of a cone of revolution in involution [2]As above, the involution having two pairs of coincident elements. Complex consists of spheres which touch a cone of revolution. Same as (111) 111, involution has two coincident double elements. As above, the involution has all its double elements coincident. Involution has three double elements coincident. Complex consists of those spheres which touch a cylinder which has a minimum axis. SURFACES...

Page 262 - It is of order and class 16, can be generated in four ways as the envelope of »2 spheres, two of which are special; it has two general developables of bitangent planes and two extraordinary. If xi = 0 is the complex of points and a-3 = 0 the complex of planes, the surface reduces to a quadric, which is tangent to K at one point, then by a Laguerre transformation this quadric inverts into a surface having a double plane. It is of order 10, class 4, has K for double line and has two finite double...

Page 273 - It was seen that the surface [1122] becomes the surface [114] when the two double spheres become consecutive, in a similar manner the surface [222] becomes [24] when two of the double spheres become consecutive. Thus the surface [6] has three consecutive double spheres which have a minimum line in common ; it is of order 12 and can be generated in one (special way as the envelope of oo!

Page 278 - Complex consists of spheres which touch aDupincyclide. (111)(12). Same as (111) 111 involution has two coincident double elements. (111)(12). As above, involution has all its double elements coincident. (111)3. Involution has three double elements coincident. (222). Complex consists of those spheres which touch a ruled cyclide. Cz Counted Twice. The complex consists of those spheres which touch paired planes and of a cone of revolution in involution [2]As above, the involution having two pairs of...

Page 265 - If 3^ = 0 is the complex of points and x% =• 0 the complex of planes, the surface reduces to a paraboloid, which is tangent to the plane at infinity at a point of K. By a Laguerre transformation, this inverts into a surface of order 9, class 4, containing K as simple line ; it has two double conies, 2. [(11)4]. a,=a4. The cyclide [114] has a focal line on the sphere a^ consisting of a twisted cubic and a minimum line tangent to it.

Page 274 - The minimum line enveloped by ar, = j-3 = x3 = 0 is a part of the surface. The Laguerre surface is of order 8, class 4. XII. The following table contains a complete list of those surfaces which appear as surfaces of singularities of the quadratic complex. The cyclide and spheroquartic are not included here, since they have been classified by Loria in the paper referred to. The following symbols are used: v, = complex of planes.

Page 270 - The minimum line common to a;, = x3 = x5 = 0 is also a part of the surface, since these spheres are all singular spheres. The presence of this line does not reduce the order nor class of the residual surface. By an (/), this inverts into the general surface of singularities, which is, therefore, of order and class 12, passes six times through K. The minimum line enveloped by xt = fy = x6 = 0 is also a part of this surface.