30 SEC. 2. GEOMETRY. 



141. Series of Cardboard Models of Surfaces, of the 

 second order, in a cardboard box. The sections are not 

 attached to each other. Prof. Dr. A. Brill, Munich. 



These models are distinguished from those in common use by their mobility, 

 by means of which each one represents not only a single ellipsoid or hyperbo- 

 loid, but also a host of surfaces of one or the other kind. For when the 

 angle of inclination of the circular sections is altered, in a direction easily re- 

 cognised by pressing or drawing out the model, there will be obtained a simple 

 but infinite system, the individual forms of which can be converted from a flat 

 figure through gradually-changing solid bodies to just such another figure 

 with a different relation of axes, without, however, losing its properties. 



The equations representing these systems of surfaces are in rectangular 

 co-ordinates : 



For central surfaces : 



* + J? (I - 1) + * - 1, or = (cone). 

 a 2 cos 2 f> \a kj ksinty 



For the elliptic paraboloid : 



a 



Where 2^ is the inclination of the circular sections, and a and k are real con- 

 stants. From the first equation it appears that among the series of ellipsoids 

 there will always be a sphere. 



142. Model of a Surface of the third order, made in 

 plaster of Paris, with 27 real right lines. 



Prof. Dr. Christian Wiener, Carlsruhe. 

 The construction of the model is described on a placard fixed to the model. 



143. Model of the same surface of the third order, in 

 discs of card-board, with 27 real right lines. 



Prof. Dr. Christian Wiener, Carlsruhe. 



144. Foinsot's Star Polyhedra. Max Doll, Carlsruhe. 



These models show the star dodecahedron with 20 points, the star 

 dodecahedron with 12 points, the icosahedron and dodecahedron. 



148. Curvilinear centre surface of the Ellipsoid, in 



four separate pieces. Proportions of the axes of the ellipsoid, 

 3 : 4 : 5. Ludwig Lohde, Berlin. 



149. Dupin's Cyclide, according to the calculation of Pro- 

 fessor Kummer, at Berlin. Model 0*094 m. diameter. 



(See Monatsbericht der Akademie der Wissenschaften zu Berlin, 

 1863, pp. 328 and 336.) Ludwig Lohde, Berlin. 



150. Kummer 5 s Cyclide. Ludwig Lohde, Berlin. 



151. Minimum-surface in a recurring number of tetra- 

 hedral surfaces. 



(Submitted to the Berlin Academy of Sciences by Professor 

 Kummer, on the 6th April 1865.) Ludwig Lohde, Berlin 



