III. MODELS. 37 



The equations representing these systems of surfaces are in rectangular 

 co-ordinates : 



For central surfaces : 



^ . v 2 /I 1\ . * 2 



a 2 cos-V \a hi ksm^ 

 For the elliptic paraboloid : 



For the hyperbolic paraboloid : 



a 2 cos 2 ^ ~ a^in 2 ^ ~ It = 



WTiere 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. Poinsot's Star Polyhedra. Dr. M. Doll, Carlsruhe. 



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

 cahedron with 12 points, the icosahedron, and dodecahedron. 



148. Curvilinear central 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. 



(Sec Monatsbericht der Akademie der Wissenschaf ten zu Berlin, 

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



150. Hummer'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. 



152. Maximum of Attraction of the Earth's Surface. 



Ludwig Lohde, Berlin. 



153. Geometric Body, executed in plaster of Paris, called 

 "Podoid"; a transcendental curved surface, which Js deter- 

 mined by the variable parallel co-ordinates p., <j>, and K, whose 

 equation represents the elliptic function 



f* ^ 



