III. - MODELS. 35 



The equation to this surface is xyz = k* (x + y + 2 I) 3 . There are 3 bi- 

 planar nodes as shown on the model. The 27 straight lines on the surface 

 are all real, but coincide 9 to each with the 3 black lines^ drawn on the model. 



123c. Sylvester's Amphigenous Surface, a surface of 

 the ninth order. Prof. O. Henrici, F.R.S. 



This surface is connected with the reality of the roots of equations of the 

 ninth degree. 



123d. Model representing the Right Lines on a 

 Surface of the Third Class, having a tangent-plane touching 

 along a conic (the singularity dualistically corresponding to a 

 double-point of the second order). 



Elling B. Hoist, Stipendiary of the University of 

 Christiania. 



The model is composed of twenty-one wires, six of which, painted light 

 red, lie in the same plane and touch the conic in points painted dark red. 

 Through the fifteen points of intersection of these six lines the others, 

 painted white, pass, again intersecting three and three, and are the lines in 

 which the surface cuts itself. All points on these lines have therefore two 

 tangent-planes ; where the latter are imaginary the lines are black. The 

 black is in part laid on schematically, especially where the black part contains 

 the point at infinity. The parabolic curve consists of the conic aforesaid 

 and two species of cuspidal curves, viz. : 



1. One curve passing the dark red points and having cusps in those six 

 limiting-points between black and white which are nearest to the conic, the 

 curve therefore having a zig-zag course. 



2. Four closed branches having cusps in the other twelve limiting-points. 

 All these parabolic curves together separate ten distinct ellipsoidally 



curved parts from the surface everywhere else hyperboloidally curved. 



124. Models. A series illustrative of Pliicker's Researches 

 in Geometry of Three Dimensions. See explanation No. 123. 



Prof. Hennessy, Dublin. 



126. Model of the ruled cubic surface called the Cylindroid. 



Dr. Robert S. Ball, LL.D., F.R.S. 



This surface was discussed by Pliicker in connexion with the theory of 

 the linear-complex. The kinematical and physical significance of the sur- 

 face will be found in the "Theory of Screws." The equation of the surface 

 is z (x* + ?/ 2 ) '2mxy = O. 



125. Diagrams (48) showing the Fundamental Principles 

 of the exhibitor's " Organic Geometry of Form." 



Prof. Franz Tilser, Prague. 



The above work demonstrates the necessity for a reform in geometry, and 

 furnishes the necessary basis for establishing a new system adapted to satisfy 

 the requirements of an exact science. To the above are added 7 " Paragram " 

 Tablets, representing in natural organic connexion a synopsis of the principal 

 elements to be observed in every graphical representation. 



127. Models (6) illustrating the relative bases of Descriptive 

 Geometry and the Organic Geometry of Form. 



Prof. Franz Tilser, Prague. 

 C 2 



