28 SEC. 2. GEOMETRY. 



Professor Cuyley's paper is printed in vol. iii. of the London Mathematical 

 Society's Proceedings, pp. 281-285. 



123a. Bough Model of Steiner's Surface. 



Prof. Caylcij. 



Steiner's surface is the quartic surface represented by the equation 

 A/OT+ A / y+ Vz+ Vw = ; where the co-ordinates x, y, z,w of a point are 

 proportional to arbitrary multiples of the perpendicular distances from four 

 given planes ; in the model, x, #, z, w are proportional to the perpendicular 

 distances from the faces of a regular tetrahedron, the co-ordinates being 

 positive for a point inside the tetrahedron. 



The surface may be regarded as inscribed in the tetrahedron, touching each 

 face along the circle inscribed in the face. The general form is that of the 

 tetrahedron with its summits rounded off, and with the portions within the 

 inscribed circles scooped away down to the centre of the tetrahedron, in such 

 wise that the surface intersects itself along the lines drawn from the centre to 

 the mid- points of the sides (or what is the same thing, the lines joining the 

 mid-points of opposite sides). The lines in question produced both ways to 

 infinity are nodal lines of the surface, but as regards the portions outside the 

 tetrahedron, they are acnodal lines, without any real sheet through them ; and 

 these portions of the lines are not represented in the model. 



The sections by a plane parallel to a face of the tetrahedron are trenodal 

 quartics, which (as the' position of the plane is varied) pass successively 

 through the forms : 



1. Four acnodes. 



2. Trigonoid, with three acnodes. 



3. Tricuspidal. 



4. Trifoliate, with three crunodes, cis-centric. 



5. Do. with triple point at centre. 



6. Do. with three crunodes, trans-centric. 



7. Twice-repeated circle. 



The three nodes being in each case the intersections of the plane by the nodal 

 lines, and the twice-repeated circle being the circle inscribed on the face of 

 the tetrahedron. 



123b. Model of a Cubic Surface. 



Prof. O. Henrici, F.fi.S. 



The equation to this surface is xyz = k* (x+y + z 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 jdrawn on the model. 



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

 the ninth order. Prof. 0. Ilenrici, F.R.S. 



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

 ninth degree. 



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

 in Geometry of Three Dimensions. Prof. Henncssy, Dublin. 



125. Diagrams (48) showing the Fundamental Principles 

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



Prof. Franz Tilser, Prague. 



