34 SEC. 2. GEOMETRY. 



tote in that fixed direction. The other two singular lines are on opposite sides 

 of this asymptote and parallel to it. When the plane of the variable parabola 

 passes through one of these lines, the parameter vanishes and changes sign. 

 When it passes through the above-mentioned asymptote, the parabola, reduces 

 to the line at infinity and the plane becomes asymptotic to the surface. The 

 latter consists of four parts, two on opposite sides of the asymptotic plane 

 between this and one of the singular lines respectively, the other two extending 

 from the singular lines to infinity. 



The remaining three models, D, E, F, represent twisted surfaces. Of the 

 four singular lines two are in each case imaginary. The remaining two are 

 real on the first, coincident on the second, and imaginary on the third. 

 Model D consists, therefore, of three, Model E of two, and Model F of one part. 



The models are copies from some constructed by Epkens of Bonn. They 

 were presented to the London Mathematical Society by Dr. Hirst, F.ll.S. 

 They have been remounted under the direction of Prof. Henrici, by M. Nolet, 

 u student of University College, London. 



Some account of complexes and complex surfaces will be found in Dr. 

 Salmon's Geometry of Three Dimensions (3rd edition, pp. 405, 493, 566, 570). 



123a. Hough Model of Steiner's Surface. 



Prof. Cayley. 



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

 Vx+ \'y + Vz + \/w*=Q ; 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, y, 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 trinodal 

 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 are in each case the intersections of the plane by the nodal 

 lines, and the twice-repeated circle is the circle inscribed in the face of the 

 tetrahedron. 



123b. Model of a Cubic Surface. 



Prof. O. Henrici, F.R.S. 



