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556. 
ON STEINER’S SURFACE. 
616, 620 
[From the Proceedings of the London Mathematical Society, vol. v. (1873—1874), 
pp. 14—25. Read December 11, 1873.] 
I have constructed a model and drawings of the symmetrical form of Steiner’s 
Surface, viz. that wherein the four singular tangent planes form a regular tetrahedron, 
and consequently the three nodal lines (being the lines joining the mid-points of 
opposite edges) a system of rectangular axes at the centre of the tetrahedron. Before 
going into the analytical theory, I describe as follows the general form of the surface : 
take the tetrahedron, and inscribe in each face a circle (there will be, of course, two 
circles touching at the mid-point of each edge of the tetrahedron; each circle will 
contain, on its circumference at angular distances of 120°, three mid-points, and the 
lines joining these with the centre of the tetrahedron, produced beyond the centre, 
meet the opposite edges, and are in fact the before-mentioned lines joining the mid 
points of opposite edges). Now truncate the tetrahedron by planes parallel to the 
faces so as to reduce the altitudes each to three-fourths of the original value, and 
from the centre of each new face round off symmetrically up to the adjacent three 
circles; and within each circle scoop down to the centre of the tetrahedron, the 
bounding surface of the excavation passing through the three right lines, and the 
sections (by planes parallel to the face) being in the neighbourhood of the face nearly 
circular, but as they approach the centre, assuming a trigonoidal form, and being close 
to the centre an indefinitely small equilateral triangle. We have thus the surface, 
consisting of four lobes united only by the lines through the mid-points of opposite 
edges, these lines being consequently nodal lines; the mid-points being pinch-points 
of the surface, and the faces singular planes, each touching the surface along the 
inscribed circle. The joining lines, produced indefinitely both ways, belong as nodal 
lines to the surface; but they are, outside the tetrahedron, mere acnodal lines not 
traversed by any real sheet of the surface. 
C. IX. 1