1894.] of 27 Straight Lines upon a Cubic Surface. 245 



Art. 2. The form of the surface, except near and within the 

 tetrahedron of reference, approximates very closely to a cone of 

 two sheets. 



I. and XII. shew the shape of two horizontal sections of this 

 cone at equal distances above and below the plane of the lines 

 4, 9, 13. 



XIII. and XIV. are vertical sections through the origin making 

 angles 45° and 150° with the plane of 4, 2, 12. 



One sheet of the surface formed by joining all points on the 

 oval to the origin is in the shape of an ordinary oblique cone. 



The second sheet is tr-aced out by the thick line drawn in 

 XIII. and XIV. If we consider a vertical section through the 

 origin first as passing through 4, and then revolving round axis z, 

 we find this line coincides at first with 4, then traces out a portion 

 of the surface above the plane 4, 9, IS till the section has revolved 

 through an angle 90°, the line now coincides with 13, it then 

 passes below the plane of 4, 9, 13. At 135° the line coincides 

 with 9, 



Between 135° and 180° it is found above the plane of 4, 9, 13. 

 At 180° it coincides with 4. At the same time the line produced 

 backwards through the origin is tracing out similar portions 

 respectively below, above, and below the plane of 4, 9, 13 from 

 180° to 360°. 



The surface only coincides with this cone when the tetrahedron 

 of reference is so small that it may be regarded as a point, but as 

 before stated, the sections of the surface and the cone have very 

 much the same shape except near the tetrahedron of reference. 



In general then, horizontal sections of the surface consist of 

 an oval and three infinite hyperbolic branches touching asymptotes 

 parallel to 4, 9, 13. Vertical sections parallel to the planes of 

 (4, 2, 12), (9, 20, 25), (13, 14, 15), except in the vicinity of the 

 tetrahedron consist of one infinite branch touching one horizontal 

 asymptote (e.g. parallel to 4) corresponding to the second sheet of 

 the surface, and two hyperbolic branches touching asymptotes 

 parallel to the remaining sides (e.g. 2, and 12) caused by intersec- 

 tion with the first sheet of the surface. 



Next to consider the shape of the surface in the vicinity of the 

 tetrahedron of reference. 



Figs, numbered II. to XL represent sections parallel to the 

 plane 4, 9, 13. The right-angled triangle in each figure represents 

 the projection of these lines upon the plane of the section. 



Dotted lines represent asymptotes. 



