8 Mr Blythe, On the forms of [Nov. 11, 



A case of great interest is that in which one of the tangent 

 planes to the surface is at an infinite distance. 

 The general equation in Cartesians is 



xyz = Ax 2 + By- + Gz 2 + 2Fx + 2Gy + 2Hz + K. 



The sections parallel to the coordinate planes are conies. 



Four of these sections in each case become pairs of straight 

 lines and two are parabolas. It is clear that we get twenty-seven 

 straight lines, namely three sets of four pairs together with three 

 at infinity. 



Examining the meaning of this as regards the model we find 

 that the three openings become infinitely elongated, we therefore 

 get a central solid joined to four infinite solid conical shaped 

 branches. These are so placed that a plane cannot cut both the 

 central solid and the four branches at the same time. 



A correct idea of the shape of this model may be obtained by 

 placing four cones with their vertices at the angular points of a 

 tetrahedron, their axes being in the line joining these points to 

 the centre of the tetrahedron. If this construction be made in 

 wood or wire it can be covered over with wax or plaster of Paris to 

 shew the surface exactly. 



If the equation is 



xyz = Ax 2 + By- + Gz 2 + K, 



we still get four sections parallel to the coordinate planes contain- 

 ing straight lines, but in each case two pairs consist of parallel 

 straight lines and the parabolic sections do not appear, for they 

 have coincided with them. 



The symmetrical surface mentioned by Mr H. M. Taylor having 

 four conical points, the equation to which is 



x 2 + y 2 + z 2 + 2xyz -1=0, 



is here given. [Plate II.] 



A, B, G, D are the conical points and are angular points of a 

 regular tetrahedron, a, b, c are the middle points of BG, GA, AB. 

 If we put x, y, or z = we get circular sections, each bisecting four 

 edges of ABGD. 



Portions of these are shewn passing through a, b ; b, c ; c, a. 



Next the curve passing through A and a is a parabola which 

 also goes through D, having its vertex at a. 



All sections at right angles to this parabola (that are also 

 parallel to AD) are conies, and the extremities of their minor or 

 major axes are upon it. 



There is another parabola passing through B, G and the middle 

 point of AD at right angles to this one, and is such that if the 

 major axis of a section has its extremities on the first parabola, 

 the extremities of its minor axis are on the second. 



