246 3/r Blythe, On the Constymction of a model [Oct. 29, 



In II. the section is above the plane 4, 9, 13, and when moved 

 a little nearer to this plane is as in III. ; it is found that a node 

 has taken place at the point marked by the asterisk and the oval 

 exists no longer but has become part of one of the hyperbolic 

 branches. 



The changes are easily traced as the section moves nearer to 

 4, 9, 13. An asterisk is used to indicate that a node is about to 

 take place. 



In X. the section coincides with 4, 9, 13. 



The oval again appears for any section below this plane, 

 as in XI. 



The equation of the surface in Cartesian coordinates can be 

 found. For if we denote the equation to the plane 4, 9, 13 by 

 (4, 9, 13)... the equation to the surface becomes 



if (4, 12, 2) (13, 14, 15) (9, 7, 8) = (4, 9, 13) (12, 7, 15) (2, 14, 8). 



Take 4 and 13 as axes x and y, using rectangular coordinates 

 We can either by measuring the intercepts of these planes on the 

 axes or the coordinates of three points in each plane determine 

 the equations to the planes. M can then be found by substituting 

 in the equation the coordinates of any point on any other of the 

 twenty-seven straight lines. 



If we take an edge of the tetrahedron in 4 or 13 as being 

 12^ inches the equation to the surface in the case under con- 

 sideration becomes — approximately — 



The equation to the cone to which the surface approximates is 



mxy{x^-y + m)z)=z{Ux + \Qy + {ill)z){lOx+lAy-{l2^)z). 



It will be observed that in either case the equation to any 

 section found by putting z constant reduces to the form 



{x — A){y — B){x + y — G) + terms of the first degree = 0, 



which shews that the asymptotes are parallel to 4, 9, 13. A and 

 B vary with z, and the sum of A, B, C, in the case of the surface 

 is very nearly 12^, unless z is very large, and in the case of the 

 cone is nearly zero. 



It is interesting to trace the points in which the twenty-seven 

 straight lines intersect the different sections, and how nodes take 

 place to allow of the intersection of lines on different sheets and 

 branches with each other. Space does not permit of these being 

 given. 



