26 SEC. 2. GEOMETRY. 



The principal conies are one of them ; an ellipse, the other imaginary ; for 

 values of x between y and 8, the variable conic has two real vertices, or it is 

 a hyperbola ; for any other values it is imaginary, so that the surface lies 

 wholly between the planes z = y, x = S. The surface contains the real lines 

 y = o, z=y, and y = 0, x = 8. 



Model 9. The form of the equation is 



y 2 z 2 



(*- 7 )(*-) + *"(*-70(*-0 + 1 = 



where, say the values 7, 8, lie between the values y, 8', the principal conies 

 are each of them an ellipse, the vertices (on the axis or line y = 0, z = 0) of 

 the one ellipse lying between those of the other ellipse. The variable conic 

 for values of x between y and 8 has four real vertices, or it is an ellipse ; for 

 values beyond these limits, but within the limits Y, 8' say, from y to y' and 

 from 8 to 8' there are two real vertices, or the conic is a hyperbola ; and 

 for values beyond the limits y', 5', the variable conic is imaginary. 



There are four real lines (y = 0, x = 7) , (y = 0, x = 8), (z = 0, x = 7') , (z = , 

 ar = 8 / ). The surface consists of a central pillow-like portion, joined on by 

 two conical points to an upper portion, and by two conical points to an under 

 portion, the whole being included between the planes x=y t x = $'. 



Model 13. The form of the equation is 



+ 1 = 0; 



the values y', 8', lying between 7, S ; the principal conies are one of them a 

 hyperbola, the other an ellipse, the vertices (on the axis or line # = 0, 2 = 0) 

 of the hyperbola lying between those of the ellipse. The variable conic, for 

 values of x between 7', 8', has two real vertices, or it is a hyperbola ; for 

 the values, say, from y' to 7, and S' to 8, there are four real vertices, or 

 the conic is an ellipse ; for values beyond the limits 7, 8, there are two 

 real vertices, and the conic is a hyperbola. There are the four real lines 

 (y=0, x = y), (y = 0, a;=S), and (z=0, ar=7 / ), (z=0, ar=8 / ). The surface 

 consists of eight portions joined to each other by eight conical points, but the 

 form can scarcely be explained by a description. 

 Model 32. The form of the equation is 



^- ? ^;: i ' 



viz., the principal conies are each of them a line-pair, the variable conic is 

 always an ellipse. 



There are the two real nodal lines (?/ = 0, x=y} and (z=0, #=7'), each of 

 these being in the neighbourhood of the axis crunodal, and beyond certain 

 limits acnodal ; the surface is a scroll, being, in fact, the well-known surface 

 which is the boundary of a small circular pencil of rays obliquely reflected, 

 and consequently passing through two fbca .ines. 



Model 34. The equation is 



+ 



where a: =8 is not intermediate between the values x=y and x y'; say the 

 order is 8, 7, 7'. The surface is thus a cubic surface ; the principal conies 

 are ellipses, having on the axis a common vertex, at the point x= S, and the 

 remaining two vertices on the same side of the last-mentioned one. The 

 variable conic for values between S and 7 has four real vertices, or it is an 

 ellipse ; for values between 7 and 7' two real vertices, or it is a hyperbola ; 

 and for values beyond the limits 8, y f , it is imaginary. There are on the 

 surface the two real lines (z/ = 0, # = 7) and (2 = 0, # = 7'). The surface 



