32 SEC. 2. GEOMETKY. 



62 to 73. One principal conic, a pair of parallel lines. 

 74 to 76. Principal conies, each a parabola. 

 77 and 78. Principal conies, one of them a parabola, the other a pair of 



parallel lines. 

 Model 2. The form of the equation is here, 



viz., the principal conies are one of them a hyperbola, the other imaginary ; 

 hence the generating conic has always two, and only two, real vertices, viz., 

 it is always a hyperbola. There are no real lines. 

 Model 3. The form of the equation is 



/ 2 [(;t- a) 2 + /8 2 ] 7 //2[(#_ a ')2 + 0/2-| ~ 



viz., the principal conies are each of them a hyperbola ; the generating conic 

 has four real vertices, viz., it is always an ellipse. There are no real lines. 

 Model 4. The form of the equation is 



+ 1 = 0. 



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 ^ = 7, a; = 5. The surface contains the real lines 

 y r=o, x = y, and y = Q, x = 5. 



Model 9. The form of the equation is 



/(*- 7 )(*-) + /' 2 (* /)<>- 80 + 



where, say the values 7, 5, lie between the values y, 8', the principal conies 

 are each of them an ellipse, the vertices (on the axis or line ,y = 0, 2 = 0) of 

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

 for values of x between 7 and 5 has four real vertices, or it is an ellipse ; for 

 values beyond these limits, but within the limits 7', 8' say, from 7 to 7' and 

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

 for values beyond the limits y, 8', the variable conic is imaginary. 



There are four real lines (y = 0,z = y), (?/ = 0, x = 8), (z = Q, x = y r ), (2 = 0, 

 or = 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, x = 8'. 



Model 13. The form of the equation is 



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

 hyperbola, the other an ellipse, the vertices (on the axis or line z/ = 0, z = 0) 

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

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

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

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

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

 (# = 0, x=y\ (# = 0, x=8~), and (z=0, #=7'), (*=0, # = 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 



r2 



= 1 



