III. - MODELS. 33 



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

 always an ellipse. 



There are the two real nodal lines (z/ = 0, #=7) and (2=0, #=/), 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 focal lines. 



Model 34. The equation is 



where x= Sis 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 5 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 5, 7', it is imaginary. There are on the 

 surface the two real lines 0/ = 0, ^ = 7) and (z = 0, x = 7')- The surface 

 consists of a finite portion joined on by two conical points to the remaining 

 portion. 



Model 40. The form of equation is 



y 2 z 2 



zV-7X*-5) + r*&=vy* + 



The surface is thus a cubic surface ; the principal conies are, one of them 

 an ellipse, the other a pair of imaginary lines intersecting on the ellipse ; for 

 values of x between 7 and 5, the variable conic has thus two real vertices, and 

 it is a hyperbola ; for values beyond these limits it is imaginary, and the 

 whole surface is thus included between the planes #=7 and x=8. There are 

 the two real lines (y = 0, x = 7) and (2=0, x=5). 



Taking / 2 =/ /2 = i } the surface is 



y 2 2 2 



Gr-7)(*-5) + (*=W 2 + l = ( 

 which is a particular case of the parabolic cy elide. 



The equatorial surfaces, not included in the preceding 78 cases, Pliicker 

 distinguishes (vol. ii. p. 363) as " gedrehte " or " tordirte," say, as twisted 

 equatorial surfaces, the equation of such a surface is 



by 2 + 2hyz + az 2 + ab - A 2 = 

 where b = Fx 2 - 2R.r + B 



A=K:r 2 Oar G (or in particular = O-r G). 

 Model A. is such a surface, being a twisted form of Model 9. 

 Model B. belongs to the case a = ; viz., the form of the equation is 



The variable conic is a hyperbola, the direction of one of the asymptotes 

 being constant (vol. ii. p. 368). 



There are, moreover, (p. 372) equatorial surfaces in which the variable conic 

 is always a parabola, and where there are on the surface four real or imaginary 

 singular lines. 



In Model C the singular lines are all four real, but two of them coincide with 

 the nodal line at infinity. Consequently, the variable parabola has its axis in 

 a fixed direction. Its vertex moves along a hyperbola which has one asymp- 

 40075. 



