III. MODELS. 27 



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

 portion. 

 Model 40. The form of equation is 



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 xy and x=S. There are 

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



Taking Z 2 =/' 2 = 1, the surface is 



which is & particular case of the parabolic cyclide. 



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

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

 equatorial surfaces, the equation of such a surface is 



2 + 2hyz + az 2 + ab h?= 



- 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- 

 tote in that fixed direction. The other two singular lines are on opposite sides 

 of this asymptote and parallel to it. When the plane of the variable parabola 

 passes through one of these lines, the parameter vanishes and changes sign. 

 When it passes through the above-mentioned asymptote, the parabola reduces 

 to the line at infinity and the plane becomes asymptotic to the surface. The 

 latter consists of four parts, two on opposite sides of the asymptotic plane 

 between this and one of the singular lines respectively, the other two extending 

 from the singular lines to infinity. 



The remaining three models, D, E, F, represent twisted surfaces. Of the 

 four singular lines two are in each case imaginary. The remaining two are 

 real on the first, coincident on the second, and imaginary on the third. 

 Model D consists, therefore, of three, Model E of two, and Model F of one part. 



The models are copies from some constructed by Epkens of Bonn. They 

 were presented to the London Mathematical Society by Dr. Hirst, F.R.S. 

 They have been remounted under the direction of Prof. Henrici, by M. Nolet, 

 a student of University College, London. 



Some account of complexes and complex surfaces will be found in Dr. 

 Salmon's Geometry of Three Dimensions -(3rd edition, pp. 405, 493, 566, 570). 



