111. MODELS. 25 



Iii the particular case where the nodal line is at infinity, the complex 

 surface becomes an equatorial surface ; viz. (attending to the first mode of 

 generation), we have here a series of parallel planes each containing a conic, 

 and the locus of these conies is the equatorial surface. 



It is convenient to remark that, taking a, b, h, to be homogeneous functions 

 of O, w} of the order 2 ;/, $, of the order 1 ; and c of the order (a constant) ; 

 then the equation of a complex surface is 



y a h g 

 z h b f 



0; 



and that, writing w=l, or considering a, h, b ; /, g ; r, as functions of ^ of 

 the orders 2, 1, respectively, we have an equatorial surface. 



A particular form of equatorial surface is thus, bey 2 j-caz 2 + ab = 0, or 

 taking c= 1, this is by 2 + az 2 + ab = 0, where a, b, are quadric functions of x. 



The surface is still, in general, of the fourth order ; it may, however, 

 degenerate into a cubic surface, or even into a quadric surface ; the last case 

 is, however, excluded from the enumeration. The section by any plane 

 parallel to that of yz is a conic ; the section by the plane y is made up of 

 the pair of lines a=0, and of the conic z 2 + b=Q ; that by the plane z=0 is 

 made up of the pair of lines 6=0, and of the conic # 2 + a = ; the last-men- 

 tioned planes may be called the principal planes, and the conies contained in 

 them principal conies. The surface is thus the locus of a variable conic, the 

 plane of which is parallel to that of yz, and which has for its vertices the 

 intersections of its plane with the two principal conies respectively. And we 

 have thus the particular equatorial surfaces considered by Pliicker, vol. ii. 

 pp. 346-363 (as already mentioned), under the form 



H 



and of which he enumerates 78 kinds, viz.: these are 

 1 to 17. Principal conies, each proper. 

 18 to 29. One of them a line-pair.. 

 30 to 32. Each a line-pair. 



33 to 39. Principal conies, each proper, but having a common point. 

 40 to 43. One of them a line-pair, its centre on the other principal conic. 

 44 to 61. One principal conic, a parabola. 

 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 



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 



if _z*_ 



+ * a ~* + 



