III. MODELS. 31 



These, all of them, represent, I believe, equatorial surfaces, viz., eight repre- 

 sent cases of the 78 forms of equatorial surfaces, " deren Breiten-Curven 

 " eine feste Axenrichtuug besitzen," vol. ii. pp. 352-363 ; the remaining 

 models, A, B, C, D, E, F, I have not completely identified. I propose to go 

 into the theory only so far as is required for the explanation of the models. 



In a " complex," or triply infinite system of lines, there is, in any plane 

 whatever, a singly infinite system of lines enveloping a curve ; and if we 

 attend only to the curves the planes of which pass through a given fixed line, 

 the locus of these curves is a " complex surface." Similarly, there is through 

 any point whatever a single infinite series of lines generating a cone ; and if 

 we attend only to the cones which have their vertices in the given fixed line, 

 then the envelope of these cones is the same complex surface. In the case 

 considered of a complex of the second degree, the curves and cones are, each 

 of them, of the second order ; the fixed line is a double line on the surface, 

 so that (attending to the first mode of generation) the complete section by 

 any plane through the fixed line is made up of this line twice, and of a conic. 

 The surface is thus of the order 4 ; it is also of the class 4 ; the surface has, 

 in fact, the nodal line, and also 8 nodes (conical points), and we have thus 

 a reduction = 32 in the class of the surface. 



In 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, //., to be homogeneous functions 

 of (r, w>) of the order 2 ; f, <jr, of the order 1 ; and c of the order (a constant) ; 

 then the equation of a complex surface is 



y z 1 1=0; 

 y a h g 

 z k b f 



1 9 f 



and that, writing w?= 1, or considering a, h, b; f, y c, as functions of x of 

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



A particular form of equatorial surface is thus, bcy~ + caz" + ab = Q, or 

 taking c= 1, this is fy/ 2 + a* 2 + a6 = 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 = Q is made up of 

 the pair of lines a=0, and of the conic z 2 + &=0 ; 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 pi'incipal conies respectively. And we 

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

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



Ex- + 2U.r + C + For 2 2RrTl3 + l '' 

 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. 



