24 SEC. 2. GEOMETRY. 



intersection, but the plans of the intersection and of the generating 

 lines. 



121. Helix or Screw-thread. 



Model showing the transformation of the right line genera- 

 tors of a right cylinder into screw threads of various pitch or 

 obliquity. 



The pitch of a screw is the distance between two successive 

 turns, measured in a direction parallel to the axis. When this 

 distance is small, the screw is said to have a fine pitch ; when 

 great, a coarse pitch or high pitch. 



COLLECTION OP MODELS CONTRIBUTED BY THE LONDON 

 MATHEMATICAL SOCIETY. 



123. Flicker's Models (14) of certain quartic surfaces, 

 representing the equatorial form of complex surfaces. 



London Mathematical Society* 



At the meeting of the British Association at Nottingham, in 1866, Prof. 

 Pliicker read a paper on " Complexes of the Second Order." On this occasion 

 he showed a series of models constructed by Epkens, of Bonn, of which the 

 above are copies made for Dr. Hirst, and presented to the London Mathe- 

 matical Society. 



The following is Prof. Cayley's description of the models, extracted from 

 Nos. 37 and 38 of the Mathematical Society's Proceedings, vol. iii., pp. 281- 

 285, supplemented by a description of models A, B, C, D, E, F, drawn up 

 by Prof. Henrici. 



The Society possesses a series of 14 wooden models of surfaces, constructed 

 under the direction of the late Prof. Pliieker, in illustration of the theory 

 developed in his posthumous work " Neue Geometric des Raumes gegriindet 

 " auf die Betrachtung der geraden Linie als Raum-elemente," Leipzig, 1869. 

 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 Axenrichtung 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. 



