43 SCIENTIFIC APPARATUS. 



gents/' as they are called by the German writers, are represented 

 by the rectilinear generators ; on surfaces of the third order we 

 have the simplest examples of the general case in which they are 

 not straight, but curved. The eye can just recognise these curves 

 upon a surface, though of course in an approximate manner. The 

 two which pass through any point are equally inclined to the lines 

 of curvature at that point; they have a theoretical importance, 

 even greater than that of the lines of curvature, because their 

 definition is graphical, and not metrical. 



(c.) As a general rule, three lines of curvature pass through an 

 umbilic. On the surfaces of the second order this property of the 

 umbilics does not exist; and the first example of it occurs on 

 surfaces of the third order. It is clearly seen in Professor Hen- 

 rici's model of the cubic surface x y z = / 3 (x -f- y -f z i) 3 . 



(d.) A model of a surface of the third order ought to exhibit to 

 the eye one of the most characteristic properties of these surfaces; 

 i.e. that of containing a certain finite number of straight lines. The 

 maximum number of these lines is twenty-seven ; and the model 

 exhibited by Dr. Wiener has this maximum number. A model, 

 showing the distribution in space of the lines themselves, unac- 

 companied by the surface on which they lie, has been constructed 

 by Professor Henrici. 



In the model (c] to which we have just referred, the twenty-seven 

 lines are all real, but are coincident in sets of nine ; so that there 

 appear to be three only. 



(3.) Models of ruled, or rectilinear surfaces. 



Ruled, or rectilinear surfaces are those which may be generated 

 by the continuous motion of a straight line in space, and which 

 therefore may be said to consist of an infinite number of straight 

 lines. Models of these surfaces exist in great variety, because 

 they can be constructed with threads or wires, instead of being 

 carved out of a solid material, or moulded out of a plastic one. 

 They are, of course, only approximate or diagrammatic, the gene- 

 rating lines being represented in such number only as may suffice 



