50 SCIENTIFIC APPARATUS. 



curve. It will be found that these tangents all form a developable 

 surface, and that the twisted curve is an edge-curve or cuspidal line 

 upon the surface. This edge-curve is characteristic of a deve- 

 lopable : in the cone it dwindles to a point, and in the cylinder 

 this point lies at an infinite distance. Since a developable surface, 

 when made to roll on a plane or on another developable surface, 

 has a line of contact with it, whereas in general two surfaces made 

 to roll on one another have only a point of contact, it is easy to 

 see that these surfaces are of great importance in the arts of con- 

 struction. But, in a theoretical point of view, it is even more 

 important to notice what is rendered visible by any model of such 

 a surface : (i) that at each point of the surface one of the two 

 curvatures is infinite, the generating line being always one of the 

 lines of curvature ; (2) that, whereas in other surfaces each tan- 

 gent plane has an infinite number of tangent planes lying near to 

 it (because we can travel from any point on the surface to an 

 adjacent point in an infinite number of different directions), in 

 the case of developable surfaces this is not so, but each tangent 

 plane is preceded and followed by only one other tangent plane ; 

 these planes, in fact, forming a singly, instead of a doubly, infinite 

 series. It follows from this, that in the duality of space there 

 answers to a developable surface a curve line, whereas to any 

 surface not developable there answers a surface not developable. 

 As an example, easy to understand and to remember, of a de- 

 velopable surface, we may mention the " Developable Helixoid " 

 of M. Fabre de Lagrange. 



Most of the models of ruled surfaces to which we have referred 

 are so arranged as to be capable of deformation ; i.e., their shape 

 can be changed by altering the form, or the relative position, of 

 the director curves or straight lines, which serve to regulate the 

 motion of the generating straight line. Thus the same model is 

 rendered capable of assuming in succession the forms of several 

 different surfaces ; and the study of the transformations by which 

 we pass from one of these forms to another is of great interest and 



