GEOMETRICAL INSTRUMENTS AND MODELS. 49 



to convey to the eye an accurate impression of the form of the 

 surface. Rectilinear surfaces are of two very different kinds, 

 being termed skew, or developable, according as the successive 

 generating lines intersect or not. Of skew surfaces the hyper- 

 boloid may be taken as an example : it may be defined as the 

 surface generated by the motion of a straight line, which moves 

 so as always to intersect three fixed straight lines which do not 

 meet in space. In the hyperbolic paraboloid the generating line 

 always intersects two fixed straight lines, and is always parallel 

 to a fixed plane : this surface is the simplest example of the family 

 of skew surfaces called conoids. The series of M. Fabre de La- 

 grange contains several models of conoidal surfaces ; they are all 

 generated by the motion of a straight line, which (i) continually 

 remains parallel to a fixed plane, and which also (2) continually 

 intersects a fixed straight line, and (3) some other fixed line in 

 space. The surface of the thread of a square cut screw (or, more 

 precisely, the surface formed by drawing lines from all the points 

 of a screw curve perpendicular to the axis of the screw), affords a 

 familiar instance of a conoidal surface. In the " Skew Helixoid " 

 of M. Fabre de Lagrange the lines drawn from the points of the 

 screw curve to the axis are not perpendicular to the axis, but are 

 inclined to it at a constant angle. The recent progress of geo- 

 metry has led to a careful study of the skew surfaces of the third, 

 fourth, and fifth orders : of some of these, models have been 

 already made ; one of the cubic surface, called the cylindroid, 

 is exhibited by Dr. Ball, Royal Astronomer of Ireland. 



Developable surfaces form a class of surfaces entirely sui generis. 

 They are called developable because, if such a surface consist of a 

 flexible and inexlensible membrane, it can be "developed," or 

 flattened out, upon a plane, without any tearing or crumpling. 

 Cones and cylinders are the simplest instances of developable 

 surfaces, but they are far from giving a complete idea of the 

 general character -of these formations. A more typical instance 

 may be obtained by considering all the lines tangent to any twisted 



