159 
intersection will form a polygon mm'm"..., or rather a con 
tinuous curve to which all the generating lines are tangents. 
Also AN and A'N', and similarly every pair of consecutive 
generating lines, will include a sectorial area Am A' of inde 
finite length, but infinitely small angle, which may be re 
garded as a plane element of the surface. If now the first 
of these elements be turned about its line of intersection with 
the second, till they are in the same plane; and then the 
system formed by these two be turned about the line of 
intersection of the second and third till they are in the same 
plane; and if this operation be continued through all the 
plane elements, they will all thus be brought into one plane, 
and the given surface will be developed without rumpling or 
tearing. 
179. We have already seen (Art. 174) that the plane 
which touches a developable surface in any point M' is the 
tangent plane at every point in the generating line passing 
through M'; and that to construct the tangent plane to any 
point M\ we have only to draw a tangent line M'T to any 
curve on the surface passing through that point, then TM'N' is 
the tangent plane required. 
180. As the generating lines are all tangents to the 
curve mmm"... formed by their perpetual intersection, the 
surface may be supposed to be generated by a moveable 
straight line which is always a tangent to a fixed curve; 
the curve must of course be of double curvature, otherwise 
the surface generated would be a plane. Hence it is suf 
ficient to assign one fixed directrix (to which the generating 
line must be always a tangent) to completely determine a 
developable surface. 
If the equations to the fixed curve be 
® = 0 (*)> V = i' (*)> 
the equation to the surface will result from eliminating a from 
the equations to the line touching the curve at a point for 
which z = a, which (Art. 123) are 
a - <p (a) = <p' (a) (z - a), y - \// (a) = \[/' (a) (z - a) ;