158 
It may be observed that the various other modes of gene 
rating this sort of surface, as for instance when we replace one 
or more of the directrices by a surface which the moveable 
line is always to touch, are reducible to this method; for in 
every case we may take for the three directrices, any three 
sections of the generated surface made at pleasure. 
Cor. To obtain the differential equation independent 
of the directrices, we must, as before, obtain by successive 
differentiations of the above system a sufficient number of 
equations to eliminate a, and the functions tp, \js, 7r and their 
derived functions; the result is a complicated equation of 
the third order. 
When in the two preceding cases, the directrices all 
become straight lines, the surfaces generated become re 
spectively the hyperbolic paraboloid, and the hyperboloid 
of one sheet, as is seen in the Appendix ; they are the only 
twisted surfaces whose equations do not rise above the second 
degree. 
Developable Surfaces. 
178. We next come to the consideration of the second 
class of surfaces which admit of being generated by a straight 
line, the characteristic law of the motion being that two 
consecutive positions are always in the same plane. Before 
proceeding to point out the various modes in which this 
condition may be satisfied, we shall shew that surfaces ge 
nerated in this manner are developable; that is, supposing 
them flexible but inextensible, they may without rumpling 
or tearing be made to coincide with a plane in all their 
points. 
Let fig. 57 represent a surface of this sort, and let AN, 
A'N', A'N", &c. be positions of the generating line inde 
finitely near to one another; then from the definition of the 
surface, AN will be intersected by A'N' in some point m, 
A'N' by A"N" in m, and the latter by the next generating 
line in m'\ and so on ; so that these successive points of