161 
of infinite length, we may regard these elements as forming 
parts of the successive positions of a plane subject to move 
according to a given law. This method will usually be found 
the most convenient in practical applications; and it leads 
easily to the general differential equation to developable 
surfaces, and to other results, which we shall now obtain by 
means of it. 
184. To find the general equation to developable sur 
faces, considered as generated by the consecutive intersections 
of the positions assumed by a plane subject to move after 
a given law. 
The law according to which the plane moves will vary 
with each particular surface; but in order that the motion 
may be completely determined, and that a single surface 
and not an infinite number of surfaces may be generated, 
it must leave only one arbitrary parameter in the equation 
to the moveable plane. The successive positions assumed 
by the plane by virtue of the infinitely small variations of 
the parameter, will cut one another consecutively in straight 
lines, which taken two and two are in the same plane; 
these lines will therefore form a developable surface touched 
by all the planes. Hence the law of succession of the planes 
which generate a developable surface requires that two of 
the three arbitrary constants which enter into the equation 
to a plane should be functions of the third, or, which is the 
same thing, that they should all be functions of the same 
parameter a; let therefore 
* = Xf t> («) + Vf(o) + 0 («) 
be the equation to the generating plane in one of its positions 
depending upon the parameter a ; then the equation to the 
plane, which differs insensibly from it in position, will be 
* = at i0(«) + 0'(a) • + &c.} + y{/(a) + /' (a) . Set + &C.j 
+ 0 (a) + \js' (a) . S a + &c.; 
and the co-ordinates of the points in which they intersect 
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