166 
polygon pp'p" &c. Then in turning the first plane element 
about A'm to bring it into the same plane with the second, 
and in turning the system formed by the two about A 'm to 
bring it into the same plane with the third, and so on, it is 
evident that neither the lengths of PN, PN', Sic., nor the 
angles which they form with PP’, P'P", &c., nor the lengths 
of the latter, are altered, but that the angles RP R', RP'R', 
&c. will be changed ; and as this continues true, however 
small we take the plane elements, the properties announced 
above for the surface and the curve traced upon it, are 
established. 
189. The shortest line on a developable surface has the 
property that its osculating plane at every point is perpen 
dicular to the plane touching the surface at that point, or 
contains the normal to the surface at that point. 
If the polygon be such that, upon bringing all the plane 
elements into the same plane, it becomes a straight line, then 
two consecutive sides must always make the same angle with 
the intermediate generating line ; that is, for every point we 
must have 
A m P'R = m P R'; 
for when this condition is fulfilled, in the developed surface 
every two consecutive sides will be the prolongment of one 
another. Also since RP', R'P', make equal angles with 
P'N', they may be regarded as two generating lines of a 
conical surface whose axis is P'N', and therefore the plane 
RP'R' will ultimately be the tangent plane to this surface 
along P’R, and therefore perpendicular to RP'N', which is a 
meridian plane of the cone containing P'R; hence, passing 
to the curve, the plane RP'R' is its osculating plane at P', 
and RP'N' is the tangent plane to the surface at the same 
point, and these planes are perpendicular to one another; 
also since, when the surface is developed, the curve becomes 
a straight line, it is the shortest line which can connect two 
points on the surface through which it passes, and it has the 
property announced at every point.