191 



contains the normal. And, as it has been proved already that 

 the quantity PD is the same for both of them, we are now in 

 possession of a proof that, in passing from one element to the 

 adjacent one, along a geodetic line traced on a central surface 

 of the second order, the quantity PD remains unaltered. Let 

 us next inquire what becomes oi L and L' in the extreme case 

 under consideration. These lines may be regarded as ultimately 

 the elements of two lines making equal angles with the lines 

 of curvature passing through V. We have, therefore, the 

 theorem that if two right lines touch a surface of the second 

 order at the same point, making there equal angles with 

 a line of curvature passing through it, the quantity PD is 

 the same for both. And, as a particular case, we have the 

 theorem that, in passing from one element to the adjacent one, 

 along a line of curvature traced on a central surface of the 

 second order, the quantity PD remains unaltered. 



Returning now to the case in which the vertex of the cone 

 is supposed to be at a finite distance from the surface, we see 

 that the four sides, L, L\ L'\ U\ are tangents to geodetic lines 

 for which the quantity PD is the same, and which, therefore, 

 touch the same line of curvature; and conversely, if two rec- 

 tilinear tangents to the same geodetic line, or to geodetic 

 lines for which the quantity PD is the same, intersect each 

 other, they make equal angles with the axes of the cone which, 

 from this point of intersection, envelopes the surface. 



When the enveloping cone becomes a right one, we have 

 PL or PD the same for all its sides ; the geodetic lines to 

 which they are tangents must, therefore, all converge to the 

 same umbilic. 



From what has been already stated, it is easy to deduce 

 the following theorems : 



If a closed flexible and inextensible cord be kept stretched 

 by a style, being partly in free contact with a surface of the 

 second order along a geodetic line, and partly free in space. 



