74 
ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS. 
[653 
dcf> remain unaltered in the curve of double curvature; and the radius of absolute 
curvature is given by the equation pdcf) = ds. In particular when, as above, the arc 
is a circular one, say of radius = a, then, however the paper is bent, the edge of 
regression has at each point thereof the radius of absolute curvature = a. 
Consider on any given surface, at a given point P thereof, and in a given 
direction, an element of length PP', then (under the restrictions presently mentioned) 
we can determine the consecutive element P'P", such that the curve PP'P”... shall 
have at P a radius of absolute curvature = a; in fact, r being the radius of 
-curvature of the normal section of the surface through the element PP', the radius 
of curvature of the section inclined at an angle 6 to the normal section is = r cos 6; 
so that we have only to take the section at the inclination 6, = cos -1 - to the 
normal section, and we have the consecutive element P'P" such that the radius of 
absolute curvature of the curve PP'P" is =a. The necessary restriction, of course, is 
that r> a; thus, if at the given point P the two principal radii of curvature are 
of the same sign (to fix the ideas, let the two principal radii and also a be each 
of them positive), then we may on the surface determine a direction PQ, for which 
the radius of curvature of the normal section is = a; and then the direction of the 
element PP' may be any direction between PQ and the direction PR, corresponding 
to the greatest of the two principal radii. 
Having obtained the element P'P", we may, if the radius of absolute curvature 
-at P' be given, construct the next element P'P", and so on; that is to say, on a 
given surface starting from a given point P and given initial direction PP', we can 
(under a restriction, as above, as to the curvature at the different points of the 
surface) construct a curve having at the successive points thereof given values of the 
radius of absolute curvature; viz., the value may be given either as a function of 
the coordinates of the point on the surface, or as a function of the length of the 
curve measured say from the initial point P; it is in this last manner that in what 
follows the value of the radius of absolute curvature is assumed to be given. 
We may thus, taking on paper an arc PQ with its half-tangents, apply it to a 
given surface, the point P to a given point, and the infinitesimal arc PP' to an 
element PP' in a given direction from the given point; and we thus obtain the 
half-sheet of a torse having for its edge of regression a determinate curve upon the 
surface. In particular, the arc PQ may be circular of the radius a, and the surface 
be a circular cylinder of radius a ; and we thus obtain the torse having for edge of 
regression a curve on the cylinder radius a, and such that the radius of absolute 
curvature is at each point = a. There are three cases according as a > a, a = a, 
or a <a; it is to be remarked that if a > a, then the curve must at each point 
cut the generating line of the cylinder at an 
angle not exceeding cos -1 -, but that 
in the other two cases the angle may have any value whatever; and, further, that 
in every case when the angle is = 0, viz. when the curve touches a generating line 
of the cylinder, then the osculating plane of the curve coincides with the tangent 
plane of the cylinder.