653] 
73 
653. 
ON A TORSE DEPENDING ON THE ELLIPTIC FUNCTIONS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877), 
pp. 235—241.] 
On attempting to cover with paper one half-sheet of the foregoing sextic torse, 
[652], I found that the paper assumed approximately the form of a circular annulus of 
an angle exceeding 360°, and this led me to consider the general theory of the 
construction of a torse in paper, and, in particular, to consider the torses such that 
when developed into a plane the edge of regression becomes a circular arc. It is 
scarcely necessary to remark that, to construct in paper a circular annulus of an 
angle exceeding 360°, we have only to take a complete annulus, cut it along a radius, 
and then insert (gumming it on to the two terminal radii) a portion of an equal 
circular annulus; drawing from each point of the inner circular boundary a half 
tangent, and considering these half-tangents as rigid lines, the paper will bend round 
them so as to form the half-sheet of a torse having for its edge of regression this 
inner boundary, which will assume the form of a closed curve with two equal and 
opposite maxima and two equal and opposite minima, described on a cylinder, and 
being approximately such as the curve given by the equations 
x = cos 6, y = sin 6, z = m cos 26. 
Considering, in general, an arc PQ (without inflexions) of any curve, and drawing 
at the consecutive points P, P', P", &c. the several half-tangents PT, P'T', P"T”,..., 
then, considering these as rigid lines and bending the paper round them, we have 
the half-sheet of a torse, having for its edge of regression the curve in question 
now bent into a curve of double curvature. It is, moreover, clear that the edge 
of regression has at each point thereof the same radius of absolute curvature as the 
original plane curve; in fact, if in the plane curve PP'=ds, and the angle T'PT 
between the consecutive half-tangents PT and P'T' be =d(J), these quantities ds and 
10 
C. X.