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NOTE ON CONES OF THE THIRD ORDER. 
together the two radial edges, be converted into a cone of a single sheet; the gene 
rating lines being all finite lines equal in length, the curve formed by the circular 
edge is, it is clear, the spherical curve which is the intersection of the cone by 
a concentric sphere. It is shown by Mobius (stating his result with respect to cones 
instead of spherical curves) that a cone of an odd order must have at least one single 
sheet; a cone of the third order consists (1) of a single sheet, or else (2) of a single 
sheet and a twin-pair sheet. These are the two general forms of cones of the third 
order. But there are two special forms and one subspecial form, making in all five 
forms: viz., the two special forms are, (3) the cone has a nodal line; (4) the cone 
has an isolated line ; and the subspecial form is, (5) the cone has a cuspidal line. The 
relation of the different forms may be explained as follows. 
Starting from the form (1), as the constants of the equation change, the cone 
gathers itself up together so as to have a nodal line; this is the form (3). The 
loops of this form then detach themselves so as to form a twin-pair sheet, the 
remaining part of the surface reverting to a form similar to that of (1); we have 
thus a single sheet and twin-pair sheet, which is the form (2). The twin-pair sheet 
then dwindles away into an isolated line, giving the form (4); and lastly, the isolated 
line disappears and the cone resumes the form (1): these four forms constitute, there 
fore, a complete cycle. The constants may be such that the loop of the form (3) is 
evanescent, or, what is the same thing, that the forms (3) and (4) arise simultaneously; 
there is in this case a cuspidal line, or we have the form (5). It may be added 
that for the general forms (1) and (2) there are always three lines of inflexion. This 
is also the case with the form (4), where there is an isolated line; but in the form (3), 
C. IV. 16