440 Mr. A. Cayley on Cones of the Third Order. 



for a cone of the second order) necessarily less than 360°. But 

 suppose that OP, starting from the position Op, and before it 

 again comes to coincide therewith, comes to coincide with Op', 

 then at the same time OP' (without having first come again to 

 coincide with Op') will come to coincide with Op ; the generation 

 is here complete, and we have a sinffle sheet, which, if the motion 

 were continued until OP came to coincide with Op, would only 

 be generated over again. The conical angle of a single sheet is 

 necessarily greater than 360°; for OP in coming to coincide 

 with Op' must describe an angle greater than 180°, and OP' de- 

 scribing an equal angle, the entire angle is therefore greater than 

 360° ; in the limiting case, where the entire angle is precisely 

 360°, the conical surface is a plane. It is easy to cut out in 

 paper and join together two sectors of a circle so as to form 

 therewith a sector the angle whereof exceeds 360° ; such a sector 

 can then, by joining together the two radial edges, be converted 

 into a cone of a single sheet ; the generating 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 (I) : 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). where there is a nodal line. 



