Mr. A. Cayley on Cones of the Third Order. 439 



plaue ^^ = with the cones P = 0, Q= respectively. But writing 

 « + Z> v/ — l=A(cos«+ \/— lsin«), 



ic + i/ 'Z — l = r{eos6+ v' — Isin^), 

 we have 



Ar"{cos {ne + cc) + v/^ sin (n^ + «) } = P'o + Q'o v/^, 



so that 



P'o = Ar"cos {nd + u), Q'o=Ar"sin {nd + oc). 



Or the intersections with the cone P = are the n lines given in 

 direction by the equation 



and the intersections with the cone Q=0 are the n hnes given 

 in direction by the equation 



nd + »=■ miT, 



in each of which equations m is any integer number from to 

 n— 1. Hence the plane 2 = meets the cones in two sets of 

 lines succeeding each other alternately, as required by the lemma, 

 and the two cones intersect in at least n lines. And it is thus 

 shown that the given equation of the nth degree has n roots. 



2 Stone Buildings, W.C, 

 September 26, 1859. 



LXVIII. Note on Cones of the Third Order. 

 By A. Cayley, Esq.* 



THE distinction adverted to in the preceding paper between 

 the twin-pair sheets and single sheets of an algebraic cone 

 is made (with respect to spherical curves, which is the same thing) 

 by Mobius, in the interesting memoir "Ueber die Grundformeu 

 derLinien der dritten Ordnung," Abh. derK. Sachs. Ges. zu Leipzig, 

 vol. i. (1849). Consider the generating line POP' of a cone, 

 vertex O, and let joO/j' be any position of this line, the points 

 P, P', and in like manner the points p, p', being on opposite sides 

 of the vertex ; then if OP originally coincides with Op (and there- 

 fore OP' with 0^'), and if, in the course of the generation of the 

 surface, OP (without having first come to coincide with Op') 

 comes to coincide with Op, at the same time OP' (without having 

 first come to coincide with Op) will come to coincide with Op', 

 and we have a twin-pair sheet, viz. one twin- sheet generated by 

 OP, and the other twin-sheet generated by OP'. This is the 

 ordinary case of a cone of the second order, and requires no 

 further explanation. It is proper to remark that, for cones of 

 superior orders, the conical angle of each twin-sheet is not (as 

 ♦ CoinnmnicateJ by the Autlior. 



