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



441 



there is but one line of inflexion ; and in the form (5), where 

 there is a cuspidal line, there is not any line of inflexion : the 

 equivalent theorem for the spherical curves is given by Mobius*. 

 It was remarked long ago by Sir I. Newton, that all curves of 

 the third order could be generated as the shadows of the five 

 cubical parabolas; these are, in fact, sections in a particular 

 manner of the above-mentioned five forms of cones of the thii'd 

 order : the existence of five essentially distinct forms of cones 

 of the third order is noticed by M. Chasles in the Aperqu His- 

 torique, 1837. The analytical distinction between the forms (1) 



64S3 

 and (2) depends on the sign of the function 1 =ir> where S, T 



are the quartinvariant and sextinvariant of the cubic form. I 

 annex stereoscopic representations of the cones of the third order 



of the general form (l),and of the form with a nodal line (3). The 



* It is liardly necessai-y to mention that, according to the general theory 

 of cones of the tliinl order, there are always nine lines of inflexion,— three 

 real and six imaginary. Six of the lines of inflexion disappear when there 

 is a double line, viz., in the case of a nodal line, two real and four imaginary 

 lines of inflexion ; hut in the case of an isolated line, the six imaginary 

 lines of inflexion. When there is a cuspidal line, eight lines of inflexion, 

 viz. two real lines and the six imaginary lines, disappear. 



