332 



perceived, and printed (in the Elements), the strikingly simple, and 

 yet complete equation, 



I'ap + V/^0/> = 0, 



which represents all twisted cuhics, if only a point of the curve be taken, 

 for convenience, as the origin : 0/^ denoting that linear and vector func- 

 tion of a vector, which has formed the subject of many former studies 

 of mine, and a being a constant vector, while /> is a variable one. 



III. It is known that a twisted cubic can in general be so chosen, 

 as to pass through any six points of space. It is therefore natural to 

 inquire, what is the Osculating Twisted Cubic to a given curve of double 

 curvature, or the one which has, at any given place, a six-point contact 

 with the curve. Yet I have not hitherto been able to learn, from any 

 book or friend, that even the conception of the j)roblem of the determi- 

 nation of such an osculatrix, had occurred to any one before me. Eut 

 it presented itself naturally to me lately, in the course of writing out a 

 section on the application of quaternions to curves ; and I conceive that 

 I have completely resolved it, in three distinct ways, of which two seem 

 to admit of being geometrically described, so as to be understood with- 

 out diagrams or calculation. 



lY. It is known that the cone of chords of a twisted cabic, having 

 its vertex at any one point of that curve, is a cone of the second order, or 

 what Dr. Salmon calls briefly a quadric cone. If, then, a point p of a 

 given curve in space be made the vertex of a cone of chords of that 

 curve, the quadric cone which has its vertex at p, and has five-side con- 

 tact with that cone, must contain the osculating cubic sought. I have 

 accordingly determined, by my own methods, the cone which is thus one 

 locus for the cubic : and may mention that I find fifth differentials to 

 enter into its equation, only through the second differential of the second 

 curvature, of the given curve in space. This may perhaps have not 

 been previously perceived, although I am aware that Mr. Cayley and 

 Dr. Salmon, and probably others, have investigated the problem of five- 

 point contact of a plane conic with a plane curve. 



Y. It is known also that three quadric cylinders can be described, 

 having their generating lines parallel to the three (real or imaginary) 

 asymptotes of a twisted cubic, and wholly containing that gauche curve. 

 My first method, then, consisted in seeking the (necessarily real) direc- 

 tion of one such asymptote, for the purpose of determining a cylinder 

 which, as a second locus, should contain the osculating cubic sought. 

 And I found a culic cone, as a locus for the generating line (or edge) of 

 such a cj^inder, through the given point p of osculation : and proved 

 that of the six right lines, common to the quadric and the cubic cones, 

 three were absorbed in the tangent to the given curve at p. 



YI. In fact, I found that this tangent, say pt, was a nodal side (or 

 ray) of the cubic cone ; and that one of the two tangent planes to that 

 cone, along that side, was the osculating plane to the curve, which plane 

 also touched the quadric cone along that common side: while the same 



