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side was to be counted a third time, as being a line of intersection, namely, 

 of the quadric cone with the second hranch of the cubic cone, the tangent 

 plane to which branch was found to cut the first branch, or the quadric 

 cone, or the osculating plane to the curve, at an angle of which the tri- 

 gonometric cotangent was equal to half the differential of the radius of 

 second curvature, divided ly the differential of the arc of the same given 

 curve. 



YII. It might then have been thus expected that a cubic equation 

 could be assigned, of an algebraical form, but involving fifth differentials 

 in its coefficients, which should determine the three planes, tangential to 

 the curve, which are parallel to the three asymptotes of the sought 

 twisted cubic : and then, with the help of what had been previously 

 done, should assign the three quadric cylinders which wholly contain that 

 cubic. 



YIII. Accordingly, I succeeded, by quaternions, in forming such a 

 cubic equation, for curves in space generally: and its correctness was 

 tested, by an application to the case of the helix, the fact of the six-point 

 contact of my osculating cubic with which well-known curve admitted of 

 a very easy and elementary verification. I had the honour of commu- 

 nicating an outline of my results, so far, to Dr. Hart, a few weeks ago, 

 with a permission, or rather a request, which was acted on, that he 

 should submit them to the inspection of Dr. Salmon. 



IX. Such, then, may be said briefly to have been my first general 

 method of resolving this new problem, of the determination of the twisted 

 cubic which osculates, at a given point, to a given curve of double cur- 

 vature. Of my second method it may be sufftcient here to say, that it 

 was suggested by a recollection of the expressions given by Professor 

 Mobius, and led again to a cuhic equation, but this time for the determi- 

 nation of a coefficient, in a development of a comparatively algebraical 

 kind. For the moment I only add, that the second method of solution, 

 above indicated, bore also the test of verification by the helix; and gave 

 me generally fractional expressions for the co-ordinates of the osculating 

 twisted cubic, which admitted, in the case of the helix, of elementary 

 verifications. 



X. Of my third general method, it may be sufficient at this stage of 

 my letter to you to say, that it consists in assigning the locus of the ver- 

 tices of all the quadric cones, which have six-point contact with a given 

 carve in space, at a given point thereof. I find this locus to be a ruled 

 cubic surface, on which the tangent pt to the curve is a singular line, 

 counting as a double line in the intersection of the surface with any 

 plane drawn through it ; and such that if the same surface be cut by a 

 plane drawn across it, the plane cubic which is the section has generally a 

 node, at the point where the plane crosses that line : although this node 

 degenerates into a cusp, when the cutting plane passes throughjthe point 

 p itself. 



XI. And I find, what perhaps is a new sort of result in these ques- 

 tions, that the intersection of this new cubic surface with the former 



E. I. A. PROC, — VOL. VIII. 2 Y 



