494 THE THEORY OF SCREWS. 



can be drawn through it. This is borne out in the case of the two cylindroids. The 

 two conies have but one point common, and the only quadric through both consists 

 of their planes. The line of intersection of the planes intersects the conies in three 

 distinct points, and hence another quadric cannot be drawn through the conies. 



As regards the ruled surfaces generated by the axes of a three-system which 

 are parallel to the edges of a cone of degree m, the degree of the surface is 

 evidently 3m. For the axes of the enclosing system which meet any assumed line 

 are parallel to the edges of a cubic cone, and there are 3m directions common to 

 this cone and the director cone. Again the locus of the feet of perpendiculars on 

 the generators from any point is a curve of degree 2m which viewed from the 

 point appears to have three multiple points of order m situated on the axes of the 

 reciprocal three-system passing through the point. For if we take any plane and 

 consider its intersections with the curve, we find easily that the axes of the 

 enclosing system whose feet of perpendiculars from the point lie in the plane are 

 parallel to the edges of a quadric cone. The theorem about the apparent 

 multiple points follows from consideration of the cylindroids of the enclosing system 

 whose double lines pass through the assumed point. 



We also note this construction for Art. 180. Assume a radius of the pitch 

 quadric, draw the tangent plane and let fall the central perpendicular on the 

 plane. Measure off on the radius and on the perpendicular the reciprocals of their 

 lengths, thus determining a triangle. Through the centre draw a normal to the 

 plane of the triangle equal in length to double the area of the triangle multiplied 

 by the product of the three principal pitches. This is the perpendicular to the 

 required axis if we consider rotation round the line from the perpendicular to the 

 radius as positive. 



One more point may be mentioned. If we take the cone reciprocal to the 

 director cone, that is the cone whose edges are perpendicular to the tangent planes, 

 and if we use this new cone for selecting the generators of a ruled surface from 

 the reciprocal three-system, the two ruled surfaces have a common line of striction 

 and they touch one another along this line. This is the extension of the theorem 

 that the reciprocal screw at right angles to the generators of a cylindroid coincides 

 with the axis. 



NOTE VII. 



Note on homographic transformation, 246. 



That there must be in general linear relations between the co-ordinates of the 

 screws of an instantaneous system, and the co-ordinates of the corresponding 

 impulsive screws is proved as follows. 



Let a be an impulsive screw and let ft be the corresponding instantaneous screw 

 with respect to a free body. 



Let A, B, C, D, E, F be six independent screws which we shall take as screws 



