CHAPTER XVI. 



FREEDOM OF THE FOURTH ORDER. 



212. Screw System of the Fourth Order. 



The most general type of a screw system of the fourth order is exhibited 

 by the set of screws which are reciprocal to an arbitrary cylindroid ( 75). 

 To obtain certain properties of this screw system it is, therefore, only 

 necessary to re-state a few results already obtained. 



All the screws which belong to a screw system of the fourth order and 

 which can be drawn through a given point are generators of a certain cone 

 of the second degree ( 23). 



All the screws of the same pitch which belong to a screw system of the 

 fourth order must intersect two fixed lines, viz. those two screws which, 

 lying on the reciprocal cylindroid, have pitches equal in magnitude but 

 opposite in sign to the given pitch ( 22). 



One screw of given pitch and belonging to a given screw system of the 

 fourth order can be drawn through each point in space ( 123). 



As we have already seen that two screws belonging to a screw system 

 of the third order can be found in any plane ( 178), so we might expect 

 to find that a singly infinite number of screws belonging to a screw system 

 of the fourth order can be found in any plane. We shall now prove that 

 all these screws envelope a parabola. A theorem equivalent to this has 

 been already proved in a different manner in 162. 



Take any point P in the plane, then the screws through P reciprocal to 

 the cylindroid form a cone of the second order, which is cut by the plane 

 in two lines. Thus two screws belonging to a given screw system of the 

 fourth order can be drawn in a given plane through a given point. But 

 it can be easily shown that only one screw of the system parallel to a 

 given line can be found in the plane. Therefore from the point at infinity 

 only a single finite tangent to the curve can be drawn. Therefore the other 



