DYNAMICS OF A RIGID BODY. 147 



expect to find that a singly infinite number of screws 

 belonging to a screw complex of the fourth order can be 

 found in any plane. We shall now prove that all these 

 -screws envelope a parabola. 



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 complex 

 of the fourth order can be drawn in a given plane through 

 a given point. From the last article it follows that only 

 one screw of the complex parallel to a given line can be 

 found in the plane. Therefore, the envelope must be a 

 parabola. 



134. Property of the Pitches of Six Co-reciprocals. 

 We may here introduce an important property of the 

 pitches of a set of co-reciprocal screws selected from a 

 .screw complex. 



There is one screw on a cylindroid of which the pitch 

 is a maximum, and another screw of which the pitch is 

 a minimum. These screws are parallel to the principal 

 axes of the pitch conic ( 20). Belonging to a screw 

 complex of the third order we have, in like manner, three 

 screws of maximum or miminum pitch, which lie along 

 the three principal axes of the pitch quadric ( 1 1 1). The 

 general question, therefore, arises, as to whether it is 

 always possible to select from a screw complex of the 

 n th order a certain number of screws of maximum or 

 minimum pitch. 



Let 0j, . . . . O n be the n co-ordinates of a screw re 

 ferred to n co-reciprocal screws belonging to the given 

 screw complex. Then the function p^ or 



is to be a maximum, while, at the same time, the co-ordi- 

 nates satisfy the condition ( 37) 



L 2 



