DYNAMICS OF A RIGID BODY. 1 1 9 



This is proved as follows : Draw three screws/, q y r y 

 of pitch + k belonging to S. Draw three screws /, m y n y 

 each of which intersects the three screws p y q y r, and 

 attribute to each of /, m y n y a pitch - k. Since two inter- 

 secting screws of equal and opposite pitches are recipro- 

 cal, it follows that/, q y r y must all be reciprocal to /, m y n. 

 Hence, since the former belong to S y the latter must be- 

 long to S'. Every other screw of pitch + k intersecting 

 /, m y n y must be reciprocal to S' y and must therefore be- 

 long to S. 



But the locus of a straight line which intersects three 

 given straight lines is an hyperboloid of one sheet,* and 

 hence the required theorem has been proved. 



in. The Pitch Quadric. There is one member of 

 this family of hyperboloids which is of exceptional in 

 terest. We allude to that which is the locus of the 

 screws' of zero pitch belonging to the screw complex. 

 As the quadric under consideration possesses a very 

 important property ( 112) besides that of being the locus 

 of the screws of zero pitch, it is desirable to denote it 

 by the special phrase pitch quadric. 



We shall now determine the equation of the pitch 

 quadric. Let one of the principal axes of the pitch 

 quadric be denoted by x y this will intersect the surface 

 in two points through each of which a pair of generators 

 can be drawn. One generator of each pair will belong 

 to S y and the other to S r . Each pair of generators will 

 be parallelf to the asymptotes of the section of the pitch 

 quadric made by the plane containing the remaining 

 principal axes y and z. Let //, v be the two generators 



Salmon's Analytic Geometry of Three Dimensions, p. 77. 

 t Salmon, loc. cit., p. 72. 



