1 28 DYNAMICS OF A RIGID BODY, 



The tangent planes at P ly P* each intersect the sur- 

 face in lines parallel to OL, OM. Let us call these lines 

 PII, 1 ly M l through the point P ly and P Z L Z , P 2 M 2 

 through the point jP 2 . Then PiL ly P z M 2y are screws 

 belonging to the complex, and PiM ly P^L Z are reciprocal 

 screws. 



Since OL is a tangent to the pitch quadric, it there- 

 fore must be intersected by two rectilinear generators,, 

 one of each system. These two generators lie in a 

 plane which contains OL ; but since OL touches the 

 hyperboloid at infinity, the lines on the surface must be 

 parallel to OL, and therefore their projections on the 

 plane of A must be S' T, S'T'. Similarly for ST, S'T'; 

 hence ST' and S'T' are the projections of two screws 

 belonging to the complex, and therefore the centre of 

 the cylindroid is at T '. In a similar way it is proved 

 that the centre of the reciprocal cylindroid is at T. 



Having thus determined the centre of the cylindroid, 

 the remainder of the construction is easy. The pitches 

 of two screws on the surface must be proportional to 

 the inverse square of the parallel diameters of the ^sec- 

 tion of the pitch quadric made by A. Therefore, the 

 greatest and least pitches will be on screws parallel 

 to the principal axes of the section. Hence, lines drawn 

 through T' parallel to the external and internal bisectors 

 of the angle between the asymptotes are the two rectan- 

 gular screws of the cylindroid. Thus the problem of 

 finding the cylindroid is completely solved. 



It is easily seen that each cylindroid touches each of 

 the quadrics in two points. 



115. Miscellaneous Remarks. It follows from the last 

 article that any plane which contains a pair of screws 

 belonging to the complex which intersect at right angles 

 must pass through the centre of the pitch quadric. 



We are now in a position to determine the actual 



