120 DYNAMICS OF A RIGID BODY. 



belonging to S, then lines bisecting internally and ex- 

 ternally the angle between two lines in the plane of 

 y and z, parallel to p, v will be two of the principal 

 axes of the pitch quadric. Draw the cylindroid (p v) 

 Now the two screws of zero pitch on the cylindroid are 

 equidistant from the centre of the cylindroid, and the 

 two rectangular screws of the cylindroid bisect inter- 

 nally and externally the angle between the lines parallel 

 to the screws of zero pitch. Hence it follows that 

 the two rectangular screws of the cylindroid (^ v) must 

 be on the axes of y and z of the pitch quadric. We 

 shall denote these screws by )3 and y, and their pitches 

 pft and/y. From the properties of the cylindroid ( 15) 

 it appears that a, the semiaxis of the pitch quadric, must 

 be determined from the equations 



a = (pft -/ y ) sin / cos /, 

 /p cos 2 / + / y sin 2 / = o ; 

 whence eliminating /, we deduce 



If I, c be the remaining semiaxes of the pitch quadric, 

 then we must have 



cos 2 / sin 2 / 

 + = > 



because the screws fi, v are parallel to the asymptotes of 



whence we find 



