( 146 



CHAPTER XII. 



THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM 

 OF THE FOURTH ORDER. 



131. Screw Complex of the Fourth Order. The most 

 general type of a screw complex of the fourth order is 

 merely a group of screws which are reciprocal to an 

 arbitrary cylindroid ( 49). To obtain a clear idea of 

 this screw complex it is, therefore, only required to 

 re-state a few results already obtained. 



All the screws belonging to a screw complex of the 

 fourth order which can be drawn through a given point 

 lie on a cone of the second degree ( 25). 



All the screws of given pitch belonging to a screw 

 complex of the fourth order must intersect two fixed 

 lines, viz., the two screws on the reciprocal cylindroid of 

 pitch equal in magnitude, and opposite in sign, to the 

 given pitch ( 24). 



One screw of given pitch belonging to a screw com- 

 plex of the fourth order can be drawn through each point 

 in space ( 97). 



132. Screws Parallel to a Given Line. It is required to 

 determine the locus of the screws parallel to a given 

 straight line Z, which belong to a screw complex of the 

 fourth order. This easily appears from the principle 

 that each screw of the screw complex must intersect one 

 screw of the reciprocal cylindroid at right angles ( 24). 

 Take, therefore, that one screw on the cylindroid 

 which is perpendicular to L. Then a plane through 

 parallel to L is the required locus. 



133. Screws in a Plane. As we have already seen that 

 two screws belonging to a screw complex of the third 

 order can be found in any plane ( 113), so we might 



