84 DYNAMICS OF A RIGID BODY. 



from a standard position to any other position which it 

 is capable of attaining. As examples of a body which 

 has freedom of the first order, we may refer to the case of 

 a body free to rotate about a fixed axis, but not to slide 

 along it, or of a body free to slide along a fixed axis, but 

 not to rotate around it. In the former case the screw com- 

 plex consists of one screw, whose pitch is zero ; in the 

 latter case the screw complex consists of one screw, 

 whose pitch is infinite. 



80. The Reciprocal Screw Complex. The integer which 

 denotes the order of a screw complex, and the ( integer 

 which denotes the order of the reciprocal screw complex, 

 will, in all cases, have the number six for their sum 

 ( 46). Hence a screw complex of the first order will 

 have as its reciprocal a screw complex of the fifth order. 



We shall, therefore, be obliged to discuss in the pre- 

 sent chapter some properties of the screw complex of 

 the fifth order, and so far to anticipate what would more 

 naturally fall under Chapter XIII. 



For a screw 9 to belong to a screw complex of the 

 fifth order, the necessary and sufficient condition is, 

 that 6 be reciprocal to one given screw a. This con- 

 dition is thus expressed : 



(fa +fo) cos O - d'sin O = o, 



where O is the angle, and d the perpendicular distance 

 between the screws 6 and a. 



We can now show that every straight line in space, 

 when it receives an appropriate pitch, constitutes a 

 screw of a given screw complex of the fifth order. For 

 the straight line and a being given, d and O are de- 

 termined, and hence the pitch p e can be determined 

 by the linear equation just written. 



Consider now a point A y and the screw a. Every 

 straight line through A> when furnished with the proper 



