CHAPTER V. 



GENERAL CONSIDERATIONS ON THE EQUILIBRIUM OF A 

 RIGID BODY. 



43. The Screw Complex. To specify with precision the 

 nature of the freedom enjoyed by a rigid body, it is 

 necessary to ascertain all the screws about which the 

 constraints will permit the body to be twisted. When the 

 attempt has been made for every screw in space, the re- 

 sults will give us all the information conceivable with 

 reference to the freedom of the body, and also with re- 

 ference to the constraints by which the movement may 

 be hampered. 



Suppose that n screws ^4 b &c., A n have been found by 

 these trials, about each of which the body can receive a twist. 

 It is evident, without further trial, that twisting about an 

 infinite number of other screws must also be possible 

 (n > i) : for suppose the body receive any n twists about 

 A i, &c., A n the position attained could have been reached 

 by a twist about some single screw A . It follows that 

 the body must be free to twist about A . Now since the 

 amplitudes of the n twists may have any magnitude (each 

 not exceeding an infinitely small quantity), A is merely 

 one of an infinite number of screws, about which twist- 

 ing must be possible. All these screws, together with 

 A &c., A , we call a screw complex of the n ih order. 



If it be found that the body cannot be twisted about 

 any screw which does not belong to the screw complex 

 of the n th order, then the body is said to have freedom of 

 the n th order. It may be necessary to remark that A &c., 

 A ny must not be themselves members of a screw complex 

 of order lower than n. If this were the case, the screws 



