CHAPTER VI. 



THE EQUILIBRIUM OF A RIGID BODY. 



69. A Screw System. 



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 results will give us all the information conceivable 

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

 constraints by which the movement may be hampered. 



Suppose that by these trials, n screws A 1} ... A n have been found 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 (w&amp;gt;l): for suppose the body receive any n twists about A^, ... 

 A n the position attained could have been reached by a twist about some 

 single screw A. The body can therefore twist about A. 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 

 twisting must be possible. All these screws, together with A,, ... A n , will in 

 general form what we call a screw system of the nth, order. 



If it be found that the body cannot be twisted about any screw which 

 does not belong to the screw system of the nth order, then the body is said 

 to have freedom of the nth order. It is assumed that A lf ... A n are 

 independent screws, i.e. not themselves members of a screw system of order 

 lower than n. If this were the case, the screws about which the body could 

 be twisted would only consist of the members of that lower screw system. 



Since the amplitudes of the n twists about AI, ... A n are arbitrary, it 

 might be thought that there are n disposable quantities in the selection of a 

 screw S from a screw system of the ?ith order. It is, however, obvious from 

 14 that the determination of the position and pitch of S depends only upon 



