69-72] THE EQUILIBRIUM OF A RIGID BODY. 63 



the ratios of the amplitudes of the twists about A 1} ... A n and hence in the 

 selection of a screw from the screw system of the nth order, we have n 1 

 disposable quantities. 



70. Constraints. 



An essential feature of a system of constraints consists in the number of 

 independent quantities which are necessary to specify the position of the 

 body when displaced in conformity with the requirements of the constraints. 

 That number which cannot be less than one, nor greater than six, is the 

 order of the freedom. To each of the six orders of freedom a certain type of 

 screw system is appropriate. 



The study of the six types of screw system as here defined is a problem 

 of kinematics, but the statical and kinematical properties of screws are so 

 interwoven that we derive great advantages by not attempting to relegate 

 the statics and kinematics to different chapters. We shall not require any 

 detailed examination of the constraints. Every conceivable condition of 

 constraints must have been included when the six screw systems have been 

 discussed in their most general form. Nor does it come within our scope, 

 except on rare occasions, to specialize the enunciation of any problem, further 

 than by mentioning the order of the freedom permitted to the body. 



71. Screw Reciprocal to a System. 



If a screw X be reciprocal to n independent screws, A l} ... A n , of a screw 

 system of order n, then X is reciprocal to every other screw A which belongs 

 to the same screw system. For, by the property of the screw system, it 

 appears that twists of appropriate amplitudes about A 1} ... A n , would 

 compound into a twist about A. It follows ( 69) that wrenches on A l , ... 

 A n , of appropriate intensities ( 30) compound into a wrench on A. Suppose 

 these wrenches on A 1} ... A n&amp;gt; were applied to a body only free to twist 

 about X, then since X is reciprocal to A lt ... A H , the equilibrium of the 

 body would be undisturbed. The resultant wrench on A must therefore be 

 incapable of moving the body, therefore A and X must be reciprocal. 



72. The Reciprocal Screw System. 



All the screws which are reciprocal to a screw system P of order n 

 constitute a screw system Q of order 6 n. This important theorem is thus 

 proved : 



Since only one condition is necessary for a pair of screws to be reciprocal, 

 it follows, from the last section, that if a screw X be reciprocal to P it will 

 fulfil n conditions. The screw X has, therefore, 5 n elements still dis 

 posable, and consequently (n &amp;lt; f&amp;gt;) an infinite number of screws Q can be 



