1 1 8 DYNAMICS OF A RIGID BODY. 



and as a change in either of these ratios changes v y the- 

 number of v screws is doubly infinite. 



All the screws of which v is a type form what we 

 call a screw complex of the third order. We shall often 

 denote this screw complex by the symbol S. 



109. The Reciprocal Screw Complex. A wrench which 

 acts on a screw i\ will not be able to disturb the equili- 

 brium of M y provided ?j be reciprocal to x y y, z. If, 

 therefore, r\ be reciprocal to three screws of the complex 

 S 9 it will be reciprocal to every screw of S. Since i? has 

 thus only three conditions to satisfy in order that it may 

 be reciprocal to ,5*, and since five quantities determine a 

 screw, it follows that r\ may be anyone of a doubly infinite 

 number of screws which we may term the reciprocal 

 screw complex S'. Remembering the property of recipro- 

 cal screws (22) we have the following theorem (47). 



A body only free to twist about all the screws of S 

 cannot be disturbed by a wrench on any screw of S' ; 

 and, conversely, a body only free to twist about the 

 screws of S' cannot be disturbed by a wrench on any 

 screw of S. 



The reaction of the constraints by which the freedom 

 is prescribed constitutes a wrench on a screw of S'. 



no. Distribution of the Screws. To present a clear 

 picture of all the movements which the body is com- 

 petent to execute, it will be necessary to examine the 

 mutual connexion of the doubly infinite number of 

 screws which form the screw complex. It will be most 

 convenient in the first place to classify the screws in the 

 complex according to their pitches; the first theorem to 

 be proved is that all the screws of given pitch + k lie upon 

 a hyperboloid of which they form one system of generator s y 

 while the other system of generators with the pitch - k 

 belong to the reciprocal screw complex S'. 



