92 THE THEORY OF SCREWS. [102, 



The parameter v a may be contrasted with the parameter u a considered 

 in 89. Each is a linear magnitude, but the latter depends only upon the 

 co-ordinates of a, and the distribution of the material of the rigid body. 

 Both quantities may be contrasted with the pitch p a , which is also a linear 

 magnitude, but depends on the screw, and neither on the rigid body nor the 

 forces. 



If a body receive a twist of small amplitude a? about one of the principal 

 screws of the potential, then the intensity of the wrench evoked on the 

 same screw is (99) : 



1 dV a 

 + 2p a da 



but we have just seen that V=Fv a -a 2 , whence we have the following 

 theorem : 



If a body which has freedom of the nth order be displaced from a position 

 of equilibrium by a twist about a screw a, of which the co-ordinates with 

 respect to the principal screws of the potential arc i, ... a n , then a reduced 

 wrench ( 96) is evoked on a screw with co-ordinates proportional to 



/tj 2 rt 2 



1 -!,... a n , where v l} ... , p l} ... , are the values of the quantity v, and 



PI Pn 



the pitch p, for the principal screws of the potential. 



We can now express with great simplicity the condition that two screws 

 and (f&amp;gt; shall be conjugate screws of the potential. For, if 6 be reciprocal 

 to the screw whose co-ordinates are proportional to 



we have : 



Vi*0i&amp;lt;l&amp;gt; l +...+v n *0 n &amp;lt;f&amp;gt; n = 0. 



The expression for the potential assumes the simple form 



Fv 1 *a 1 3 +... + Fv n *a n s . 



If the function V be proportional to the product of the pitch of the 

 displacement screw and the square of the amplitude, then every displacement 

 screw will coincide with the screw about which the wrench is evoked. 



103. Form of the Potential. 



The n principal screws of the potential form a unique group, inasmuch 

 as they are co-reciprocal, as well as being conjugate screws of the potential. 

 They therefore fulfil n (n 1) conditions, being the total number available in 

 the selection of n screws in an n system. 



