430 



THE THEORY OF SCREWS. 



+ 0A (Pi 2 - p 3 2 + ba - be) 



+ 0A (/&amp;gt; 2 2 - pi +cb- GO) 

 + 0A (I? - fyo) - 3 0i (li 2 



[391- 



+ 0* 



+ 01 



The coefficients of #/, # 2 , 0s respectively each equated to zero will give 

 three conies 7=0, F= 0, Tf = 0. These conies have three common points 

 which are of course the three permanent screws. 



If we introduce a new quantity O we can write the three equations 



(I/ - 



+ (l*-ax*)0 2 + (ab - p-&amp;gt; + H) 3 = 0. 



The elimination of ft between each pair of these equations will produce 

 the three equations U = Q, V0, W = 0. If therefore we eliminate 1} 2 , 3 

 from the three equations just written the resulting determinant gives a 

 cubic for H. The solution of this cubic will give three values for fl which 

 substituted in the three equations will enable the corresponding values of 

 0i &amp;gt; 0zy 0s to be found. We thus express the co-ordinates of the three per 

 manent screws in terms of the nine co-ordinates of the rigid body and their 

 determination is complete. 



It may be noted that the same permanent screws will be found for any 

 one of the systems of rigid bodies whose co-ordinates are 



whatever h may be. 



392. Case of Two Degrees of Freedom. 



We have already shown that there is a single permanent screw in every 

 case where the rigid body has two degrees of freedom. We can demonstrate 

 this in a different manner as a deduction from the case of the three- 

 system. 



Consider a cylindroid in a three-system, that is of course a straight 

 line in the plane representation ( 200). Let this line be AB (tig. 44). If 

 the movements of the body be limited to twists about the screws on the 

 cylindroid, there may be reactions about the screw which corresponds to the 

 pole P of this ray with respect to the pitch conic, in addition to the reactions 

 of the three-system. 



