426 THE THEORY OF SCREWS. [388- 



We notice here the somewhat remarkable circumstance, that if 



/! 2 - ajojpi = 0&amp;gt; and I? + y*P* = 0, 

 then all the screws on the cylindroid are permanent screws. 



It hence appears that if two screws on a cylindroid are permanent, then 

 every screw on the cylindroid is permanent. 



389. Three Degrees of Freedom. 



Let us now specially consider the case of a rigid body which has freedom 

 of the third order. On account of the evanescence of the emanant we have 



*dT * dT , dT _ 



*W 4 * W 4 *W? 



It is well known that if U, V, W be three conies whose equations submit to 

 the condition 



those conies must have three common intersections. 

 It therefore follows that the three equations 



must have three common screws. These are, of course, the permanent 

 screws, and, accordingly, we have the theorem : 



A rigid system which has freedom of the third order has, in general, three 

 permanent screws. 



There will be a special convenience in taking these three screws as the 

 screws of reference. We shall use the plane representation of the three- 

 system, and the equations of the conies will be 



AJA + BAOi + CAO^O, or 7=0, 



AAo s + BAOi + cAo, = o, F=O, 

 AAh + BA^ + cAe^o, w=o-, 

 but, as e l u + e 2 v+0 3 w = o, 



identically, we must have 



^ = 0; A, = Q; A 3 = 0; 

 (7 1= 0; C 2 =0; B 3 =0; 



and also A : + B 2 + C 3 = 0. 



For symmetry we may write 



