388] THE THEORY OF PERMANENT SCREWS. 423 



Then 



01 = @i + OiOt, 

 &amp;lt;j&amp;gt;2 ~ $2 + d z ot, 



whence 



which is, accordingly, the rate at which P will change its position. If we 

 substitute for 0j and 6 2 their values already found in the last article, we 

 obtain for the velocity of P the expression 



N being the position of the permanent screw, let p be the length of the 

 chord P^, then the expression just written assumes the form 



where k is a constant. 



This expression illustrates the character of the screw corresponding to N. 

 If p be zero, then the expression for this velocity vanishes. This means that 

 P has no tendency to abandon N ; in other words, that the screw correspond 

 ing to N is permanent. 



388. Another method. 



It is worth while to investigate the question from another point 

 of view. 



Let us think of any cylindroid 8 placed quite arbitrarily with respect to 

 the position of the rigid body. A certain restraining screw i) will corre 

 spond to each screw 6 on S. As 6 moves over the cylindroid, so must the 

 corresponding screw 17 describe some other ruled surface S . The two 

 surfaces, S and 8 , will thus have two corresponding systems of screws, 

 whereof every two correspondents are reciprocal. One screw can be dis 

 covered on S , which is reciprocal, not alone to its corresponding 6, but to 

 all the screws on the cylindroid. A wrench on this ij can be provided by the 

 reactions of the constraints, and, consequently, the constraints will, in this 

 case, arrest the tendency of the body to depart from 6 as the instantaneous 

 screw. It follows that this particular 6 is the permanent screw. 



The actual calculation of the relations between 77 and the cylindroid is as 

 follows : 



A set of forces applied to a rigid system has components X, T, Z at 

 a point, and three corresponding moments F, G, H in the rectangular planes 

 of reference. 



