140 THE THEORY OF SCREWS. [150- 



In the limit we allow P and Q to coalesce, in which case, of course, P and 

 Q coalesce, and p and q become coincident ; but obviously we have then 



PQ : ML :: PX : LX, 

 P Q : ML :: P Y : LY- 



P Q P Y LX 

 whence PQ = PX = LY&amp;gt; 



and as P Foc -L and PX oc , 



we have finally p oc - T v . 



Li 



The result is, of course, the same as that of 141. Being given the 

 impulsive screw corresponding to P, we find P by drawing PXL and LYP ; 

 and then to produce a unit twist velocity on P , the intensity of the impul 

 sive wrench on P must be proportional to LX -r- LY. It is obvious that by 

 a proper adjustment of the units of length, force and twist velocity, LX 

 may be the intensity of the impulsive wrench, and LY the acquired twist 

 velocity. 



151. Principal Screws of the Potential. 



Let us suppose that a body having two degrees of freedom is in a position 

 of stable equilibrium under the influence of a conservative system of forces. 

 If the body be displaced by a small twist, it will no longer be in a position of 

 equilibrium, and a wrench has commenced to act upon it. This wrench can 

 always, by suitable composition with the reactions of the constraints, be 

 replaced by an equivalent wrench on a screw of the cylindroid (see 96). 



For every point H, corresponding to a displacement screw, we have a 

 related point, H , corresponding to the screw about which the wrench is evoked. 

 The relation is of the one-to-one type, and it will now be proved that the 

 system of screws H is homographic with the corresponding system H . The 

 proof is obtained in the same manner as that already given in 137, for 

 impulsive and instantaneous screws. 



Let E be a displacement screw about which a twist of unit magnitude 

 evokes a wrench of intensity e on E ; let f be the similar quantity for 

 another pair of screws, F and F . 



A twist of unit amplitude about H may be decomposed into components, 



HF HE 

 EF EF 



about E and F, respectively. 



