CHAPTER III. 



RECIPROCAL SCREWS. 



20. Reciprocal Screws. 



If a body only free to twist about a screw a be in equilibrium, though 

 acted upon by a wrench on the screw ft, then conversely a body only free to 

 twist about the screw ft will be in equilibrium, though acted upon by a wrench 

 on the screw a. 



The principle of virtual velocities states, that if the body be in equili 

 brium the work done in a small displacement against the external forces 

 must be zero. That the virtual coefficient should vanish is the necessary and 

 the sufficient condition, or ( 10) 



(p a + pp) cos d sin = 0. 



The symmetry shows that precisely the same condition is required 

 whether the body be free to twist about a, while the wrench act on ft, or 

 vice versa. A pair of screws are said to be reciprocal when their virtual co 

 efficient is zero. 



21. Particular Instances. 



Parallel or intersecting screws are reciprocal when the sum of their pitches 

 is zero. Screws at right angles are reciprocal either when they intersect, 

 or when one of the pitches is infinite. Two screws of infinite pitch are 

 reciprocal, because a couple could not move a body which was only sus 

 ceptible of translation. A screw whose pitch is zero or infinite is reciprocal 

 to itself*. 



22. Screw Reciprocal to Cylindroid. 



If a screw t] be reciprocal to two given screws 9 and (f&amp;gt;, then 77 is reciprocal 

 to every screw on the cylindroid (6, &amp;lt;). 



* See also Professor Everett, F.R.S., Messenger of Mathematics, New Series (1874), No. 39. 



