376] THE THEORY OF PERMANENT SCREWS. 413 



It has been shown in 265 that if lt 2 , ... 6 are the co-ordinates of 

 a screw of zero pitch, then the co-ordinates of a screw of pitch p x on the 

 same axis are respectively 



* p x dR A p x dR * p x dR 



&quot;l~l ---- - ~ , t/2 &quot;1 -- -- T~ , . . . t/u T -- ~ - . 



^ d0 1 4p 2 d0 2 4&amp;gt;p 6 d0 G 



In these expressions p x denotes an arbitrary pitch, while R is the 

 function 



es + fa ... + 6 2 + 2 (12) eA + 2 (23) e,e 3 , &c., 



where (12) is the cosine of the angle between the first and second screws of 

 reference, and similarly for (23), &c. 



The principle just stated asserts that T must remain unchanged if we 

 substitute for #/, #2 , & c - the expressions 



We thus obtain the formula 



p^dR^dT , A p,dRy^ = Q 



^dejde, 4&amp;gt;p 6 deJ d0 6 



As this must be true for every value of p x , we must have, besides the 

 vanishing ernanant, the condition 



^dR dT^ 1^?^ =0 



p l dO, dffi p 6 d0 6 d0 6 ~ 



It is plain that this is equivalent to the statement that the screw whose 

 co-ordinates are 



l^dR T_dR ^dR 

 ~Pi d0\ P* d0 2 p* d6 6 



must be reciprocal to the screw defined by the co-ordinates 



I dT_ l^dT^ \dT 



Pl d0 l &quot; p,d0:r &quot; p 6 d0 6 &quot; 



The former denotes a screw of infinite pitch parallel to and hence it 

 follows that the restraining screw must be perpendicular to 6. Remember 

 ing also that the restraining screw is reciprocal to 0, it follows that the 

 restraining screw must intersect 0. We thus obtain the following result : 



If be the screw about which a free rigid body is twisting, then to check 

 the tendency of the body to depart from twisting about a restraining wrench 

 on a screw which intersect* at right angles must in general be applied. 



