104 THE THEORY OF SCREWS. [ill- 



reference, we shall have for the intensity of the component of the resultant 

 wrench on &&amp;gt; n 



6&quot; 6 n + 4&amp;gt;&quot;&amp;lt;j&amp;gt; n - 



Hence the co-ordinates of the resultant wrench are proportional to 



For equilibrium this screw must be reciprocal to a, whence we have 

 Pl a, (ff 0, + f &amp;gt;0 +...+P*. (ff Ot + f &) = 0, 



Or, 0&quot;^a6 + $&quot;1*04 = 0. 



This equation merely expresses that the sum of the works done in a 

 small twist about a against the wrenches on and &amp;lt;f&amp;gt; is zero. 



We also perceive that a given wrench may be always replaced by a 

 wrench of appropriate intensity on any other screw, in so far as the effect 

 on a body only free to twist about a is concerned. 



It may not be out of place to notice the analogy which the equation just 

 written bears to the simple problem of the determination of the condition 

 that two forces should be unable to disturb the equilibrium of a particle 

 only free to move on a straight line. If P, Q be the two forces, and if I, m 

 be the angles which the forces make with the direction in which the particle 

 can move, then the condition is 



P cos I + Q cos in = 0. 



This suggests an analogy between the virtual co-efficient of two screws, 

 and the cosine of the angle between two lines. 



112. Particular Case. 



If a body having freedom of the first order be in equilibrium under the 

 action of gravity, then the vertical through the centre of inertia must lie 

 in the plane of reciprocal screws of zero pitch, drawn through the centre 

 of inertia. 



113. Impulsive Forces. 



If an impulsive wrench of intensity 77 &quot; act on the screw &amp;gt;/, while the 

 body is only permitted to twist about a, then we have seen in 90 how the 

 twist velocity produced can be found. We shall now determine the impulsive 

 reaction of the constraints. This reaction must be an impulsive wrench of 

 intensity X &quot; on a screw X, which is reciprocal to a. The determination of X 

 may be effected geometrically in the following manner : Let /* be the screw, 

 an impulsive wrench on which would, if the body were perfectly free, cause 

 an instantaneous twisting motion about a ( 80). Draw the cylindroid (77, fi). 



