220 THE THEORY OF SCREWS. [213, 



We can also investigate the conditions under which five forces applied 

 to a free rigid body can neutralize each other. The five forces must, as 

 the body is free, belong to a screw system of the fourth order. Draw 

 the cylindroid reciprocal to the system. The five forces must, therefore, 

 intersect both the screws of zero pitch on the cylindroid. We thus prove 

 the well-known theorem that if five forces equilibrate two straight lines can 

 be drawn which intersect each of the five forces. Four of the forces will 

 determine the two transversals, and therefore the fifth force may enjoy any 

 liberty consistent with the requirement that it also intersects the same two 

 lines. 



If A l ,...A 5 be the five forces, the ratio of any pair, let us say for 

 example, A : A z is thus determined. 



Let P, Q be the two screws of zero pitch upon the cylindroid, i.e. the 

 two common transversals of A ly ... A 5 . 



Choose any two screws X and Y reciprocal to both A! and J. 2 , but not 

 reciprocal to A 3 , A 4 or A 5 . 



Choose any screw Z reciprocal to A 3 , A^, A 5 , but not reciprocal to A l 

 or A 2 . 



Construct ( 25) the single screw / reciprocal to the five screws 



X, 7, P, Q, Z. 



The four screws X, Y, P, Q are reciprocal to the cylindroid A 1} A. 2 \ 

 therefore /, which is reciprocal to X, Y, P, Q, must lie upon the cylindroid 

 (A lt Aj($2Q 



Since P, Q, Z are all reciprocal to A 3 , A t , A 5 , it follows that / being 

 reciprocal to P, Q, Z must belong to the screw system A 3 , A t , A 5 . Hence 

 / is found on the cylindroid (A 1} A 3 ), and it must also belong to the system 

 (A 3 , AI, A s ). If, therefore, forces along A 1 ,...A 5 are to equilibrate, the 

 forces along A 1&amp;gt; A 3 must compound into a wrench on /. 



But / being known by construction the angles A Z I and A^I are known, 

 and consequently the ratio of the sines of these angles, i.e. the relative 

 intensities of the forces on A 1 and A 2 are determined ( 14). 



If a free rigid body is acted upon by five forces, the preceding con 

 siderations will show in what manner the body could be moved so that it 

 shall not do work against nor receive energy from any one of the forces. 



Let A!, ... A 5 be the five forces. Draw two transversals L, M intersecting 

 A 1 ,...A i . Construct the cylindroid of which L, M are the screws of zero 

 pitch ; find, upon this cylindroid, the screw X reciprocal to A 5 . Then the 



