186 THE THEORY OF SCREWS. [184- 



the equation becomes 



= p a -p , R p cos (a/3), + Q p cos (017) 



R -p cos (aft), Pfi-p 



- Q - P cos ( a 7) &amp;gt; + P - p cos 



By expanding this as a cubic for p we see that the coefficient of p 2 

 divided by that of p 3 with its sign changed is 



p a sin 2 (ffy) +jp/3 sin 2 (yq) + jo Y sin 2 (off) 



sii4[(y) + ( 7 a) + (a/3)]sin|[^ 



This is accordingly the constant sum of the three pitches of the screws of 

 the system which can be drawn through any point. 



185. Equilibrium of Four Forces applied to a Rigid Body. 



If the body be free, the four forces must be four wrenches on screws of 

 zero pitch which are members of a screw system of the third order. The 

 forces must therefore be generators of a hyperboloid, all belonging to the 

 same system ( 132). 



Three of the forces, P, Q, R, being given in position, S must then be a 

 generator of the hyperboloid determined by P, Q, R. This proof of a 

 well-known theorem (due to Mobius) is given to show the facility with 

 which such results flow from the Theory of Screws. 



Suppose, however, that the body have only freedom of the fifth order, 

 we shall find that somewhat more latitude exists with reference to the 

 choice of S. Let X be the screw reciprocal to the screw system by which 

 the freedom is defined. Then for equilibrium it will only be necessary that 

 S belong to the system of the fourth order defined by the four screws 



P, Q, R, X. 



A cone of screws can be drawn through every point in space belonging 

 to this system, and on that cone one screw of zero pitch can always be 

 found ( 123). Hence one line can be drawn through every point in space 

 along which S might act. 



If the body have freedom of the fourth order, the latitude in the choice 

 of S is still greater. Let X ly X 2 be two screws reciprocal to the system, 

 then S is only restrained by the condition that it belong to the screw system 

 of the fifth order defined by the screws 



P, Q, R, X 1} X. 



