108 DYNAMICS OF A RIGID BODY. 



and therefore also must be Y ly Y z , Y 3 . The nine wrenches 

 X ly X 2y X 3y R ly R zy R z , - Y ly - Y zy - Y, must equilibrate ; 

 but if X ly Xtj X z equilibrate, then the twist velocities 

 about A i, A 2 , A 3 must neutralize, and therefore the 

 wrenches about Y ly Y z , Y^ must equilibrate. Hence 

 RI, Rty R 3 equilibrate, and are therefore cocylindroidal. 



Following the same line of proof used in the last 

 section, we can show that 



If impulsive wrenches on any four cocylindroidal 

 screws act upon a partially free rigid body, the four 

 corresponding initial reactions of the constraints also 

 constitute wrenches about four cocylindroidal screws ; 

 and, further, the anharmonic ratios of the two groups of 

 four screws are equal. 



10 1. Principal Screws of Inertia. If a quiescent body 

 with freedom of the second order receive impulsive 

 wrenches on three screws X ly X 2y X z on the cylindroid which 

 expresses the freedom, and if the corresponding instan- 

 taneous screws on the same cylindroid be A l9 A Z) A^ then 

 the relation between any other impulsive screw X on 

 the cylindroid and the corresponding instantaneous 

 screw A is completely defined by the condition that the 

 anharmonic ratio of X, X ly X Zy X z is equal to the anhar- 

 monic ratio of A, A ly A z A s . 



Now, if three rays parallel to X ly X- 2y X 3 be drawn 

 from a point, and also three rays parallel to A ly A 2y A z , 

 then it is well known* that the problem to determine a 

 ray Z such that the anharmonic ratio of Z, A ly A 2 , A z is 

 equal to that of Z, X ly X 2y X 3y admits of two solutions. 

 There are, therefore, two screws on a cylindroid which 

 possess the property that an impulsive wrench on one of 

 these screws will cause the body to commence to twist 

 about the same screw. 



* Chasles, passim. See alsoTownsend's Modern Geometry, vol. ii., p. 246, 



