106 DYNAMICS OF A RIGID BODY. 



the instantaneous screw would have been A 2 , or A$. Then 

 we have the following theorem : 



If X ly X Zy X* lie upon a cylindroid S (which we may 

 call the impulsive cylindroid), then A l9 A 2y A* lie on a 

 cylindroid S' (which we may call the instantaneous 

 cylindroid). 



For if the three wrenches have suitable intensities 

 they may equilibrate, since they are cocylindroidal : 

 when this is the case the three instantaneous twist velo- 

 cities must, of course, neutralize ; but this is only possi- 

 ble if the instantaneous screws be cocylindroidal ( 63). 



If we draw a pencil of four lines through a point 

 parallel to four generators of a cylindroid, the lines 

 forming the pencil will lie in a plane. We may define 

 the anharmonic ratio of four generators on a cylindroid to 

 be the anharmonic ratio of the parallel pencil. We shall 

 now prove the following theorem* : 



The anharmonic ratio of four screws on the impul- 

 sive cylindroid is equal to the anharmonic ratio of the 

 four corresponding screws on the instantaneous cylin- 

 droid. 



Before commencing the proof we remark that, 



If an impulsive wrench of intensity F acting on the 

 screw X be capable of producing the unit of twist 

 velocity about A y then a wrench of intensity F<a on X 

 will produce a twist velocity w about A . 



Let Jfi, X^ X^ X be four screws on the impulsive 

 cylindroid, the intensities of the wrenches appropriate to- 

 which are FXO\, F^, -F 3 eo 3 , Fu. Let the four corres- 

 ponding instantaneous screws be A l9 A 2y A-^ A^ and 

 the twist velocities be ai, w 2 , w 3 > ^i- Let <p m be the angle 



* This theorem is an illustration of the important bearings of the Theory 

 of Correspondence on the Theory of Screws. 



