12 5] FREEDOM OF THE SECOND ORDER. 113 



If we draw a pencil of four lines through a point parallel to four gene 

 rators of a cylindroid, the lines forming the pencil will lie in a plane. We 

 may define the (inharmonic ratio of four generators on a cylindroid to be 

 the anharmonic ratio of the parallel pencil. We shall now prove the follow 

 ing theorem : 



The anharmonic ratio of four screws on the impulsive cylindroid is equal 

 to the anharmonic ratio of the four corresponding screws on the instantaneous 

 cylindroid. 



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, then an impulsive wrench 

 of intensity Fta on X will produce a twist velocity w about A. 



Let X 1} X 2 , X 3) X^ be four screws on the impulsive cylindroid, the 

 intensities of the wrenches appropriate to which are F^, F 2 a) 2 , F 3 to 3t Fw. 

 Let the four corresponding instantaneous screws be A 1} A 2 , A s , A^ and the 

 twist velocities be a&amp;gt; 1; &&amp;gt; 2 , a&amp;gt; 3 , a&amp;gt; 4 . Let &amp;lt;j) m be the angle on the impulsive 

 cylindroid defining X m , and let 6 m be the angle on the instantaneous 

 cylindroid defining A m . 



If three impulsive wrenches equilibrate, the corresponding twist velocities 

 neutralize by the second law of motion : hence ( 14) certain values of 

 &&amp;gt;!, &amp;lt;B.;,, 6) :! , ft) 4 must satisfy the following equations: 



to, 



sin (0 2 - 3 ) sin (0 3 - 0,) sin (0, - a ) 



sin (fa (j&amp;gt; 3 ) sin (&amp;lt;/&amp;gt; 3 fa) sin (fa 



too 



sin (0 3 - 4 ) sin (0 4 - 2 ) sin (0 2 - :! ) 



_ 



sin (0 :j - fa) ~ sin (fa - fa) sin (fa - fa*) 

 whence 



sin (0j 0) sin (0 3 4 ) _ sin (fa fa) sin (&amp;lt;ft :i fa) 

 sin (0 3 0!&amp;gt; sin (0 4 - 2 ) ~ sin (fa fa) sin (fa fa,) 



which proves the theorem. 



If we are given three screws on the impulsive cylindroid, and the 

 corresponding three screws on the instantaneous cylindroid, the connexion 

 between every other corresponding pair is, therefore, geometrically deter 

 mined. 



B. $ 



