301] DEVELOPMENTS OF THE DYNAMICAL THEORY. 321 



Let a be the instantaneous screw and d a the length of the perpendicular 

 thereon from the centre of gravity. If cos A,, cos /i, cos v be the direction 

 cosines of d a then 



d a cos X = ( 5 - a 6 ) (a, + 4 ) c - ( 5 + a 6 ) ( 3 - o 4 ) b, 

 d a cos /*=(!- 2 ) (a, + a e )a- fa + o^) (a 5 - 6 ) c, 

 d a cos v = ( 3 4 ) (ttj + a 2 ) & ( 3 + 4 ) (i 2 ) a. 



But if 77 is the impulsive screw corresponding to a as the instantaneous 

 screw we have 



! = - ^-\*hi -C& 2 = - ^ N^&quot;&amp;gt; &C., &C., 

 cos (a?;) cos (a?;) 



whence 



*) 



da COS X = CQS Y . ((T/., + 77 fi ) (173 + 4 ) - ( 5 + e) (173 + 1; 4 )), 

 2) 



rf a cos /^ = / ?. ((% + 7? 2 ) ( 5 + 6 ) - (! + 2 ) (775 + 77 6 )), 



d COS &quot; = ^T~\ (^s + ^4&amp;gt; (i + On) - (s + 4 ) (77! + 77 2 )). 

 COo \Jjiij} 



But 



(% + %) ( 3 + 4 ) - ( B + 6&amp;gt; (?3 + 7 4 ) = sin (077) cos X , 



with similar expressions for sin (0(77) cos /* and sin (377) cos v where cos X , 

 cos p, and cos v are the direction cosines of the common perpendicular to 

 and 77. We have therefore 



v 



d a cos X = v r sin (an) cos X , 

 cos 



in 



d a coKu,= ~ . sin (77)cos /A 

 cos (at}) 



rn 



d a cos v = *? - sin (a77) cos v, 

 cos (277) 



whence 



cos X = cos X ; cos /it = cos pf ; cos v cos v ; 



and d a = p a tan (77), 



which proves the theorems. 



B. 21 



