THE THEORY OF PERMANENT SCREWS. 421 



Let us take any two screws on the cylindroid a and ft, and let their co 

 ordinates, when referred to the absolute screws of inertia, be 



ls ... 6 , and &, ... /8. 



Then any other screw on the cylindroid, about which the body has been 

 displaced by a twist, by components #/ on a. and 2 on ft will have, for co 

 ordinates, 



^ *MV...%tf*M . 



and the screw about which the body is twisting, with a twi^t velocity 0, will 

 have, for co-ordinates, 



It readily appears that, so far as the terms involving #/and #./are concerned, 

 the kinetic energy is the expression 



where 



A = + be (! + 2 ) [( 3 - 6 ) (& - &) - (3 - 4&amp;gt; (A - &)] 



+ (6 2 - c 2 ) (a, + 04) (a, + a 6 ) (A + & 



+ ca (a g + 4 ) [(a, - a.,) (& - /3 6 ) - ( 5 - 6 ) (A - A)] 



+ (c 2 - a 2 ) ( B + 6 ) (! + 2 ) (/3 S 



+ a6 (a s + 6 ) [(a, - 4 ) (A - &) - ( ai - a 2 ) (/3 3 - /3 4 )] 



+ (a 2 - 6 2 ) (! + a a ) ( 3 + 4 ) (A 



= + be (& + /8 8 ) [( 5 - 6 ) 09, - A) - (a, - 4 ) (A - 



- (Z&amp;gt; 2 - c 2 ) (a : + a, 



ca 3 + ! - 2 / B - / 6 - 5 - 



- (c 2 - a 2 ) ( 3 + 4 ) (/S B 



- a,) (A - /3 2 ) - (i - 



- (a 2 - 6 2 ) (a, + .) (A + A) (A + A). 



386. The Permanent Screw. 



We now write the equations of motion for a body which has two degrees 

 of freedom, and is unacted upon by any force, the screws of reference being 

 the two principal screws of inertia. 



We have, from the general equations (367) 



M&quot; 2/3 ^^ 



= 



