SCREW CO-ORDINATES. 33 



reciprocal, thirty of these quantities disappear, and the 

 remainder have for their sum* 



2/ l n 1 '/3i"+&C. + 



36. Screw Co-ordinates. A wrench on the screw a, 

 of which the intensity is one unity has for its compo- 

 nents, on six co-reciprocal screws, wrenches of which 

 the intensities may be said to constitute the co-ordinates 

 of the screw a. These co-ordinates may be denoted by 

 eti, &c., a 6 . 



When the co-ordinates of a screw are given, the screw 

 itself may be thus determined. Let i be any small 

 quantity. Take a body in the position A, and impart to 

 it successively twists about each of the screws of refe- 

 rence of amplitudes mi, ia zy &c., ia 6 . Let the position thus 

 attained be B ; then the twist which would bring the 

 body directly from A to B is about the required screw a. 



37. Identical Relation. The six co-ordinates of a 

 screw are not independent quantities, they fulfil one 

 relation, the nature of which is suggested by the relation 

 between three direction cosines. 



When two twists are compounded by the cylindroid 

 ( 1 7), it will be observed that the amplitude of the result- 

 ant twist, as well as the direction of its screw, depend 

 solely on the amplitudes of the given twists, and the 

 directions of the given screws, and not at all upon either 

 their pitches or their absolute situations. So also when 

 any number of twists are compounded, the amplitude and 

 direction of the resultant depend only on the amplitudes 

 and directions of the components. We may, therefore, 

 state the following general principle. If n twists neu- 

 tralize (or n wrenches equilibrate) then a closed polygon 



* That the work done can be represented by an expression of this kind was 

 stated by Dr. Klein, Math. Ann., Band iv., p. 413. 



D 



