Ball — Contributions to the Tlimri/ of Hcreivx. 10 



(2) That if wo conceived the vector to be lyhig upon the Eaitli's axis, 

 and pointing in the same direction as the vector from the Earth's 

 centre to the North Pole, then a right-handed rotation of the body 

 will tnrn it in the same direction as that in which tlio liarth is 

 rotating. 



As a convenient method of rememliering the relation of a right-handed 

 rotation to the vector about which the rotation has been performed, we may 

 note that the direction in which the numbers increase on the face of a watch 

 is right-handed with regard to a vector from the dial to the hack. 



The conception due to Chasles, that any movement of a rigid body is a 

 twist about a screw, is, of course, a fundamental principle in the Theory of 

 Screws. The word " twist " was defined with regard to a screw at the 

 beginning of the original memoir* with which the present series commenced. 

 At present we are concerned with vector-screws rather than screws ; and it 

 has become necessary to explain how the vector-screw introduced in the 

 present paper for the first time enables an absolutely precise specification of 

 a twist to be made, and thus attention is called to the fact that any 

 specification of a twist with regard to a screw {i.e., not a vector-screw) must 

 be necessarily defective in one detail. 



A twist may be completely represented by a vector-screw a (of pitch p^ 

 and a scalar a which is termed the amplitude of the twist. The twist is 

 produced by compounding — (1) a rotation, and (2) a translation. 



1. The rotation is right-handed or left-handed about the vector, 

 according as the given scalar a is positive or negative. 



The angular magnitude of the rotation is o' radians. 



2. The translation is to be in the same direction as the vector, or in the 

 opposite direction according as a''p„, is positive or negative. 



The linear magnitude of the translation is 



If ]y^ = 0, the twist is simply a rotation. If p^ = » and the twist is 

 to be finite, then a = 0, and tlie twist is a translation. 



In the Theory of Screws, a is always regarded as a small quantity. 

 When this is the case, the result of the composition of any number of twists 

 is independent of the order of their application. 



If a rigid body is not at rest, its movement must be at every movement a 

 twisting motion about some instantaneous vector-screw. 



An instantaneous twisting motion may be completely represented by a 

 vector-screw a of pitch p^, which may be either positive or negative, and a 



Trans. Itoy. Iv. Auad., vol. xxv., p. 159 (1S71). See also "Treatise," p. 



