34 THE THEORY OF SCREWS. [31- 



The facilities presented by rectangular axes for questions connected with 

 the dynamics of a particle have perhaps their analogues in the conveniences 

 which arise from the use of co-reciprocal sets of screws in the present 

 theory. 



If the six screws of reference be co-reciprocal, then the formula of the 

 last section assumes the very simple form 



32. Co-ordinates of a Wrench. 



We shall henceforth usually suppose that the screws of reference are 

 co-reciprocal. We may also speak of the co-ordinates of a wrench*, meaning 

 thereby the intensities of its six components on the six screws of reference. 

 So also we may speak of the co-ordinates of a twist, meaning thereby the 

 amplitudes of its six components about the six scretvs of reference. 



The co-ordinates of a wrench of intensity a&quot; on the screw a are denoted 

 by / , ... fi &quot;. The co-ordinates of a twist of amplitude a about a are 

 denoted by a/, . . . a B . 



The co-ordinates of a twist-velocity a about a are denoted by a 1? cL, ... d ti . 

 The actual motion of the body is in this case a translation with velocity ap a 

 parallel to a and a rotation around p with the angular velocity d. 



33. The Work done in a twist of amplitude a about a screw a, by 

 a wrench of intensity ft&quot; on the screw ft, can be expressed in terms of the 

 co-ordinates. 



Replace the twist and the wrench by their respective components about 

 the co-reciprocals. Then the total work done will be equal to the sum of 

 the thirty-six quantities of work done in each component twist by each 

 component wrench. Since the screws are co-reciprocal, thirty of these 

 quantities disappear, and the remainder have for their sumf 



34. Screw Co-ordinates. 



A wrench on the screw a, of which the intensity is one unit, has for its 

 components, 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 a lt ... a 6 . 



* Pliicker introduced the conception of the six co-ordinates of a system of forces Phil. Trans., 

 Vol. CLVI. p. 362 (1866). See also Battaglini, &quot; Sulle dinami in involuzione,&quot; Atti di Nnpoli iv., 

 (1869); Zeuthen, Math. Ann., Band i. p. 432 (1869). 



t That the work done can be represented by an expression of this type was announced by 

 Klein, Math. Ann. Band iv. p. 413 (1871). 



