A Treatise on the Theory of Screws

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 pa.a"i=piafi + ... + pX'2- 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 screws of reference. The co-ordinates of a wrench of intensity a" on the screw a are denoted by a/', ... a,". The co-ordinates of a twist of amplitude a about a. are denoted by «/, ... aB'. The co-ordinates of a twist-velocity å about a are denoted by a,, d.,, ... The actual motion of the body is in this case a translation with velocity åpa parallel to a and a rotation around p with the angular velocity a. 33. The Work done in a twist of amplitude a about a screw a, by a wrench of intensity ß" on the screw ß, 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 surnf ‘ip^ß "+ ... + Zp^ßf. 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 an ... a6. * Pliicker introduced the conception of the six co-ordinates of a system of forces—Phil. Trans., Vol. CLVi. p. 362 (1866). See also Battaglini, “ Sulle dinami in involuzione,” Atti di Napoli iv., (1869); Zeuthen, Math. Ann., Band i. p. 432 (1869). * That the work done can be represented by an expression of this type was announced by Klein, Math. Ann. Band iv. p. 413 (1871).