Ball — Contributions In the Tlieori) of Screws. 25 



moment at J' is cancelled by the third term in tlie viitual moment at Q, 

 and that the final result is 



«'/3"{(i'<. + Pp)cosO - dHinlll. (1) 



We have next to consider the efl'ect of an interchange which, instead fif 

 assigning the wrench to /3 and the twist to a, makes the twist belong to /3 

 and the wrench to a. The problem may be enunciated as follows : — 



It is required to find the virtual moment when a biody receives a twist of 

 small amplitude j3' about the vector-screw (3 ; while at the same time it is 

 acted upon by a wrench of intensity a" about the vector screw a. 



The answer is obtained by the interchange of a and (5 in the expression 

 just proved for the virtual moment. It is first of all to be noticed that this 

 interchange does not alter the expression 



(/'« + P^) '^os Q - tJ sin Q 



in any way whatever, for {p^ x pf^) is of course unchanged, and remembering 

 the definition of the right-handed angle between two screws, it is easily seen 

 that the right-handed angle between a and /3 is also the right-handed angle 

 between /3 and o. Thus the interchange of a and /3 is devoid of effect on 0, 

 nor is d altered, for this is a signless quantity. As the quantity 



(^)„ + i^p)cos & - (7 sin 9 



is unaltered by the interchange of a and /3, the only alteration in the virtual 

 moment caused by the interchange of a and j3 lies in the factor outside the 

 bracket, which becomes a"/3' instead of a'/3". Thus the required result is 



a"l3'i(i'»+i^^)cos0-rfsinOl. 



The virtiial coefficient is the name given to that symmetrical frmction of 

 two vector-screws which is expressed by 



'^ap- + AfCl'a +2'/3)cos0- f^sinflj, (2) 



where is the right-handed angle between the two screws. 



This has of course been the expression so long used in the Theory of Screws. 

 The particular point now brought out is that to make the virtual coefficient 

 so written universally valid exactly as it stands, it is necessary that the two 

 screws shall be vector-screws, and that Q shall be the right-handed angle 

 between them. 



If the vector-screw /3 coincides with the vector-screw a, then p^ =• pa, 

 6 = 0, d = 0; and, consequently, the virtual coefficient reduces simply to ^;,. 

 That this reduction shall take place is the principal reason why the factor | 

 has been introduced into the function which defines the vii-tual coefficient. 



