26 Proceedings of the Royal Irish Academy. 



The convenience arising from our adoption of the right-haiided aiu/le 

 between two vector-screws, as one of the elements indicating their relative 

 position, may be illustrated by showing the inconvenience that would arise if 

 we adopted another angle which may or may not be the right-handed 

 angle, namely, the acute angle between the two screws. 



In fig. 5 the acute angle between a and /3 is 9, whether a be above /3 or 

 /3 above a. But, in the former case, the wtual coefficient would be 



5 \(pa + Pp) cosO - d sin 0\ , 



and, in the latter, it can easily be proved that the virtual coefifi- 

 cient would be 

 F'°- 5- i ! (/?« + Pp) cos + rf sin e}. 



We avoid this change of sign in the second term by agreeing that the angle 

 to be employed in the expression of the virtual coefficient shall be the right- 

 handed angle. When a is above /3, this angle is 6 ; but when a is below fi, 

 the angle is 360' - 6. Thus the sign of the second term in the virtual 

 coefficient is always to be negative, and ambiguity is escaped when the 

 angle is understood to be the right-handed angle between the two vector- 

 screws. Of coui-se ambiguity would have been equally escaped by consistently 

 taking the loft-handed angle for 0. Wc have preferred to take the right- 

 handed angle, because the form that it gives to the virtual coefficient is that 

 with which wc liad already been familiar. 



Owing to the symmetry of the virtual coefficient as respects the two 

 screws to which it relates, we are enabled to make the following statement : — 



If a wrench on a screw /3 does not disturb the equilibrium of a body only 

 free to twist about a screw a, then, conversely, a wrench on a screw a will not 

 disturb the equilibrium of a body only free to twist about /3. 



In both cases bj.^ = 0, and the screws a and /3 arc said to be reciprocal.* 

 It is abundantly shown in the precetling memoirs that the doctrine of 

 reciprocal screws is fundamental in the present theory. 



We have hitherto been discussing the right-handed angle between two 

 wrfar-screws; but we may sometimes find it ctmvenient to introduce tlie 

 notion of the right-handed angle between two screws (i.e. not necessarily 

 vector-screws), presuming only that in this more general case the right- 

 handed angle may Ije ambiguous to the extent of any integral number of 

 multiples of 180' ; whereas the right-handed angle between two vcdor-Bcrev/H 

 never becomes ambiguous so long as they do not intersect. (Of course we 

 do not now count integral multiples of 360°.) 



• "Treatise," p. 26. 



