22 Proceedings of the Uoijal Irish Academy. 



right-handed angle and the left-handed angle between two vector-screws 

 will be obvious from the following considerations : — 



The angle between two screws (i.e., not vector-screws) is an ambiguous 

 expression ; the angle may mean B, or 180° - 6, or 180° + Q, or 360° - d, just 

 as in the case of the angle between two lines. 



The angle between two vectors is not ambiguous to the same extent ; it 

 can only be or 360° - 0, if we agree tliat we are to measure the angle 

 between vectors diverging from their point of intersection. 



But it is worthy of special note that when the two vectors are not 

 in the same plane, as, for example, when they are the vectors on two non- 

 intersecting vector-.screws, we can distinguish geometrically one of the two 

 angles whose sum is 360° as the riglit-handed angle, and the other as the 

 left-handed angle. Thus the right-lianded angle between two vector-screws 

 which do not intersect is free from all ambiguity; and it may be any angle 

 between 0° and 360°. 



We are now to reconsider the fundamental problem which introduces the 

 virtual coefficient. This lies, indeed, at the commencement of our subject ; 

 but the expression of the virtual coefiBcient as it has been used Iiitherto lias, 

 unfortunately, Iwen sometimes haunted by ambiguity as to wliich of tlie two 

 angles between the two screws was to be understood. This ambiguity is 

 henceforth removed. Tlie two screws involved each leceives the addition of 

 the unit- vector, by which they are transformed into vector-screws ; and then 

 the convention is established that the right-handed angle between the two 

 vector-screws is the angle to be employed in the expression of the virtual 

 coefficient. No doubt this had been to some extent implied in the paper* 

 already referred to ; but it is now for the first time explicitly stated. The 

 deduction of the expression for the virtual coefiicient will here be set forth 

 not as it was originally given.t nor as it was given many years later in a 

 general treatise^ on the Theory of Screws, but as it should have been 

 given. 



The subject had been always troublesome ; and when I saw at last what I 

 ought to have seen at first, it was plain that a difficulty had been removed 

 from the foundation of the Theory of Screws. I therefore desire that this 

 emendation shall have a place in the series of memoiis which the Academy 

 have so kindly received from me for so many years. 



The problem is as follows : — It is required to find an expression for the 

 work done when a body makes a small twist of amplitude a about one 



* Trans. Eojr. Ir. Acnd., Tol. xxxii., pp. 109-115 (1902). 

 + Trans. Roy. Ir. Acad., vol. xxv., p. 167 (1871). 

 X "Treatise," p. 17 (1900). 



