70 
Proceedings of Royal Society of Edinburgh. [sess. 
Quaternion Binaries: an Extension of Quaternions to 
give an Eight-element System applicable to Ordinary 
Space. By Dr W. Peddie. 
(Read November 18, 1901.) 
{Abstract.) 
In this system vectors are regarded as translators only. A 
special operator, R, transforms them into rotors. A second appli- 
cation of the operator retransforms rotors into translators. The 
system is essentially Hamilton’s, with the removal of the restric- 
tion that vectors shall act as translators in addition, and as rotors 
in multiplication. The quantities i, j, k being unit rectangular 
vectors, the fundamental equations may be wiitten 
R ij = k , V\jk = i , R hi =j ; 
^2 = J2 = 7 ,2 = _ i . R2 = i . 
In the present paper, the fundamental properties of a quaternion 
binary B = q + Rr , q and r being quaternions, are investigated, the 
applications being restricted to the theory of screws, — in particular, 
of screws upon a cylindroid. The related binary RB, and the 
corresponding cylindroid, are discussed specially. 
The relation of the above system to that of Clifford, as developed 
by Macaulay in his Octonions, is evident from the consideration 
that Macaulay’s operator fi, while it transforms a rotor into a 
translator, destroys a translator, so that the transforming action of 
O is restricted to rotors. In other respects, O and R both act as 
scalars, the former as an infinitesimal scalar, the latter as a unit 
scalar. 
(Issued separately March 8. 1902.) 
