macfarlane: algebra of physics 



335 



axes of the first four orders. These are given in the first place 

 by the homogeneous products: 



and in the second place by inserting the conjugate forms. At 

 that time I supposed these expressions to denote compound axes 

 and the elements to be orthogonal. It was only after much subse- 

 quent study that I discovered the true nature of these expres- 

 sions as compound units. 



In connection with this I used a pole notation; a/3 was used to 

 denote the axis which is perpendicular to a and to /3, or with greater 

 truth the conjugate axis of their plane. As the vinculum is 

 inconvenient to print, it is better to use a special form of bracket 

 as [a/3]. This notation expresses unambiguously any complex 

 pole, such as [[a/8] 7] or [[a/3] [y8]]. 



A quaternion was defined, not as the ratio of two vectors sim- 

 ply, but as the ratio of two vectors when they are free to rotate 

 in their own plane. The former definition — the ratio of two vec- 

 tors — was held to be the true definition of Gibb's dyad. A qua- 

 ternion was analysed into tensor, axis, and amount of angle, and 

 the relation of these elements was expressed, for example, by 



#i = a a q 2 = b /3 6 



The axis was supposed to be defined by two elements, and in 

 contrast to Hamilton (Elem. Bk. II, Art. 130) the angle was not 

 restricted to being less than w, but on the contrary might have 

 any magnitude. 



As a consequence the square of a quaternion was defined by 

 q 2 = a 2 or a , which Hamilton could not logically do, because if a 

 were greater than t/2, then 2a would be impossible. Similarly 

 the square-root was defined definitely by 



f = V' a 



a 



a/2 



