180 



QUATERNIONS AND VECTOR ANALYSIS. 



There is another scalar quantity connected with the quaternion and 

 represented by the notation Sq. It has the important property 

 expressed by the equation, 



S (qrs) = S (rsq) = S (sqr), 



and so for products of any number of quaternions, in which the cyclic 

 order remains unchanged. In the theory of the linear vector operator 

 there is an important quantity which I have represented by the 

 notation < B , and which has the property represented by the equation 



where the number of the factors is as before immaterial. <1> S may be 

 defined as the sum of the latent roots of <1, just as 2Sg may be 

 defined as the sum of the latent roots of q. 



The analogy of these notations may be further illustrated by 

 comparing the equations 



and 



* 



I do not see why it is not as reasonable for the vector analyst to 

 have notations like |< and 3? s , as for the quaternionist to have the 

 notations Tq and Sq. 



This is of course an argumentum ad quaternionisten. I do not 

 pretend that it gives the reason why I used these notations, for the 

 identification of the quaternion with a matrix was, I think, unknown 

 to me when I wrote my pamphlet. The real justification of the 

 notations | 3> | and 3? g is that they express functions of the linear 

 vector operator qua quantity, which physicists and others have 

 continually occasion to use. And this justification applies to other 

 notations which may not have their analogues in quaternions. Thus 

 I have used 3> x to express a vector so important in the theory of the 

 linear vector operator, that it can hardly be neglected in any treatment 

 of the subject. It is described, for example, in treatises as different as 

 Thomson and Tait's Natural Philosophy and Kelland and Tait's 

 Quaternions. In the former treatise the components of the vector 

 are, of course, given in terms of the elements of the linear vector 

 operator, which is in accordance with the method of the treatise. In 

 the latter treatise the vector is expressed by 



Voo'4-W+Vyy. 



As this supposes the linear vector operator to be given not by a 

 single letter, but by several vectors, it must be regarded as entirely 

 inadequate by any one who wishes to treat the subject in the spirit of 

 multiple algebra, i.e. to use a single letter to represent the linear 

 vector operator. 



But my critic does not like the notations | < |, <l s , < x - His ridicule, 





