QUATERNIONS AND VECTOR ANALYSIS. 179 



which lends itself so well to such broader treatment, is worthy of it. 

 Of course, when vectors are treated by the methods of ordinary 

 algebra, linear vector operators will naturally be treated by the same 

 methods, but in an algebra formed for the sake of expressing the 

 relations between vectors, and in which vectors are treated as multiple 

 quantities, it would seem an incongruity not to apply the methods 

 of multiple algebra also to the linear vector operator. 



The dyadic is practically the linear vector operator regarded as 

 quantity. More exactly it is the multiple quantity of the ninth order 

 which affords various operators according to the way in which it is 

 applied. I will not venture to say what ought to be included in a 

 treatise on quaternions, in which, of course, a good many subjects 

 would have claims prior to the linear vector operator; but for the 

 purposes of my pamphlet, in which the linear vector operator is one 

 of the most important topics, I cannot but regard a treatment like 

 that in Hamilton's Lectures, or Elements, as wholly inadequate on 

 the formal side. To show what I mean, I have only to compare 

 Hamilton's treatment of the quaternion and of the linear vector 

 operator with respect to notations. Since quaternions have been 

 identified with matrices, while the linear vector operator evidently 

 belongs to that class of multiple quantities, it seems unreasonable to 

 refuse to the one those notations which we grant to the other. Thus, 

 if the quaternionist has e q , log q, sin q, cos q, why should not the vector 

 analyst have e*, log <!>, sin <, cos <, where <1> represents a linear vector 

 operator ? I suppose the latter are at least as useful to the physicist. 

 I mention these notations first, because here the analogy is most 

 evident. But there are other cases far more important, because more 

 elementary, in which the analogy is not so near the surface, and 

 therefore the difference in Hamilton's treatment of the two kinds of 

 multiple quantity not so evident. We have, for example, the tensor 

 of the quaternion, which has the important property represented by 

 the equation: T(^r) = T^Tr. 



There is a scalar quantity related to the linear vector operator, 

 which I have represented by the notation |$| and called the 

 determinant of 3?. It is in fact the determinant of the matrix by 

 which 4> may be represented, just as the square of the tensor of q 

 (sometimes called the norm of q) is the determinant of the matrix by 

 which q may be represented. It may also be defined as the product 

 of the latent roots of <3?, just as the square of the tensor of q might be 

 defined as the product of the latent roots of q. Again, it has the 

 property represented by the equation 



|$. = *||| 



which corresponds exactly with the preceding equation with both 

 sides squared. 



