104 MULTIPLE ALGEBRA. 



where x is any quantity or function. We may also have occasion to 

 write A(x)+B(x) = D(x), or (A+B)(x) = D(x). 



But it is almost impossible to resist the tendency to express these 

 relations in the form T>A _ n 



A+B=D, 



in which the operators appear in a sense as quantities, i.e., as subjects 

 of functional operation. Now since these operators are often of such 

 nature that they cannot be perfectly specified by a single numerical 

 quantity, when we treat them as quantities they must be regarded as 

 multiple quantities. In this way certain formulae which essentially 

 belong to multiple algebra get a precarious footing where they are 

 only allowed because they are regarded as abridged notations for 

 equations in ordinary algebra. Yet the logical development of such 

 notations would lead a good way in multiple algebra, and doubtless 

 many investigators have entered the field from this side. 



One might also notice, to show how the ordinary algebra is be- 

 coming saturated with the notions and notations which seem destined 

 to turn it into a multiple algebra, the notation so common in the 

 higher algebra (a> ^ c)(x> y> g) 



for ax + by + cz. 



This is evidently the same as Grassmann's internal product of the 

 multiple quantities (a, b, c) and (x, y, z), or, in the language of 

 quaternions, the scalar part, taken negatively, of the product of the 

 vectors of which a, b, c and x, y, z are the components. A similar 

 correspondence with Grassmann's methods might, I think, be shown 

 in such notations as, for example, 



(a, b, c, d)(x, y) B . 



The free admission of such notations is doubtless due to the fact that 

 they are regarded simply as abridged notations. 



The author of the celebrated " Memoir on the Theory of Matrices " 

 goes much farther than this in his use of the forms of multiple 

 algebra. Thus he writes explicitly one equation to stand for several, 

 without the use of any of the analytical artifices which have been 

 mentioned. This work has indeed, as we have seen, been characterized 

 as marking the commencement of multiple algebra, a view to which 

 we can only take exception as not doing justice to earlier writers. 



But the significance of this memoir with regard to the point which 

 I am now considering is that it shows that the chasm so marked 

 in the second quarter of this century is destined to be closed up. 

 Notions and notations for which a Cayley is sponsor will not be 



