96 MULTIPLE ALGEBRA. 



of many remarkable papers on the same subject (which might be 

 more definitely expressed as the algebra of matrices) in various 

 foreign journals. 



It is not an accident that this century has seen the rise of multiple 

 algebra. The course of the development of ideas in algebra and in 

 geometry, although in the main independent of any aid from this 

 source, has nevertheless to a very large extent been of a character 

 which can only find its natural expression in multiple algebra. 



Our Modern Higher Algebra is especially occupied with the theory 

 of linear transformations. Now what are the first notions which we 

 meet in this theory ? We have a set of n variables, say x, y, z, and 

 another set, say x', y\ z', which are homogeneous linear functions of 

 the first, and therefore expressible in terms of them by means of a 

 block of n 2 coefficients. Here the quantities occur by sets, and invite 

 the notations of multiple algebra. It was in fact shown by Grass- 

 mann in his first Ausdehnungslehre and by Cauchy nine years later, 

 that the notations of multiple algebra afford a natural key to the 

 subject of elimination. 



Now I do not merely mean that we may save a little time or space 

 by writing perhaps p for x, y and z ; p for x', y' and z' ; and <3? for a 

 block of 7i 2 quantities. But I mean that the subject as usually treated 

 under the title of determinants has a stunted and misdirected develop- 

 ment on account of the limitations of single algebra. This will appear 

 from a very simple illustration. After a little preliminary matter 

 the student comes generally to a chapter entitled " Multiplication of 

 Determinants," in which he is taught that the product of the deter- 

 minants of two matrices may be found by performing a somewhat 

 lengthy operation on the two matrices, by which he obtains a third 

 matrix, and then taking the determinant of this. But what signifi- 

 cance, what value has this theorem ? For aught that appears in the 

 majority of treatises which I have seen, we have only a complicated 

 and lengthy way of performing a simple operation. The real facts 

 of the case may be stated as follows : 



Suppose the set of n quantities p to be derived from the set p by 

 the matrix <1>, which we may express by 



/>'=$./; 



and suppose the set p" to be derived from the set p by the matrix "&, i.e., 



and p" = ^^.p', 



it is evident that p" can be derived from p by the operation of a 



single matrix, say 9, i.e., 



i i* 



so that 9 = ^.$. 



