418 Dr. T. Muir on an overlooked Discoverer 



symbols for which we now should use 



the letter A being accompanied by two suffixed numbers, the 

 second of which is itself a letter with a suffix. 



A determinant with elements of this kind is denoted by the 

 principal term enclosed within the brackets 



ii ) 5 



thus 



»i «2 a i *a »2 a i 



| Aj A 2 ) ze Ai Ag — A| A 2 . 



This notation, and no other, is used throughout the whole 

 treatise. As the name " Producte mit Yersetzungen " implies, 

 the view r taken of a determinant is as an aggregate of terms 

 one of which gives rise to all the others by permutation of 

 suffixes : naturally, therefore, determinants with elements 

 other than those specified above do not occur, and "rows" 

 and " columns " are things unthought of. 



The definition adopted is a statement of the recurring law 

 of formation — that is to say, the law of development of a de- 

 terminant of the nth order in terms of determinants of the 

 (n — l)th order; a definition, by the way, recently employed 

 by Professor Cayley in his article on Determinants in the 

 Encyclopcedia Britannica. 



Immediately following the definition are given careful 

 enunciations and proofs of the following theorems : — 



I. The suffixes being allowed to remain unchanged, the 

 superfixes {die oberen Elemente) are interchanged in every 

 possible way to obtain the full development. 



II. The sign preceding each term is dependent upon the 

 number of interchanges of places necessary to arrive at the 

 term. 



III. If two adjacent superfixes be interchanged, the sign of 

 the determinant is altered. 



IV. If a superfix be moved a number of places to the right 

 or left, the sign of the determinant is changed or not accord- 

 ing as the number of places is odd or even. 



V. If several superfixes change places, the sign of the de- 

 terminant is altered or not according as the number is odd or 

 even which indicates how often a smaller superfix follows a 

 greater. 



VI. If in any term the number which indicates how often 

 a smaller superfix follows a greater be even, the sign preceding 

 the term must be positive, and if the number be odd the sign 

 must be negative. 



VII. Instead of keeping the suffixes unchanged and inter- 



