462 



Notices respecting New Books. 



in view of a second edition of their book being called for. They 

 will find all the needful passages of Leibnitz, Gauss, Cauchy, &c, 

 •carefully reprinted in the ' Proceedings of the Royal Society of 

 Edinburgh,' vol. xiii. pp. 547-590, xiv. pp. 452-518.' 



The next section or " book " consists of eight chapters — the first 

 dealing with the Adjugate determinant, the second with Symmetric 

 determinants, the third with Skew determinants, the fourth with 

 " determinates multiples" the fifth with Circulants, the sixth with 

 Alternants, the seventh with Continuants, and the eighth with 

 Functional determinants. Most of these chapters are very complete 

 and satisfactory, indeed in no foreign text-book is the treatment of 

 special forms more methodical and exhaustive. The only chapter 

 to which objection can be taken is the fourth, but the objectionable 

 part is so fundamental that no mere improvement in the details 

 would do much good. As the term " multiple " determinant is 

 unknown to English-speaking and German mathematicians, the 

 definition with which the chapter opens must be turned to for in- 

 formation. It is as follows : — " A determinant which has each of 

 the elements of a single row or of a single column equal to unity is 

 called a multiple determinant." Whatever doubt may exist about 

 the convenience of the new name thus introduced, there can be 

 absolutely none about the inconvenience of the notation ushered 

 in along with it. The determinants 



1 



a * \ 



1 a x 

 1 \ 

 1 c. 



we are told, are denoted by 



a \ ^i 



\ 





"l 



<*> 



2 



> 



\ 



h 







C l 



C 2 



As an abridgment this notation is clearly not of the slightest 

 consequence ; but a much more serious objection is to be found in 

 the fact that it has been used for many years by the best mathe- 

 maticians in a totally different sense. Still, bad as this may be, 

 it is as nothing compared with the almost incredible blunder of 

 using the notation in the new sense on one page and in another 

 sense on the next. Eor example, according to the definition we 

 have 



-{ 



«, 



&1 





c, 



a. 





\ 



c i 



1 



I 



+ 



1 





+ 



i 



K 





a t 



\ 





c* 



% 





c* 



}', 



