A Special Determinant. ^11 



the number of terms in the final development must be 2"~'. Further, 

 since the second determinant on the right is got from the first by 

 interchanging the suffixes 1 and 2, it follows that when we have got 

 2"~' terms of the development the other half of the terms may be 

 obtained by making the said interchange. We thus derive the 

 following rule for writing out currente calamo the full development. 

 Write first the principal term ajj^c-^ ... ijn-i^n • interchange n — 1 and 

 n and a second term is got; iiiterchange n-2 andn-1 in the pre- 

 ceding tivo terms and other tiuo are obtained : interchange n-3 and 

 n-2 and the number of terms is again doubled : and so on. (xx.) 



Of course with every interchange there is a change of sign. Thus 

 the four-line determinant above is equal to 



a^b^c^d^': —^aib^c^d^: —ajb^c^d^-^-a^b^c^d^l 

 : —ajj^c-^d^^ -{-aj)^c^d^ -{-ajj^c^d^ — a^b^c^d^ \ . 



Again, we may view the terms of the development according to the 

 number of the elements of the minor diagonal a^, b^, c^, ... which 

 they contain. The term containing none of these elements is 



aAc/J^, 

 the terms containing only one are 



— a^.b^-c/l^ — bya^c^d^ — c^.ajj^d^ 

 the terms containing only tivo are 



+ a^hyC^d^ + a^c^.b^d^ -\- b^c^.a^d^, 

 and the term containing all three is 



— a2b^c^.d^. 

 There being only one term for each of the combinations 



a^, h., c^ ; a^by a,c^, b^c^ ; a,bx^, 

 the total number of terms is thus 



i+a_,,+c„_,3+ ... +c„-,„-. 



i.e., 2«-' 

 as before. 



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