AND RELATED VANISHING, AGGREGATES OF DETERMINANT MINORS. 313 



terms |BC 4 |.|BF 6 | + 



of three, on which the identity * 



I i «A 



Those twenty-seven terms may be grouped into five sets 



aid 



a x d 3 j 



h c 2 I ! M2 



! <Y l -2 



a x | b. 2 c 3 d i J 



may be used, and six sets of two, on which the identity 



I h°3 I • I C A d i I ~ I 6 3 C 4 i • ! C 2 d S I = C 3 I b 2 C A I 



may be used ; and as one of the three-line determinants thus obtained vanishes, and the 

 others all have the cofactor B, the result is 



BriBC 5 E 6 | - jBC 4 F 6 | + jBC 4 G 5 l - 1 BE 3 F ti | + j BE 3 G. | - | BF 3 G 4 



+ | CE 2 F 6 | - j CE,G 5 1 + | CF,G 4 1 - | EF 2 G 3 



[' 



] 



When the sixty terms which form the final expansion of the cofactor of — B here 

 are regrouped into five sets of twelve, so as to give the cofactors of B, C, E, F, G, 

 Pfaff's second form of denominator is reached. 



(3) 



The determinant aggregate 



Do o 



«3&4 I - I «2 C 4 i + I h \ C i 



in which the sign of each determinant is dependent on the number of inverted pairs 

 in the row of integers consisting of the row-numbers of the determinant followed 

 immediately by the column-numbers, has been found t to be conveniently represent- 

 able by 



2 



1 2 

 3 4 



* It does not seem to have been noted before that 

















«i 



a 2 a 3 a i 



= llo 



a & 2 1 | a 



A ' 1 a A l 





















h 



b-2 b 3 b i 



2 3 4 



1 \c 



A I 1 C A 1 



| Co'-ll 1 























d 2 d 3 d 4 













and that therefore 



























1 a l b 2 c 3 d i 



= 1 



a t b 2 I i a x b 3 \ 



\ cA I 



1 a A 1 



i C 2 f? 4 



+ I \e 



A 1 f °A l 1 C A ' 1 



1 a 2 b 3 1 1 a 2 6 4 1 1 









similarly, that 













Cod^ 





a 3 b i \ \ ; 











a, a 2 a 3 



«4 



«5 



a 6 



"I 



a A 1 



a l b 3 | 1 a 1 b i \ 



1 a A 1 a A 1 



+ 1 I a 1 6 2 1 I a A 1 



V4I 



I aA | 



1 a A 1 





&! b. 2 b 3 



h 



b. 

 



h 





c 2 d 3 1 I c 2 d i 1 



c 2 d s | 1 c 2 d e 1 



' ' hf 3 ' 



02A 1 



1 ^2/5 1 



1 « 2 /6 1 





■ H C 3 



C 4 



c- 



c fi 







1 c 3 d i | 



1 c 3 d s l ' C A ' 





S3/4I 



l*s/sl 



1 eJa 1 





. d 2 d 3 



d, 



d- 



d e 









\ej & \ \ej s \ 







1 c 4 rf 5 1 



1 c A 1 





e, e 3 



«4 



e s 



e e 









1 Hft 1 









1 c 5 d 6 I 





• /. /a 



/* 



h 



/« 



















and therefore that 





















I a-JjoCgd^ 



5/6 1 



= 



1] | ajjo | 



1 <A 1 



' ?A 1 1 e J 



si KA | + l| 



\ « A 1 1 e 2 f 3 | | e 3 / 4 



"4^6 1 



1 <>A 1 



1 









+ II 1 cA I 



1 a A 1 



1 a-h e J 



> 1 1 «s/6 1 | + '| 



1 cA 1 1 02/3 1 1 ^3/4 1 



1 «4°5 1 



1 a B 6 e 1 



1 











+ l l 



«i/»l 



1 a 



A\ 



1 a 3 b i | | c±d 



, 1 1 C A 



\ + \ 



1 hfl 1 ' C 2<^3 



I I c 3 d t 1 



1 a A 1 



1 a h 1 



h 



and generally that a determinant of order 2m is expressible as a sum of m ! Pfaffians. When the Pfamans are expanded 

 the terms obtained are those given by Laplace's expansion-theorem. 



t Philos. Magazine (1884), xviii. pp. 416-427 ; (1902) (6), iii. pp. 411-416. 



