620 



DR THOMAS MUIR ON THE GENERATING FUNCTION 



and as the expansion of each of the four fractions under the sign of summation contains 

 in every term a positive power of x-y , it follows that the cofactor of x~ x~ x~ x~ in the 

 expansion of | a 1 b 2 c 3 d i \ lu^u^u^u^ is 1, as was to be proved. 



(7) The construction of the vanishing determinant on which the proof rests is easily 

 completed when the diagonal elements are given ; in every case, however, there is 

 more than one set of diagonal elements suitable for the purpose. The requirements are 

 (1) that the elements chosen from | a-J) 2 c 3 • • • | to appear in the diagonal of the 

 vanishing determinant shall form a positive or negative term of | a 1 b 2 c s . . . j according 

 as the number of these elements is odd or even : and (2) that no suffix shall be in the 

 place which it occupied in the original diagonal. 



The first requirement is due to the need for having the sign of | a-\b 2 c% • • • ! - x \ x 2 x z ■ ■ • 

 negative at the end of the expansion of the vanishing determinant : and the second to 

 the need for having a positive power of one of the x's iu every term, except one, of 

 the expansion of the subsequent fractions. 



Both requirements are seen to be satisfied by taking the suffixes 1, 2, 3, . . . , n and 

 moving either the last to the first place or the first to the last place : for then the 

 number of inverted-pairs in the suffixes is n — 1 and no element is in its original place. 

 In the case where n = 3, these are the only possibilities : in the case where n = 4 there are, 

 however, four others, viz., a 2 ,b 4 , c : , d s ; a s , b x , c 4 , d 2 ; a z ,b i ,c 2 ,d l ; a 4 , b 3 , c 1 , d 2 . 



(8) For the purpose of clearing up Jacobi's mode of procedure, let us examine the 

 new proof as applied to his second case. The two vanishing determinants are 



u l 



a % X 2 



-b 2 x 2 



-c,x 2 





U \ ~ a 3 X 3 



"3 X 3 



c s x 3 





- a z x 3 



U 'l ~ "Z X 3 



- fft 





- a x x x 



U 2 - ^ 



-Vi 





- afr 



— b-iX-y 



"■i - < : v'\ 



3 



- Cl 2 X 2 



- b 2 x. 2 



M 3 ~" c r l '-2 



their expansions being 



o o 



M 1«2 M 3 - 2"l M 2 • C \ X l + 2 M 1 ' I ^3 C 1 I • X 3 X \ - I a 2 b B C l I • X 2 X 3 X \ » 



o o 



WjW.^g - ^U X U 2 ■ <\,.r 2 + 2 mi U \ ' I ^'l C 2 I • X \ X 2 ~ I a "J'\ C 2 I ' X Z X \ X -2 > 



and the resulting identities 



"iV 3 1 



u,u.,u. 



l"^ 



i/y-i tls,fljt} 



2 



+ 



2^ 



'.) IA/Q 



a^A 1 = _J ^ c 2 + ^C? 1 \ C 2 1 



U i' t 2 U 3 X A X 2 X^ ^m U 3 ' X l X Z J*=A U 2 U 3 ' X 3 



the one being as suitable as the other for establishing the theorem regarding the co- 

 efficient of x^x^Xg 1 . If now, however, we combine the last two terms of each, there 

 results 



! «iV 3 



tC-l UJC) tin 



U-.U.M., 



1 



■>\ X -> X 3 



x 2 z s ■ u 2 u 3 



o 



2' 



