626 



DR THOMAS MUIR ON THE GENERATING FUNCTION 



We note that here as in the five other cases (1) the letters of the coefficients are b, c, d, 

 (2) that the suffixes of these letters are the suffixes of the associated x's ; and we note 

 that (3) in this particular case, as a glance at the elements suffices to show, b s , c 2 , d± 

 do not occur. Now the diagonal elements of the quadrate array of the equivalent 

 bipartite form are the coefficients of 





O* • T* 1"* 



X, • X,Xa 



and therefore in accordance with the conditions just noted must be 



\e 1 d l , a x h 2 d 2 , b^d^ , d 1 b i c i . 

 Next we note that because of the in verso-symmetry (4) the product of any two 

 conjugate elements of the quadrate array is equal to the product of the two 

 diagonal elements which occupy the same rows and columns. Using this we take the 

 product of any two diagonal elements, say the 3rd and 4th, viz., 



and rearrange the suffixes so as to have two sets of 1,3,4, these being the suffixes of 

 x 1 ' x z x± , x x * a; 4 a?3 . The result b-^c^d^ , b^d-^ and others similarly obtained gives us 



whence we have as before 



x l 



x 2 



x g 



x i 



b^di 



6 1 c 1 cf 2 



b l c A 



bs^ 



b c 1 d 1 



h 2 C \ d -2 



\ e i d s 



h C l d 2 



Vs (/ i 



b x c 3 d 2 



VA 



b i c i'h 



VA 



h c A 



Vs d i 



\ Ci d x 



•"1 

 x 



x„ 



1 



'L 



d -l 

 d, 



d„ 



d„ 



d, 



d z 



b l 

 b. 



b^c^d-y 



b 2 c x d 2 



x 3 ■ bjcjl 

 x, • lx,di 



(19) It deserves to be noted in passing that in no case is the inverso-symmetry 

 unique, any one of the non-diagonal elements of the first row of the array being 

 replaceable by another quotient, provided the corresponding change be made in the 

 first column. Thus, instead of d 2 /d 1 in the case just dealt with, we may write bjb-^ 

 provided we alter the conjugate element into bjb 2 . 



(20) Jfu 1 , u 2 , . . . , u 5 be used to stand for a^ + . . . + a b x 5 , \x x + . . . + b b x 5 , . . . 

 6^+. . . +e 5 x 5 , there are twenty-four determinants of the type 



u 2 ~ "2 X 2 



— b^x 3 



- h X i 





-h x 5 



- c 2 x _ 



U 3 ~ C 3 X 3 



" 'k X 4 





~ C b X 5 



— CC^Uj^) 



— d 3 x s 



"i ~ d 4 .<: 4 





~ d 5 X 5 



~ e 2 X 2 



~ e 3 r 3 



- e 4 * 4 



U o 



~ e b X 5 



