628 



DR THOMAS MUJR ON THE GENERATING FUNCTION 



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where the square array is inverso-symmetric. Each of the four other sets can be 

 similarly represented : but it is curious to note that the five arrays so obtained are 

 neither all alike nor all different. In fact, if we use E to stand for that just written, the 

 second and third sets of twenty-five terms are respectively 



V ! 2 





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C 2 ( *i e i 

 C 2**2 e 2 



c 2 d 3 e 3 



C 4^2 6 4 

 «5^5 e 2 » 



b 3 z 3 - 



E 



while the fourth and fifth are 



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a; 2 • c 1 a 2 e 2 



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 *^4 * ^4^*j^4 



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 X3 • CjWgeg 



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the square array of the fourth differing from E in the place (2 , 5) and that of the fifth 

 in the places (1 , 2) , (2 , 3) , (2 , 4). 



The cofactors of & 1 a? 1 and b H x 3 , it will be seen, are identical, — a fact which would 

 have appeared in the preceding paragraph if the expressions for them as determinants 

 of the third order had been given in the case of both. 



(22) The general character of the theorem which includes §§ 17, 20 as special cases is 

 readily apparent, the number of terms in F 1 for the next case being 6 4 , and generally 

 n' 1-2 ; and this is exactly the number which the supposed theorem of Jacobi required, as 

 the following considerations will show : — The expansion of u x u 2 . . . u n is an aggregate 

 of n n terms all positive : and as in j « 1 ?> 2 c 3 . . . | • x x x 2 . . . x n there are \{nf) positive terms 



