TIIKORY OF THE I'AI.TITK >NS OF NUMBERS. 

 prbs, equal to the determinant 



365 



and this determinant is clearly 



A 3 



B 3 



a;C 3 

 C,(l+2) 



Art. 27. This is evidently a perfectly general process and suffices to establish that 

 a solution of the functional equation of order n is a determinant of order u of which 

 the constituent in the s lh row and t ili column is 



r \ } (at +s-t+l) ... K-f n-t) . (l-s+t+1) ... (l+t-1) ; 

 and when this determinant is divided by 



we have, as will be proved, the expression of 



Art. 28. To establish this we may apply a series of tests. 

 Thus, take the expression of IL (I ; a, b, c, d) 



x(l)(a+2)(a+3), (l+l)(b+l)(b+2), (l+2)(o) (c+1) , x (1+3) (d-1) (d) 



divided by 



It clearly reduces to IL ( oo ; a, b, c, d) when I is put equal to < . 



Moreover, it can be shown that when d = c = b = a, the determinant involves 

 I and a always in the combination l + a; for consider the determinant in question 

 when I is put equal to a. 



