234 CHAPTER X. 



The matrix of the coefficients of H is made up of (r + I) 2 rectangles, 

 of which the general one EIJ is of height a f and of base %. Let t 

 be the smaller of the integers i, j or their common value if i = j. 

 Then EIJ includes at its left or bottom a square array S t of coeffi- 

 cients a t to a side. The coefficients in its diagonal are all equal; 

 likewise those in any parallel to the diagonal. All the coefficients 

 in Rij which lie above or to the right of the diagonal of the square 

 S f are zeros. 



219. The results of 218 will be applied only in such simple 

 cases that the determinant D of H Q can be simplified by inspection. 

 It will therefore be sufficient to state without proof 1 ) the simplest 

 expression which can be given to D. Our notations may be fixed 

 so that a, *> > > & !> > &/-4-1 - Let 



where 



The determinant D equals DfcDf* . . -D^*, where, if (i,j} = 



'< -i\ f-\ A I l\ /-j o A I 1^ ("i 5 A A I 1 ^ 



X* J. J \^1 "^1 ~l / \ 7 "^1 "T" / \"^1 *M. ^1 -^J-1 "|" Xy 



-4 + 1, i) (1,4-4 + 1, i) ... (1,4-4 + 1, 1,4-4 



. . . (1,4+1,4-4+1, 1,4+1,4-4+1) 

 (1,4+1,4+1, 1,4+124+1) . . .(1,4+1,4+1, 1,4+1,4+1,4-4+1) 



,-4+1,1,4+1,4+1) . . . (1,4+1,4+1,4-4+1, I^^A^I^-A^ 



Since the coefficients dy are functions of K Q , a root of an 

 equation of degree Jc belonging to and irreducible in the 6rF[jp*], 

 the number of sets of values for the $ coefficients entering DI O for 

 which this determinant is not zero is ( 99) 



Excluding the coefficients of H Q which are always zero, there 

 remains the following number of distinct coefficients d/^: 



1) A method of proof is given by the author in the American Journal, 

 vol. 22, pp. 133134. 



