A Special Determinant. 371 



From this it follows that the co-factor of ( - a^y' in | N^D.+i | is 



a,.+^bl + a,{a„ ..., a,.J/t,., ..., a)-a,[h„ ..., h,.^,\h,.j^„ ..., h) (iii) 



— an expression which we may denote by 9,., so that 



Q, = aJ)\ + a\-2a,hfi,, 



9, = a^h\ + Sa'a, - 2a,b,b^ - afil, 



We can then establish the interesting result that 



|N,DJ=-ao9„ 

 |N,D3|=-r., 



INAh 



and consequently that 



fto 



e, 



fto 







e. 



a. 



> 



«o 



e, 



a. 



. 





9, 



a, 



«o 





©3 



a. 



a, 



+ a^92 -= I N.Dj I - 6i,|N,D,|, 



-^6^93 = |N,DJ- ajN,D3|+ 6^,a,|N,DJ, 



Viewed as a condition, therefore, the vanishing of 

 |NAI, |N,D3l, ..., |N,DJ 

 is exactly equivalent to the vanishing of 



9i, 9,, 93, ..., 9,i_i. 



e„ 



e, 



a. 





e, 



a. 



. 





03 



a. 



) 



63 



03 



a. 



a, 

 a. 





(IV) 



(V) 



(VI) 



But it is equally true that the vanishing of 9^, 9,, ..., 9„_i is 

 exactly equivalent to the vanishing of the determinants 



(VII) 



which differ from those in (IV.) in that a,, now everywhere takes the 

 place of a^_i except in the 9's : and as the latter determinants can 

 be shown to be equal to 



b, a, 

 b^ <f6o 



a. 



a, 

 b. b. 



&. 



a. 



. 



b. 



a. 



a. 



63 



b., 



b. 



6. 



a. 







b. 



dj 



a. 





*-^ 



«3 



«2 



a. 



a.3 



t: 



a. 



«2 



(VIII) 



