A Special Determinant. 



are obtainable for all the other merabers of the array 

 1 N,D, !, 



iN.DjI, |N,D,5l, 



|N.D,|, |N,D,|, IN3DJ, 



IN.DJ, iN.D.i, |N3D,.|, |N._.D.|, 



the full equivalent array being 



375 



^0 



. 



a. 







a. 



e. 



a^ 



J 





ao 



. 



a. 



. 



al 



a. 



ex 



a^ 



ao 



a. 





e. 



a. 



a. 



> 



e. 



ao 





al 

 a. 



a. 



ao 





a^ 



a. 



, 



Bi 



a. 



ao 



a^ 



a. 



a, 



<%2 



a^ 

 a^ 

 a, 

 a. 



ao 

 a. 



. 



al 



. 



a, 



ao 





a. 



i 



Ox 



03 



03 a. 



ao 

 a^ 

 a^ 

 a. 



The first and last member of any row in this array being already 

 known from (IV.) and (XI.), we have only to ascertain how the 

 other members of the row are formable from either of those. As an 

 example let us take the (s — l)th now, and start with the last member 

 of the row, our knowledge from (XI.) being that the expression for 



N,_xD, I is 



a-. 



. 



a. 



ao 



. 



... 





0x 



a^ 



a. 



ao 



... 





0. 



a. 



a^ 



a. 



... 



• 



0.-3 



as-. 



as-^ 



as-, ... 



... a^ 



a. 



0.-Z 



a. 



as-. 



as-, ... 



... a. 



a^ 



where each a is confined to one diagonal. Lowering each element 

 of the last column of this by one place we obtain the preceding 

 determinant, that is to say, the determinant in the expression for 

 j N5_2D5 1 : lowering in addition each element of the second-last 

 column we obtain the determinant in the expression for | N^.jD. | : 

 and so on. Quite generally, therefore, we have 



N,D.| = g 



a, 

 a^ 



0S-I as 



(xvii) 



