(369) 



A SPECIAL DETEEMINANT HAVING {r, s) EQUAL TO 

 ZEEO WHEN s>r + l. 



By Thomas Muie, LL.D. 



(Bead August 28, 1907.) 



1. In connection with a certain differential equation there has 

 recently come to light ''' a peculiar determinant, which, when of the 

 7ith. order, has 271 — 1 distinct elements 



COq, (Xj, CL2} •••) CLn—i 



b,, 6„ ..., b^_. 



so placed that a^_^ is the element common to the first column and 

 last row, that the complementary minor of a^-^ is the persymmetric 

 determinant 



<^o 



. 



. 



... 



a. 



a. 



. 



... 



a. 



a, 



«o 





««-3 



an-. 



an-s ••• 



... (Xo 



CI'n-2 



^«-3 



an-, ... 



. . . a, a 



and that the 6's occupy the remaining part of the first column, and 

 in the reverse order the remaining part of the last row. When 

 7^ = 2, 3, 4, ... the determinants are thus 



b, a^ 

 a^ b. 



a. 





a, 



a. 



K 



bX 





and may be conveniently denoted by N^, N,, N3, .... 



The requirement in the above-mentioned connection was to show 

 that a special case of such a determinant vanishes, and Professor 



* BuRGATTi, p., SuUe condizioni per 1' equivalenza di un equazione differenziale 

 lineare e della sua aggiunta. Renclic. 1st. Lomb. (2) xi. (1907). 



