THE HESSIAN OF A GENERAL DETERMINANT. 



207 



where n = 3 ; and thus the product determinant comes to be transformable into 3 times 

 a determinant whose first four columns are 



A, 



A 2 . 



A 3 . 



A 4 



"A 2 

 "A 3 

 "A 4 



-A, 



-A 2 



-A 3 



-A 4 



• "A, 

 . -A 2 

 . -A, 



• "A 4 



these being followed by four columns containing B's similarly placed but all negative, 

 these again by four columns containing negative C's, and these by four columns contain- 

 ing negative D's. Interchanging columns as before, with the necessary accompaniment 

 of (3 + 2 + 1) + 3 changes of sign, we obtain 



and therefore 



H ( I ai b,c 3 d 4 | ) • ! a.b.^d, j* = ( - )93 | AjB^L^ |* , 



= (-y3\ ai hc 3 d,\v, 



H ( | a 1 b 2 c s d i | ) = - 3 | a l b i c 3 d i | 8 . 



(4) The natural generalisation for an originating determinant of order n is thus fully 

 legitimised, the number of requisite changes of sign then being 



{{n-\) + (n-2) + ... + 2 + l} + (n-1), 



which of course is the same as 



(n - 2) + (n - 3) + . . . + 2 + 1, 



i.e., 



\{n - 1) (n - 2) : 

 and the resulting equation is 



H ( I «i„ I ) • | flu, |" = ( - )*'"- 1,( "- 2 » • (n - 1) • I A 1B |- , 

 so that 



H(|a lH |) = ( _ )i(»-D(«-2) . ( w _l) . | ain |»(n-l)-» 



= (_l)i(n-l)(ft-S) . ( w _!) . | ai . 



|n(«-2) 



VOL. XL. PART I. (NO. 10). 



2 H 



