THE HESSIAN OF A GENERAL DETERMINANT. 



205 



have — A ls — A 2 , — A g ; — B l5 - B 2 , — B 3 ; — C l9 — C 2 , — C 3 respectively in their second 

 columns, and B l5 B 2 , B g , in a column of each ; and so in the remaining case. The 

 product-determinant has thus two kinds of columns, viz., those with three non-zero 

 elements, and those with six. Of the former kind there are six, and of the latter three. 

 Further, when there are only three non-zero elements in a column they are all negative, 

 and when there are six they are all positive. Adding together the three columns which 

 have six non-zero elements, we can remove the factor 2 from the resulting column : and 

 if we then diminish the two other columns by this altered column, we shall change 

 them into columns with only three non-zero elements like the rest. Our result then 

 will take the form 



-Pi • 



-0, 



-c. 



A, 





-»i 



A 2 





~ B 2 



A 3 



. 



-Bi 



Bi 



-A x 





B 2 



-A 2 . 





B a 



-A s . 





"B 2 

 -B. 



C, 



-A, 



-B 1 



-B, 

 -B„ 



-C 2 



-c. 



-c 2 

 -a 



which is easily changed further into 



(-)3 2 



Ax 



A 2 

 A 3 

 Ba 

 B 2 

 B 3 

 Oi 



Co 



c. 



B t 

 B 2 

 B Q 



Oi 



Ax 



-B, 



Cx 



A 2 



-B 2 



c 2 



A 2 



-B 2 



c 3 



Ax 

 A, 



B 2 

 Bo 



"Cx 



-c 2 

 -a 



by interchanging the 2nd, 3rd, 6th columns with the 4th, 7th, 8th respectively. We are 

 thus led to the equation 



H ( | a, \ H | ) • | ai b 2 c 3 j" = ( - ) 5 2 | A^Cg i 3 , 



and therefore finally to 



H (lajVsk) = -2 |aA c s |3 - 



(3) Taking next the case of n = 4, the originating determinant being 



a, a a* 



h h 



t?j d 2 d 3 d i 



we obtain, from single differentiation with respect to the sixteen elements, 



