( 203 ) 



X. — The Hessian of a General Determinant. 

 By Thomas Muir, LL.D. 



(Read January 21, 1901.) 



(1) There being n 2 independent variables in a general n-line determinant, the Hessian 

 of the determinant with respect to the said variables must be a determinant with n 2 

 lines : and as a general w-line determinant has all its terms linear in the elements 

 involved, it follows that each element of the Hessian being a second differential- 

 quotient cannot be of a higher degree in the variables than the (n — 2) th , and that 

 consequently the degree of the Hessian itself cannot exceed n 2 (n — 2). The object of 

 the present short paper is to show that this degree is attained by the n(n— 2) th power 

 of the given determinant being a factor of the Hessian. 



(2) Beginning with the case of n = 3, let the originating determinant be 



Co 



a., 



so that single differentiation with respect to the nine elements produces 



he* 



l 3H 



- I a 2 e s I , 



a bo 



- I h t- I 



I a i C 3 I ' 



- | a x b 3 | , 



- a.&, 



I «AU 



"1 » 2 ' 3 



Bj, B,, B 3 say. 



Cj , C. 2 , C 3 



These being each differentiated in the same way, i.e. with respect to each of the nine 



original elements, we obtain the eighty-one elements of the Hessian : and if we agree 



to take the independent variables in the order, a lt a 2 , a z ; b v b 2 , b 3 ; c v c 2 , c 3 , the 

 Hessian itself will be 



-c Q 



-Ci 



c 2 

 -c. 



-h 

 I 





-h 



h 



h 





~h 



h 



h 







a 3 



-a 2 



a 3 





«i 



a., 



-a. 





h - h 



a 3 

 -a<, 



By way of remembering its constitution we should think of it as divided into three 

 equal parts by two left-to-right lines, and at the same time into three equal parts by 

 VOL. XL. PART I. (NO. 10). 2 G 



