206 



DR THOMAS MUIR ON 



or say 



1 h 2 c a d i 1 . 



- 1 v 3 rf 4 1 , 



1 V 2 < ? 4 1 » 



1 "l C 2^3 ' 



- 1 a 2 c 3 rf 4 | , 



Iw^li 



- | ajC 2 rf 4 1 , 



1 a l C 2 3 1 > 



1 « 2 M4 1 > 



- i <h b 4 l i i . 



1 «i& 2 ^4 ! » 



-KM 3 |. 



- | aj)^ | , 



1 « A C 4 1 . 



- 1 « A C 4 1 » 



1 «A C 3 1 » 





1 2 



A 3 A 4 







B l B 2 



Bg B 4 







C l C 2 



Cg C 4 







Di I> 2 



Dg D,. 





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

 original elements, we obtain the 256 elements of the Hessian ; and find the latter, when 

 partitioned into 4x4 minor matrices of the 4th order, to be of the form 



where (a,/3), (a, r ), 



(a,(3) = 



(y,S) 



-(AS) (Ay) 



- (y,S) 



(a,S) - (a,y) 



(AS) "(a,8) 



(«,/?) 



-(Ay) (<*,y) 



-<»,/*) 



1 exactly alike,- 



— for example 



1 «3^4 1 



- | a. 2 b 4 1 j a 2 b 3 



- 1 «3^4 1 



1 «i & 4 1 - 1 a A 



i «2 & 4 1 - 1 a A 1 



1 a A 



- 1 «2^3 1 I a A 1 



- 1 »A ! 



— being thus all zero-axial skew like their parent matrix.* 



Multiplying the Hessian by | a-fi^d^ | 4 we obtain as before a product which, when 

 partitioned like the multiplicand, is of the form 



where 



- (yi a s)' - {ySaf 

 -(W -(W 



(yi a s)' 

 (y 2 A)' 



(V4)' 



(W 



(/W'= \ I 2 

 I --^3 



l-B 4 



A, 



A 2 



A 3 • 



A 4 • 





- 



the suffix of a indicating the column in which the positive A's appear, and the suffix of 

 /3 the column in which the negative B's appear. Any one of the four lines of matrices, 

 sav the line 



(/V2)' (yi a s)' (Sia 4 )'» 



may consequently be described in language exactly similar to that employed in the case 



Note, too, that the determinant of every matrix vanishes, being, in the case exemplified, the square of the vanishing 

 Pfaffian 



aA I I «3 6 4 I - I a A I a Jh I + I a A I I «2 6 3 1 ■ 



