512 On the Jacobian of the Primary Minors. 



and as 



a 1 A 1 -\-b 2 B 2 + . . . + 2a 2 A 2 = a 1 A l + a 2 A 2 + a d A z -\-a i A± 



+ a 2 A 2 + b 2 B 2 + 6 3 B 3 + 6 4 B 4 

 4- a d A 3 + 63B3 + c 3 C 3 + c 4 C 4 

 + a 4 A 4 + ^Bi + C4C4 + d 4 D 4 

 = 4 A, 

 the final result is 



J = A 5 (l-4). 



(6) The process, it is not difficult to see, is perfectly 

 general, the fundamental part of it being the multiplication 

 of the Jacobian by A w+1 expressed in a particular form, and 



the transformation of the product into A* n(w+1) ( 1 — j 



This gives the equation 



JA" +1 = A^(nWl--^\ 



whence there follows 



Since I Ai„ | = | a\ n \ n ~ l , and therefore 



I A ln I -r- I a u I = I a,„ I -» 



the theorem may be more neatly enunciated thus : — If 



J am I be an axisymmetric determinant, then 



3 (An, A 22 , . . ., An n , A n - hn ,..., A 12 ) = r | A lw 1 1 §(•+») 



^(a n , <2 2 2? • • •? a »»3 a «-l,n >•"••> ^12) (. I a l« I J 



(7) Putting n = 3 we obtain a result in which both Sylvester 

 and Cayley were interested from a totally different point of 

 view*, and which on account of its seemingly unique 

 character has attracted considerable attention since their 

 time. 



Capetown, South Africa, 

 16th July, 1902. 



* Sylvester J. J., "Examples of the Dialytic Method of Elimination 

 as applied to Ternary Systems of Equations," Cambridge Math. Journ. ii. 

 pp. 232-236 (1841) ; Cayley A., "Note upon a Eesult of Ehmination/' 

 Phil. Mag. [4] xi. pp. 378-379 (1856). 



