Axisymmetric Determinants. 



451 



Consequently, by performing the operations 



156 



row 2 + : 156 : row n col 2 + 



156 

 156 



col r , 



row 2 -f- a, col 2 -=- a, 



we obtain 



N(flx/A + eJS +/VU) 



a' 



. , 



a 



e 2 



/ 2 



a 2 : 



156 



■ ; 15 ; 



; 16; 



156 



; ; 15 ; 



; 16; 



e 2 



; 15 ; 

 ; 15 ; 



- 



A 



f 2 



: 16! 



; 16; 



A 



• 



Further, having the knowledge from §2 that the four-line deter- 

 minant here is divisible by A, and seeing that it would remain 

 unaltered although the sign of / were changed, we conclude that it 

 is also divisible by the determinant resulting from A by substituting 

 -./ for /. We thus have the theorem : If any axisymmetric deter- 

 minant A which has (lr) equal to zero for all values of r except 

 1, n-1, n, the cof actor of A in the norm of 



(1, 1) n/[TJT+ (1, n-1) J[n-l,n-l] + (1, n) J[n^n] 



is a 2 A', where A' is the determinant differing from A merely in the 

 sign of (1,4. (VIII.) 



Also, if A be any axisymmetric determinant ivhich has (l,r) equal 

 to zero for all values of r except 1, n-1, n, and A' be the determinant 

 differing from A merely in the sign of (1, n) 



AA' = 



■ l,n,n-l ■ 



(i,i) 



: 1, n • 



; 1,^-1 



: l,n, n-1 : 



: 1, n : 



! l,n-l 



(i,i) 



: 1, n : 



\l,n\ 



(l,n-iy 



(1, n-1) 2 



(l,n) 2 



'• i ; 

 ; i ; 



:I,»-1 i 



; i,n-i ; 



(l,n) a 



■ 1 ' 

 ': 1 ': 



• 



(IX.) 



It would be interesting to see this last theorem proved directly, 

 that is to say, to see the right-hand side evolved from the left-hand 

 side by mere application of the elementary properties of determi- 

 nants. In default of such a proof the theorem may be verified as 

 follows, the return to the special A with which we started being 

 made to save space in writing : — ? 



A = ak - C 1 



15 

 15 



P 



16 ; , o x 



16 ; + 2e f 



15 

 16 



31 



