OF THE RECIPROCAL OF A DETERMINANT. 



619 



for 1 22 and 2 12 in the first equation produces an equation like the second, it follows that 

 there are not more than four solutions possible for the triad. These are 



1 22 = 0, 



l h c -i d i 



2 12 = c 2 d i a 1 + c 2 d 1 a 2 + (' x d % a 2 c 2 d 2 a x + c x d % a 2 



= h c \ d i 



VA 



C l d i a \ + c 2^i a 2 



b l c 1 d 2 + b 2 c 1 d 



b 1 c 2 d 1 + b 2 c 1 d 1 

 2 1X = ajCjt^ + ajCgcZj JajC^g 



The solutions of the next three triads are got by cyclic substitution. 

 The fifth triad in the same way gives us 



b 2 c. 2 d 1 + b 2 c 1 d 2 



c 2 d 2 a x 



b x c x d 2 + & 1 c 2 f? 1 + b 2 c x d x 



0. 



I33 = 



3 13 = a 3 b 1 d 3 + a r b 3 d 3 + 

 3 U = a 1 b 1 d 3 + a-Jj^ 



\ hc 3 d 3 \ Vv'i i /; i c 8 rf 3 + Wi 



b l Cl d 3 + b^ c 8 j 6^ c 3 + c x d x 6, j 6j c^g + ft^Cj, + V A 

 a i'h^-i ) a \bid 3 JO. 



and the solutions of the sixth are got therefrom by cyclical substitution. 



Here, however, a hitch occurs, for if we proceed with the cyclical substitution we 

 find that while in the matter of the equations we are led back, as we ought to be, from 

 the sixth triad to the fifth, the same is not true of the solutions. In fact, the solutions 

 of the fifth triad alone are immediately convincing of the hopelessness of our quest ; 

 for pairs of the unknowns in that triad, e.g., 1 33 , 3 n belong to the same cycle, and the 

 value of the second of the pair should therefore be obtainable from that of the first by 

 cyclical substitution, which is not the case. 



(6) A different mode of procedure from Jacobi's being thus necessary for the 

 establishment of his theorem, I would substitute the following, which, though 

 ostensibly concerned only with the special case which we have just been examining, 

 is of perfectly general application. 



Since the sum of the elements of every one of its columns is zero, the determinant 



1 2 ° 2 2 Co**' 9 



• a 3 x 3 u. 2 — b 3 x 3 





-a,x 



d B x 3 



1 »t 1 ~~ * L/'iJL'-i ■*■ C-ifcC-i Wi ™" CI -I JL 1 



J 4ri » 3 -- 4 ^ 4 



vanishes identically. Expanding it in a series arranged according to products of the 

 u's we have therefore 



= u l u. 2 u 3 u i - £ju x u 2 u 3 ■ d x x x + /^.WjWg • I c i d i I • x i x i 



~ 2l M l ' I ^3 C 4^1 i * X ?, X i X l + I a 2^3 C 4^1 I " X 2 X 2 X r'\ 



and consequently 



o 



I «A C 3^4 I ' ^fl^i = u l u 2 u S U i ~ 2, M 1 M 2 M 3 " d : X \ + 2< W 1^2 * I C 4«l I ' ®&\ 



+ 2, M i w 3 * I h d \ I • x 3 x i -2t u i\ h c i d i • x z x i x - 

 On dividing by x^x^x^ • u^u^u^ this is changed into 



IW-^I = 1 _ X^ d i + X 1 l^i I + X 1 I Mi _ ^ I he A 



M 1*W4 X^X^ / |M, • X,X n X. /iO, -X„X n / } UM. -X„X. / \ 1MIM. • X, 



