OF THE RECIPROCAL OF A DETERMINANT. 



617 



and in (b) 



/ a b c \ / 1 2 3 \ 



\,b c a) , ti2 3 l). 



This, of course, is as it should be, because the remaining terms of the two identities 

 are unaltered by the substitution : but the observation derives additional importance 

 from the fact that the general application of it does away with the necessity for any 

 stipulation as to the order of the terms in u x , u 2 , . . . , for, having Wj = a x x x + a 2 x 2 + a z x z 

 cyclical substitution gives % = b 2 x 2 + b 3 x 3 + b 1 x l and not = 6^ + 6 2 jc 2 + 6 3 a; 3 . The 

 identities (a) and (b) are thus more appropriately written 



i a \b 



2 I 



1 



h 



'i 

 i x u 2 



y 







\a y b.f. i \ 1 V'/Yy'i + 'i'v'i + 'VV'a. 



UjU 2 u s ' a-^Wg £± r.,w, ••'• 3 M 3 



or if we fall back to the stage which Jacobi views in the second case as being 

 fundamental 



o 



I a A I oh?* = u \ u -i - 2,^i w i ' & i > 



o 



\ u 1 b. 2 c. i \x l x. 2 x z «= u x u, z u z - y,a , 1 M 1 • (^W^ + c^/y;., + b,crfc a ) . 



The simple question with which we are therefore brought face to face is as to the 

 existence of a general identity of the latter kind. 



(5) By way of answer to this an examination of the next case may be found 

 sufficient. Is there, then, a quadric function, Fj , of x x , x 2 , x 3 , x i such that 



o 

 I «A C 3^4 I * X l X 2 X Z X A " *Wi»4 _ 2*1*1 • F l ' 



So probable is it made by analogy, by enumeration of terms and by actual experiment, 

 that a searching examination cannot be dispensed with. 



Denoting the ten possible coefficients of F T by l n , 1 ]2 , . . 



\ u so that 



2 a i M i ' F i - x i x \ * 



x, x. 



Ml M2 M3 

 1 

 1 



*22 ^23 



33 



H~ XtyUty 



X 2 



X 2 



a-, 



x i 



*5 



9 



22 







J 23 



2 24 



221 





2 



-"33 



2 3 4 



2 31 







•2 



^44 



2« 



2„ 



we have for the determination of the 40 unknowns the set of 35 equations obtained by 

 taking the coefficients on the right-hand side of the supposed identity and equating 

 to zero the coefficients of x x , x 2 , . . . , x^x 2 , . . . , and to ] afi 2 c 3 d± j the coefficient of 

 x^x^x^* The first four equations, — a cyclic set, — 



«lVl fZ l - «1M1 = ° . a A C 2 d 2 - «2 2 22 = ° » ■ • • (!)l ' (^2 



* In view of the insufficiency in the number of equations it may be noted that in the preceding case there is a 

 redundancy, viz., ten equations to nine unknowns, but no inconsistency. 



