OF THE RECIPROCAL OF A DETERMINANT. 621 



one of which, as might have been expected, is identical with Jacobi's, and therefore not 

 less suitable than in its previous form. In the other case combination has proved 

 harmful, the reason beiug that then it is not the same x which occurs with positive 

 index in the expansions of the two fractions concerned.* 



An examination of the corresponding fractions in § 6 will attest the accuracy of the 

 result reached in § 5. 



(9) Jacobi's theorem that 



the coefficient of x~ x~ x~ ■ ■ • in the expansion of u~ u~ u~ ■ • ■ is \ a i b. 2 c 3 ■ • • | _1 , 



has a curious analogue in the theorem that 



■*■ + 



the coefficient of x-^x^x^ ■ ■ • in the expansion of u l u. 2 u. i ■ ■ ■ is | n/v-3 ' ' ' I > 



+ + 



where the permanent | a-J). 2 c z • • • I is a function differing from the determinant 



I a i^2 c 3 • • • I m having all its terms positive, f I now propose to show that this is not a 



solitary instance of such dualism, — a dualism which surely warrants the two theorems 



being called per -reciprocals of one another : but before doing so it is desirable to change 



the point of view somewhat, so as to have the theorems in a wider field. 



(10) If to each linear homogeneous function u r of x x , x 2 , . • . there be prefixed as a 

 factor another letter, t say, with the same suffix, the terms in r r u r consist of one in 

 which the suffix of t is the same as the suffix of the x with which it is associated, and 

 others where this similarity does not exist : in other words, r r u r is partitionable 

 into a r T r x r and r r (u r — a r x r ). Terms in which each x is accompanied by a t with the 

 same suffix may be conveniently spoken of as conjugal terms, and the operation of 

 taking the aggregate of the conjugal terms existing in the expansion of any function 

 of TjWj , r 2 u 2 , . . . may be denoted by T. It is of course clear that 



Tfa x + <£,+ •••) = T(^) + T(^ 2 )+ •••, 

 T(C<£) = CT(*), 



where is a constant with respect to the t's and x's : and it is convenient to consider 



T(C) = C. 



Now, returning to the first of the two identities in § 8, and dividing by ^t"^, 

 we have 



o o 



LV¥jsJ = i _ ^ _cj + XT * I ft 3 c i I . 



T l«l • T-2 U 2 • T 3 M 3 T 1 X 1 • T -2 X -2 ■ T i X i / \U„ ■ Tj • T^ • Tg^ ^Jt^ -2(,.,U 3 ■ T^X., ' 



and, as in the expansion of the fractions under the sign of summation t 1 is negative 

 and x x positive, it follows that the aggregate of the conjugal terms in the expansions on 

 the right is I/t^j ' r 2 x 2 ' r 3 rr 3 , and therefore that the aggregate of the conjugal 



* Had we taken as our vanishing determinant that corresponding to the only other positive term of | a^c., [ we 

 should have found the resulting identity objectionable in both forms, 

 t Proceedings Roy. Soc. Edin., xi. pp. 409-418 (Sess. 1881-2). 



