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XXV. — The Generating Function of the Reciprocal of a Determinant. 



By Thomas Muir, LL.D. 



(MS. received 8th September 1902. Read 3rd November 1902. Issued separately March 6, 1903.) 



(1) This is a subject to which very little study has been directed. The first to 

 enunciate any proposition regarding it was Jacobi ; but the solitary result which he 

 reached received no attention from mathematicians, — certainly no fruitful attention, — 

 during seventy years following the publication of it. 



Jacobi was concerned with a problem regarding the partition of a fraction with 

 composite denominator (u x — t^) (u 2 — t 2 ) ... into other fractions whose denominators 

 are factors of the original, where Wj , u 2 , ... are linear homogeneous functions of one 

 and the same set of variables. The specific chtiracter of the partition was only definable 

 by viewing the given fraction (u 1 — t 1 )' 1 (u 2 — t 2 )~ 1 ... as expanded in series form, it 

 being required that each partial fraction should be the aggregate of a certain set of 

 terms in this series. Of course the question of the order of the terms in each factor of 

 the original denominator had to be attended to at the outset, since the expansion for 

 [a l x-\-b l y J rC l z — t)~ l is not the same as for (b x y + c x z + a x x — t)' 1 . Now one general 

 proposition to which Jacobi was led in the course of this investigation was that the 

 coefficient of x~ l x~ x~ ... in the expansion of u~ u~ u~ . . . , ivhere 



Uj = afa + a 2 x 2 + a 3 x 3 + . . . 

 u 2 = b^x x + b 2 x 2 + b 3 x' 3 + . . . 



is |a 1 b 2 c 3 . . . | 1 , provided that in every case the first term of u r is that containing x r . 

 This is the proposition with which we are here concerned. 



(2) Jacobi gave no proof of it. He considered first the case of (ax + by)' 1 

 (b'y + a'x)~ l , passing thence to (ax + by — t)' 1 (b'y + a'x — f)' 1 : next he dealt with 

 (ax + by + cz)~ 1 (b r y + c'z + a / x)~ 1 (c"z + a"x + b"y)~ 1 , from which he passed readily as 

 before to (ax + by + cz-t)~ 1 . . . . : and then he found the work so forbidding that he 

 turned to other things, — " Ad quatuor pluresve variabiles haec extendere non lubet, cum 

 iam pro tribus tarn prolixa exstiterint. Progredimur ad alia" (p. 354).* 



(3) Doubtless it is true that often in the theory of determinants the treatment of the 

 first two cases of a theorem is not only a sufficient guide to the generalisation of the 



* C. G. J. Jacobi, " Exercitatio algebraica circa discerptionem singularem fractionum, quae plures variabiles 

 involvunt," Crelle's Journ., v. pp. 344-364 (year 1829). 



TRANS. ROY. SOC. EDIN., VOL. XL. PART III. (NO. 25) 4 y 



