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LXII. On the Expre8sib'dity of a Determinant in Terms of its 

 Coaxial Minors. By Thomas Mum, LL.D* 



1. TN a memoir on "A certain Class of Generating Func- 

 A tions in the Theory of Numbers," recently published 

 in the Philosophical Transactions f, Major MacMahon, F.R.S., 

 establishes the following noteworthy theorem : — 



In t/ie case of every determinant of even order greater than 

 the second there are two special relations between its coaxial 

 minors, and each of these tivo relations can be thrown into a 

 form which exhibits the determinant as an irrational function 

 of its coaxial minors : in the case of a determinant of odd order, 

 on the other hand, no such relations exist, and it is not possible 

 to express the determinant as a function of its coaxial minors. 



He deduces the theorem readily from another to the effect 

 that— 



There are 2 n — n 2 + n — 2 relations between the coaxial minors 

 of any determinant of the n th order. 



His proof of this latter theorem, however, is not by any 

 means simple, occupying as many as eight pages (pp. 133-140) 

 of the memoir. By reason of the importance of the theorem 

 a simpler proof is much to be desired, and part of my object 

 at present is to supply the want. 



2. I start from the familiar proposition, that if the rows of 

 a determinant of the n th order bs multiplied by #!, x 2 , x z ..., x n 

 respectively, and the columns be then divided by the same 

 quantities, the determinant is unaltered in value ; but I prefer 

 to include it in a more general but equally evident theorem, 

 viz. : — 



If the rows of a determinant of the n th order be multiplied 

 by x 1: x 2 , x 3 , . . . , x n respectively, and the columns be then divided 

 by Xj, x 2 , x 3 , . . . , x n respectively, the determinant is unaltered in 

 value, and each of the minors of the transformed determinant is, 

 to a factor pres, equal to the corresponding minor of the original 

 determinant, tlie connecting multiplier being x h Xk Xi . . ./x r x 9 x t . . . 

 if the minor belong to the h th , k th , 1 th , . . . rows, and r th , s* h , t th , . . . 

 columns of the original. 



From this we have manifestly the corollary : — 



Tlie connecting multiplier in the case of the coaxial minors, 

 as in the case of the whole determinant, is 1 : in other words, 

 the coaxial minors remain unaltered by the transformation. 



* Communicated by the Author, 

 t Vol. clxxxv. (1894) pp. 111-160. 



Phil. Mag. S. 5. Vol. 38. No. 235. Dec. 1804. 2 





