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X. — The Relations between the Coaxial Minors of a Determinant of the 

 Fourth Order. By Thomas Mum, LL.D. 



(Read January 31, 1898.) 



1. The existence of relations between the coaxial minors of a determinant was 

 first discovered by MacMahon in 1893. The whole literature of the subject is 

 comprised in three papers, viz. : — 



MacMahon, Phil. Trans., clxxxv. pp. 111-160. 

 Mctir, Phil. Mag., 5th series, xli. pp. 537-541. 

 Nanson, Phil. Mag., 5th series, xhv. pp. 362-367. 



My present object is to continue the investigation of the relations in question, and 

 more particularly to draw attention to an explicit expression for a determinant of the 

 4th order in terms of its own coaxial minors. At the outset some fresh considerations 

 regarding determinants in general will be found useful. 



2. As is well known, the coaxial minors of a determinant of the nth order are 2" — 1 

 in number, the determinant itself and each of the elements of its primary diagonal being 

 counted. For example, the coaxial minors of | a 1 b 2 c 3 d i | are 



I ajfadi |, 



Ojb^ |, 



(JO-lOtyLL A | j 



\cA I. 



a,c. 



ctp o 2 > 



VS l> 



Cg, d v 



I $i<*4 I j 



i \c 3 I, I b 2 d 4 j, 



C 3^4 I 



Of these the first 2 n — 1 — n may be devertebrated, if we may say so, by substituting 

 zeros for the elements of their primary diagonals ; and the determinants thus resulting 

 are found to be of considerable interest. They appear in Cayley's well-known 

 expansion-theorem, which for a determinant of the 3rd order is 



a l 



a 2 



a 3 





\ 



h 



h 



= 



c l 



C 2 



C 3 





= \ 



C 2 



+ a x ' 



+ ftj 



+ C 3 



h 



+ a i\h- 



Indeed this theorem may be described as giving an expression for a determinant in 

 terms of its own devertebrated coaxial minors and its primary diagonal elements. 



Now, if we use Cayley's expansion in connection with each of the first 2 n - 1 — n 

 coaxial minors, we obtain 2 n — 1 - n equations, linear in respect to the devertebrated 

 minors. So that, on solving for the latter, there must result an expression for each 



VOL. XXXIX. PART II. (NO. 10). 3 C 



