﻿Determinant in Terms of its Coaxial Minors. 531) 



Calling these A, B, 0, D respectively, and denoting the 

 original determinant by A, we have the five relations 



> 1 



A = a- 2 4- -» 



B = 7,-2 + f, 

 C = c-2+-, 



o ac 



y 



A _ (l-a)(l-6)(l-e)(ae-6) 



abe 



Clearly from these a, b, c may be eliminated, and five rela- 

 tions found, viz. connecting ABCD, ABCA, ABDA, ACDA, 

 BCDA : manifestly, however, only two of the relations can 

 be independent. 



5. To find the simplest of the five, viz. that which connects 

 ABCD, let us call the two values of a in the first equation a 



and - , the two values of b in the second equation /3 and -x , 



the two values of c in the third equation 7 and -, and let us 



substitute these values in the fourth equation. The elimi- 

 nant is thus seen to be 



{5- (2+D)+ f}{?- (2+D)+ ^}{^- (2+D)+ ^} 



(ay 

 1/3 



•( 2+D > + £}U-( 2+D >-4 7 } = 0; 



that is, 



WK.Vf)}>— (■*♦£)}■; 



202 



