the Coaxial Minors of a Determinant. 



365 



four rows and replacing A by \ A+l &c. we get the result 

 given by Dr. Muir*. 



Again, rejecting in turn each of the first four rows in (11), 

 we obtain four different formulae each expressing A as a 

 rational function of the coaxial first minors. 



6. The relation connecting A with any three of the four 

 quantities A, B, (J, D may readily be found. Thus from (8), 

 (9) we have 



(l-AK+(l-B)^ + (l-% + ^0, . . (12) 



6 = /j, — X 



= /r-BC-OA-AB-D + ABC 



= (A-l)(B-l)(C-l)-iA. 



Now from (12) we deduce, by the method previously applied 

 to (8), that 



where 







/(1-C) 



l(l-B) 



1-A 



»/(l-C) n(l-B) wm(l-A) 

 n(l-A) ?z/(l -B) 



ro(l-A) 

 1-B 







1-C 



lm[l-0) 



= 0. 



This is the equation connecting; A, A, B, C. It is of the 

 fourth order in A and symmetrical in A, B, C. Three similar 

 relations are found by replacing any one of the letters A, B, 

 C, by D. 



7. Thus nine different relations have been found between 

 the coaxials of an inversely symmetric determinant of the 

 fourth order. Five of these are unique and connect A, B, C, D ; 



B, C, D, A ; C, A, D, A ; A, B, C, A respectively. The re- 

 maining four each express A as a rational function of A, B, 



C, D. The nine relations are of course equivalent to not 

 more than two independent equations. 



8. Next consider the general determinant of the fourth 

 order, 



a 



h 



<)' 



X 



h' 



b 



f 



V 



9 



f 



c 



z 



x' 



!/ 



j 



d 



Denoting this determinant by A and its coaxial minors by 



* Phil. Mag. Dec. 1894, p. 540. 

 Phil. Mag. S. 5. Vol. 44. No. 269. Oct. 1897. 



2 D 



