the Coaxial Minors of a Determinant. 



i » I 



a 



1 

 / 

 1111 



363 



9 



f 



1 



Denoting this determinant by A and its coaxial first minors 



by 



we have 



2A-2, 2B-2, 2C-2, 2D-2, 

 /+1=2A, #+^ = 2B, A+issSO, 



»*+^-2D, 



A = p^ 1 ~ /)(l ~^ )(1 ~ 7i)(1 -^ /4); 



(1) 

 (2) 

 (3) 



and the relations in question are to be found by eliminating 

 the three quantities/, <?, h from the five equations (1), (2), (3). 

 3. Let 



f=e ia > 9 = ^, h = e { y; 

 tlien from (1), (2) we have 



cosa = A, cos/3 = B, cos7=C; . . . (4) 



cos(<*+/3 + 7 )=D; (5) 



and from (3) we find 



cos (ft + y) 4- cos (y + a) + cos (« + /3) =k, . 



where 



Now let 



* = A + B + C + D-iA-l. 



£= sin ce, rj= sin ft, £= sin y ; 

 then from (5) we get 



A^+BC|+C^f\=0, . . 

 where 



X = D-ABC; 



and from (6) we have 



(6) 



(7) 



(8) 

 (9) 



