COAXIAL MINORS OF A DETERMINANT OF THE FOURTH ORDER. 327 



5. Taking the first four equations of the set of five, and using <y u y 2 , 73 as just 

 indicated, we have 



B 1 B 6 +B 2 B 5 +B 3 B 4 -D=(B 2 B 6 ^ + BA^)-(B3 ri y3 + B 1 BA^J + (B 2 B 5 ^+B3B 4 ^y 

 C, = — B,y, + B.B. — , 



C 2 = ~ B s72 + B 1 B 5~' 

 72 



C 3 = - B 373 + B 2 B 6~- 

 73 



Now, by means of each pair of the last three of these equations, the y's may be 

 eliminated from a corresponding one of the bracketed expressions in the first equation, 

 the results of this action in fact being 



BB 7 2 + BB 73 _ -C 2 C 3 + JC,' + 4B 1 B s B 6 J(y+4BAB, 



2 °73 1 5 72 2B 3 



R , K1 ,1 _ C 1 C 3 + N /C 1 H4B 1 B 2 B 4N /C3H4B 2 B 3 B 6 



- b 37i73 "+" D x D *P€^ZZ.~ — 2B ~~ 



7l73 Zi3 2 



T, H tt + T* T* * _ -C X C 2 + JC 1 » + 4B 1 B,B 4 ,/C i » + 4B 1 B,B B 

 2 5 7 2 3 4 7i 2B X 



We thus have 



D = BjB 6 + B 2 B 5 + B 3 B 4 + j~ + ^jr 1 + -^ 



- ~ VC 2 2 +4B 1 B 3 B 5 V /C 3 2 +4B 2 B 3 B 6 + ^ N /C 3 2 +4B 2 B 3 B 6 JC?+4B^B 



- 2^- > /C 1 2 +4B 1 B 2 B 4 7C 2 2+4B X B 3 B 5 , 



— a relation among ten of the eleven devertebrated coaxial minors of | a-Jj^di \- Then 

 as for each of the ten there is an expression in terms of the vertebrate coaxial minors, 

 and, in the case of one of them, viz., D, this expression involves the original 

 determinant | ai& 2 c 3 c? 4 1, it is clear that we may deduce from this the result foreshadowed 

 by MacMahon, viz., an expression for | a^b^di | in terms of its coaxial minors. 



Making the actual substitutions in places where subsequent simplification is readily 

 possible,* we find 



«Av* 4 | = SoJ&Ml + 2kAl I«AI - 2Za 1 \\c 3 d i \ + Ga^cA + ^ + ^ + ^ 



1 



Z^JCf+tB^JCt + iB&B,. + 2B g JC^+^B.B, VC 3 2 +4B 2 B 3 B 6 



_JL_ 

 2B 



~ ^C,«+4B 1 B,B B 70^+46^36, , 



In the case of each expression under a root-sign a certain amount of simplification is also possible, e.g., we find 

 C, 2 + 4B 1 B 3 B 5 = \ajb 2 d 4 \ 2 - %2\a 1 i^l i \ IM4K + 4|OiM4l«iM4 + 4I«AI l«AI IM4I 



+ 2 a l 2 I M4 I 2 - 2| 2a A I «1^4 I I M4 1 • 



