COAXIAL MINORS OF A DETERMINANT OF THE FOURTH ORDER. 325 



Again, we have 



• (to CCn C&A 



\ ■ h h 

 d x d 2 d 3 rf 4 



— c. 



• Ct-rt Cvo (jVA 



h ■ h l i 



C l C 2 C 3 C 4 



CL. COn CCo CC. 



= | a x b 2 c 3 d A | — aj h. 2 Cvd i \ — b 2 \ a^d^ | + a x b 2 \ e 3 d^ 1 



— c s {\a 1 b 2 d i \ — a 1 |6 2 ^ 4 | — b 2 \a x d^\ + ^4}, 





a 2 



a 4 



&1 



. 



&4 



tf x 



^2 



^4 



= I a x b 2 c 3 d i I — aj Z> 2 c 3 d 4 1 — b 2 \ a 1 c 3 d i \ — c 3 \ a 1 b 2 d i j 

 + a^lcgrifj + OjCglSg^l + ^Cgla^l 



(A 3 ) 



• 



«2 



a 3 



a 4 



61 





\ 



K 



c l 



C 2 



• 



c i 



^1 



^2 



^3 



- 



And lastly, by proceeding in exactly the same way, we have the theorem of the 

 preceding section, viz.: — 



= I a 1 b 2 c 3 d i I — 2aJ b 2 c 3 d i | + 2a 4 & 2 ! c 3 d i \ — 3a 1 b 2 e 3 d i , 



where the 2 refers to combinations of the four elements, a u b 2 , c 3 , d t . 



4. MacMahon's problem of expressing the determinant of the 4th order in terms of 

 its coaxial minors may thus be transformed into something apparently simpler, viz., 

 expressing the determinant in terms of its devertebrated coaxial minors and the 

 primary diagonal elements. 



In the case of the determinant \a 1 b 2 c 3 d i \ the eleven (i.e., 2 i — 1 — 4) devertebrated 

 coaxial minors are 



\ 



Ct/A CCn CO a 



■ h h 



CVh G/n Ctn 



i.e., a 2 b x c lj d 3 + a 3 b 4 c x d 2 + a^> 3 c 2 d x — 



a 2 b 4 c x d 3 + a 3 b x c^d 2 

 + a 2 b 3 c A d x + ap x c 2 d 3 

 + ap 3 c x d 2 + a 3 & 4 c 2 (^ 1 



- = D say , 



h 





a 2 



a i 



\ 



. 



h 



d x 



d 2 



. 



WO w 'a 



d l d 3 



°i 



■ h 



h 



C 2 • 



d 2 d 3 



c i 



i.e., a 2 b 3 c x + a 3 b x c 2 = CjSay, 



i.e., a 2 btd x + ap x d 2 = C 2 say, 



i.e., a 3 c^d x + a^d^ = C 3 say, 



i.e., b 3 c^d 2 + b£ 2 d z = C 4 say, 



