COAXIAL MINORS OF A DETERMINANT OP THE FOURTH ORDER. 



329 



Now this determinant is easily seen to be the same as 



DLMN CQMN BBMN AQBLMN+2BCDMN 



CPNL DMNL AENL BBPMNL+2CADNL 



BPLM AQLM DNLM CPQNLM + 2 ABDLM 



AL BM CN DLMN +2ABC 



Taking BC/L times each element of the last row from the corresponding element of the 

 1st row, CA/M times each element of the last row from the corresponding element of the 

 2nd row, and AB/N times each element of the last row from the corresponding element 

 of the 3rd row, we transform this new determinant into 



DLMN- ABC 



CPNL-A 2 C^ 



BPLM-A 2 B^ 

 AL 



CQMN - B 2 C^ BEMN - BC 2 ^ AQELMN + BCDMN - 



DLMN- ABC 

 n M 



AQLM-AB 2 

 BM 



N 



ABNL - AC 2 ^ BEPLMN + CADLN 



DLMN- ABC 



■ CN 



CPQLMN+ABDLM 



DLMN +2 ABC 



2AB 2 C 2 



L 

 2A 2 BC 2 



M 

 2A 2 B 2 C 



N 



Diminishing now each element of the last column by BC/L times the corresponding 

 element of the 1st column, by CA/M times the corresponding element of the 2nd 

 column, and by AB/N times the corresponding element of the 3rd column, we change 

 the last column into 



A R 2 n 2 ""i 

 AQRLMN - AC 2 QN - AB 2 EM + ^^- 



A 2 "Rf! 2 

 BEPLMN - BA 2 EL - BC 2 PN + — 



^(NLQ - B 2 ) (LME - C 2 ) 

 B. 



CPQLMN - CB 2 PM - CA 2 QL + 

 DLMN -ABC 



M 

 A 2 B 2 C 



N 



Y or <{ M 



c 



(LME- C 2 ) (MNP -A 2 ) 



^(MNP-A 2 ) (NLQ -B 2 ) 

 DLMN -ABC 



ind if, merely for shortness' sake, we put 



he determinant becomes 



DLMN -ABC CN/* 2 BM* 2 AL^V 



CNX 2 DLMN -ABC AL* 2 BM„ 2 A 2 



BM\ 2 AL M 2 DLMN -ABC CNXV 2 



AL BM CN DLMN -ABC 



