330 



DR THOMAS MUIR ON THE 



Dividing the columns by X, m, ", \/jlv respectively, and multiplying the rows in order by 

 the same, we obtain 



DLMN-ABC CNX M BM„X ALfxv 



CNX M DLMN-ABC AL MJ , BM^X 



BMi/X ALfiv DLMN-ABC CNX M 



AL M i/ BMi/\ CNXm DLMN-ABC 



— a determinant which is seen to have all the elements of the primary diagonal alike, 

 all the elements of the secondary diagonal alike, and to be symmetric with respect to 

 both diagonals. Such a determinant, when of the 4th order, must clearly be a 

 function of the four elements which necessarily recur in every line ; and, as a matter of 

 fact, it is known to be expressible as the product of four factors, the first of which is the 

 sum of the said four elements, and differs from each of the others in the sign of two of 

 its last three terms. The biquadratic we began with is thus the same as 



(DLMN - ABC + CNX M + BM„X + AL M „) 

 . (DLMN - ABC + CNXm - BMvX - AL^) 

 . (DLMN - ABC - CNX/x + BMvX - ALpv) 

 . (DLMN - ABC - CNXm - BMvX + AL Ml >) 



so that if we put back the values of X, /m, v and solve, we have 



= 



D = LMN{ ABC ± C^-MNP^-NLQ ± B y C 2 -LMR J A'-MN P 



± CVA 2 -MNPVB 2 -NLQJ. 



and this, on the required changes being made, will be found to be identical with the 

 result of section 6. 



9. The other biquadratic referred to is 



e 



(l-C)Vl-B 2 



(1-B) N /1-C 2 



(l-A^l-BVl-C 2 



(1-C)J1-A* 



e 



(l-A)Vl-C 2 



(l-B)Vl-CVl-A 2 



(1-B) N /1-A 2 



(l-A) v /l-B 2 



d 



(1-C) X /1-A 2 V /1-B 2 



1-A 



1-B 



1-C 



e 



= o, 



where 9 stands for (A-l) (B-l) (C-l)-|-A. It is the biquadratic not for the 

 general determinant | a 1 6 2 c 3 c?4 1 but for the very special instance 



1 / 

 1 1 



or 



In this case the required transformation is very easy. All that is necessary is to divide 

 the first three rows by Jl-B 2 Jl^C 2 , ^/T^C 2 Jl ^A?, x/NAVl - B 2 respectively, 

 and then multiply in order the first three columns by the same. The result is 



