328 



DR THOMAS MUIR ON THE 



where B 1( B 2 , . . . B 6 , C^ C 2 , C 3 have the significations given to them in section 4, but 

 are to be replaced by using the theorem of sections 2, 3. 



6. Again, taking the last four of the set of five equations in section 4, and bearing 

 in mind that 7^3 = 7 2 ?4> all that is necessary for elimination is to put 





or - 



2B X B 4 



_ -C,+ ^C g »+4B 1 B,B 6 

 y2 ~ 2B Q 



-C 3 +VC 3 2 + 4B 2 B 3 B 6 

 y *-- 2B, 



or 



or 



Ci+ VC 1 2 +4B 1 B 2 B 4 ' 

 C 2 +VC 2 2+4B 1 B 3 B 5 ' 



2 B 2 B 6 



C 3 +VC 3 2 + 4B 2 B 3 B 6 ' 



in the equation 



C 4 = 



B y& + b 4 b b -^ 



y2 



7i73 



The result of this action is 



46^30, + C^C, - C 1N /C 2 2 +4B 1 B 3 B 5 VC 3 2 +4B 2 B 3 B 6 

 + C 2 J C 3 2 + 4B 2 B 3 B 6 y C^B^B, 

 - C 8> /C 1 «+4B 1 B 1 B 4 ^C,«+4B 1 B lt B 6 = 0; 



and similar equations can be got for C 3 in terms of C l5 C 2 , C 4 ; for C 2 in terms of 

 C 1} C 3 , C 4 ; and for Cx in terms of C 2 , C 3 , C 4 . 



7. On comparison of these results with those of Professor Nanson it will be found 

 that instead of an explicit expression for | a^c^ | in terms of its coaxial minors, and 

 an explicit expression for one of the coaxial minors of the 3rd order in terms of the 

 three others and those of lower order, he obtains in each case an unsolved biquadratic 

 equation. The presumption therefore is that each of his biquadratics must be 

 resolvable into linear factors. This will now be shown to be the case. The series of 

 necessary transformations is among the most interesting of the kind, and therefore 

 well worthy of attention apart altogether from the problem with which they are here 

 connected. 



8. The latter of the two biquadratics is 



DL CQ BR AQRL + 2BCD 

 CP DM AR BRPM + 2CAD 

 BP AQ DN CPQN + 2ABD 

 AL BM CN DLMN + 2ABC 

 where 



D, C, B, A ; R, Q, L, P, M, N 



correspond to but are not identical with the 



C v C 2 , C 3 , C 4 ; B 1; B 2 , B 3 , B 4 , B 5 , B e 

 of the present paper. 



0, 



