COAXIAL MINORS OF A DETERMINANT OF THE FOURTH ORDER. 



331 



(1 



■C^l-AVl-B* 



(i - c) vi - a* /i _ b« (i - b) jt^j CT (1 _ A) n/i ^ 2 jT ^ 2 



(1 - A) v / 1 - B* V /T^C2 (1 _ B) jy^ ji^ 



(lB)yi-cyi-A^ (l-Ajyi-Bvi^c^ 



(l|.A)Vl-BVl-C2 (1-B) JT=&JT=D (l-OVr-AVl-B* 



where again the determinant has the elements of the primary diagonal all alike, the 

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

 diagonals. As before, therefore, it resolves into four factors, and we have on sub- 

 stituting the value of 9 



(A-1)(B-1)(C-1) - iA ± (1-C) N /T3AVT^B 2 ± (1-B)/L^C\/T^ 



± (1-A) N /1-A\/1-B2 = 0, 

 or 



A = 2ABC - 22AB + 22A - 2 ± 2(1 - C) JI=I* JT=& ± 2(l-B)yi-BVl-A^ 



± 2(1-A) X /1-AV1-B2, 

 which is readily shown to be in agreement with the more general result in section 5.* 



10. Not only does the determinant 



DL CQ BE AQEL+2BCD 



CP DM AE BEPM+2CAD 



BP AQ DN CPQN + 2ABD 



AL BM CN DLMN + 2ABC 



resolve into factors, but each of the two determinants into which it may be partitioned 

 is also so resolvable. For, multiplying the columns in order by VMNQR, V^LRP 

 v/LMPQ, 1, and then dividing the rows in order by JLQR, /s /WRp > JKPQ, JTMn' 

 we obtain the new form ' ' 



= 



DVLMN C/NPQ B S /MEP A /LQE + 

 CVNPQ DVLMN ATLQB BVMEP + 



2BCD 



v/LQE 

 2GDA 



B VMEP A VLQE D JLMN C JWPQ + 



A/LQE B^/MEP C V 'NPQ DVLMN + 



VMEP 

 2DAB 



VNPQ 

 2ABC 



JUffi 



where M + Vl^BVf- C* + VI - CVl^A 2 + Vl^AVT^B 2 = 0, 



yu = A + B + C + D-U-l-BC-CA-AB. 

 Observe also that this equation gives a much simpler expression for a, viz.:- 



A = - 2 + 2^A - 2^AB + 25 VI - B 2 VI - C 2 . 

 VOL. XXXJX. PART II. (NO. 10) 



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